Designing Buildings - User contributions [en]

User:Nicky nguyen 91

2013-01-15T13:46:39Z

Nicky nguyen 91:

'''NGOC NGUYEN QUANG'''

International Halls, Lansdowne Terrace, London, WC1N 1AS

Email: [mailto:nicky_nguyen_91@hotmail.co.uk nicky_nguyen_91@hotmail.co.uk]

'''Education'''
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'''2012-2013'''

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University College London (UCL), MSc Civil Engineering

'''''Modules taken''''': Structural Dynamics and Mitigation, Advanced Structural Analysis; Seismic Design of Structures; Roads and Underground Infrastructure; Advanced Finite Element Modeling

 

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'''2009-2012'''

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University of Southampton, BEng Civil Engineering, Upper-second class 2.1 (69%)

· Modules: Mechanics, Structures & Materials, Engineering Science, Mathematics, Soil Mechanics, Hydraulics, Numerical Methods, Statistics achieving '''higher than 80%; '''with''' Structural Design & Materials '''and''' Structures & Dynamics '''achieving''' 97% and 94%'''

· Dissertation: Using Timoshenko beam theory for resonant column testing (67%)

 

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'''2007-2009'''

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Castle College Nottingham, A-Levels

Mathematics (A), Further Mathematics (A), Physics (A), Chemistry (A)

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'''Work Experience'''
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'''2012 (July -September)'''

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'''Designer and CAD technician at Vi.N.Vi Co Ltd for Design, Construction and Commerce, Vietnam '''

'''''Duties included''''': Complete Architectural visualisation and Structural design for a modern residential housing project of more than 500 m2 plan area including garden.

'''''Skills developed''''':

· '''''Teamwork and Leadership''''': coordinate the design team in 3D modeling and offer suggestion to improve the workflow; thus result in the design being approved by the clients a week earlier.

· '''''Communication''''': Negotiate with the clients about different approaches and changes'''.'''

· '''''Attention to details''''': carry out all structural analysis and calculations; also assist in producing more than 10 x A3 detailed structural drawings; bring back the profit worth equally 2000 GBP in just one and a half months.

 

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'''2011 (June - September'''

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'''Trainee Site Supervisor at Hoa Binh Construction and Real Estate Corporation, Vietnam'''

'''''Duties included''''': Worked with chartered engineers to supervise construction of a 15-storey commercial center project

'''''Skills developed''''':

· '''''Leadership''''': Act as a ‘zone leader’ to guide the labourers in making reinforcements for bearing pad foundations and structural frames.

· '''''Communication''''': Maintain regular communication with other zone leaders; as well as report daily progress to section manager and project leader by oral presentation and written reports.

· '''''Organizing and Time management''''': Assigned tasks to everyone at the start of the day; produced a ‘zone’s working plan’ by Microsoft Project to ensure it keeps up with the overall project’s plan.

 

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'''2011 (throughout 2nd year)'''

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'''Structural and Sustainability Design Coordinator of the Ferrari car showroom Design Project'''

· Lead a dynamic team of 5 students to produce '''one of the best 2nd year design projects'''.

· Coordinate the team in designing entire structural framings and geotechnical components

· Split the work to complete '''4 technical reports''' and '''a presentation video''' within a limited three month timeframe. The video is available at [http://www.youtube.com/watch?v=QS1fpW_9FaE http://www.youtube.com/watch?v=QS1fpW_9FaE].

 

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'''2010'''

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'''Summer Student Presenter at the UK Alumni ''''''organised'''''' by British Council in Vietnam'''

· Share study skills and life experiences in the Public Talk about Engineering held by Sterling Group

· Motivate Vietnamese students to pursue an Engineering degree in the UK

· Develop excellent public speaking skills and enhance recognition among international students

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'''2009 (June to September)'''

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'''Translator and Renting Adviser at Vi.N.Vi Co Ltd'''

'''''Duties included''''': focus on consumer relations with international clients who are interested in renting complete residential housing projects.

'''''Skills developed''''':

· '''''Strategic thinking''''': Contribute ideas in turning a residential house for family use to a Room-for-rent model, to attract national and international renters; it results in obtaining various short term Contracts and a 5-year long Contract from a Japanese Kimono Production Company

· '''''Multitasking''''': be in charge of answering telephone for customer’s requests while producing monthly rent bills and translating rent contracts.

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'''2007-2009'''

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'''A-Levels Personal Academic Tutor'''

· Teach Advanced Maths and English for two high school students every summer, who eventually got offers from LSE and UCL in London.

· Organize weekly presentation and ‘debate showcase’ for them to improve their English and confidence

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'''Activities and Interests'''
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· Since 2009: Piano performer in annual Christmas show of Vietnamese Society at University of Southampton.

· Intermediate level in classical piano

· Sports: Active member of Southampton University’s Badminton society

· Founder and Event-coordinator of Vietnamese Society’s Swimming Group

· Highly enthusiastic in performing magic and be an active member of UCL Magic Society

 

'''Membership and Skills'''
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· Fluent in English and Vietnamese (native)

· Membership of Institution of Civil Engineers (ICE), Institution of Structural Engineers (IStructe)

· Competent in Microsoft Word, Excel, Power Point and Project

· Obtained 80% in Autocad Professional Certificate Test in 2009

· Familiar with BIM software including Revit Architecture, Revit Structure and Robot Structural Analysis

· Matlab, Maple, Ansys, Oasys, Google Sketchup and all other subject-related IT packages

Resonant column method

2012-12-14T16:46:46Z

Nicky nguyen 91: Protected "The resonant column method" ([edit=author] (indefinite) [move=author] (indefinite))

= introduction =

== Background ==

The resonant column method was initially developed by Japanese engineers: Ishimoto & Iida (1937). It was made popular in the 1960s by authors such as Hall & Richart (1963), Drnevich et al. (1967) and Hardin & Black (1968). The resonant column apparatus has been used to measure the dynamic response of soils including the shear and elastic modulus based on the theory of wave propagation in prismatic rods. 

== Shear modulus (G) ==

The resonant column method was conventionally used in torsion to measure the shear modulus (G) of the material. In most cases, the clamped-free configuration has been chosen for research purpose as its mathematical derivation is more straightforward. In the clamped-free test, a cylindrical specimen is fixed at the base and excited via a drive mechanism attached to its free end. The resonant frequency (omega) is measured from which the velocity of the propagating wave is derived. Based on the derived velocity and the sample’s density, the low-strain shear modulus (G) of the material can be computed from the basic equation for torsional vibration.

== Young modulus (E) ==

The resonant column can also be used in flexural excitation to determine the material’s Young modulus (E). The conventional method with long samples allowing the application of Rayleighod’s energy method and Euler-Bernoulli beam theory, disregarded the shear strain energy and rotary inertia effect. When the tested specimen is short in length compared to its diameter, the effects of rotation and shear deformation of the samples during flexure can be substantial. These effects can be significant in interpreting data from flexural test, especially at high frequencies. Therefore, the Euler-Bernoulli theory of flexural vibration of elastic beam is found to be inadequate for short specimens and also for the prediction of higher modes of vibration. To be more accurate, Timoshenko beam theory is used as a model for this interpretation. The theory was developed by Ukrainian scientist Stephen Timoshenko in the 20th century which takes into account the shear deformation and rotary inertia. Different frequency equations for the clamped-free Timoshenko beam with an end mass in flexural vibration are solved to compute the value of elastic stiffness (E).

= The resonant column method =

== Resonant column for torsional excitation ==

In the standard torsional resonant column (Stokoe cell SBEL D1128) as mentioned in Allen and Stokoe (1982), the specimen is rigidly fixed at the base while torsional oscillation is applied to the free end by a drive head. The basic equations for the clamped-free resonant column subjected to torsion are:

[[File:Eq1.JPG|RTENOTITLE]]

The derivation of these equations is based on the assumption that the rotation is small and each transverse section remains plane and rotates about its centre. All the terms expressed in equation (2.1) are functions of the geometric properties of the specimen, except omega n. Treating the system as a single degree of freedom system, the resonant frequency measured in the resonant column apparatus is the damped natural frequency (omega d) but is sufficiently close to the natural frequency (omega n). In this case, the error can be tolerable as omega d is within 1% of omega n. Solving equation (2.1) and (2.2) with omega n, the shear wave velocity (Vs) can be found from which the shear modulus of the material (G) can be derived by rearranging equation (2.3).

== Resonant colum for flexural excitation ==

=== ===

=== Finding the Young modulus by Euler-Bernoulli beam theory (short samples) ===

The RCA can also be used to measure the Young modulus (E) of the material. Cascante et al. (1998) modified the standard Stokoe torsional resonant column (Stokoe cell SBEL D1128) to include flexural vibration mode. The schematic view of the apparatus is shown in Figure 1:

[[File:Schematic view.JPG|RTENOTITLE]] 

''Figure 1: Schematic view of the modified Stokoe RCA.''

''Image taken from Cascante et al. (1998)''

In the original configuration, four pairs of excitation coils are connected in series to produce a net torque at the top of the sample (Figure 2A). In Cascante’s modified version, the coils are reconnected so that only two magnets are used to produce a net horizontal force on top of the specimen (Figure 2B).

[[File:Coil.JPG|RTENOTITLE]]

''Figure 2'''': Coil-magnet arrangements for torsional and flexural RCA. Images taken from Cascante et al. (1998). ''''' ''A: Torsional excitation. ''''' '''''B: Flexural excitation''''' '''

In the reduction of data for flexural excitation, the specimen and its drive head can be idealised as an elastic column with a rigid point mass at the top free end (Fig 3). The behaviour of the system is assumed to be elastic. Cascante et al. (1998) has developed a general mathematical formulation for the angular resonant frequency by using Rayleigh’s energy method and Euler Bernoulli beam theory. Based on this general equation, the Young Modulus (E) can be determined by:

[[File:Eq2.JPG|RTENOTITLE]]

In previous literatures, as the cross-sectional dimensions of the sample were small in comparison with its length, Euler-Bernoulli beam theory has been used to treat the boundary conditions and derive the frequency equation, from which E could be determined.

[[File:Resonant column.JPG|RTENOTITLE]] 

''Figure 3'''': Exaggerated view of deflected column for an ''''idealised'''' system''''' ''Image taken from Priest (2004)''

=== Finding the Young modulus by Timoshenko beam theory (long samples) ===

When the tested specimens are short in length compared to their thicknesses, the effect of shear deformation during flexure is significant which can result in possible discrepancies in interpreting data from flexural test. On the other hand, the effect of rotation is large when the curvature of the beam is large relative to its thickness. This is true when the beam is short in length compared to its thickness. Therefore, Timoshenko beam theory is used in this interpretation as it takes into consideration the effect of shear deformation and rotary inertia in which the conventional Euler-Bernoulli theory doesn’t. During vibration, a typical element of a beam not only performs translatory movement, but also rotation. With shear deformation being considered, the assumption of the elementary Euler-Bernoulli theory that ‘’plane section remains plane’’ is no longer applicable. Therefore, the angle of rotation which is equal to the slope theta of any section along the length of the beam cannot be obtained by simple differentiation of the transverse displacement y. Thus, it results in two independent motions theta(x,t) and y(x,t).

Timoshenko gave the coupled equations of motion for the beam with constant cross-section as:

[[File:Eq3.JPG|RTENOTITLE]][[File:Bruch and mitchel.JPG|RTENOTITLE]] 

''Figure 4'': ''The beam-mass system used in the analysis'' '''''Image taken from Bruch and Mitchell (1987''''')

Bruch and Mitchell (1987) investigated a particular case of a cantilevered Timoshenko beam with a tip mass (Figure 4.4). By applying the boundary conditions and using Huang’s non-dimensional variables, the solutions to the coupled equations are determined as functions of the Young modulus (E), the Shear modulus (G), material’s density (rho), the angular natural frequency (omega n) and the geometry of the specimen. Bruch and Mitchell derived the frequency equation of the beam in flexural excitation by inserting the solutions to the coupled equations (2.6) and (2.7) into the boundary conditions, from which the matrix equation can be determined. By taking the determinant of the coefficient matrix equation, the resonant frequency equation was found from which the Young modulus (E) can be calculated.

Liu (1989) suggested three ways in which the work of Bruch and Mitchell could be further extended: (i) The base condition for the beam-mass system considered in [3] should be modeled as an imperfect clamped support (or elastic support), (ii) The tip mass’s centre of gravity is not practically right at the top of the beam but usually at a distance from the beam tip, (iii) the shear coefficient depends on both the shape of the cross-section and the Poisson ratio. Liu added springs at the hub to simulate the imperfect clamped support therefore the boundary condition also includes the spring’s properties which are the rotational spring constant and translational spring constant. By substituting the general solution into the new boundary conditions, Liu gave the improvement of Bruch and Mitchell’s frequency equation for the mass-loaded clamped-free Timoshenko beam.

The shear coefficient in Timoshenko’s beam theory is a dimensionless quantity, dependent on the shape of the cross section, which accounts for the fact that the shear stress and shear strain are not uniformly distributed over the cross section of the specimen. Cowper (1966) developed a new formula for the shear coefficient from the derivation of the equations of Timoshenko beam theory. For a circular cross-section, the value of K was given in terms of the Poisson ratio as:

[[File:K.JPG|RTENOTITLE]] Farghaly (1993) offered suggestion to extend Liu’s work by applying Timoshenko beam theory in treating the boundary conditions. He realised that the use of Euler-Bernoulli theory in the boundary conditions could result in inaccurate natural frequencies calculated, particularly for high slenderness ratios and higher modes of vibration. Farghaly’s model also includes the root flexibilities and the tip mass’s eccentricity as can be shown in Figure 5:

[[File:Farghaly model.JPG|RTENOTITLE]] 

''Figure 5'''': Thick beam with tip mass and root flexibilities.''''' ''Image taken from Farghaly (1993)''

The work of Bruch and Mitchell (1987), Liu (1989) and Farghaly (1993) were in an attempt to simulate the motion of a flexible robot arm modeled as a cantilevered Timoshenko beam with a lumped mass and lumped moment of inertia at the free end. However, for the purpose of this essay, their resonant frequency equations were considered to be adequate for use in computing the material’s Young modulus from the flexural resonant column test, if the angular natural frequency is known.

= Using Timoshenko’s beam theory for resonant column testing =

== Frequency equation by Bruch and Mitchell ==

Bruch and Mitchell started with the original coupled equations of motion given by Timoshenko for the beam with constant cross section: 

[[File:Eq 3.1.3.2.JPG|RTENOTITLE]]

[[File:Eq 3.3.JPG|RTENOTITLE]][[File:Eq 3.4.JPG|RTENOTITLE]] 

From the simple harmonic motion equations: 

[[File:Eq 3.8.JPG|RTENOTITLE]]

Using the non-dimensional variables and series of equations (3.1) to (3.3), equations (3.4) reduced the problem to:

[[File:Eq3.9.JPG|RTENOTITLE]][[File:Eq3.14.JPG|RTENOTITLE]][[File:Eq3.20.JPG|RTENOTITLE]][[File:Eq3.25.JPG|RTENOTITLE]] 

Taking the determinant of the coefficient matrix equation (3.20) gives the frequency equation, from which the elastic stiffness can be computed with the natural resonant frequency (omega n) as an input.

== Frequency equation by Liu ==

Liu [16] introduced a rotational spring constant (Kr) and a translational spring constant (Kl) to model the imperfection of a clamped support. For simplicity, assuming the base of the resonant column is perfectly clamped, the values of the spring constants (Kr) and (Kl) therefore approach infinity. The distance from the beam tip to the centre of the added mass (d) was added to model the eccentricity. Moment of inertia of the added mass (J) was also included in the revised matrix equation to improve the accuracy of the original model by Bruch and Mitchel. Liu started from the single free vibration equation of a Timoshenko beam given in [16], rather than the coupled equation of motion as in Bruch and Mitchell’s. The frequency equation given by Liu is:

[[File:Eq3.29.JPG|RTENOTITLE]] 

In which 

[[File:Eq3.30.JPG|RTENOTITLE]][[File:Eq3.39.JPG|RTENOTITLE]] 

Equation (3.39) was solved using Matlab with the same inputs as in the case of Bruch and Mitchell’s model, plus the eccentricity and moment of inertia of the tip mass, to evaluate the sample’s elastic stiffness.

== Frequency equation by Farghaly ==

Liu has derived a frequency equation to further improve the work of Bruch and Mitchell. The root flexibility, eccentricity and moment of inertia of the tip mass have been taken into consideration to improve the accuracy on modeling a robot arm as a clamped-free Timoshenko beam with a lumped mass and a lumped moment of inertia at its free end. The same idea of simulating a robot arm by Timoshenko beam theory can be used to model the resonant column apparatus when the sample is short in length relative to its diameter. Farghaly commented in his published paper that in [16], Liu used Timoshenko beam theory for the system differential equation, while Euler-Bernoulli theory was applied to treat the boundary conditions. Farghaly stressed that, when using Liu’s formula to compute the resonant frequency with proper inputs, inaccurate natural frequencies maybe obtained, particularly for significant values of the slenderness ratio and higher modes of vibration.

The system frequency equation in terms of the root rigidity parameters can be written as:

[[File:Eq3.40.JPG|RTENOTITLE]] 

In which

[[File:Eq3.41.JPG|RTENOTITLE]][[File:Eq3.45.JPG|RTENOTITLE]] 

In theory, to model the perfectly clamped support condition of the resonant column, the spring constants theta and K should be made to approach infinity. However, in Matlab, for simplicity, extreme values have been assigned to theta and K to give significant values of the root rigidity parameters theta and Z. As mentioned in Farghaly (1992), when using the matrix determinant equation (3.40) to compute the resonant frequencies, inaccurate results might be obtained for large values of the slenderness ratio. According to Liu in Author’s Reply (1992), from his own practical point of view, if one can accept the idea of treating a complicated cantilever structure as a Timoshenko beam, then the discrepancies caused by non-exact boundary conditions might be considered as tolerable.

= references =

Alan, K. (2011). Stiffness and damping of sand at small strain using a resonant column. ''Civil Engineering 3rd year Individual Project, University of Southampton''.

Allen, J. C. and Stokoe, K. H. (1982). Development of resonant column apparatus with anisotropic loading. ''Geotechnical Engineering Report GR82-28, Civil Engineering Dept., University of Texas at Austin''.

Banerjee, J. R. (2001). Frequency equation and mode shape formulae for composite Timoshenko beams. ''Composite Structures 51 '', 381-388.

Bisplinghoff, R. L., and Ashley, H. (1962). ''Principles of Aeroelasticity.'' New York: Dover.

Bruch, J. C., and Mitchell, T. P. (1987). Vibrations of a mass-loaded clamped-free Timoshenko beam. ''Journal of Sound and Vibration 114'', 341-345.

Cascante, G., Santamarina, C., and Yassir, N. (1998). Flexural excitation in a standard torsional-resonant column device. ''Can. Geotech. J., 35'', 478-490.

Cowper, G. (1966). The shear coefficient in Timoshenko's beam theory. ''Journal of Applied Mechanics 33'', 335-340.

Drnevich, V. P. (n.d.). Resonant column testing - problems and solutions. ''Dynamic Geotechnical Testing, ASTM 654'', 394-398.

Drnevich, V. P., Hall, J. R., and Richart, F. (1967). Effects of amplitude vibration on the shear modulus of sand. In N. M. Albuquerque, ''Proc. of the International Symposium on Wave Propagation and Dynamic Properties of Earth Material'' (pp. 189-199).

Farghaly, S. H. (1993). On comments on ''Vibration of a mass-loaded clamped-free Timoshenko beam''. ''Journal of Sound and Vibration 164(3)'', 549-552.

Griffin, C. (2011). Understanding the modes of deformation of dry stone retaining wall. ''Civil Engineering 3rd year Individual Project, University of Southampton''.

Hall, J. R. and Richart, F. E. (1963). Dissipation of elastic waves in granular soils. ''Journal of Soil Mechanics and Foundation. Div, 89 (SM6)''.

Hardin, B. O. and Black, W. L. (1968). (n.d.). Vibration modulus of normally consolidated clays. ''Journal of Soil Mechanics and Foundation. Div., ASCE, Vol 94, No. SM2, Proc. Paper 5833'', 353-369.

Horr, A. M. and Schmidt, L. C. (1995). Closed-form solution for the Timoshenko beam theory using a computer-based mathematical package. ''Computers & Structures Vol. 5. No. 3'', 405-412.

Huang, T. C. (1961). The effect of rotatory inertia and of shear deformation on the frequency and normal mode equations of uniform beams with simple end conditions. ''Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics 28'', 579-584.

Ishimoto, M. and Iida, K. (n.d.). Determination of elastic constants of soils by means of vibration methods. ''Bulletin of the Earthquake Research Institute'', 15-67.

Liu, W. H. (1987). On the natural frequencies of restrained cantilever beams. ''Journal of Sound and Vibration 117'', 571-572.

Liu, W. H. (1989). Comments on ''Vibrations of a mass-loaded clamped-free Timoshenko beam''. ''Journal of Sound and Vibration 129'', 343-344.

Oguamanam, D. C. (2003). Free vibration of beams with finite mass rigid tip load and flexural-torsional coupling. ''International Journal of Mechanical Sciences 45'', 963-979.

Priest, J. A. (2004). ''The Effects of Methane Gas Hydrate.'' Southampton, United Kingdom: School of Civil Engineering and the Environment, University of Southampton.

Richart, F.E., Hall, J.R., and Woods, R.D. (1970). ''Vibrations of soils and foundations.'' Englewood Cliffs, N.J.: Prentice-Hall, Inc.

Rossi, R. E., Laura, P. A. A., and Gutierrez, R. H. (1990). A note on transverse vibrations of a Timoshenko beam of non-uniform thickess clamped at one end and carrying a concentrated mass at the other. ''Journal of Sound and Vibration 143'', 491-502.

Salarieh, H., and Ghorashi, M. (2006). Free vibration of Timoshenko beam with finite mass rigid tip load and flexural-torsional coupling. ''International Journal of Mechanical Sciences 48'', 763-779.

Sniady, P. (2008). Dynamic Response of a Timoshenko Beam to a Moving Force. ''Journal of Applied Mechanics Vol. 75''.

Timoshenko, S. P. (1955). Vibration Problems in Engineering. In S. P. Timoshenko. New York: D. Van Nostrand Company, third edition.

White, M. W. D., and Heppler, G. R. (1995). Vibration Modes and Frequencies of Timoshenko Beams with Attached rigid bodies. ''Journal of Applied Mechanics Vol. 62''

[[Category:Student_engineer_essay_competition]]

Resonant column method

2012-12-14T16:45:38Z

Nicky nguyen 91:

Resonant column method

2012-12-14T16:32:23Z

Nicky nguyen 91:

= introduction =

== Background ==

The resonant column method was initially developed by Japanese engineers: Ishimoto & Iida (1937). It was made popular in the 1960s by authors such as Hall & Richart (1963), Drnevich et al. (1967) and Hardin & Black (1968). The resonant column apparatus has been used to measure the dynamic response of soils including the shear and elastic modulus based on the theory of wave propagation in prismatic rods. 

== Shear modulus (G) ==

The resonant column method was conventionally used in torsion to measure the shear modulus (G) of the material. In most cases, the clamped-free configuration has been chosen for research purpose as its mathematical derivation is more straightforward. In the clamped-free test, a cylindrical specimen is fixed at the base and excited via a drive mechanism attached to its free end. The resonant frequency (omega) is measured from which the velocity of the propagating wave is derived. Based on the derived velocity and the sample’s density, the low-strain shear modulus (G) of the material can be computed from the basic equation for torsional vibration.

== Young modulus (E) ==

The resonant column can also be used in flexural excitation to determine the material’s Young modulus (E). The conventional method with long samples allowing the application of Rayleighod’s energy method and Euler-Bernoulli beam theory, disregarded the shear strain energy and rotary inertia effect. When the tested specimen is short in length compared to its diameter, the effects of rotation and shear deformation of the samples during flexure can be substantial. These effects can be significant in interpreting data from flexural test, especially at high frequencies. Therefore, the Euler-Bernoulli theory of flexural vibration of elastic beam is found to be inadequate for short specimens and also for the prediction of higher modes of vibration. To be more accurate, Timoshenko beam theory is used as a model for this interpretation. The theory was developed by Ukrainian scientist Stephen Timoshenko in the 20th century which takes into account the shear deformation and rotary inertia. Different frequency equations for the clamped-free Timoshenko beam with an end mass in flexural vibration are solved to compute the value of elastic stiffness (E).

= The resonant column method =

== Resonant column for torsional excitation ==

In the standard torsional resonant column (Stokoe cell SBEL D1128) as mentioned in Allen and Stokoe (1982), the specimen is rigidly fixed at the base while torsional oscillation is applied to the free end by a drive head. The basic equations for the clamped-free resonant column subjected to torsion are:

[[File:Eq1.JPG|RTENOTITLE]]

The derivation of these equations is based on the assumption that the rotation is small and each transverse section remains plane and rotates about its centre. All the terms expressed in equation (2.1) are functions of the geometric properties of the specimen, except omega n. Treating the system as a single degree of freedom system, the resonant frequency measured in the resonant column apparatus is the damped natural frequency (omega d) but is sufficiently close to the natural frequency (omega n). In this case, the error can be tolerable as omega d is within 1% of omega n. Solving equation (2.1) and (2.2) with omega n, the shear wave velocity (Vs) can be found from which the shear modulus of the material (G) can be derived by rearranging equation (2.3).

== Resonant colum for flexural excitation ==

=== ===

=== Finding the Young modulus by Euler-Bernoulli beam theory (short samples) ===

The RCA can also be used to measure the Young modulus (E) of the material. Cascante et al. (1998) modified the standard Stokoe torsional resonant column (Stokoe cell SBEL D1128) to include flexural vibration mode. The schematic view of the apparatus is shown in Figure 1:

[[File:Schematic view.JPG|RTENOTITLE]] 

''Figure 1: Schematic view of the modified Stokoe RCA.''

''Image taken from Cascante et al. (1998)''

In the original configuration, four pairs of excitation coils are connected in series to produce a net torque at the top of the sample (Figure 2A). In Cascante’s modified version, the coils are reconnected so that only two magnets are used to produce a net horizontal force on top of the specimen (Figure 2B).

[[File:Coil.JPG|RTENOTITLE]]

''Figure 2'''': Coil-magnet arrangements for torsional and flexural RCA. Images taken from Cascante et al. (1998). ''''' ''A: Torsional excitation. ''''' '''''B: Flexural excitation''''' '''

In the reduction of data for flexural excitation, the specimen and its drive head can be idealised as an elastic column with a rigid point mass at the top free end (Fig 3). The behaviour of the system is assumed to be elastic. Cascante et al. (1998) has developed a general mathematical formulation for the angular resonant frequency by using Rayleigh’s energy method and Euler Bernoulli beam theory. Based on this general equation, the Young Modulus (E) can be determined by:

[[File:Eq2.JPG|RTENOTITLE]]

In previous literatures, as the cross-sectional dimensions of the sample were small in comparison with its length, Euler-Bernoulli beam theory has been used to treat the boundary conditions and derive the frequency equation, from which E could be determined.

[[File:Resonant column.JPG|RTENOTITLE]] 

''Figure 3'''': Exaggerated view of deflected column for an ''''idealised'''' system''''' ''Image taken from Priest (2004)''

=== Finding the Young modulus by Timoshenko beam theory (long samples) ===

When the tested specimens are short in length compared to their thicknesses, the effect of shear deformation during flexure is significant which can result in possible discrepancies in interpreting data from flexural test. On the other hand, the effect of rotation is large when the curvature of the beam is large relative to its thickness. This is true when the beam is short in length compared to its thickness. Therefore, Timoshenko beam theory is used in this interpretation as it takes into consideration the effect of shear deformation and rotary inertia in which the conventional Euler-Bernoulli theory doesn’t. During vibration, a typical element of a beam not only performs translatory movement, but also rotation. With shear deformation being considered, the assumption of the elementary Euler-Bernoulli theory that ‘’plane section remains plane’’ is no longer applicable. Therefore, the angle of rotation which is equal to the slope theta of any section along the length of the beam cannot be obtained by simple differentiation of the transverse displacement y. Thus, it results in two independent motions theta(x,t) and y(x,t).

Timoshenko gave the coupled equations of motion for the beam with constant cross-section as:

[[File:Eq3.JPG|RTENOTITLE]][[File:Bruch and mitchel.JPG|RTENOTITLE]] 

''Figure 4'': ''The beam-mass system used in the analysis'' '''''Image taken from Bruch and Mitchell (1987''''')

Bruch and Mitchell (1987) investigated a particular case of a cantilevered Timoshenko beam with a tip mass (Figure 4.4). By applying the boundary conditions and using Huang’s non-dimensional variables, the solutions to the coupled equations are determined as functions of the Young modulus (E), the Shear modulus (G), material’s density (rho), the angular natural frequency (omega n) and the geometry of the specimen. Bruch and Mitchell derived the frequency equation of the beam in flexural excitation by inserting the solutions to the coupled equations (2.6) and (2.7) into the boundary conditions, from which the matrix equation can be determined. By taking the determinant of the coefficient matrix equation, the resonant frequency equation was found from which the Young modulus (E) can be calculated.

Liu (1989) suggested three ways in which the work of Bruch and Mitchell could be further extended: (i) The base condition for the beam-mass system considered in [3] should be modeled as an imperfect clamped support (or elastic support), (ii) The tip mass’s centre of gravity is not practically right at the top of the beam but usually at a distance from the beam tip, (iii) the shear coefficient depends on both the shape of the cross-section and the Poisson ratio. Liu added springs at the hub to simulate the imperfect clamped support therefore the boundary condition also includes the spring’s properties which are the rotational spring constant and translational spring constant. By substituting the general solution into the new boundary conditions, Liu gave the improvement of Bruch and Mitchell’s frequency equation for the mass-loaded clamped-free Timoshenko beam.

The shear coefficient in Timoshenko’s beam theory is a dimensionless quantity, dependent on the shape of the cross section, which accounts for the fact that the shear stress and shear strain are not uniformly distributed over the cross section of the specimen. Cowper (1966) developed a new formula for the shear coefficient from the derivation of the equations of Timoshenko beam theory. For a circular cross-section, the value of K was given in terms of the Poisson ratio as:

[[File:K.JPG|RTENOTITLE]] Farghaly (1993) offered suggestion to extend Liu’s work by applying Timoshenko beam theory in treating the boundary conditions. He realised that the use of Euler-Bernoulli theory in the boundary conditions could result in inaccurate natural frequencies calculated, particularly for high slenderness ratios and higher modes of vibration. Farghaly’s model also includes the root flexibilities and the tip mass’s eccentricity as can be shown in Figure 5:

[[File:Farghaly model.JPG|RTENOTITLE]] 

''Figure 5'''': Thick beam with tip mass and root flexibilities.''''' ''Image taken from Farghaly (1993)''

The work of Bruch and Mitchell (1987), Liu (1989) and Farghaly (1993) were in an attempt to simulate the motion of a flexible robot arm modeled as a cantilevered Timoshenko beam with a lumped mass and lumped moment of inertia at the free end. However, for the purpose of this essay, their resonant frequency equations were considered to be adequate for use in computing the material’s Young modulus from the flexural resonant column test, if the angular natural frequency is known.

= Using Timoshenko’s beam theory for resonant column testing =

== Frequency equation by Bruch and Mitchell ==

Bruch and Mitchell started with the original coupled equations of motion given by Timoshenko for the beam with constant cross section: 

[[File:Eq 3.1.3.2.JPG|RTENOTITLE]]

[[File:Eq 3.3.JPG|RTENOTITLE]][[File:Eq 3.4.JPG|RTENOTITLE]] 

From the simple harmonic motion equations: 

[[File:Eq 3.8.JPG|RTENOTITLE]]

Using the non-dimensional variables and series of equations (3.1) to (3.3), equations (3.4) reduced the problem to:

[[File:Eq3.9.JPG|RTENOTITLE]][[File:Eq3.14.JPG|RTENOTITLE]][[File:Eq3.20.JPG|RTENOTITLE]][[File:Eq3.25.JPG|RTENOTITLE]] 

Taking the determinant of the coefficient matrix equation (3.20) gives the frequency equation, from which the elastic stiffness can be computed with the natural resonant frequency (omega n) as an input.

== Frequency equation by Liu ==

Liu [16] introduced a rotational spring constant (Kr) and a translational spring constant (Kl) to model the imperfection of a clamped support. For simplicity, assuming the base of the resonant column is perfectly clamped, the values of the spring constants (Kr) and (Kl) therefore approach infinity. The distance from the beam tip to the centre of the added mass (d) was added to model the eccentricity. Moment of inertia of the added mass (J) was also included in the revised matrix equation to improve the accuracy of the original model by Bruch and Mitchel. Liu started from the single free vibration equation of a Timoshenko beam given in [16], rather than the coupled equation of motion as in Bruch and Mitchell’s. The frequency equation given by Liu is:

[[File:Eq3.29.JPG|RTENOTITLE]] 

In which 

[[File:Eq3.30.JPG|RTENOTITLE]][[File:Eq3.39.JPG|RTENOTITLE]] 

Equation (3.39) was solved using Matlab with the same inputs as in the case of Bruch and Mitchell’s model, plus the eccentricity and moment of inertia of the tip mass, to evaluate the sample’s elastic stiffness.

== Frequency equation by Farghaly ==

Liu has derived a frequency equation to further improve the work of Bruch and Mitchell. The root flexibility, eccentricity and moment of inertia of the tip mass have been taken into consideration to improve the accuracy on modeling a robot arm as a clamped-free Timoshenko beam with a lumped mass and a lumped moment of inertia at its free end. The same idea of simulating a robot arm by Timoshenko beam theory can be used to model the resonant column apparatus when the sample is short in length relative to its diameter. Farghaly commented in his published paper that in [16], Liu used Timoshenko beam theory for the system differential equation, while Euler-Bernoulli theory was applied to treat the boundary conditions. Farghaly stressed that, when using Liu’s formula to compute the resonant frequency with proper inputs, inaccurate natural frequencies maybe obtained, particularly for significant values of the slenderness ratio and higher modes of vibration.

The system frequency equation in terms of the root rigidity parameters can be written as:

[[File:Eq3.40.JPG|RTENOTITLE]] 

In which

[[File:Eq3.41.JPG|RTENOTITLE]][[File:Eq3.45.JPG|RTENOTITLE]] 

In theory, to model the perfectly clamped support condition of the resonant column, the spring constants theta and K should be made to approach infinity. However, in Matlab, for simplicity, extreme values have been assigned to theta and K to give significant values of the root rigidity parameters theta and Z. As mentioned in Farghaly (1992), when using the matrix determinant equation (3.40) to compute the resonant frequencies, inaccurate results might be obtained for large values of the slenderness ratio and for higher modes of vibration. However, in our case, the length of the specimen is only about more than twice the diameter and vibration was limited to the 1st fundamental mode. According to Liu in Author’s Reply (1992), from his own practical point of view, if one can accept the idea of treating a complicated cantilever structure as a Timoshenko beam, then the discrepancies caused by non-exact boundary conditions might be considered as tolerable.

= references =

Alan, K. (2011). Stiffness and damping of sand at small strain using a resonant column. ''Civil Engineering 3rd year Individual Project, University of Southampton''.

Allen, J. C. and Stokoe, K. H. (1982). Development of resonant column apparatus with anisotropic loading. ''Geotechnical Engineering Report GR82-28, Civil Engineering Dept., University of Texas at Austin''.

Banerjee, J. R. (2001). Frequency equation and mode shape formulae for composite Timoshenko beams. ''Composite Structures 51 '', 381-388.

Bisplinghoff, R. L., and Ashley, H. (1962). ''Principles of Aeroelasticity.'' New York: Dover.

Bruch, J. C., and Mitchell, T. P. (1987). Vibrations of a mass-loaded clamped-free Timoshenko beam. ''Journal of Sound and Vibration 114'', 341-345.

Cascante, G., Santamarina, C., and Yassir, N. (1998). Flexural excitation in a standard torsional-resonant column device. ''Can. Geotech. J., 35'', 478-490.

Cowper, G. (1966). The shear coefficient in Timoshenko's beam theory. ''Journal of Applied Mechanics 33'', 335-340.

Drnevich, V. P. (n.d.). Resonant column testing - problems and solutions. ''Dynamic Geotechnical Testing, ASTM 654'', 394-398.

Drnevich, V. P., Hall, J. R., and Richart, F. (1967). Effects of amplitude vibration on the shear modulus of sand. In N. M. Albuquerque, ''Proc. of the International Symposium on Wave Propagation and Dynamic Properties of Earth Material'' (pp. 189-199).

Farghaly, S. H. (1993). On comments on ''Vibration of a mass-loaded clamped-free Timoshenko beam''. ''Journal of Sound and Vibration 164(3)'', 549-552.

Griffin, C. (2011). Understanding the modes of deformation of dry stone retaining wall. ''Civil Engineering 3rd year Individual Project, University of Southampton''.

Hall, J. R. and Richart, F. E. (1963). Dissipation of elastic waves in granular soils. ''Journal of Soil Mechanics and Foundation. Div, 89 (SM6)''.

Hardin, B. O. and Black, W. L. (1968). (n.d.). Vibration modulus of normally consolidated clays. ''Journal of Soil Mechanics and Foundation. Div., ASCE, Vol 94, No. SM2, Proc. Paper 5833'', 353-369.

Horr, A. M. and Schmidt, L. C. (1995). Closed-form solution for the Timoshenko beam theory using a computer-based mathematical package. ''Computers & Structures Vol. 5. No. 3'', 405-412.

Huang, T. C. (1961). The effect of rotatory inertia and of shear deformation on the frequency and normal mode equations of uniform beams with simple end conditions. ''Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics 28'', 579-584.

Ishimoto, M. and Iida, K. (n.d.). Determination of elastic constants of soils by means of vibration methods. ''Bulletin of the Earthquake Research Institute'', 15-67.

Liu, W. H. (1987). On the natural frequencies of restrained cantilever beams. ''Journal of Sound and Vibration 117'', 571-572.

Liu, W. H. (1989). Comments on ''Vibrations of a mass-loaded clamped-free Timoshenko beam''. ''Journal of Sound and Vibration 129'', 343-344.

Oguamanam, D. C. (2003). Free vibration of beams with finite mass rigid tip load and flexural-torsional coupling. ''International Journal of Mechanical Sciences 45'', 963-979.

Priest, J. A. (2004). ''The Effects of Methane Gas Hydrate.'' Southampton, United Kingdom: School of Civil Engineering and the Environment, University of Southampton.

Richart, F.E., Hall, J.R., and Woods, R.D. (1970). ''Vibrations of soils and foundations.'' Englewood Cliffs, N.J.: Prentice-Hall, Inc.

Rossi, R. E., Laura, P. A. A., and Gutierrez, R. H. (1990). A note on transverse vibrations of a Timoshenko beam of non-uniform thickess clamped at one end and carrying a concentrated mass at the other. ''Journal of Sound and Vibration 143'', 491-502.

Salarieh, H., and Ghorashi, M. (2006). Free vibration of Timoshenko beam with finite mass rigid tip load and flexural-torsional coupling. ''International Journal of Mechanical Sciences 48'', 763-779.

Sniady, P. (2008). Dynamic Response of a Timoshenko Beam to a Moving Force. ''Journal of Applied Mechanics Vol. 75''.

Timoshenko, S. P. (1955). Vibration Problems in Engineering. In S. P. Timoshenko. New York: D. Van Nostrand Company, third edition.

White, M. W. D., and Heppler, G. R. (1995). Vibration Modes and Frequencies of Timoshenko Beams with Attached rigid bodies. ''Journal of Applied Mechanics Vol. 62''

[[Category:Student_engineer_essay_competition]]

Resonant column method

2012-12-14T16:30:10Z

Nicky nguyen 91:

= introduction =

== Background ==

The resonant column method was initially developed by Japanese engineers: Ishimoto & Iida (1937). It was made popular in the 1960s by authors such as Hall & Richart (1963), Drnevich et al. (1967) and Hardin & Black (1968). The resonant column apparatus has been used to measure the dynamic response of soils including the shear and elastic modulus based on the theory of wave propagation in prismatic rods. 

== Shear modulus (G) ==

The resonant column method was conventionally used in torsion to measure the shear modulus (G) of the material. In most cases, the clamped-free configuration has been chosen for research purpose as its mathematical derivation is more straightforward. In the clamped-free test, a cylindrical specimen is fixed at the base and excited via a drive mechanism attached to its free end. The resonant frequency (omega) is measured from which the velocity of the propagating wave is derived. Based on the derived velocity and the sample’s density, the low-strain shear modulus (G) of the material can be computed from the basic equation for torsional vibration.

== Young modulus (E) ==

The resonant column can also be used in flexural excitation to determine the material’s Young modulus (E). The conventional method with long samples allowing the application of Rayleighod’s energy method and Euler-Bernoulli beam theory, disregarded the shear strain energy and rotary inertia effect. When the tested specimen is short in length compared to its diameter, the effects of rotation and shear deformation of the samples during flexure can be substantial. These effects can be significant in interpreting data from flexural test, especially at high frequencies. Therefore, the Euler-Bernoulli theory of flexural vibration of elastic beam is found to be inadequate for short specimens and also for the prediction of higher modes of vibration. To be more accurate, Timoshenko beam theory is used as a model for this interpretation. The theory was developed by Ukrainian scientist Stephen Timoshenko in the 20th century which takes into account the shear deformation and rotary inertia. Different frequency equations for the clamped-free Timoshenko beam with an end mass in flexural vibration are solved to compute the value of elastic stiffness (E).

= The resonant column method =

== Resonant column for torsional excitation ==

In the standard torsional resonant column (Stokoe cell SBEL D1128) as mentioned in Allen and Stokoe (1982), the specimen is rigidly fixed at the base while torsional oscillation is applied to the free end by a drive head. The basic equations for the clamped-free resonant column subjected to torsion are:

[[File:Eq1.JPG|RTENOTITLE]]

The derivation of these equations is based on the assumption that the rotation is small and each transverse section remains plane and rotates about its centre. All the terms expressed in equation (2.1) are functions of the geometric properties of the specimen, except omega n. Treating the system as a single degree of freedom system, the resonant frequency measured in the resonant column apparatus is the damped natural frequency (omega d) but is sufficiently close to the natural frequency (omega n). In this case, the error can be tolerable as omega d is within 1% of omega n. Solving equation (2.1) and (2.2) with omega n, the shear wave velocity (Vs) can be found from which the shear modulus of the material (G) can be derived by rearranging equation (2.3).

== Resonant colum for flexural excitation ==

=== ===

=== Finding the Young modulus by Euler-Bernoulli beam theory (short samples) ===

The RCA can also be used to measure the Young modulus (E) of the material. Cascante et al. (1998) modified the standard Stokoe torsional resonant column (Stokoe cell SBEL D1128) to include flexural vibration mode. The schematic view of the apparatus is shown in Figure 1:

[[File:Schematic view.JPG|RTENOTITLE]] 

''Figure 1: Schematic view of the modified Stokoe RCA.''

''Image taken from Cascante et al. (1998)''

In the original configuration, four pairs of excitation coils are connected in series to produce a net torque at the top of the sample (Figure 2A). In Cascante’s modified version, the coils are reconnected so that only two magnets are used to produce a net horizontal force on top of the specimen (Figure 2B).

[[File:Coil.JPG|RTENOTITLE]]

''Figure 2'''': Coil-magnet arrangements for torsional and flexural RCA. Images taken from Cascante et al. (1998). ''''' ''A: Torsional excitation. ''''' '''''B: Flexural excitation''''' '''

In the reduction of data for flexural excitation, the specimen and its drive head can be idealised as an elastic column with a rigid point mass at the top free end (Fig 3). The behaviour of the system is assumed to be elastic. Cascante et al. (1998) has developed a general mathematical formulation for the angular resonant frequency by using Rayleigh’s energy method and Euler Bernoulli beam theory. Based on this general equation, the Young Modulus (E) can be determined by:

[[File:Eq2.JPG|RTENOTITLE]]

In previous literatures, as the cross-sectional dimensions of the sample were small in comparison with its length, Euler-Bernoulli beam theory has been used to treat the boundary conditions and derive the frequency equation, from which E could be determined.

[[File:Resonant column.JPG|RTENOTITLE]] 

''Figure 3'''': Exaggerated view of deflected column for an ''''idealised'''' system''''' ''Image taken from Priest (2004)''

=== Finding the Young modulus by Timoshenko beam theory (long samples) ===

When the tested specimens are short in length compared to their thicknesses, the effect of shear deformation during flexure is significant which can result in possible discrepancies in interpreting data from flexural test. On the other hand, the effect of rotation is large when the curvature of the beam is large relative to its thickness. This is true when the beam is short in length compared to its thickness. Therefore, Timoshenko beam theory is used in this interpretation as it takes into consideration the effect of shear deformation and rotary inertia in which the conventional Euler-Bernoulli theory doesn’t. During vibration, a typical element of a beam not only performs translatory movement, but also rotation. With shear deformation being considered, the assumption of the elementary Euler-Bernoulli theory that ‘’plane section remains plane’’ is no longer applicable. Therefore, the angle of rotation which is equal to the slope theta of any section along the length of the beam cannot be obtained by simple differentiation of the transverse displacement y. Thus, it results in two independent motions theta(x,t) and y(x,t).

Timoshenko gave the coupled equations of motion for the beam with constant cross-section as:

[[File:Eq3.JPG|RTENOTITLE]][[File:Bruch and mitchel.JPG|RTENOTITLE]] 

''Figure 4'': ''The beam-mass system used in the analysis'' '''''Image taken from Bruch and Mitchell (1987''''')

Bruch and Mitchell (1987) investigated a particular case of a cantilevered Timoshenko beam with a tip mass (Figure 4.4). By applying the boundary conditions and using Huang’s non-dimensional variables, the solutions to the coupled equations are determined as functions of the Young modulus (E), the Shear modulus (G), material’s density (rho), the angular natural frequency (omega n) and the geometry of the specimen. Bruch and Mitchell derived the frequency equation of the beam in flexural excitation by inserting the solutions to the coupled equations (2.6) and (2.7) into the boundary conditions, from which the matrix equation can be determined. By taking the determinant of the coefficient matrix equation, the resonant frequency equation was found from which the Young modulus (E) can be calculated.

Liu (1989) suggested three ways in which the work of Bruch and Mitchell could be further extended: (i) The base condition for the beam-mass system considered in [3] should be modeled as an imperfect clamped support (or elastic support), (ii) The tip mass’s centre of gravity is not practically right at the top of the beam but usually at a distance from the beam tip, (iii) the shear coefficient depends on both the shape of the cross-section and the Poisson ratio. Liu added springs at the hub to simulate the imperfect clamped support therefore the boundary condition also includes the spring’s properties which are the rotational spring constant and translational spring constant. By substituting the general solution into the new boundary conditions, Liu gave the improvement of Bruch and Mitchell’s frequency equation for the mass-loaded clamped-free Timoshenko beam.

The shear coefficient in Timoshenko’s beam theory is a dimensionless quantity, dependent on the shape of the cross section, which accounts for the fact that the shear stress and shear strain are not uniformly distributed over the cross section of the specimen. Cowper (1966) developed a new formula for the shear coefficient from the derivation of the equations of Timoshenko beam theory. For a circular cross-section, the value of K was given in terms of the Poisson ratio as:

[[File:K.JPG|RTENOTITLE]] Farghaly (1993) offered suggestion to extend Liu’s work by applying Timoshenko beam theory in treating the boundary conditions. He realised that the use of Euler-Bernoulli theory in the boundary conditions could result in inaccurate natural frequencies calculated, particularly for high slenderness ratios and higher modes of vibration. Farghaly’s model also includes the root flexibilities and the tip mass’s eccentricity as can be shown in Figure 5:

[[File:Farghaly model.JPG|RTENOTITLE]] 

''Figure 5'''': Thick beam with tip mass and root flexibilities.''''' ''Image taken from Farghaly (1993)''

The work of Bruch and Mitchell (1987), Liu (1989) and Farghaly (1993) were in an attempt to simulate the motion of a flexible robot arm modeled as a cantilevered Timoshenko beam with a lumped mass and lumped moment of inertia at the free end. However, for the purpose of this essay, their resonant frequency equations were considered to be adequate for use in computing the material’s Young modulus from the flexural resonant column test, if the angular natural frequency is known.

= Using Timoshenko’s beam theory for resonant column testing =

== Frequency equation by Bruch and Mitchell ==

Bruch and Mitchell started with the original coupled equations of motion given by Timoshenko for the beam with constant cross section: 

[[File:Eq 3.1.3.2.JPG|RTENOTITLE]]

[[File:Eq 3.3.JPG|RTENOTITLE]][[File:Eq 3.4.JPG|RTENOTITLE]] 

From the simple harmonic motion equations: 

[[File:Eq 3.8.JPG|RTENOTITLE]]

Using the non-dimensional variables and series of equations (3.1) to (3.3), equations (3.4) reduced the problem to:

[[File:Eq3.9.JPG|RTENOTITLE]][[File:Eq3.14.JPG|RTENOTITLE]][[File:Eq3.20.JPG|RTENOTITLE]][[File:Eq3.25.JPG|RTENOTITLE]] 

Taking the determinant of the coefficient matrix equation (3.20) gives the frequency equation, from which the elastic stiffness can be computed with the natural resonant frequency (omega n) as an input.

== Frequency equation by Liu ==

Liu [16] introduced a rotational spring constant (Kr) and a translational spring constant (Kl) to model the imperfection of a clamped support. For simplicity, assuming the base of the resonant column is perfectly clamped, the values of the spring constants (Kr) and (Kl) therefore approach infinity. The distance from the beam tip to the centre of the added mass (d) was added to model the eccentricity. Moment of inertia of the added mass (J) was also included in the revised matrix equation to improve the accuracy of the original model by Bruch and Mitchel. Liu started from the single free vibration equation of a Timoshenko beam given in [16], rather than the coupled equation of motion as in Bruch and Mitchell’s. The frequency equation given by Liu is:

[[File:Eq3.29.JPG|RTENOTITLE]] 

In which 

[[File:Eq3.30.JPG|RTENOTITLE]][[File:Eq3.39.JPG|RTENOTITLE]] 

Equation (3.39) was solved using Matlab with the same inputs as in the case of Bruch and Mitchell’s model, plus the eccentricity and moment of inertia of the tip mass, to evaluate the sample’s elastic stiffness.

== Frequency equation by Farghaly ==

Liu has derived a frequency equation to further improve the work of Bruch and Mitchell. The root flexibility, eccentricity and moment of inertia of the tip mass have been taken into consideration to improve the accuracy on modeling a robot arm as a clamped-free Timoshenko beam with a lumped mass and a lumped moment of inertia at its free end. The same idea of simulating a robot arm by Timoshenko beam theory can be used to model the RCA when the sample is short in length relative to its diameter. Farghaly commented in his published paper that in [16], Liu used Timoshenko beam theory for the system differential equation, while Euler-Bernoulli theory was applied to treat the boundary conditions. Farghaly stressed that, when using Liu’s formula to compute the resonant frequency with proper inputs, inaccurate natural frequencies maybe obtained, particularly for significant values of the slenderness ratio and higher modes of vibration.

The system frequency equation in terms of the root rigidity parameters can be written as:

[[File:Eq3.40.JPG|RTENOTITLE]] 

In which

[[File:Eq3.41.JPG|RTENOTITLE]][[File:Eq3.45.JPG|RTENOTITLE]] 

In theory, to model the perfectly clamped support condition of the resonant column, the spring constants theta and K should be made to approach infinity. However, in Matlab, for simplicity, extreme values have been assigned to theta and K to give significant values of the root rigidity parameters theta and Z. As mentioned in Farghaly (1992), when using the matrix determinant equation (3.40) to compute the resonant frequencies, inaccurate results might be obtained for large values of the slenderness ratio and for higher modes of vibration. However, in our case, the length of the specimen is only about more than twice the diameter and vibration was limited to the 1st fundamental mode. According to Liu in Author’s Reply (1992), from his own practical point of view, if one can accept the idea of treating a complicated cantilever structure as a Timoshenko beam, then the discrepancies caused by non-exact boundary conditions might be considered as tolerable.

= references =

Alan, K. (2011). Stiffness and damping of sand at small strain using a resonant column. ''Civil Engineering 3rd year Individual Project, University of Southampton''.

Allen, J. C. and Stokoe, K. H. (1982). Development of resonant column apparatus with anisotropic loading. ''Geotechnical Engineering Report GR82-28, Civil Engineering Dept., University of Texas at Austin''.

Banerjee, J. R. (2001). Frequency equation and mode shape formulae for composite Timoshenko beams. ''Composite Structures 51 '', 381-388.

Bisplinghoff, R. L., and Ashley, H. (1962). ''Principles of Aeroelasticity.'' New York: Dover.

Bruch, J. C., and Mitchell, T. P. (1987). Vibrations of a mass-loaded clamped-free Timoshenko beam. ''Journal of Sound and Vibration 114'', 341-345.

Cascante, G., Santamarina, C., and Yassir, N. (1998). Flexural excitation in a standard torsional-resonant column device. ''Can. Geotech. J., 35'', 478-490.

Cowper, G. (1966). The shear coefficient in Timoshenko's beam theory. ''Journal of Applied Mechanics 33'', 335-340.

Drnevich, V. P. (n.d.). Resonant column testing - problems and solutions. ''Dynamic Geotechnical Testing, ASTM 654'', 394-398.

Drnevich, V. P., Hall, J. R., and Richart, F. (1967). Effects of amplitude vibration on the shear modulus of sand. In N. M. Albuquerque, ''Proc. of the International Symposium on Wave Propagation and Dynamic Properties of Earth Material'' (pp. 189-199).

Farghaly, S. H. (1993). On comments on ''Vibration of a mass-loaded clamped-free Timoshenko beam''. ''Journal of Sound and Vibration 164(3)'', 549-552.

Griffin, C. (2011). Understanding the modes of deformation of dry stone retaining wall. ''Civil Engineering 3rd year Individual Project, University of Southampton''.

Hall, J. R. and Richart, F. E. (1963). Dissipation of elastic waves in granular soils. ''Journal of Soil Mechanics and Foundation. Div, 89 (SM6)''.

Hardin, B. O. and Black, W. L. (1968). (n.d.). Vibration modulus of normally consolidated clays. ''Journal of Soil Mechanics and Foundation. Div., ASCE, Vol 94, No. SM2, Proc. Paper 5833'', 353-369.

Horr, A. M. and Schmidt, L. C. (1995). Closed-form solution for the Timoshenko beam theory using a computer-based mathematical package. ''Computers & Structures Vol. 5. No. 3'', 405-412.

Huang, T. C. (1961). The effect of rotatory inertia and of shear deformation on the frequency and normal mode equations of uniform beams with simple end conditions. ''Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics 28'', 579-584.

Ishimoto, M. and Iida, K. (n.d.). Determination of elastic constants of soils by means of vibration methods. ''Bulletin of the Earthquake Research Institute'', 15-67.

Liu, W. H. (1987). On the natural frequencies of restrained cantilever beams. ''Journal of Sound and Vibration 117'', 571-572.

Liu, W. H. (1989). Comments on ''Vibrations of a mass-loaded clamped-free Timoshenko beam''. ''Journal of Sound and Vibration 129'', 343-344.

Oguamanam, D. C. (2003). Free vibration of beams with finite mass rigid tip load and flexural-torsional coupling. ''International Journal of Mechanical Sciences 45'', 963-979.

Priest, J. A. (2004). ''The Effects of Methane Gas Hydrate.'' Southampton, United Kingdom: School of Civil Engineering and the Environment, University of Southampton.

Richart, F.E., Hall, J.R., and Woods, R.D. (1970). ''Vibrations of soils and foundations.'' Englewood Cliffs, N.J.: Prentice-Hall, Inc.

Rossi, R. E., Laura, P. A. A., and Gutierrez, R. H. (1990). A note on transverse vibrations of a Timoshenko beam of non-uniform thickess clamped at one end and carrying a concentrated mass at the other. ''Journal of Sound and Vibration 143'', 491-502.

Salarieh, H., and Ghorashi, M. (2006). Free vibration of Timoshenko beam with finite mass rigid tip load and flexural-torsional coupling. ''International Journal of Mechanical Sciences 48'', 763-779.

Sniady, P. (2008). Dynamic Response of a Timoshenko Beam to a Moving Force. ''Journal of Applied Mechanics Vol. 75''.

Timoshenko, S. P. (1955). Vibration Problems in Engineering. In S. P. Timoshenko. New York: D. Van Nostrand Company, third edition.

White, M. W. D., and Heppler, G. R. (1995). Vibration Modes and Frequencies of Timoshenko Beams with Attached rigid bodies. ''Journal of Applied Mechanics Vol. 62''

[[Category:Student_engineer_essay_competition]]

Off-site prefabrication of buildings: A guide to connection choices

2012-12-14T16:26:43Z

Nicky nguyen 91:

= Introduction =

Off-site prefabrication is a possible solution to many issues regarding the underachievement of the UK’s construction industry including: safety record, public perception, client satisfaction, profitability, delays, skilled workforce and overall contribution to the national economy. Prefabrication is believed to be advantageous over traditional construction methods in the followings:
*Quality – Higher-quality finishes with defects eliminated prior to completion.
*Safety – Safer working environment under factory conditions.
*Cost – Repeated use of moulds through standardisation reduces formwork materials, preliminaries, site storage and on-site facilities.
*Waste – Reduced off-cuts from formwork and the introduction of prefabricated rebars.
*Programme – Increased predictability due to reduced external factors such as weather.
*Local disruption – less environmental impacts such as dust and noise pollution.
*Accuracy – Increased accuracy since templates produced using Computer Aided Design (CAD) systems.
*Timescale – Components built off-site leads to reduced on-site construction time.

Among those mentioned, a driving factor for using prefabrication is to improve both quality and safety, as are rated 4.3 and 3.9 respectively on a five point Likert scale (Pann et al, 2008).

[[File:Laings O Rourke precast factory.JPG|RTENOTITLE]]

Figure 1: Laing O'Rourke's Explore precast manufacturing facility, Steetley, UK (Croxon, 2010) 

= History =

The World War One (1910s) and World War Two (1940s) stimulated research into newer methods than traditional brick construction. At that time, the requirement for rapid construction and the willingness to pay for it were the driving forces, in contrast to the major shortages of skilled labour and building materials. In the 1940s, UK government promoted methods of newer construction from the industries. This eventually led to the construction of hundreds of prefabricated concrete tower blocks and thousands of schools in the 1950s and 1960s which were often poorly designed. These were of low cost and often built without the lifetime of the buildings considered. Volumetric construction, the construction technique involving the production of buildings as a number of boxes connected on site, was used throughout the 1960s and 1970s. The collapse of the Ronan Point tower block in East London in 1968 is well known and attributed as one of the reasons for continued suspicion, fear and decline of prefabrication in this country.

Following the above downturn, there is currently a shift towards prefabrication within the industry. Many UK major construction firms are starting to see the benefits of prefabrication. Kier, Interserve, NG Bailey, Arup, Capita Symonds and Laing O’Rourke (LOR) are some of the market leaders registering their interest. LOR is recognised as having the largest interest, investing £100m in its Design for Manufacture and Assembly facility located at its Explore industrial park, which is “the most advanced facility of its type in Europe”. They aim to take advantage of public and private sector clients including BAA, Premier Inn, the Department of Education and both the Ministry of Justice and the Ministry of Defence (Wright, 2010), who all consider off-site prefabrication solutions.

Although a current shortage of a skilled workforce is said to be the cause, the increased uptake of prefabrication is believed to be a permanent move, as opposed to the short-lived uptake seen in the 60s and the 70s.

[[File:Ronan point tower block.JPG|RTENOTITLE]] 

Figure 2: Ronan Point tower block failure, East London 1968 (Daily Telegraph, 1968)

= Precast construction method =

As with in-situ reinforced concrete construction, precast construction lends itself to a variety of different construction techniques, layouts and sequences.

== Frame and Deck Construction ==

[[File:Frame and deck construction.JPG|RTENOTITLE]] 

Figure 3: Frame and deck systems, a) single storey columns, b) multi storey columns (Task Group 6.2 F.I.B., 2008, p. 5).

A precast deck supported by precast beams and columns form the building’s structural system. This form is frequently used in the construction of multi storey car parks with up to 16m spans to reduce columns between car parking spaces. It can also be used where floor to beam soffit height does not need to be minimised. The overall column height within a frame and deck system may correspond to greater than one storey.

The following connections are utilised in frames and deck construction:
*column to column
*column to base
*beam to column
*beam to base

[[File:Diagram.jpg|RTENOTITLE]] 

Figure 4: Diagram of the precast elements and connections used in frame and deck construction (Irish Precast Concrete Association, 2003)

== Crosswall Construction ==

Crosswall is a modern method where load bearing walls provide the primary vertical support for precast floors and lateral stability. External wall panels, lift cores or staircases are used to provide the required longitudinal stability. Bridging components such as floors, roofs and beams are supported by the load bearing walls or façade wall. The system is ideal for buildings with cellular and orthogonal grids, with rooms of up to 4mx9m as standard. Thus it leads to a structurally efficient building with high levels of sound and fire insulation between adjacent rooms.

Crosswall construction utilises the following connections:
*wall to wall at vertical joints
*wall to wall at horizontal joints
*wall to base/foundation

[[File:Crosswall construction.JPG|RTENOTITLE]] 

Figure 5 - Crosswall systems, a) load bearing crosswall, b) load bearing facade wall (Task Group 6.2 F.I.B., 2008, p. 8). 

The precast elements are brought to site just in time allowing them to be lifted from the transport vehicle and installed in place. Hidden joints and ties, both horizontally and vertically are grouted in place as the work develops, allowing progressive collapse criteria of the Building Regulations to be met. With the possibility of incorporated mechanical and electrical components and minimal finishing needed, following trades can start prior to the completion of precast erection. 

== Volumetric Construction ==

The term volumetric construction is given when concrete modules (constructed in the factory) are installed on site to form a cellular system or used independently as a self-contained cell. The modules can be cast as a room or as panels which are subsequently joined together in the factory prior to site delivery. For a cellular system, the ground floor cells are laid on pre-prepared ground floor slabs with individual modules lowered into place usually forming the roof of the unit below.

[[File:Volumetric construction.JPG|RTENOTITLE]] 

Figure 6 - Examples of volumetric construction used for prison construction with an example of a possible finish (Oldcastle Precast Inc. ). 

Cellular systems are advantageous when being incorporated with a repetitive design (Figure 6). Common uses include hotels, prison cells, student halls and residential buildings. Self-contained cells are used mostly for specialised purposes where services are needed such as wet rooms, bathroom pods (Figure 7) and service utility rooms. Once lifted into place, the modules are secured by a number of methods including bolted and doweled connections. 

[[File:Volumetric bathroom.JPG|RTENOTITLE]]

Figure 7 - Volumetric construction in the form of a bathroom pod lifted into a crosswall frame (The Concrete Centre, 2007a). 

== Hybrid Construction ==

Hybrid construction is not a standalone method of precast construction. It is mentioned for its use in conjunction with the above construction methods. Precast elements can be used to provide permanent formwork for in-situ concrete. The combination of in-situ and precast concrete allows the benefits of both to be utilised. Figure 8 shows how safe working platforms are created by the precast floors, which increases safety on site and omits the need for in-situ concrete formwork, both factors significantly decrease the construction time. Greater spans can be achieved with hybrid construction as composite action is achieved by using different structural materials for the upper and lower areas of the element. The interface between the two materials will have to withstand shear stresses which can be overcome through the use of shear studs or precast reinforcement in the floor slab. 

[[File:Hybrid construction.JPG|RTENOTITLE]]

Figure 8 - Construction site utilising hybrid construction. a) Temporary props in place to support precast lattice floor slabs b) Installed precast lattice floor slabs awaiting reinforcement c) Concrete curing and binding with precast slab beneath. (webbaviation on behalf of Laing O'Rourke, 2011) 

= Precast Connections =

== Classification of connections ==

[[File:Precast connection.jpg|RTENOTITLE]] 

Figure 9 - Generic forms of beam-column connections, A: hidden beam end connection, B: corbel / haunch bearing connection, C: continuous column connection, D: continuous beam connection (Task Group 6.2, Fédération internationale du béton., 2008, p. 289)

For structural design, the connection’s stiffness is important in designing for moment distribution. There are three classes of connections based upon their degree of rigidity:
*Rigid connection – This connection can sustain vertical and horizontal actions as well as bending moment. The relative angle between connected members is maintained due to the stiffness of the connection.
*Pinned connection – This connection can sustain vertical and horizontal actions but not bending moment. The connected members are free to rotate in one direction with the connection having no degree of stiffness.
*Semi-rigid connection – This connection is between rigid or pinned as it is able to sustain vertical and horizontal actions and some amount of moment.

With in-situ reinforced concrete construction, a monolithic rigid connection is usually produced through design and provided on site. Precast connections range in their level of rigidity, from fully rigid to a completely pinned connection. A true pinned connection containing zero moment capacity is rare. In fact, many connections have some degree of rigidity but are conservatively assumed pinned. The steel connection shown in Figure 10 will retain some degree of rigidity, yet is usually modelled in design as a pinned connection. This is a conservative measure as beams spanning pinned connections are subject to the full action moment. Due to the connection having some degree of stiffness and therefore moment capacity, the negative bending moment acting upon the beam will be overestimated. 

[[File:Connectin classification.JPG|RTENOTITLE]] 

Figure 10 - Steel shear plate connection to CHS column (Kurobane et al., 2005)

Within the steel industry, research has shown cost reductions of between 10 to 20% for semi-rigid frames over rigid frames (Kurobane et al., 2005). Therefore the level of rigidity is an important consideration when choosing a method of connecting precast concrete elements.

== Continuous column with Corbel connections ==

[[File:Corbel connection.JPG|RTENOTITLE]]
<div>Figure 11 -Examples of concrete corbel connection with continuity reinforcement. A. (The Concrete Centre, 2007a) B. (Task Group 6.2 F.I.B., 2008, p. 49) </div>
Corbel connections, as shown in Figure 11 are most often used to support long span beams or heavy loads. Due to the visual and physical intrusion caused by the corbel or haunch widening the column, this connection is not widely used in the construction of multi-storey concrete frames. The basic corbel connection is designed as simply supported, dowel bars and/or fixing cleats. This type of connection can be used to prevent lateral movement and provide some joint fixity, although research has proven that the basic dowelled connection is best modelled as pinned. In-situ, structural screed can be used to increase continuity of the connection, thus allowing the tension reinforcement to resist the forces arising from beam movements. This can either be at the end, or across the whole length of the beam or floor slab. It was shown that a corbel / haunch connections with small amounts of cast in place reinforced concrete, although designed as a simply supported pinned connection, can improve strength and stiffness resulting in a semi or often fully rigid connection.

[[File:Corbel connection real example.JPG|RTENOTITLE]] 

Figure 12 - Photo of example beam to column corbel connection (General Precast Concrete Ltd, 2008). 

== Continuous beam connection ==

[[File:Continuous beam connection.JPG|RTENOTITLE]] 

Figure 13 - Discontinuous column with continuous beam, Left: (Elliot, 1992, p.67), Right: (Task Group 6.2 F.I.B., 2008, p. 299)

This type of connection is mainly used in portal frames or in skeletal frames when beams need to be continuous over supports, as is required for a cantilever. The beams are seated on dry pack mortar on top of the vertical members and reinforcing starter bars are projected through sleeves in the beam from the lower column up into the upper column. These sleeves are subsequently grouted to provide vertical continuity. Once the beam is lowered into place, this connection requires no additional formwork providing the grout is poured through vents in the upper column. Therefore, provided the remaining beam end is secured, loads for construction access can be placed upon the beam. This enhances the simplicity of installation and therefore safety on site.

== Wall and Column shoes ==

[[File:Wall and Column shoe connection 1.JPG|RTENOTITLE]] 

Figure 14 - Photos of example hidden corbel systems used with a precast column and precast beam (Peikko concrete connections, 2009)

Investment into modern technologies has resulted in the production of the hidden corbel connection. This is the most popular type of precast connection used in the UK so far. This type produces fireproof connections which are architecturally advantageous as they minimise visual intrusion whilst maximising floor to soffit height.

The connection area is minimal, protecting the reinforcement steel used in the connection. The connection also benefits from superior adjustability with the modern connection utilising a small adjustable plate, allowing fine tuning of the column corbel prior to installation of the beam.

[[File:Wall and Column shoe connection 2.JPG|RTENOTITLE]] 

Figure 15 – (Left) Inside detailed view of the anchorage of the corbel connection into the precast column and beams (Peikko concrete connections, 2009). (Right) Labelled example of an alternative hidden corbel system (JVI inc.)

== Ground foundation column connections ==

There are three main methods of connecting columns to foundations:
*Projecting starter bars – The in-situ foundation houses cast in starter bars which the precast column is later lowered onto and grouted to provide continuity.
*Pocket connection – This is the most rigid connection and is utilised when the moment resisting capacity of the connection is required for the lateral stability of the structure. A pocket is provided within the foundation into which the precast column is lowered. The surrounding area is grouted or filled with in-situ concrete.
*Baseplate connection – The base of the precast column contains steel base plates which cast-in bolts are fed through and bolted into place. The surrounding area to the holding down bolts is then filled with non-shrink grout to complete the connection.

The three types above are conservatively modelled as pinned connections resulting in an underestimate of the moments transferred to the columns and beams above. The foundation column connection is subjected to certain degree of variability such as possible rotations due to ground conditions.

= Connection’s discussion and evaluation =

There exists a variety of precast connection types within each big group above. The connections are assessed against different criteria including: the amount of additional materials; aesthetic/space intrusion of the finished connections; allowable tolerance; amount of wet work formwork required; possibility of future reuse/dismantle; operative involvement on site; level of rigidity; safety; skills required; amount of temporary works; time of assembly; tools required; weather sensitivity and level of wet casting needed.

== Continuous beam connection ==

[[File:Discussion - continuous beam.JPG|RTENOTITLE]] 

Figure 16 - Bolted steel shoe diagram (Halfen GmbH, 2011)

The bolted steel shoe is considered to be the most favourable of the continuous beam connection type as it is simple to produce and quick to assemble on site. This connection requires no structural in-situ works compared to other sub-types. The connection can have a number of different bolt arrangements depending on the size and shape of the column. When used correctly the anchor bolts can be utilised to transfer both tensile and compressive load through to the column below, thus minimising stress on the beam / slab in between. Alternatively the beam / slab can be suitably designed to transfer the load to the column below.

Due to the presence of a continuous beam, the large hogging moments generated at the connection will be transferred to the column. The moment capacity of this connection is high due to the high tensile capacity of the steel holding down bolts resisting the rotation of the column due to buckling, which may result from the hogging moments transferred from the beams.

The amount and positioning of the holding down bolts will determine the connections rigidity. The closer the bolts are to the centre point in the plane of rotation the more the connection will represent a pinned connection between the columns and the beam. As shown in Figure 16, the bolts have been positioned to give the maximum lever arm against any point of pivot and thus maximises rotational resistance.

== Corbel/Billet connection ==

[[File:Discussion-corbel.JPG|RTENOTITLE]] 

Figure 17 - Bearing only connection diagram (Pujol Group)

The bearing only is the favoured connection due to its simple straightforward design which facilitates quick assembly time on site. This fully pinned connection type will therefore transfer vertical and horizontal loads into the column, but all moment will be contained within the beam. For this reason, the beam will be designed to resist greater moment. The connection is therefore less efficient than moment sustaining connections.

The bearing with bolted dowel bar which has been taken as the preferred connection method for this connection type. The bearing with bolted dowel bar allows reduced connection width due to dowel action acting as horizontal restraint. The connection is fixed at the top using a bolt which extends through the column. This is positioned to resist maximum bending moment as the point of pivot will be within the underside of the beam. It is common practice in design to ensure that fixing elements of connections are not the limiting element and therefore the bolt will be able to transfer a considerable amount of moment to the column. The flat landing of the corbel, although unsightly, when combined with the dowel, acts as a torsional restraint. This can be further improved by using a cleat which extends the width of the beam. As the column is continuous, the beam will be required to sustain the majority of bending moment. This connection is semi rigid; therefore it can sustain vertical and horizontal loads with a degree of hogging moment transferred to the column.

[[File:Discussion-corbel 2.JPG|RTENOTITLE]]

Figure 18 - Bearing with bolted dowel, the preferred connection from billet / corbel type as bearing only connection disregarded

== Concealed fixing ==

The concealed bolted steel billet, which is one of the most modern connections, is favoured within this category. It utilises the advantages of other connections with an aesthetically pleasing and simple design, which allows minor adjustments to be made to the plate rather than the beam. There are other connections which show greater rigidity but require greater installing time on site, therefore are less favoured. This connection transfers horizontal and vertical loads to the column through the bolted connection. Its moment resisting capacity is small when compared to the bolted doweled corbel. However, as the connection extends to full height of the beam it is well positioned to sustain some moment. As there are just two bolts per beam to column connection, the level of moment transfer will be limited. For this reason the connection will act as a semi-rigid connection.

[[File:Concealed fixing.JPG|RTENOTITLE]]

Figure 19 - Preferred concealed connection, bolted steel corbel connection (Peikko concrete connections, 2009)

= Conclusion =

== Vertical load resistance ==

All three connections are capable of transferring the vertical load both from the column above and from the beams. As all three of the preferred connections from each group utilise a bolted mechanism to provide fixity, it is feasible that a structure constructed using only these connections would be able to resist against vertical loading without using any in-situ casting.

== Horizontal lateral restraint ==

The connections identified above, unless suitably designed for using excessive sized members and reinforcement, will struggle to resist lateral loading. The lateral loading will need to be taken by shear walls and/or concrete cores such as lift shafts or steel bracing to create a hybrid structure. But as with the initial problem of over-engineered connections through the neglect of their moment capacity, the lateral loading capacity of the preferred connections would need to be assessed and accounted for in order to produce the most efficient design.

Combination of continuous column and continuous beam joints can be used to help transfer moments to stiffer areas of the structure. It thus also allows for a more efficient design with only critical members designed to facilitate the load transfer.

== Frame analysis ==

In-situ frames have fully rigid connections. Should a precast connection be capable of transferring moment to the columns and thus down to the supports, then it can be assed as a complete frame or a series of sub frames. Moments, either hogging or sagging are attracted to stiffer members. Should the connection be capable of transferring these moments, the moments at the columns will then be in hogging and will need to be accounted for. Many published papers (Gorgun, 1997; Aguiar et al., 4 June 2012; Baharuddin et al., 2006) have discovered that some precast connections (including the ones mentioned above) can sustain hogging moment, and are therefore over engineered using the current design process. Therefore the structural frame should be modelled similar to a steel frame, where if almost no moment can be sustained then the connections are designed as pinned.

== Disproportionate Collapse ==

Since 2004, the Building Regulations in England and Wales have been revised to ensure all buildings are designed against disproportionate collapse. The connections above have been analysed with moment capacity as the desired attribute, but they will also provide some tensional resistance which would inherently provide resistance against disproportionate collapse.

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UK National Audit Office. (2005). ''NAO MMC - Assumptions and Scenarios 20th October 2005''. Recuperado el 28 de November de 2011, de NAO: output from Google search of 'NAO MMC lead time offsite'

webbaviation on behalf of Laing O'Rourke. (14 de June de 2011). fb17148. ''aerial photographs''. Barnsley: www.webbaviation.co.uk.

Wright, E. (26 de March de 2010). Fast Build Nation: Richard Ogden on offsite construction. ''Building Magazine''.

=== ===

[[Category:Construction_management]]
[[Category:Health_and_safety_/_CDM]]
[[Category:Products_/_components]]
[[Category:Project_types]]
[[Category:Sustainability]]
[[Category:Student_architect_essay_competition]]
[[Category:Student_engineer_essay_competition]]

Resonant column method

2012-12-14T16:25:02Z

Nicky nguyen 91:

= introduction =

== Background ==

The resonant column method was initially developed by Japanese engineers: Ishimoto & Iida (1937). It was made popular in the 1960s by authors such as Hall & Richart (1963), Drnevich et al. (1967) and Hardin & Black (1968). The resonant column apparatus has been used to measure the dynamic response of soils including the shear and elastic modulus based on the theory of wave propagation in prismatic rods. 

== Shear modulus (G) ==

The resonant column method was conventionally used in torsion to measure the shear modulus (G) of the material. In most cases, the clamped-free configuration has been chosen for research purpose as its mathematical derivation is more straightforward. In the clamped-free test, a cylindrical specimen is fixed at the base and excited via a drive mechanism attached to its free end. The resonant frequency (omega) is measured from which the velocity of the propagating wave is derived. Based on the derived velocity and the sample’s density, the low-strain shear modulus (G) of the material can be computed from the basic equation for torsional vibration.

== Young modulus (E) ==

The resonant column can also be used in flexural excitation to determine the material’s Young modulus (E). The conventional method with long samples allowing the application of Rayleighod’s energy method and Euler-Bernoulli beam theory, disregarded the shear strain energy and rotary inertia effect. When the tested specimen is short in length compared to its diameter, the effects of rotation and shear deformation of the samples during flexure can be substantial. These effects can be significant in interpreting data from flexural test, especially at high frequencies. Therefore, the Euler-Bernoulli theory of flexural vibration of elastic beam is found to be inadequate for short specimens and also for the prediction of higher modes of vibration. To be more accurate, Timoshenko beam theory is used as a model for this interpretation. The theory was developed by Ukrainian scientist Stephen Timoshenko in the 20th century which takes into account the shear deformation and rotary inertia. Different frequency equations for the clamped-free Timoshenko beam with an end mass in flexural vibration are solved to compute the value of elastic stiffness (E).

= The resonant column method =

== Resonant column for torsional excitation ==

In the standard torsional resonant column (Stokoe cell SBEL D1128) as mentioned in Allen and Stokoe (1982), the specimen is rigidly fixed at the base while torsional oscillation is applied to the free end by a drive head. The basic equations for the clamped-free resonant column subjected to torsion are:

[[File:Eq1.JPG|RTENOTITLE]]

The derivation of these equations is based on the assumption that the rotation is small and each transverse section remains plane and rotates about its centre. All the terms expressed in equation (2.1) are functions of the geometric properties of the specimen, except omega n. Treating the system as a single degree of freedom system, the resonant frequency measured in the resonant column apparatus is the damped natural frequency (omega d) but is sufficiently close to the natural frequency (omega n). In this case, the error can be tolerable as omega d is within 1% of omega n. Solving equation (2.1) and (2.2) with omega n, the shear wave velocity (Vs) can be found from which the shear modulus of the material (G) can be derived by rearranging equation (2.3).

== Resonant colum for flexural excitation ==

=== ===

=== Finding the Young modulus by Euler-Bernoulli beam theory (short samples) ===

The RCA can also be used to measure the Young modulus (E) of the material. Cascante et al. (1998) modified the standard Stokoe torsional resonant column (Stokoe cell SBEL D1128) to include flexural vibration mode. The schematic view of the apparatus is shown in Figure 1:

[[File:Schematic view.JPG|RTENOTITLE]] 

''Figure 1: Schematic view of the modified Stokoe RCA.''

''Image taken from Cascante et al. (1998)''

In the original configuration, four pairs of excitation coils are connected in series to produce a net torque at the top of the sample (Figure 2A). In Cascante’s modified version, the coils are reconnected so that 

only two magnets are used to produce a net horizontal force on top of the specimen (Figure 2B). 

[[File:Coil.JPG|RTENOTITLE]]

''Figure 2'''': Coil-magnet arrangements for torsional and flexural RCA. Images taken from Cascante et al. (1998). ''''' ''A: Torsional excitation. ''''' '''''B: Flexural excitation''''' '''

In the reduction of data for flexural excitation, the specimen and its drive head can be idealised as an elastic column with a rigid point mass at the top free end (Fig 3). The behaviour of the system is assumed to be elastic. Cascante et al. (1998) has developed a general mathematical formulation for the angular resonant frequency by using Rayleigh’s energy method and Euler Bernoulli beam theory. Based on this general equation, the Young Modulus (E) can be determined by:

[[File:Eq2.JPG|RTENOTITLE]]

In previous literatures, as the cross-sectional dimensions of the sample were small in comparison with its length, Euler-Bernoulli beam theory has been used to treat the boundary conditions and derive the frequency equation, from which E could be determined.

[[File:Resonant column.JPG|RTENOTITLE]] 

''Figure 3'''': Exaggerated view of deflected column for an ''''idealised'''' system''''' ''Image taken from Priest (2004)''

=== Finding the Young modulus by Timoshenko beam theory (long samples) ===

When the tested specimens are short in length compared to their thicknesses, the effect of shear deformation during flexure is significant which can result in possible discrepancies in interpreting data from flexural test. On the other hand, the effect of rotation is large when the curvature of the beam is large relative to its thickness. This is true when the beam is short in length compared to its thickness. Therefore, Timoshenko beam theory is used in this interpretation as it takes into consideration the effect of shear deformation and rotary inertia in which the conventional Euler-Bernoulli theory doesn’t. During vibration, a typical element of a beam not only performs translatory movement, but also rotation. With shear deformation being considered, the assumption of the elementary Euler-Bernoulli theory that ‘’plane section remains plane’’ is no longer applicable. Therefore, the angle of rotation which is equal to the slope theta of any section along the length of the beam cannot be obtained by simple differentiation of the transverse displacement y. Thus, it results in two independent motions theta(x,t) and y(x,t).

Timoshenko gave the coupled equations of motion for the beam with constant cross-section as:

[[File:Eq3.JPG|RTENOTITLE]][[File:Bruch and mitchel.JPG|RTENOTITLE]] 

''Figure 4'': ''The beam-mass system used in the analysis'' '''''Image taken from Bruch and Mitchell (1987''''')

Bruch and Mitchell (1987) investigated a particular case of a cantilevered Timoshenko beam with a tip mass (Figure 4.4). By applying the boundary conditions and using Huang’s non-dimensional variables, the solutions to the coupled equations are determined as functions of the Young modulus (E), the Shear modulus (G), material’s density (rho), the angular natural frequency (omega n) and the geometry of the specimen. Bruch and Mitchell derived the frequency equation of the beam in flexural excitation by inserting the solutions to the coupled equations (2.6) and (2.7) into the boundary conditions, from which the matrix equation can be determined. By taking the determinant of the coefficient matrix equation, the resonant frequency equation was found from which the Young modulus (E) can be calculated.

Liu (1989) suggested three ways in which the work of Bruch and Mitchell could be further extended: (i) The base condition for the beam-mass system considered in [3] should be modeled as an imperfect clamped support (or elastic support), (ii) The tip mass’s centre of gravity is not practically right at the top of the beam but usually at a distance from the beam tip, (iii) the shear coefficient depends on both the shape of the cross-section and the Poisson ratio. Liu added springs at the hub to simulate the imperfect clamped support therefore the boundary condition also includes the spring’s properties which are the rotational spring constant and translational spring constant. By substituting the general solution into the new boundary conditions, Liu gave the improvement of Bruch and Mitchell’s frequency equation for the mass-loaded clamped-free Timoshenko beam.

The shear coefficient in Timoshenko’s beam theory is a dimensionless quantity, dependent on the shape of the cross section, which accounts for the fact that the shear stress and shear strain are not uniformly distributed over the cross section of the specimen. Cowper (1966) developed a new formula for the shear coefficient from the derivation of the equations of Timoshenko beam theory. For a circular cross-section, the value of K was given in terms of the Poisson ratio as:

[[File:K.JPG|RTENOTITLE]] Farghaly (1993) offered suggestion to extend Liu’s work by applying Timoshenko beam theory in treating the boundary conditions. He realised that the use of Euler-Bernoulli theory in the boundary conditions could result in inaccurate natural frequencies calculated, particularly for high slenderness ratios and higher modes of vibration. Farghaly’s model also includes the root flexibilities and the tip mass’s eccentricity as can be shown in Figure 5:

[[File:Farghaly model.JPG|RTENOTITLE]] 

''Figure 5'''': Thick beam with tip mass and root flexibilities.''''' ''Image taken from Farghaly (1993)''

The work of Bruch and Mitchell (1987), Liu (1989) and Farghaly (1993) were in an attempt to simulate the motion of a flexible robot arm modeled as a cantilevered Timoshenko beam with a lumped mass and lumped moment of inertia at the free end. However, for the purpose of this essay, their resonant frequency equations were considered to be adequate for use in computing the material’s Young modulus from the flexural resonant column test, if the angular natural frequency is known.

= Using Timoshenko’s beam theory for resonant column testing =

== Frequency equation by Bruch and Mitchell ==

Bruch and Mitchell started with the original coupled equations of motion given by Timoshenko for the beam with constant cross section: 

[[File:Eq 3.1.3.2.JPG|RTENOTITLE]]

[[File:Eq 3.3.JPG|RTENOTITLE]][[File:Eq 3.4.JPG|RTENOTITLE]] 

From the simple harmonic motion equations: 

[[File:Eq 3.8.JPG|RTENOTITLE]]

Using the non-dimensional variables and series of equations (3.1) to (3.3), equations (3.4) reduced the problem to:

[[File:Eq3.9.JPG|RTENOTITLE]][[File:Eq3.14.JPG|RTENOTITLE]][[File:Eq3.20.JPG|RTENOTITLE]][[File:Eq3.25.JPG|RTENOTITLE]] 

Taking the determinant of the coefficient matrix equation (3.20) gives the frequency equation, from which the elastic stiffness can be computed with the natural resonant frequency (omega n) as an input.

== Frequency equation by Liu ==

Liu [16] introduced a rotational spring constant (Kr) and a translational spring constant (Kl) to model the imperfection of a clamped support. For simplicity, assuming the base of the resonant column is perfectly clamped, the values of the spring constants (Kr) and (Kl) therefore approach infinity. The distance from the beam tip to the centre of the added mass (d) was added to model the eccentricity. Moment of inertia of the added mass (J) was also included in the revised matrix equation to improve the accuracy of the original model by Bruch and Mitchel. Liu started from the single free vibration equation of a Timoshenko beam given in [16], rather than the coupled equation of motion as in Bruch and Mitchell’s. The frequency equation given by Liu is:

[[File:Eq3.29.JPG|RTENOTITLE]] 

In which 

[[File:Eq3.30.JPG|RTENOTITLE]][[File:Eq3.39.JPG|RTENOTITLE]] 

Equation (3.39) was solved using Matlab with the same inputs as in the case of Bruch and Mitchell’s model, plus the eccentricity and moment of inertia of the tip mass, to evaluate the sample’s elastic stiffness.

== Frequency equation by Farghaly ==

Liu has derived a frequency equation to further improve the work of Bruch and Mitchell. The root flexibility, eccentricity and moment of inertia of the tip mass have been taken into consideration to improve the accuracy on modeling a robot arm as a clamped-free Timoshenko beam with a lumped mass and a lumped moment of inertia at its free end. The same idea of simulating a robot arm by Timoshenko beam theory can be used to model the RCA when the sample is short in length relative to its diameter. Farghaly commented in his published paper that in [16], Liu used Timoshenko beam theory for the system differential equation, while Euler-Bernoulli theory was applied to treat the boundary conditions. Farghaly stressed that, when using Liu’s formula to compute the resonant frequency with proper inputs, inaccurate natural frequencies maybe obtained, particularly for significant values of the slenderness ratio and higher modes of vibration.

The system frequency equation in terms of the root rigidity parameters can be written as:

[[File:Eq3.40.JPG|RTENOTITLE]] 

In which

[[File:Eq3.41.JPG|RTENOTITLE]][[File:Eq3.45.JPG|RTENOTITLE]] 

In theory, to model the perfectly clamped support condition of the resonant column, the spring constants theta and K should be made to approach infinity. However, in Matlab, for simplicity, extreme values have been assigned to theta and K to give significant values of the root rigidity parameters theta and Z. As mentioned in Farghaly (1992), when using the matrix determinant equation (3.40) to compute the resonant frequencies, inaccurate results might be obtained for large values of the slenderness ratio and for higher modes of vibration. However, in our case, the length of the specimen is only about more than twice the diameter and vibration was limited to the 1st fundamental mode. According to Liu in Author’s Reply (1992), from his own practical point of view, if one can accept the idea of treating a complicated cantilever structure as a Timoshenko beam, then the discrepancies caused by non-exact boundary conditions might be considered as tolerable.

= references =

Alan, K. (2011). Stiffness and damping of sand at small strain using a resonant column. ''Civil Engineering 3rd year Individual Project, University of Southampton''.

Allen, J. C. and Stokoe, K. H. (1982). Development of resonant column apparatus with anisotropic loading. ''Geotechnical Engineering Report GR82-28, Civil Engineering Dept., University of Texas at Austin''.

Banerjee, J. R. (2001). Frequency equation and mode shape formulae for composite Timoshenko beams. ''Composite Structures 51 '', 381-388.

Bisplinghoff, R. L., and Ashley, H. (1962). ''Principles of Aeroelasticity.'' New York: Dover.

Bruch, J. C., and Mitchell, T. P. (1987). Vibrations of a mass-loaded clamped-free Timoshenko beam. ''Journal of Sound and Vibration 114'', 341-345.

Cascante, G., Santamarina, C., and Yassir, N. (1998). Flexural excitation in a standard torsional-resonant column device. ''Can. Geotech. J., 35'', 478-490.

Cowper, G. (1966). The shear coefficient in Timoshenko's beam theory. ''Journal of Applied Mechanics 33'', 335-340.

Drnevich, V. P. (n.d.). Resonant column testing - problems and solutions. ''Dynamic Geotechnical Testing, ASTM 654'', 394-398.

Drnevich, V. P., Hall, J. R., and Richart, F. (1967). Effects of amplitude vibration on the shear modulus of sand. In N. M. Albuquerque, ''Proc. of the International Symposium on Wave Propagation and Dynamic Properties of Earth Material'' (pp. 189-199).

Farghaly, S. H. (1993). On comments on ''Vibration of a mass-loaded clamped-free Timoshenko beam''. ''Journal of Sound and Vibration 164(3)'', 549-552.

Griffin, C. (2011). Understanding the modes of deformation of dry stone retaining wall. ''Civil Engineering 3rd year Individual Project, University of Southampton''.

Hall, J. R. and Richart, F. E. (1963). Dissipation of elastic waves in granular soils. ''Journal of Soil Mechanics and Foundation. Div, 89 (SM6)''.

Hardin, B. O. and Black, W. L. (1968). (n.d.). Vibration modulus of normally consolidated clays. ''Journal of Soil Mechanics and Foundation. Div., ASCE, Vol 94, No. SM2, Proc. Paper 5833'', 353-369.

Horr, A. M. and Schmidt, L. C. (1995). Closed-form solution for the Timoshenko beam theory using a computer-based mathematical package. ''Computers & Structures Vol. 5. No. 3'', 405-412.

Huang, T. C. (1961). The effect of rotatory inertia and of shear deformation on the frequency and normal mode equations of uniform beams with simple end conditions. ''Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics 28'', 579-584.

Ishimoto, M. and Iida, K. (n.d.). Determination of elastic constants of soils by means of vibration methods. ''Bulletin of the Earthquake Research Institute'', 15-67.

Liu, W. H. (1987). On the natural frequencies of restrained cantilever beams. ''Journal of Sound and Vibration 117'', 571-572.

Liu, W. H. (1989). Comments on ''Vibrations of a mass-loaded clamped-free Timoshenko beam''. ''Journal of Sound and Vibration 129'', 343-344.

Oguamanam, D. C. (2003). Free vibration of beams with finite mass rigid tip load and flexural-torsional coupling. ''International Journal of Mechanical Sciences 45'', 963-979.

Priest, J. A. (2004). ''The Effects of Methane Gas Hydrate.'' Southampton, United Kingdom: School of Civil Engineering and the Environment, University of Southampton.

Richart, F.E., Hall, J.R., and Woods, R.D. (1970). ''Vibrations of soils and foundations.'' Englewood Cliffs, N.J.: Prentice-Hall, Inc.

Rossi, R. E., Laura, P. A. A., and Gutierrez, R. H. (1990). A note on transverse vibrations of a Timoshenko beam of non-uniform thickess clamped at one end and carrying a concentrated mass at the other. ''Journal of Sound and Vibration 143'', 491-502.

Salarieh, H., and Ghorashi, M. (2006). Free vibration of Timoshenko beam with finite mass rigid tip load and flexural-torsional coupling. ''International Journal of Mechanical Sciences 48'', 763-779.

Sniady, P. (2008). Dynamic Response of a Timoshenko Beam to a Moving Force. ''Journal of Applied Mechanics Vol. 75''.

Timoshenko, S. P. (1955). Vibration Problems in Engineering. In S. P. Timoshenko. New York: D. Van Nostrand Company, third edition.

White, M. W. D., and Heppler, G. R. (1995). Vibration Modes and Frequencies of Timoshenko Beams with Attached rigid bodies. ''Journal of Applied Mechanics Vol. 62''

[[Category:Student_engineer_essay_competition]]

Resonant column method

2012-12-14T16:23:10Z

Nicky nguyen 91:

= introduction =

== Background ==

The resonant column method was initially developed by Japanese engineers: Ishimoto & Iida (1937). It was made popular in the 1960s by authors such as Hall & Richart (1963), Drnevich et al. (1967) and Hardin & Black (1968). The resonant column apparatus has been used to measure the dynamic response of soils including the shear and elastic modulus based on the theory of wave propagation in prismatic rods. 

== Shear modulus (G) ==

The resonant column method was conventionally used in torsion to measure the shear modulus (G) of the material. In most cases, the clamped-free configuration has been chosen for research purpose as its mathematical derivation is more straightforward. In the clamped-free test, a cylindrical specimen is fixed at the base and excited via a drive mechanism attached to its free end. The resonant frequency (omega) is measured from which the velocity of the propagating wave is derived. Based on the derived velocity and the sample’s density, the low-strain shear modulus (G) of the material can be computed from the basic equation for torsional vibration.

== Young modulus (E) ==

The resonant column can also be used in flexural excitation to determine the material’s Young modulus (E). The conventional method with long samples allowing the application of Rayleighod’s energy method and Euler-Bernoulli beam theory, disregarded the shear strain energy and rotary inertia effect. When the tested specimen is short in length compared to its diameter, the effects of rotation and shear deformation of the samples during flexure can be substantial. These effects can be significant in interpreting data from flexural test, especially at high frequencies. Therefore, the Euler-Bernoulli theory of flexural vibration of elastic beam is found to be inadequate for short specimens and also for the prediction of higher modes of vibration. To be more accurate, Timoshenko beam theory is used as a model for this interpretation. The theory was developed by Ukrainian scientist Stephen Timoshenko in the 20th century which takes into account the shear deformation and rotary inertia. Different frequency equations for the clamped-free Timoshenko beam with an end mass in flexural vibration are solved to compute the value of elastic stiffness (E).

= The resonant column method =

== Resonant column for torsional excitation ==

In the standard torsional resonant column (Stokoe cell SBEL D1128) as mentioned in Allen and Stokoe (1982), the specimen is rigidly fixed at the base while torsional oscillation is applied to the free end by a drive head. The basic equations for the clamped-free resonant column subjected to torsion are:

[[File:Eq1.JPG|RTENOTITLE]]

The derivation of these equations is based on the assumption that the rotation is small and each transverse section remains plane and rotates about its centre. All the terms expressed in equation (2.1) are functions of the geometric properties of the specimen, except omega n. Treating the system as a single degree of freedom system, the resonant frequency measured in the resonant column apparatus is the damped natural frequency (omega d) but is sufficiently close to the natural frequency (omega n). In this case, the error can be tolerable as omega d is within 1% of omega n. Solving equation (2.1) and (2.2) with omega n, the shear wave velocity (Vs) can be found from which the shear modulus of the material (G) can be derived by rearranging equation (2.3).

== Resonant colum for flexural excitation ==

=== ===

=== Finding the Young modulus by Euler-Bernoulli beam theory (short samples) ===

The RCA can also be used to measure the Young modulus (E) of the material. Cascante et al. (1998) modified the standard Stokoe torsional resonant column (Stokoe cell SBEL D1128) to include flexural vibration mode. The schematic view of the apparatus is shown in Figure 1:

[[File:Schematic view.JPG|RTENOTITLE]] 

''Figure 1: Schematic view of the modified Stokoe RCA.''

''Image taken from Cascante et al. (1998)''

In the original configuration, four pairs of excitation coils are connected in series to produce a net torque at the top of the sample (Figure 2A). In Cascante’s modified version, the coils are reconnected so that 

only two magnets are used to produce a net horizontal force on top of the specimen (Figure 2B). 

[[File:Coil.JPG|RTENOTITLE]]

''Figure 2'''': Coil-magnet arrangements for torsional and flexural RCA. Images taken from Cascante et al. (1998). ''''' ''A: Torsional excitation. ''''' '''''B: Flexural excitation''''' '''

In the reduction of data for flexural excitation, the specimen and its drive head can be idealised as an elastic column with a rigid point mass at the top free end (Fig 3). The behaviour of the system is assumed to be elastic. Cascante et al. (1998) has developed a general mathematical formulation for the angular resonant frequency by using Rayleigh’s energy method and Euler Bernoulli beam theory. Based on this general equation, the Young Modulus (E) can be determined by:

[[File:Eq2.JPG|RTENOTITLE]]

In previous literatures, as the cross-sectional dimensions of the sample were small in comparison with its length, Euler-Bernoulli beam theory has been used to treat the boundary conditions and derive the frequency equation, from which E could be determined.

[[File:Resonant column.JPG|RTENOTITLE]] 

''Figure 3'''': Exaggerated view of deflected column for an ''''idealised'''' system''''' ''Image taken from Priest (2004)''

=== Finding the Young modulus by Timoshenko beam theory (long samples) ===

When the tested specimens are short in length compared to their thicknesses, the effect of shear deformation during flexure is significant which can result in possible discrepancies in interpreting data from flexural test. On the other hand, the effect of rotation is large when the curvature of the beam is large relative to its thickness. This is true when the beam is short in length compared to its thickness. Therefore, Timoshenko beam theory is used in this interpretation as it takes into consideration the effect of shear deformation and rotary inertia in which the conventional Euler-Bernoulli theory doesn’t. During vibration, a typical element of a beam not only performs translatory movement, but also rotation. With shear deformation being considered, the assumption of the elementary Euler-Bernoulli theory that ‘’plane section remains plane’’ is no longer applicable. Therefore, the angle of rotation which is equal to the slope theta of any section along the length of the beam cannot be obtained by simple differentiation of the transverse displacement y. Thus, it results in two independent motions theta(x,t) and y(x,t).

Timoshenko gave the coupled equations of motion for the beam with constant cross-section as:

[[File:Eq3.JPG|RTENOTITLE]][[File:Bruch and mitchel.JPG|RTENOTITLE]] 

''Figure 4'': ''The beam-mass system used in the analysis'' '''''Image taken from Bruch and Mitchell (1987''''')

Bruch and Mitchell (1987) investigated a particular case of a cantilevered Timoshenko beam with a tip mass (Figure 4.4). By applying the boundary conditions and using Huang’s non-dimensional variables, the solutions to the coupled equations are determined as functions of the Young modulus (E), the Shear modulus (G), material’s density (rho), the angular natural frequency (omega n) and the geometry of the specimen. Bruch and Mitchell derived the frequency equation of the beam in flexural excitation by inserting the solutions to the coupled equations (2.6) and (2.7) into the boundary conditions, from which the matrix equation can be determined. By taking the determinant of the coefficient matrix equation, the resonant frequency equation was found from which the Young modulus (E) can be calculated.

Liu (1989) suggested three ways in which the work of Bruch and Mitchell could be further extended: (i) The base condition for the beam-mass system considered in [3] should be modeled as an imperfect clamped support (or elastic support), (ii) The tip mass’s centre of gravity is not practically right at the top of the beam but usually at a distance from the beam tip, (iii) the shear coefficient depends on both the shape of the cross-section and the Poisson ratio. Liu added springs at the hub to simulate the imperfect clamped support therefore the boundary condition also includes the spring’s properties which are the rotational spring constant and translational spring constant. By substituting the general solution into the new boundary conditions, Liu gave the improvement of Bruch and Mitchell’s frequency equation for the mass-loaded clamped-free Timoshenko beam.

The shear coefficient in Timoshenko’s beam theory is a dimensionless quantity, dependent on the shape of the cross section, which accounts for the fact that the shear stress and shear strain are not uniformly distributed over the cross section of the specimen. Cowper (1966) developed a new formula for the shear coefficient from the derivation of the equations of Timoshenko beam theory. For a circular cross-section, the value of K was given in terms of the Poisson ratio as:

[[File:K.JPG|RTENOTITLE]] Farghaly (1993) offered suggestion to extend Liu’s work by applying Timoshenko beam theory in treating the boundary conditions. He realised that the use of Euler-Bernoulli theory in the boundary conditions could result in inaccurate natural frequencies calculated, particularly for high slenderness ratios and higher modes of vibration. Farghaly’s model also includes the root flexibilities and the tip mass’s eccentricity as can be shown in Figure 5:

[[File:Farghaly model.JPG|RTENOTITLE]] 

''Figure 5'''': Thick beam with tip mass and root flexibilities.''''' ''Image taken from Farghaly (1993)''

The work of Bruch and Mitchell (1987), Liu (1989) and Farghaly (1993) were in an attempt to simulate the motion of a flexible robot arm modeled as a cantilevered Timoshenko beam with a lumped mass and lumped moment of inertia at the free end. However, for the purpose of this essay, their resonant frequency equations were considered to be adequate for use in computing the material’s Young modulus from the flexural resonant column test, if the angular natural frequency is known.

= Using Timoshenko’s beam theory for resonant column testing =

== Frequency equation by Bruch and Mitchell ==

Bruch and Mitchell started with the original coupled equations of motion given by Timoshenko for the beam with constant cross section: [[File:Eq 3.1.3.2.JPG|RTENOTITLE]]

[[File:Eq 3.3.JPG|RTENOTITLE]][[File:Eq 3.4.JPG|RTENOTITLE]] 

From the simple harmonic motion equations: 

[[File:Eq 3.8.JPG|RTENOTITLE]]

Using the non-dimensional variables and series of equations (3.1) to (3.3), equations (3.4) reduced the problem to:

[[File:Eq3.9.JPG|RTENOTITLE]][[File:Eq3.14.JPG|RTENOTITLE]][[File:Eq3.20.JPG|RTENOTITLE]][[File:Eq3.25.JPG|RTENOTITLE]] 

Taking the determinant of the coefficient matrix equation (3.20) gives the frequency equation, from which the elastic stiffness can be computed with the natural resonant frequency (omega n) as an input.

== Frequency equation by Liu ==

Liu [16] introduced a rotational spring constant (Kr) and a translational spring constant (Kl) to model the imperfection of a clamped support. For simplicity, assuming the base of the resonant column is perfectly clamped, the values of the spring constants (Kr) and (Kl) therefore approach infinity. The distance from the beam tip to the centre of the added mass (d) was added to model the eccentricity. Moment of inertia of the added mass (J) was also included in the revised matrix equation to improve the accuracy of the original model by Bruch and Mitchel. Liu started from the single free vibration equation of a Timoshenko beam given in [16], rather than the coupled equation of motion as in Bruch and Mitchell’s. The frequency equation given by Liu is:

[[File:Eq3.29.JPG|RTENOTITLE]] 

In which 

[[File:Eq3.30.JPG|RTENOTITLE]][[File:Eq3.39.JPG|RTENOTITLE]] 

Equation (3.39) was solved using Matlab with the same inputs as in the case of Bruch and Mitchell’s model, plus the eccentricity and moment of inertia of the tip mass, to evaluate the sample’s elastic stiffness.

== Frequency equation by Farghaly ==

Liu has derived a frequency equation to further improve the work of Bruch and Mitchell. The root flexibility, eccentricity and moment of inertia of the tip mass have been taken into consideration to improve the accuracy on modeling a robot arm as a clamped-free Timoshenko beam with a lumped mass and a lumped moment of inertia at its free end. The same idea of simulating a robot arm by Timoshenko beam theory can be used to model the RCA when the sample is short in length relative to its diameter. Farghaly commented in his published paper that in [16], Liu used Timoshenko beam theory for the system differential equation, while Euler-Bernoulli theory was applied to treat the boundary conditions. Farghaly stressed that, when using Liu’s formula to compute the resonant frequency with proper inputs, inaccurate natural frequencies maybe obtained, particularly for significant values of the slenderness ratio and higher modes of vibration.

The system frequency equation in terms of the root rigidity parameters can be written as:

[[File:Eq3.40.JPG|RTENOTITLE]] 

In which

[[File:Eq3.41.JPG|RTENOTITLE]][[File:Eq3.45.JPG|RTENOTITLE]] 

In theory, to model the perfectly clamped support condition of the resonant column, the spring constants theta and K should be made to approach infinity. However, in Matlab, for simplicity, extreme values have been assigned to theta and K to give significant values of the root rigidity parameters theta and Z. As mentioned in Farghaly (1992), when using the matrix determinant equation (3.40) to compute the resonant frequencies, inaccurate results might be obtained for large values of the slenderness ratio and for higher modes of vibration. However, in our case, the length of the specimen is only about more than twice the diameter and vibration was limited to the 1st fundamental mode. According to Liu in Author’s Reply (1992), from his own practical point of view, if one can accept the idea of treating a complicated cantilever structure as a Timoshenko beam, then the discrepancies caused by non-exact boundary conditions might be considered as tolerable.

= references =

Alan, K. (2011). Stiffness and damping of sand at small strain using a resonant column. ''Civil Engineering 3rd year Individual Project, University of Southampton''.

Allen, J. C. and Stokoe, K. H. (1982). Development of resonant column apparatus with anisotropic loading. ''Geotechnical Engineering Report GR82-28, Civil Engineering Dept., University of Texas at Austin''.

Banerjee, J. R. (2001). Frequency equation and mode shape formulae for composite Timoshenko beams. ''Composite Structures 51 '', 381-388.

Bisplinghoff, R. L., and Ashley, H. (1962). ''Principles of Aeroelasticity.'' New York: Dover.

Bruch, J. C., and Mitchell, T. P. (1987). Vibrations of a mass-loaded clamped-free Timoshenko beam. ''Journal of Sound and Vibration 114'', 341-345.

Cascante, G., Santamarina, C., and Yassir, N. (1998). Flexural excitation in a standard torsional-resonant column device. ''Can. Geotech. J., 35'', 478-490.

Cowper, G. (1966). The shear coefficient in Timoshenko's beam theory. ''Journal of Applied Mechanics 33'', 335-340.

Drnevich, V. P. (n.d.). Resonant column testing - problems and solutions. ''Dynamic Geotechnical Testing, ASTM 654'', 394-398.

Drnevich, V. P., Hall, J. R., and Richart, F. (1967). Effects of amplitude vibration on the shear modulus of sand. In N. M. Albuquerque, ''Proc. of the International Symposium on Wave Propagation and Dynamic Properties of Earth Material'' (pp. 189-199).

Farghaly, S. H. (1993). On comments on ''Vibration of a mass-loaded clamped-free Timoshenko beam''. ''Journal of Sound and Vibration 164(3)'', 549-552.

Griffin, C. (2011). Understanding the modes of deformation of dry stone retaining wall. ''Civil Engineering 3rd year Individual Project, University of Southampton''.

Hall, J. R. and Richart, F. E. (1963). Dissipation of elastic waves in granular soils. ''Journal of Soil Mechanics and Foundation. Div, 89 (SM6)''.

Hardin, B. O. and Black, W. L. (1968). (n.d.). Vibration modulus of normally consolidated clays. ''Journal of Soil Mechanics and Foundation. Div., ASCE, Vol 94, No. SM2, Proc. Paper 5833'', 353-369.

Horr, A. M. and Schmidt, L. C. (1995). Closed-form solution for the Timoshenko beam theory using a computer-based mathematical package. ''Computers & Structures Vol. 5. No. 3'', 405-412.

Huang, T. C. (1961). The effect of rotatory inertia and of shear deformation on the frequency and normal mode equations of uniform beams with simple end conditions. ''Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics 28'', 579-584.

Ishimoto, M. and Iida, K. (n.d.). Determination of elastic constants of soils by means of vibration methods. ''Bulletin of the Earthquake Research Institute'', 15-67.

Liu, W. H. (1987). On the natural frequencies of restrained cantilever beams. ''Journal of Sound and Vibration 117'', 571-572.

Liu, W. H. (1989). Comments on ''Vibrations of a mass-loaded clamped-free Timoshenko beam''. ''Journal of Sound and Vibration 129'', 343-344.

Oguamanam, D. C. (2003). Free vibration of beams with finite mass rigid tip load and flexural-torsional coupling. ''International Journal of Mechanical Sciences 45'', 963-979.

Priest, J. A. (2004). ''The Effects of Methane Gas Hydrate.'' Southampton, United Kingdom: School of Civil Engineering and the Environment, University of Southampton.

Richart, F.E., Hall, J.R., and Woods, R.D. (1970). ''Vibrations of soils and foundations.'' Englewood Cliffs, N.J.: Prentice-Hall, Inc.

Rossi, R. E., Laura, P. A. A., and Gutierrez, R. H. (1990). A note on transverse vibrations of a Timoshenko beam of non-uniform thickess clamped at one end and carrying a concentrated mass at the other. ''Journal of Sound and Vibration 143'', 491-502.

Salarieh, H., and Ghorashi, M. (2006). Free vibration of Timoshenko beam with finite mass rigid tip load and flexural-torsional coupling. ''International Journal of Mechanical Sciences 48'', 763-779.

Sniady, P. (2008). Dynamic Response of a Timoshenko Beam to a Moving Force. ''Journal of Applied Mechanics Vol. 75''.

Timoshenko, S. P. (1955). Vibration Problems in Engineering. In S. P. Timoshenko. New York: D. Van Nostrand Company, third edition.

White, M. W. D., and Heppler, G. R. (1995). Vibration Modes and Frequencies of Timoshenko Beams with Attached rigid bodies. ''Journal of Applied Mechanics Vol. 62''

[[Category:Student_engineer_essay_competition]]

Resonant column method

2012-12-14T16:13:39Z

Nicky nguyen 91:

= introduction =

== Background ==

The resonant column method was initially developed by Japanese engineers: Ishimoto & Iida (1937). It was made popular in the 1960s by authors such as Hall & Richart (1963), Drnevich et al. (1967) and Hardin & Black (1968). The resonant column apparatus has been used to measure the dynamic response of soils including the shear and elastic modulus based on the theory of wave propagation in prismatic rods. 

== Shear modulus (G) ==

The resonant column method was conventionally used in torsion to measure the shear modulus (G) of the material. In most cases, the clamped-free configuration has been chosen for research purpose as its mathematical derivation is more straightforward. In the clamped-free test, a cylindrical specimen is fixed at the base and excited via a drive mechanism attached to its free end. The resonant frequency (omega) is measured from which the velocity of the propagating wave is derived. Based on the derived velocity and the sample’s density, the low-strain shear modulus (G) of the material can be computed from the basic equation for torsional vibration.

== Young modulus (E) ==

The resonant column can also be used in flexural excitation to determine the material’s Young modulus (E). The conventional method with long samples allowing the application of Rayleighod’s energy method and Euler-Bernoulli beam theory, disregarded the shear strain energy and rotary inertia effect. When the tested specimen is short in length compared to its diameter, the effects of rotation and shear deformation of the samples during flexure can be substantial. These effects can be significant in interpreting data from flexural test, especially at high frequencies. Therefore, the Euler-Bernoulli theory of flexural vibration of elastic beam is found to be inadequate for short specimens and also for the prediction of higher modes of vibration. To be more accurate, Timoshenko beam theory is used as a model for this interpretation. The theory was developed by Ukrainian scientist Stephen Timoshenko in the 20th century which takes into account the shear deformation and rotary inertia. Different frequency equations for the clamped-free Timoshenko beam with an end mass in flexural vibration are solved to compute the value of elastic stiffness (E).

= The resonant column method =

== Resonant column for torsional excitation ==

In the standard torsional resonant column (Stokoe cell SBEL D1128) as mentioned in Allen and Stokoe (1982), the specimen is rigidly fixed at the base while torsional oscillation is applied to the free end by a drive head. The basic equations for the clamped-free resonant column subjected to torsion are:

[[File:Eq1.JPG|RTENOTITLE]]

The derivation of these equations is based on the assumption that the rotation is small and each transverse section remains plane and rotates about its centre. All the terms expressed in equation (2.1) are functions of the geometric properties of the specimen, except omega n. Treating the system as a single degree of freedom system, the resonant frequency measured in the resonant column apparatus is the damped natural frequency (omega d) but is sufficiently close to the natural frequency (omega n). In this case, the error can be tolerable as omega d is within 1% of omega n. Solving equation (2.1) and (2.2) with omega n, the shear wave velocity (Vs) can be found from which the shear modulus of the material (G) can be derived by rearranging equation (2.3).

== Resonant colum for flexural excitation ==

=== ===

=== Finding the Young modulus by Euler-Bernoulli beam theory (short samples) ===

The RCA can also be used to measure the Young modulus (E) of the material. Cascante et al. (1998) modified the standard Stokoe torsional resonant column (Stokoe cell SBEL D1128) to include flexural vibration mode. The schematic view of the apparatus is shown in Figure 1:

[[File:Schematic view.JPG|RTENOTITLE]] 

''Figure 1: Schematic view of the modified Stokoe RCA.''

''Image taken from Cascante et al. (1998)''

In the original configuration, four pairs of excitation coils are connected in series to produce a net torque at the top of the sample (Figure 2A). In Cascante’s modified version, the coils are reconnected so that only two magnets are used to produce a net horizontal force on top of the specimen (Figure 2B). [[File:Coil.JPG|RTENOTITLE]]

''Figure 2'''': Coil-magnet arrangements for torsional and flexural RCA. Images taken from Cascante et al. (1998). ''''' ''A: Torsional excitation. ''''' '''''B: Flexural excitation''''' '''

In the reduction of data for flexural excitation, the specimen and its drive head can be idealised as an elastic column with a rigid point mass at the top free end (Fig 3). The behaviour of the system is assumed to be elastic. Cascante et al. (1998) has developed a general mathematical formulation for the angular resonant frequency by using Rayleigh’s energy method and Euler Bernoulli beam theory. Based on this general equation, the Young Modulus (E) can be determined by:

[[File:Eq2.JPG|RTENOTITLE]]

In previous literatures, as the cross-sectional dimensions of the sample were small in comparison with its length, Euler-Bernoulli beam theory has been used to treat the boundary conditions and derive the frequency equation, from which E could be determined.

[[File:Resonant column.JPG|RTENOTITLE]] 

''Figure 3'''': Exaggerated view of deflected column for an ''''idealised'''' system''''' ''Image taken from Priest (2004)''

=== Finding the Young modulus by Timoshenko beam theory (long samples) ===

When the tested specimens are short in length compared to their thicknesses, the effect of shear deformation during flexure is significant which can result in possible discrepancies in interpreting data from flexural test. On the other hand, the effect of rotation is large when the curvature of the beam is large relative to its thickness. This is true when the beam is short in length compared to its thickness. Therefore, Timoshenko beam theory is used in this interpretation as it takes into consideration the effect of shear deformation and rotary inertia in which the conventional Euler-Bernoulli theory doesn’t. During vibration, a typical element of a beam not only performs translatory movement, but also rotation. With shear deformation being considered, the assumption of the elementary Euler-Bernoulli theory that ‘’plane section remains plane’’ is no longer applicable. Therefore, the angle of rotation which is equal to the slope theta of any section along the length of the beam cannot be obtained by simple differentiation of the transverse displacement y. Thus, it results in two independent motions theta(x,t) and y(x,t).

Timoshenko gave the coupled equations of motion for the beam with constant cross-section as:

[[File:Eq3.JPG|RTENOTITLE]][[File:Bruch and mitchel.JPG|RTENOTITLE]] 

''Figure 4'': ''The beam-mass system used in the analysis'' '''''Image taken from Bruch and Mitchell (1987''''')

Bruch and Mitchell (1987) investigated a particular case of a cantilevered Timoshenko beam with a tip mass (Figure 2.4). By applying the boundary conditions and using Huang’s non-dimensional variables, the solutions to the coupled equations are determined as functions of the Young modulus (E), the Shear modulus (G), material’s density (rho), the angular natural frequency (omega n) and the geometry of the specimen. Bruch and Mitchell derived the frequency equation of the beam in flexural excitation by inserting the solutions to the coupled equations (2.6) and (2.7) into the boundary conditions, from which the matrix equation can be determined. By taking the determinant of the coefficient matrix equation, the resonant frequency equation was found from which the Young modulus (E) can be calculated.

Liu (1989) suggested three ways in which the work of Bruch and Mitchell could be further extended: (i) The base condition for the beam-mass system considered in [3] should be modeled as an imperfect clamped support (or elastic support), (ii) The tip mass’s centre of gravity is not practically right at the top of the beam but usually at a distance from the beam tip, (iii) the shear coefficient depends on both the shape of the cross-section and the Poisson ratio. Liu added springs at the hub to simulate the imperfect clamped support therefore the boundary condition also includes the spring’s properties which are the rotational spring constant and translational spring constant. By substituting the general solution into the new boundary conditions, Liu gave the improvement of Bruch and Mitchell’s frequency equation for the mass-loaded clamped-free Timoshenko beam.

The shear coefficient in Timoshenko’s beam theory is a dimensionless quantity, dependent on the shape of the cross section, which accounts for the fact that the shear stress and shear strain are not uniformly distributed over the cross section of the specimen. Cowper (1966) developed a new formula for the shear coefficient from the derivation of the equations of Timoshenko beam theory. For a circular cross-section, the value of K was given in terms of the Poisson ratio as:

[[File:K.JPG|RTENOTITLE]] Farghaly (1993) offered suggestion to extend Liu’s work by applying Timoshenko beam theory in treating the boundary conditions. He realised that the use of Euler-Bernoulli theory in the boundary conditions could result in inaccurate natural frequencies calculated, particularly for high slenderness ratios and higher modes of vibration. Farghaly’s model also includes the root flexibilities and the tip mass’s eccentricity as can be shown in Figure 5:

[[File:Farghaly model.JPG|RTENOTITLE]] 

''Figure 5'''': Thick beam with tip mass and root flexibilities.''''' ''Image taken from Farghaly (1993)''

The work of Bruch and Mitchell (1987), Liu (1989) and Farghaly (1993) were in an attempt to simulate the motion of a flexible robot arm modeled as a cantilevered Timoshenko beam with a lumped mass and lumped moment of inertia at the free end. However, for the purpose of this essay, their resonant frequency equations were considered to be adequate for use in computing the material’s Young modulus from the flexural resonant column test, if the angular natural frequency is known.

= Using Timoshenko’s beam theory for resonant column testing =

== Frequency equation by Bruch and Mitchell ==

Bruch and Mitchell started with the original coupled equations of motion given by Timoshenko for the beam with constant cross section: [[File:Eq 3.1.3.2.JPG|RTENOTITLE]]

[[File:Eq 3.3.JPG|RTENOTITLE]][[File:Eq 3.4.JPG|RTENOTITLE]] From the simple harmonic motion equations: [[File:Eq 3.8.JPG|RTENOTITLE]]

Using the non-dimensional variables and series of equations (3.1) to (3.3), equations (3.4) reduced the problem to:

[[File:Eq3.9.JPG|RTENOTITLE]][[File:Eq3.14.JPG|RTENOTITLE]][[File:Eq3.20.JPG|RTENOTITLE]][[File:Eq3.25.JPG|RTENOTITLE]] 

Taking the determinant of the coefficient matrix equation (3.20) gives the frequency equation, from which the elastic stiffness can be computed with the natural resonant frequency (omega n) as an input.

== Frequency equation by Liu ==

== <o:p></o:p> ==

Liu [16] introduced a rotational spring constant (Kr) and a translational spring constant (Kl) to model the imperfection of a clamped support. For simplicity, assuming the base of the resonant column is perfectly clamped, the values of the spring constants (Kr) and (Kl) therefore approach infinity. The distance from the beam tip to the centre of the added mass (d) was added to model the eccentricity. Moment of inertia of the added mass (J) was also included in the revised matrix equation to improve the accuracy of the original model by Bruch and Mitchel. Liu started from the single free vibration equation of a Timoshenko beam given in [16], rather than the coupled equation of motion as in Bruch and Mitchell’s. The frequency equation given by Liu is:

[[File:Eq3.29.JPG|RTENOTITLE]] In which 

[[File:Eq3.30.JPG|RTENOTITLE]][[File:Eq3.39.JPG|RTENOTITLE]]Equation (3.49) was solved using Matlab with the same inputs as in the case of Bruch and Mitchell’s model, plus the eccentricity and moment of inertia of the tip mass, to evaluate the sample’s elastic stiffness.

== Frequency equation by Farghaly ==

Liu has derived a frequency equation to further improve the work of Bruch and Mitchell. The root flexibility, eccentricity and moment of inertia of the tip mass have been taken into consideration to improve the accuracy on modeling a robot arm as a clamped-free Timoshenko beam with a lumped mass and a lumped moment of inertia at its free end. The same idea of simulating a robot arm by Timoshenko beam theory can be used to model the RCA when the sample is short in length relative to its diameter. Farghaly commented in his published paper that in [16], Liu used Timoshenko beam theory for the system differential equation, while Euler-Bernoulli theory was applied to treat the boundary conditions. Farghaly stressed that, when using Liu’s formula to compute the resonant frequency with proper inputs, inaccurate natural frequencies maybe obtained, particularly for significant values of the slenderness ratio and higher modes of vibration.

The system frequency equation in terms of the root rigidity parameters can be written as:

[[File:Eq3.40.JPG]]
In which

[[File:Eq3.41.JPG]][[File:Eq3.45.JPG]]
In theory, to model the perfectly clamped support condition of the resonant column, the spring constants theta and K should be made to approach infinity. However, in Matlab, for simplicity, extreme values have been assigned to theta and K to give significant values of the root rigidity parameters theta and Z. As mentioned in Farghaly (1992), when using the matrix determinant equation (3.40) to compute the resonant frequencies, inaccurate results might be obtained for large values of the slenderness ratio and for higher modes of vibration. However, in our case, the length of the specimen is only about more than twice the diameter and vibration was limited to the 1st fundamental mode. According to Liu in Author’s Reply (1992), from his own practical point of view, if one can accept the idea of treating a complicated cantilever structure as a Timoshenko beam, then the discrepancies caused by non-exact boundary conditions might be considered as tolerable.

= references =

Alan, K. (2011). Stiffness and damping of sand at small strain using a resonant column. ''Civil Engineering 3rd year Individual Project, University of Southampton''.

Allen, J. C. and Stokoe, K. H. (1982). Development of resonant column apparatus with anisotropic loading. ''Geotechnical Engineering Report GR82-28, Civil Engineering Dept., University of Texas at Austin''.

Banerjee, J. R. (2001). Frequency equation and mode shape formulae for composite Timoshenko beams. ''Composite Structures 51 '', 381-388.

Bisplinghoff, R. L., and Ashley, H. (1962). ''Principles of Aeroelasticity.'' New York: Dover.

Bruch, J. C., and Mitchell, T. P. (1987). Vibrations of a mass-loaded clamped-free Timoshenko beam. ''Journal of Sound and Vibration 114'', 341-345.

Cascante, G., Santamarina, C., and Yassir, N. (1998). Flexural excitation in a standard torsional-resonant column device. ''Can. Geotech. J., 35'', 478-490.

Cowper, G. (1966). The shear coefficient in Timoshenko's beam theory. ''Journal of Applied Mechanics 33'', 335-340.

Drnevich, V. P. (n.d.). Resonant column testing - problems and solutions. ''Dynamic Geotechnical Testing, ASTM 654'', 394-398.

Drnevich, V. P., Hall, J. R., and Richart, F. (1967). Effects of amplitude vibration on the shear modulus of sand. In N. M. Albuquerque, ''Proc. of the International Symposium on Wave Propagation and Dynamic Properties of Earth Material'' (pp. 189-199).

Farghaly, S. H. (1993). On comments on ''Vibration of a mass-loaded clamped-free Timoshenko beam''. ''Journal of Sound and Vibration 164(3)'', 549-552.

Griffin, C. (2011). Understanding the modes of deformation of dry stone retaining wall. ''Civil Engineering 3rd year Individual Project, University of Southampton''.

Hall, J. R. and Richart, F. E. (1963). Dissipation of elastic waves in granular soils. ''Journal of Soil Mechanics and Foundation. Div, 89 (SM6)''.

Hardin, B. O. and Black, W. L. (1968). (n.d.). Vibration modulus of normally consolidated clays. ''Journal of Soil Mechanics and Foundation. Div., ASCE, Vol 94, No. SM2, Proc. Paper 5833'', 353-369.

Horr, A. M. and Schmidt, L. C. (1995). Closed-form solution for the Timoshenko beam theory using a computer-based mathematical package. ''Computers & Structures Vol. 5. No. 3'', 405-412.

Huang, T. C. (1961). The effect of rotatory inertia and of shear deformation on the frequency and normal mode equations of uniform beams with simple end conditions. ''Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics 28'', 579-584.

Ishimoto, M. and Iida, K. (n.d.). Determination of elastic constants of soils by means of vibration methods. ''Bulletin of the Earthquake Research Institute'', 15-67.

Liu, W. H. (1987). On the natural frequencies of restrained cantilever beams. ''Journal of Sound and Vibration 117'', 571-572.

Liu, W. H. (1989). Comments on ''Vibrations of a mass-loaded clamped-free Timoshenko beam''. ''Journal of Sound and Vibration 129'', 343-344.

Oguamanam, D. C. (2003). Free vibration of beams with finite mass rigid tip load and flexural-torsional coupling. ''International Journal of Mechanical Sciences 45'', 963-979.

Priest, J. A. (2004). ''The Effects of Methane Gas Hydrate.'' Southampton, United Kingdom: School of Civil Engineering and the Environment, University of Southampton.

Richart, F.E., Hall, J.R., and Woods, R.D. (1970). ''Vibrations of soils and foundations.'' Englewood Cliffs, N.J.: Prentice-Hall, Inc.

Rossi, R. E., Laura, P. A. A., and Gutierrez, R. H. (1990). A note on transverse vibrations of a Timoshenko beam of non-uniform thickess clamped at one end and carrying a concentrated mass at the other. ''Journal of Sound and Vibration 143'', 491-502.

Salarieh, H., and Ghorashi, M. (2006). Free vibration of Timoshenko beam with finite mass rigid tip load and flexural-torsional coupling. ''International Journal of Mechanical Sciences 48'', 763-779.

Sniady, P. (2008). Dynamic Response of a Timoshenko Beam to a Moving Force. ''Journal of Applied Mechanics Vol. 75''.

Timoshenko, S. P. (1955). Vibration Problems in Engineering. In S. P. Timoshenko. New York: D. Van Nostrand Company, third edition.

White, M. W. D., and Heppler, G. R. (1995). Vibration Modes and Frequencies of Timoshenko Beams with Attached rigid bodies. ''Journal of Applied Mechanics Vol. 62''

[[Category:Student_engineer_essay_competition]]

File:Eq3.45.JPG

2012-12-14T16:10:42Z

Nicky nguyen 91:

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Resonant column method

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Resonant column method

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Nicky nguyen 91: uploaded a new version of "File:Eq 3.3.JPG"

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Resonant column method

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Resonant column method

2012-12-14T02:19:39Z

Nicky nguyen 91:

= Introduction =

== Background ==

The resonant column method was initially developed by Japanese engineers: Ishimoto & Iida (1937). It was made popular in the 1960s by authors such as Hall & Richart (1963), Drnevich et al. (1967) and Hardin & Black (1968). The resonant column apparatus has been used to measure the dynamic response of soils including the shear and elastic modulus based on the theory of wave propagation in prismatic rods. file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image002.png

== Shear modulus (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image002.png) ==

The resonant column method was conventionally used in torsion to measure the shear modulus (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.png) of the material. In most cases, the clamped-free configuration has been chosen for research purpose as its mathematical derivation is more straightforward. In the clamped-free test, a cylindrical specimen is fixed at the base and excited via a drive mechanism attached to its free end. The resonant frequency (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image006.png) is measured from which the velocity of the propagating wave is derived. Based on the derived velocity and the sample’s density, the low-strain shear modulus (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.png) of the material can be computed from the basic equation for torsional vibration.

== Young modulus (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image008.png) ==

The resonant column can also be used in flexural excitation to determine the material’s Young modulus (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.png). The conventional method with long samples allowing the application of Rayleigh’s energy method and Euler-Bernoulli beam theory, disregarded the shear strain energy and rotary inertia effect. When the tested specimen is short in length compared to its diameter, the effects of rotation and shear deformation of the samples during flexure can be substantial. These effects can be significant in interpreting data from flexural test, especially at high frequencies. Therefore, the Euler-Bernoulli theory of flexural vibration of elastic beam is found to be inadequate for short specimens and also for the prediction of higher modes of vibration. To be more accurate, Timoshenko beam theory is used as a model for this interpretation. The theory was developed by Ukrainian scientist Stephen Timoshenko in the 20th century which takes into account the shear deformation and rotary inertia. Different frequency equations for the clamped-free Timoshenko beam with an end mass in flexural vibration are solved to compute the value of elastic stiffness (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.png).

= Resonant column for torsional excitation =

In the standard torsional resonant column (Stokoe cell SBEL D1128) as mentioned in Allen and Stokoe (1982), the specimen is rigidly fixed at the base while torsional oscillation is applied to the free end by a drive head. The basic equations for the clamped-free resonant column subjected to torsion are:

{| border="0" cellspacing="0" cellpadding="0" width="90%" style="width:90.66%;"
|-
| style="width:12.32%;" |
| style="width:64.64%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image012.png

| style="width:23.04%;" |
(1)

|-
| style="width:12.32%;" |
Where

| style="width:64.64%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image014.png

| style="width:23.04%;" |
(2)

|-
| style="width:12.32%;" |
| style="width:64.64%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image016.png

| style="width:23.04%;" |
(3)

|}

The derivation of these equations is based on the assumption that the rotation is small and each transverse section remains plane and rotates about its centre. All the terms expressed in equation (1) are functions of the geometric properties of the specimen, except file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image018.png. Treating the system as a single degree of freedom system, the resonant frequency measured in the resonant column apparatus is the damped natural frequency (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image020.png) but is sufficiently close to the natural frequency (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image018.png). In this case, the error can be tolerable as file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image020.png is within 1% of file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image018.png. Solving equation (1) and (2) with file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image018.png, the shear wave velocity (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image022.png) can be found from which the shear modulus of the material (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.png) can be derived by rearranging equation (3).

= Resonant column for flexural excitation =

== Finding the Young modulus by Euler-Bernoulli beam theory (short samples) ==

The RCA can also be used to measure the Young modulus (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.png) of the material. Cascante et al. (1998) modified the standard Stokoe torsional resonant column (Stokoe cell SBEL D1128) to include flexural vibration mode. The schematic view of the apparatus is shown in Figure 1:

{| border="0" cellspacing="0" cellpadding="0" align="center"
|-
| style="width:566px;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image024.jpg

''Figure 1: Schematic view of the modified Stokoe RCA.''

''Image taken from Cascante et al. (1998)''

|}
<div style="clear:both;"></div>
In the original configuration, four pairs of excitation coils are connected in series to produce a net torque at the top of the sample (Figure 2 A). In Cascante’s modified version, the coils are reconnected so that only two magnets are used to produce a net horizontal force on top of the specimen (Figure 2 B).

{| border="0" cellspacing="0" cellpadding="0"
|-
| colspan="2" style="width:450px;height:220px;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image026.jpg

''Figure ''''2''''.''''2'''': Coil-magnet arrangements for torsional and flexural RCA. Images taken from Cascante et al. (1998)''

|-
| style="width:232px;height:16px;" |
'''''A: Torsional excitation'''''

| style="width:216px;height:16px;" |
'''''B: Flexural excitation'''''

|}

In the reduction of data for flexural excitation, the specimen and its drive head can be idealised as an elastic column with a rigid point mass at the top free end (Fig 3). The behaviour of the system is assumed to be elastic. Cascante et al. (1998) has developed a general mathematical formulation for the angular resonant frequency by using Rayleigh’s energy method and Euler Bernoulli beam theory. Based on this general equation, the Young Modulus (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.png) can be determined by:

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| style="width:16.34%;" |
| style="width:66.9%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image028.png

| style="width:16.74%;" |
(4)

|}

Where

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image030.png

| style="width:22.78%;" |
(5)

|}

In previous literatures, as the cross-sectional dimensions of the sample were small in comparison with its length, Euler-Bernoulli beam theory has been used to treat the boundary conditions and derive the frequency equation, from which file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.png could be determined.

{| border="0" cellspacing="0" cellpadding="0" width="564" style="width:563px;"
|-
| style="width:563px;height:232px;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image032.jpg

''Figure ''''2''''.''''3'''': Exaggerated view of deflected column for an ''''idealised'''' system''

''Image taken from Priest (2004)''

|}

== Finding the Young modulus by Timoshenko beam theory (long samples) ==

When the tested specimens are short in length compared to their thicknesses, the effect of shear deformation during flexure is significant which can result in possible discrepancies in interpreting data from flexural test. On the other hand, the effect of rotation is large when the curvature of the beam is large relative to its thickness. This is true when the beam is short in length compared to its thickness. Therefore, Timoshenko beam theory is used in this interpretation as it takes into consideration the effect of shear deformation and rotary inertia in which the conventional Euler-Bernoulli theory doesn’t. During vibration, a typical element of a beam not only performs translatory movement, but also rotation. With shear deformation being considered, the assumption of the elementary Euler-Bernoulli theory that ‘’plane section remains plane’’ is no longer applicable. Therefore, the angle of rotation which is equal to the slope file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image034.png of any section along the length of the beam cannot be obtained by simple differentiation of the transverse displacement y (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image036.png). Thus, it results in two independent motions file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image038.png and file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image040.png.

Timoshenko gave the coupled equations of motion for the beam with constant cross-section as:

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| style="width:16.34%;height:44px;" |
| style="width:66.9%;height:44px;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image042.png

| style="width:16.74%;height:44px;" |
(6)

|-
| style="width:16.34%;" |
| style="width:66.9%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image044.png

| style="width:16.74%;" |
(7)

|}

{| border="0" cellspacing="0" cellpadding="0"
|-
| style="width:526px;height:153px;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image046.jpg

''Figure ''''2''''.''''4'': ''The beam-mass system used in the analysis

'''''Image taken from Bruch and Mitchell (1987''''')

|}

Bruch and Mitchell (1987) investigated a particular case of a cantilevered Timoshenko beam with a tip mass (Figure 2.4). By applying the boundary conditions and using Huang’s non-dimensional variables, the solutions to the coupled equations are determined as functions of the Young modulus (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.png), the Shear modulus (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.png), material’s density (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image048.png), the angular natural frequency (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image018.png) and the geometry of the specimen. Bruch and Mitchell derived the frequency equation of the beam in flexural excitation by inserting the solutions to the coupled equations (6) and (7) into the boundary conditions, from which the matrix equation can be determined. By taking the determinant of the coefficient matrix equation, the resonant frequency equation was found from which the Young modulus (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.png) can be calculated.

Liu (1989) suggested three ways in which the work of Bruch and Mitchell could be further extended: (i) The base condition for the beam-mass system considered in [3] should be modeled as an imperfect clamped support (or elastic support), (ii) The tip mass’s centre of gravity is not practically right at the top of the beam but usually at a distance from the beam tip, (iii) the shear coefficient depends on both the shape of the cross-section and the Poisson ratio. Liu added springs at the hub to simulate the imperfect clamped support therefore the boundary condition also includes the spring’s properties which are the rotational spring constant and translational spring constant. By substituting the general solution into the new boundary conditions, Liu gave the improvement of Bruch and Mitchell’s frequency equation for the mass-loaded clamped-free Timoshenko beam.

The shear coefficient in Timoshenko’s beam theory is a dimensionless quantity, dependent on the shape of the cross section, which accounts for the fact that the shear stress and shear strain are not uniformly distributed over the cross section of the specimen. Cowper (1966) developed a new formula for the shear coefficient from the derivation of the equations of Timoshenko beam theory. For a circular cross-section, the value of file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image050.png was given in terms of the Poisson ratio as:

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| style="width:13.32%;" |
| style="width:63.9%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image052.png

| style="width:22.78%;" |
(8)

|}

Farghaly (1993) offered suggestion to extend Liu’s work by applying Timoshenko beam theory in treating the boundary conditions. He realised that the use of Euler-Bernoulli theory in the boundary conditions could result in inaccurate natural frequencies calculated, particularly for high slenderness ratios and higher modes of vibration. Farghaly’s model also includes the root flexibilities and the tip mass’s eccentricity as can be shown in Figure 2.5:

{| border="0" cellspacing="0" cellpadding="0"
|-
| style="width:590px;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image054.jpg

''Figure ''''2''''.''''5'''': Thick beam with tip mass and root flexibilities.''

''Image taken from ''''Farghaly (1993)'''''

|}

The work of Bruch and Mitchell (1987), Liu (1989) and Farghaly (1993) were in an attempt to simulate the motion of a flexible robot arm modeled as a cantilevered Timoshenko beam with a lumped mass and lumped moment of inertia at the free end. However, for the purpose of this essay, their resonant frequency equations were considered to be adequate for use in computing the material’s Young modulus from the flexural resonant column test, if the angular natural frequency is known.

= Using Timoshenko’s beam theory for resonant column testing =

== Frequency equation by Bruch and Mitchell ==

Bruch and Mitchell started with the original coupled equations of motion given by Timoshenko for the beam with constant cross section:

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| style="width:16.34%;" |
| style="width:66.9%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image056.png

| style="width:16.74%;" |
(3.1)

|-
| style="width:16.34%;" |
| style="width:66.9%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image058.png

| style="width:16.74%;" |
(3.2)

|}

Where file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image060.png is the weight per unit volume, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image062.png is the slope due to bending

The boundary conditions are:

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| style="width:16.34%;" |
| style="width:66.9%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image064.png

| style="width:16.74%;" |
(3.3)

|-
| style="width:16.34%;" |
| style="width:66.9%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image066.png

| style="width:16.74%;" |
(3.4)

|-
| style="width:16.34%;" |
| style="width:66.9%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image068.png

| style="width:16.74%;" |
(3.5)

|}

Where file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image070.png is the moment of inertia of the tip body about the axis of bending, with file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image072.png is the tip mass, and file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image074.png is the radius of gyration.

Huang’s non-dimensional variables (given by Huang [13]) and defined variables were used to solve the problem:

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| colspan="2" style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image076.png

file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image078.png

| colspan="2" style="width:22.78%;" |
(3.6)

|-
| style="width:16.34%;" |
| colspan="2" style="width:66.9%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image080.png

| style="width:16.74%;" |
(3.7)

|-
|
|
|
|
|}

Where file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image082.png is the mass of the beam, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image048.png is mass per unit volume of the beam, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image084.png is non-dimensional length along the beam

From the simple harmonic motion equations:

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| style="width:16.34%;" |
| style="width:66.9%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image086.png

| style="width:16.74%;" |
(3.8)

|}

Using the non-dimensional variables and series of equations (3.1) to (3.3), equations (3.4) reduced the problem to:

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| style="width:16.34%;" |
| style="width:66.9%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image088.png

| style="width:16.74%;" |
(3.9)

|-
| style="width:16.34%;" |
| style="width:66.9%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image090.png

| style="width:16.74%;" |
(3.10)

|-
| style="width:16.34%;" |
| style="width:66.9%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image092.png

| style="width:16.74%;" |
(3.11)

|-
| style="width:16.34%;" |
| style="width:66.9%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image094.png

| style="width:16.74%;" |
(3.12)

|-
| style="width:16.34%;" |
| style="width:66.9%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image096.png

| style="width:16.74%;" |
(3.13)

|}

Eliminating file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image098.png or file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image100.png from equations (3.9) and (3.13) gives:

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image102.png

| style="width:22.78%;" |
(3.14)

|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image104.png

| style="width:22.78%;" |
(3.15)

|}

Equations (3.14) and (3.15) were solved by Huang [13] yields:

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image106.png

| style="width:22.78%;" |
(3.16)

|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image108.png

| style="width:22.78%;" |
(3.17)

|}

Where file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image110.png

For file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image112.png, it was assumed that file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image114.png. Therefore, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image116.png. This is the first branch of the Timoshenko beam dispersion relation.

The following constants were derived in terms of file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image118.png, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image120.png, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image122.png and file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image124.png :

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| style="width:16.34%;" |
| style="width:66.9%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image126.png

file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image128.png

| style="width:16.74%;" |
(3.19)

|}

The second branch of the dispersion relation file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image130.png was discussed in Huang [23].

Substituting the solutions of file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image100.png and file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image098.png provided by equations (3.16) and (3.17), into the boundary conditions, gives the following matrix equation:

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image132.png

| style="width:22.78%;" |
(3.20)

|}

Where

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| style="width:13.32%;" |
| style="width:63.9%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image134.png

| style="width:22.78%;" |
(3.21)

|-
| style="width:13.32%;" |
| style="width:63.9%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image136.png

| style="width:22.78%;" |
(3.22)

|-
| style="width:13.32%;" |
| style="width:63.9%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image138.png

| style="width:22.78%;" |
(3.23)

|-
| style="width:13.32%;" |
| style="width:63.9%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image140.png

| style="width:22.78%;" |
(3.24)

|-
| colspan="2" style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image142.png

| style="width:22.78%;" |
(3.25)

|-
| colspan="2" style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image144.png

| style="width:22.78%;" |
(3.26)

|-
| colspan="2" style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image146.png

| style="width:22.78%;" |
(3.27)

|-
| colspan="2" style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image148.png

| style="width:22.78%;" |
(3.28)

|}

Taking the determinant of the coefficient matrix equation (3.20) gives the frequency equation, from which the elastic stiffness can be computed with the natural resonant frequency (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image006.png) as input.

== Frequency equation by Liu ==

Liu [16] introduced a rotational spring constant (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image150.png) and a translational spring constant (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image152.png) to model the imperfection of a clamped support. For simplicity, assuming the base of the resonant column is perfectly clamped, the values of the spring constants (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image150.png) and (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image152.png) therefore approach infinity. The distance from the beam tip to the centre of the added mass (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image154.png) was added to model the eccentricity. Moment of inertia of the added mass (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image156.png) was also included in the revised matrix equation to improve the accuracy of the original model by Bruch and Mitchel.

Liu started from the single free vibration equation of a Timoshenko beam given in [16], rather than the coupled equation of motion as in Bruch and Mitchell’s. The equation was presented as:

{| border="0" cellspacing="0" cellpadding="0" width="93%"
|-
| style="width:77.04%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image158.png

| style="width:22.96%;" |
(3.29)

|}

Where file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image048.png is the density of the beam per unit length, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image074.png is the shear coefficient

For a free vibration problem, it can be assumed a solution of the form:

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| style="width:16.34%;" |
| style="width:66.9%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image160.png

| style="width:16.74%;" |
(3.30)

|}

Substituting equation (3.30) into equation (3.29) gives:

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image162.png

| style="width:22.78%;" |
(3.31)

|}

In which file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image164.png, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image166.png and file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image168.png. If file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image170.png and file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image172.png satisfy the condition:

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image174.png

| style="width:22.78%;" |
(3.32)

|}

The general solution for file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image176.png is:

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image178.png

| style="width:22.78%;" |
(3.33)

|}

Where file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image180.png to file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image182.png are constants to be determined, and file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image184.png and file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image186.png are defined as:

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image188.png

| style="width:22.78%;" |
(3.34)

|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image190.png

| style="width:22.78%;" |
(3.35)

|}

The boundary conditions for file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image176.png can be taken as:

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image192.png

| style="width:22.78%;" |
(3.36)

|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image194.png

| style="width:22.78%;" |
(3.37)

|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image196.png

| style="width:22.78%;" |
(3.38)

|}

Substituting equation (3.30) into equations (3.36) to (3.38) produces the frequency equation:

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image198.png

| style="width:22.78%;" |
(3.39)

|}

In which

{| border="0" cellspacing="0" cellpadding="0" width="98%" style="width:98.36%;"
|-
|
| style="width:72.0%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image200.png

| style="width:21.24%;" |
(3.40)

|-
|
| style="width:72.0%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image202.png

| style="width:21.24%;" |
(3.41)

|-
|
| style="width:72.0%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image204.png

| style="width:21.24%;" |
(3.42)

|-
|
| style="width:72.0%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image206.png

| style="width:21.24%;" |
(3.43)

|-
|
| style="width:72.0%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image208.png

| style="width:21.24%;" |
(3.44)

|-
|
| style="width:72.0%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image210.png

| style="width:21.24%;" |
(3.45)

|-
|
| style="width:72.0%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image212.png

| style="width:21.24%;" |
(3.46)

|-
|
| style="width:72.0%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image214.png

| style="width:21.24%;" |
(3.47)

|-
| colspan="2" style="width:78.76%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image216.png, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image218.png, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image220.png, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image222.png

file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image224.png, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image226.png, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image228.png, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image230.png, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image232.png, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image234.png, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image236.png

| style="width:21.24%;" |
(3.48)

|-
|
|
|
|}

Assumed the base condition of the resonant column apparatus is fully clamped, the rotational spring constant (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image150.png) and translational spring constant (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image152.png) were considered to approach infinity (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image238.png) or in other words, the attached spring at the base support has no elastic stiffness. The values of file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image240.png and file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image242.png then became 0 as file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image244.png. The matrix determinant equation (3.39) was simplified to be:

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image246.png

| style="width:22.78%;" |
(3.49)

|}

Equation (3.49) was solved using Matlab with the same inputs as in the case of Bruch and Mitchell’s model, plus the eccentricity and moment of inertia of the tip mass, to evaluate the sample’s elastic stiffness.

== Frequency equation by Farghaly ==

Liu has derived a frequency equation to further improve the work of Bruch and Mitchell. The root flexibility, eccentricity and moment of inertia of the tip mass have been taken into consideration to improve the accuracy on modeling a robot arm as a clamped-free Timoshenko beam with a lumped mass and a lumped moment of inertia at its free end. The same idea of simulating a robot arm by Timoshenko beam theory can be used to model the RCA when the sample is short in length relative to its diameter. Farghaly commented in his published paper that in [16], Liu used Timoshenko beam theory for the system differential equation, while Euler-Bernoulli theory was applied to treat the boundary conditions. Farghaly stressed that, when using Liu’s formula to compute the resonant frequency with proper inputs, inaccurate natural frequencies maybe obtained, particularly for significant values of the slenderness ratio and higher modes of vibration.

The governing differential equations of the boundary conditions were presented as:

file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image248.png, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image250.png file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image252.png

file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image254.png, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image256.png

file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image258.png

The general solutions of the coupled equations of motion are:

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image260.png

| style="width:22.78%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image262.png

|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image264.png

| style="width:22.78%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image266.png

|}

By following the same procedure as in Bruch and Mitchell [3], combining equations (3.53) and (3.54) with the boundary conditions equation (3.50) to (3.52), the system frequency equation in terms of the root rigidity parameters can be written as:

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image268.png

| style="width:22.78%;" |
(3.55)

|}

In which

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image270.png

| style="width:22.78%;" |
(3.56)

|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image272.png

| style="width:22.78%;" |
(3.57)

|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image274.png

| style="width:22.78%;" |
(3.58)

|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image276.png

| style="width:22.78%;" |
(3.59)

|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image278.png

| style="width:22.78%;" |
(3.60)

|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image280.png

| style="width:22.78%;" |
(3.61)

|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image282.png

| style="width:22.78%;" |
(3.62)

|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image284.png

| style="width:22.78%;" |
(3.63)

|}

Where

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image286.png

| style="width:22.78%;" |
(3.64)

|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image288.png

| style="width:22.78%;" |
(3.65)

|}

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image290.png, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image292.png, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image294.png, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image296.png, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image298.png

| style="width:22.78%;" |
(3.66)

|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image300.png, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image236.png, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image302.png, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image304.png, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image306.png

| style="width:22.78%;" |
(3.67)

|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image308.png, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image310.png, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image312.png, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image314.png

| style="width:22.78%;" |
(3.68)

|}

In theory, to model the perfectly clamped support condition of the resonant column, the spring constants file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image316.png and K should be made to approach infinity. However, in Matlab, for simplicity, extreme values have been assigned to file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image316.png and K to give significant values of the root rigidity parameters file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image318.png and file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image320.png. As mentioned in Farghaly (1992), when using the matrix determinant equation (3.55) to compute the resonant frequencies, inaccurate results might be obtained for large values of the slenderness ratio and for higher modes of vibration. However, in our case, the length of the specimen is only about more than twice the diameter and vibration was limited to the 1st fundamental mode. According to Liu in Author’s Reply (1992), from his own practical point of view, if one can accept the idea of treating a complicated cantilever structure as a Timoshenko beam, then the discrepancies caused by non-exact boundary conditions might be considered as tolerable.

[[Category:Student_engineer_essay_competition]]

Resonant column method

2012-12-14T02:17:38Z

Nicky nguyen 91:

Resonant column method

2012-12-14T02:15:31Z

Nicky nguyen 91: Created page with " = Introduction = == Background == The resonant column method was initially developed by Japanese engineers: Ishimoto & Iida (1937). It was made popular in the 1960s by author..."

= Introduction =

== Background ==

The resonant column method was initially developed by Japanese engineers: Ishimoto & Iida (1937). It was made popular in the 1960s by authors such as Hall & Richart (1963), Drnevich et al. (1967) and Hardin & Black (1968). The resonant column apparatus has been used to measure the dynamic response of soils including the shear and elastic modulus based on the theory of wave propagation in prismatic rods.

== Shear modulus (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image002.png) ==

The resonant column method was conventionally used in torsion to measure the shear modulus (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.png) of the material. In most cases, the clamped-free configuration has been chosen for research purpose as its mathematical derivation is more straightforward. In the clamped-free test, a cylindrical specimen is fixed at the base and excited via a drive mechanism attached to its free end. The resonant frequency (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image006.png) is measured from which the velocity of the propagating wave is derived. Based on the derived velocity and the sample’s density, the low-strain shear modulus (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.png) of the material can be computed from the basic equation for torsional vibration.

== Young modulus (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image008.png) ==

The resonant column can also be used in flexural excitation to determine the material’s Young modulus (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.png). The conventional method with long samples allowing the application of Rayleigh’s energy method and Euler-Bernoulli beam theory, disregarded the shear strain energy and rotary inertia effect. When the tested specimen is short in length compared to its diameter, the effects of rotation and shear deformation of the samples during flexure can be substantial. These effects can be significant in interpreting data from flexural test, especially at high frequencies. Therefore, the Euler-Bernoulli theory of flexural vibration of elastic beam is found to be inadequate for short specimens and also for the prediction of higher modes of vibration. To be more accurate, Timoshenko beam theory is used as a model for this interpretation. The theory was developed by Ukrainian scientist Stephen Timoshenko in the 20th century which takes into account the shear deformation and rotary inertia. Different frequency equations for the clamped-free Timoshenko beam with an end mass in flexural vibration are solved to compute the value of elastic stiffness (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.png).

= Resonant column for torsional excitation =

In the standard torsional resonant column (Stokoe cell SBEL D1128) as mentioned in Allen and Stokoe (1982), the specimen is rigidly fixed at the base while torsional oscillation is applied to the free end by a drive head. The basic equations for the clamped-free resonant column subjected to torsion are:

{| border="0" cellspacing="0" cellpadding="0" width="90%" style="width:90.66%;"
|-
| style="width:12.32%;" |

| style="width:64.64%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image012.png

| style="width:23.04%;" |
(1)

|-
| style="width:12.32%;" |
Where

| style="width:64.64%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image014.png

| style="width:23.04%;" |
(2)

|-
| style="width:12.32%;" |

| style="width:64.64%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image016.png

| style="width:23.04%;" |
(3)

|}

The derivation of these equations is based on the assumption that the rotation is small and each transverse section remains plane and rotates about its centre. All the terms expressed in equation (1) are functions of the geometric properties of the specimen, except file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image018.png. Treating the system as a single degree of freedom system, the resonant frequency measured in the resonant column apparatus is the damped natural frequency (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image020.png) but is sufficiently close to the natural frequency (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image018.png). In this case, the error can be tolerable as file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image020.png is within 1% of file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image018.png. Solving equation (1) and (2) with file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image018.png, the shear wave velocity (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image022.png) can be found from which the shear modulus of the material (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.png) can be derived by rearranging equation (3).

= Resonant column for flexural excitation =

== Finding the Young modulus by Euler-Bernoulli beam theory (short samples) ==

The RCA can also be used to measure the Young modulus (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.png) of the material. Cascante et al. (1998) modified the standard Stokoe torsional resonant column (Stokoe cell SBEL D1128) to include flexural vibration mode. The schematic view of the apparatus is shown in Figure 1:

{| border="0" cellspacing="0" cellpadding="0" align="center"
|-
| style="width:566px;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image024.jpg

''Figure 1: Schematic view of the modified Stokoe RCA.

''Image taken from Cascante et al. (1998)''

|}
<div style="clear:both;"></div>

In the original configuration, four pairs of excitation coils are connected in series to produce a net torque at the top of the sample (Figure 2 A). In Cascante’s modified version, the coils are reconnected so that only two magnets are used to produce a net horizontal force on top of the specimen (Figure 2 B).

{| border="0" cellspacing="0" cellpadding="0"
|-
| colspan="2" style="width:450px;height:220px;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image026.jpg

''Figure ''''2''''.''''2'''': Coil-magnet arrangements for torsional and flexural RCA. Images taken from Cascante et al. (1998)

|-
| style="width:232px;height:16px;" |
'''''A: Torsional excitation'''''

| style="width:216px;height:16px;" |
'''''B: Flexural excitation'''''

|}

In the reduction of data for flexural excitation, the specimen and its drive head can be idealised as an elastic column with a rigid point mass at the top free end (Fig 3). The behaviour of the system is assumed to be elastic. Cascante et al. (1998) has developed a general mathematical formulation for the angular resonant frequency by using Rayleigh’s energy method and Euler Bernoulli beam theory. Based on this general equation, the Young Modulus (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.png) can be determined by:

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| style="width:16.34%;" |

| style="width:66.9%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image028.png

| style="width:16.74%;" |
(4)

|}

Where

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image030.png

| style="width:22.78%;" |
(5)

|}

In previous literatures, as the cross-sectional dimensions of the sample were small in comparison with its length, Euler-Bernoulli beam theory has been used to treat the boundary conditions and derive the frequency equation, from which file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.png could be determined.

{| border="0" cellspacing="0" cellpadding="0" width="564" style="width:563px;"
|-
| style="width:563px;height:232px;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image032.jpg

''Figure ''''2''''.''''3'''': Exaggerated view of deflected column for an ''''idealised'''' system

''Image taken from Priest (2004)''

|}

== Finding the Young modulus by Timoshenko beam theory (long samples) ==

When the tested specimens are short in length compared to their thicknesses, the effect of shear deformation during flexure is significant which can result in possible discrepancies in interpreting data from flexural test. On the other hand, the effect of rotation is large when the curvature of the beam is large relative to its thickness. This is true when the beam is short in length compared to its thickness. Therefore, Timoshenko beam theory is used in this interpretation as it takes into consideration the effect of shear deformation and rotary inertia in which the conventional Euler-Bernoulli theory doesn’t. During vibration, a typical element of a beam not only performs translatory movement, but also rotation. With shear deformation being considered, the assumption of the elementary Euler-Bernoulli theory that ‘’plane section remains plane’’ is no longer applicable. Therefore, the angle of rotation which is equal to the slope file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image034.png of any section along the length of the beam cannot be obtained by simple differentiation of the transverse displacement y (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image036.png). Thus, it results in two independent motions file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image038.png and file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image040.png.

Timoshenko gave the coupled equations of motion for the beam with constant cross-section as:

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| style="width:16.34%;height:44px;" |

| style="width:66.9%;height:44px;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image042.png

| style="width:16.74%;height:44px;" |
(6)

|-
| style="width:16.34%;" |

| style="width:66.9%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image044.png

| style="width:16.74%;" |
(7)

|}

{| border="0" cellspacing="0" cellpadding="0"
|-
| style="width:526px;height:153px;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image046.jpg

''Figure ''''2''''.''''4'': ''The beam-mass system used in the analysis

'''''Image taken from Bruch and Mitchell (1987''''')

|}

Bruch and Mitchell (1987) investigated a particular case of a cantilevered Timoshenko beam with a tip mass (Figure 2.4). By applying the boundary conditions and using Huang’s non-dimensional variables, the solutions to the coupled equations are determined as functions of the Young modulus (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.png), the Shear modulus (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.png), material’s density (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image048.png), the angular natural frequency (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image018.png) and the geometry of the specimen. Bruch and Mitchell derived the frequency equation of the beam in flexural excitation by inserting the solutions to the coupled equations (6) and (7) into the boundary conditions, from which the matrix equation can be determined. By taking the determinant of the coefficient matrix equation, the resonant frequency equation was found from which the Young modulus (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.png) can be calculated.

Liu (1989) suggested three ways in which the work of Bruch and Mitchell could be further extended: (i) The base condition for the beam-mass system considered in [3] should be modeled as an imperfect clamped support (or elastic support), (ii) The tip mass’s centre of gravity is not practically right at the top of the beam but usually at a distance from the beam tip, (iii) the shear coefficient depends on both the shape of the cross-section and the Poisson ratio. Liu added springs at the hub to simulate the imperfect clamped support therefore the boundary condition also includes the spring’s properties which are the rotational spring constant and translational spring constant. By substituting the general solution into the new boundary conditions, Liu gave the improvement of Bruch and Mitchell’s frequency equation for the mass-loaded clamped-free Timoshenko beam.

The shear coefficient in Timoshenko’s beam theory is a dimensionless quantity, dependent on the shape of the cross section, which accounts for the fact that the shear stress and shear strain are not uniformly distributed over the cross section of the specimen. Cowper (1966) developed a new formula for the shear coefficient from the derivation of the equations of Timoshenko beam theory. For a circular cross-section, the value of file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image050.png was given in terms of the Poisson ratio as:

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| style="width:13.32%;" |

| style="width:63.9%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image052.png

| style="width:22.78%;" |
(8)

|}

Farghaly (1993) offered suggestion to extend Liu’s work by applying Timoshenko beam theory in treating the boundary conditions. He realised that the use of Euler-Bernoulli theory in the boundary conditions could result in inaccurate natural frequencies calculated, particularly for high slenderness ratios and higher modes of vibration. Farghaly’s model also includes the root flexibilities and the tip mass’s eccentricity as can be shown in Figure 2.5:

{| border="0" cellspacing="0" cellpadding="0"
|-
| style="width:590px;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image054.jpg

''Figure ''''2''''.''''5'''': Thick beam with tip mass and root flexibilities.

''Image taken from ''''Farghaly (1993)''

|}

The work of Bruch and Mitchell (1987), Liu (1989) and Farghaly (1993) were in an attempt to simulate the motion of a flexible robot arm modeled as a cantilevered Timoshenko beam with a lumped mass and lumped moment of inertia at the free end. However, for the purpose of this essay, their resonant frequency equations were considered to be adequate for use in computing the material’s Young modulus from the flexural resonant column test, if the angular natural frequency is known.

= Using Timoshenko’s beam theory for resonant column testing =

== Frequency equation by Bruch and Mitchell ==

Bruch and Mitchell started with the original coupled equations of motion given by Timoshenko for the beam with constant cross section:

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| style="width:16.34%;" |

| style="width:66.9%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image056.png

| style="width:16.74%;" |
(3.1)

|-
| style="width:16.34%;" |

| style="width:66.9%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image058.png

| style="width:16.74%;" |
(3.2)

|}

Where file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image060.png is the weight per unit volume, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image062.png is the slope due to bending

The boundary conditions are:

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| style="width:16.34%;" |

| style="width:66.9%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image064.png

| style="width:16.74%;" |
(3.3)

|-
| style="width:16.34%;" |

| style="width:66.9%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image066.png

| style="width:16.74%;" |
(3.4)

|-
| style="width:16.34%;" |

| style="width:66.9%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image068.png

| style="width:16.74%;" |
(3.5)

|}

Where file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image070.png is the moment of inertia of the tip body about the axis of bending, with file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image072.png is the tip mass, and file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image074.png is the radius of gyration.

Huang’s non-dimensional variables (given by Huang [13]) and defined variables were used to solve the problem:

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| colspan="2" style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image076.png

file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image078.png

| colspan="2" style="width:22.78%;" |
(3.6)

|-
| style="width:16.34%;" |

| colspan="2" style="width:66.9%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image080.png

| style="width:16.74%;" |
(3.7)

|- height="0"
|
|
|
|
|}

Where file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image082.png is the mass of the beam, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image048.png is mass per unit volume of the beam, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image084.png is non-dimensional length along the beam

From the simple harmonic motion equations:

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| style="width:16.34%;" |

| style="width:66.9%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image086.png

| style="width:16.74%;" |
(3.8)

|}

Using the non-dimensional variables and series of equations (3.1) to (3.3), equations (3.4) reduced the problem to:

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| style="width:16.34%;" |

| style="width:66.9%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image088.png

| style="width:16.74%;" |
(3.9)

|-
| style="width:16.34%;" |

| style="width:66.9%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image090.png

| style="width:16.74%;" |
(3.10)

|-
| style="width:16.34%;" |

| style="width:66.9%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image092.png

| style="width:16.74%;" |
(3.11)

|-
| style="width:16.34%;" |

| style="width:66.9%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image094.png

| style="width:16.74%;" |
(3.12)

|-
| style="width:16.34%;" |

| style="width:66.9%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image096.png

| style="width:16.74%;" |
(3.13)

|}

Eliminating file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image098.png or file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image100.png from equations (3.9) and (3.13) gives:

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image102.png

| style="width:22.78%;" |
(3.14)

|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image104.png

| style="width:22.78%;" |
(3.15)

|}

Equations (3.14) and (3.15) were solved by Huang [13] yields:

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image106.png

| style="width:22.78%;" |
(3.16)

|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image108.png

| style="width:22.78%;" |
(3.17)

|}

Where file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image110.png

For file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image112.png, it was assumed that file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image114.png. Therefore, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image116.png. This is the first branch of the Timoshenko beam dispersion relation.

The following constants were derived in terms of file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image118.png, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image120.png, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image122.png and file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image124.png :

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| style="width:16.34%;" |

| style="width:66.9%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image126.png

file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image128.png

| style="width:16.74%;" |
(3.19)

|}

The second branch of the dispersion relation file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image130.png was discussed in Huang [23].

Substituting the solutions of file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image100.png and file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image098.png provided by equations (3.16) and (3.17), into the boundary conditions, gives the following matrix equation:

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image132.png

| style="width:22.78%;" |
(3.20)

|}

Where

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| style="width:13.32%;" |

| style="width:63.9%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image134.png

| style="width:22.78%;" |
(3.21)

|-
| style="width:13.32%;" |

| style="width:63.9%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image136.png

| style="width:22.78%;" |
(3.22)

|-
| style="width:13.32%;" |

| style="width:63.9%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image138.png

| style="width:22.78%;" |
(3.23)

|-
| style="width:13.32%;" |

| style="width:63.9%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image140.png

| style="width:22.78%;" |
(3.24)

|-
| colspan="2" style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image142.png

| style="width:22.78%;" |
(3.25)

|-
| colspan="2" style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image144.png

| style="width:22.78%;" |
(3.26)

|-
| colspan="2" style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image146.png

| style="width:22.78%;" |
(3.27)

|-
| colspan="2" style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image148.png

| style="width:22.78%;" |
(3.28)

|}

Taking the determinant of the coefficient matrix equation (3.20) gives the frequency equation, from which the elastic stiffness can be computed with the natural resonant frequency (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image006.png) as input.

== Frequency equation by Liu ==

Liu [16] introduced a rotational spring constant (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image150.png) and a translational spring constant (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image152.png) to model the imperfection of a clamped support. For simplicity, assuming the base of the resonant column is perfectly clamped, the values of the spring constants (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image150.png) and (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image152.png) therefore approach infinity. The distance from the beam tip to the centre of the added mass (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image154.png) was added to model the eccentricity. Moment of inertia of the added mass (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image156.png) was also included in the revised matrix equation to improve the accuracy of the original model by Bruch and Mitchel.

Liu started from the single free vibration equation of a Timoshenko beam given in [16], rather than the coupled equation of motion as in Bruch and Mitchell’s. The equation was presented as:

{| border="0" cellspacing="0" cellpadding="0" width="93%"
|-
| style="width:77.04%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image158.png

| style="width:22.96%;" |
(3.29)

|}

Where file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image048.png is the density of the beam per unit length, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image074.png is the shear coefficient

For a free vibration problem, it can be assumed a solution of the form:

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| style="width:16.34%;" |

| style="width:66.9%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image160.png

| style="width:16.74%;" |
(3.30)

|}

Substituting equation (3.30) into equation (3.29) gives:

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image162.png

| style="width:22.78%;" |
(3.31)

|}

In which file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image164.png, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image166.png and file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image168.png. If file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image170.png and file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image172.png satisfy the condition:

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image174.png

| style="width:22.78%;" |
(3.32)

|}

The general solution for file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image176.png is:

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image178.png

| style="width:22.78%;" |
(3.33)

|}

Where file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image180.png to file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image182.png are constants to be determined, and file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image184.png and file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image186.png are defined as:

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image188.png

| style="width:22.78%;" |
(3.34)

|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image190.png

| style="width:22.78%;" |
(3.35)

|}

The boundary conditions for file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image176.png can be taken as:

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image192.png

| style="width:22.78%;" |
(3.36)

|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image194.png

| style="width:22.78%;" |
(3.37)

|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image196.png

| style="width:22.78%;" |
(3.38)

|}

Substituting equation (3.30) into equations (3.36) to (3.38) produces the frequency equation:

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image198.png

| style="width:22.78%;" |
(3.39)

|}

In which

{| border="0" cellspacing="0" cellpadding="0" width="98%" style="width:98.36%;"
|-
|

| style="width:72.0%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image200.png

| style="width:21.24%;" |
(3.40)

|-
|

| style="width:72.0%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image202.png

| style="width:21.24%;" |
(3.41)

|-
|

| style="width:72.0%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image204.png

| style="width:21.24%;" |
(3.42)

|-
|

| style="width:72.0%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image206.png

| style="width:21.24%;" |
(3.43)

|-
|

| style="width:72.0%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image208.png

| style="width:21.24%;" |
(3.44)

|-
|

| style="width:72.0%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image210.png

| style="width:21.24%;" |
(3.45)

|-
|

| style="width:72.0%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image212.png

| style="width:21.24%;" |
(3.46)

|-
|

| style="width:72.0%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image214.png

| style="width:21.24%;" |
(3.47)

|-
| colspan="2" style="width:78.76%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image216.png, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image218.png, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image220.png, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image222.png

file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image224.png, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image226.png, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image228.png, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image230.png, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image232.png, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image234.png, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image236.png

| style="width:21.24%;" |
(3.48)

|- height="0"
|
|
|
|}

Assumed the base condition of the resonant column apparatus is fully clamped, the rotational spring constant (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image150.png) and translational spring constant (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image152.png) were considered to approach infinity (file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image238.png) or in other words, the attached spring at the base support has no elastic stiffness. The values of file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image240.png and file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image242.png then became 0 as file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image244.png. The matrix determinant equation (3.39) was simplified to be:

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image246.png

| style="width:22.78%;" |
(3.49)

|}

Equation (3.49) was solved using Matlab with the same inputs as in the case of Bruch and Mitchell’s model, plus the eccentricity and moment of inertia of the tip mass, to evaluate the sample’s elastic stiffness.

== Frequency equation by Farghaly ==

Liu has derived a frequency equation to further improve the work of Bruch and Mitchell. The root flexibility, eccentricity and moment of inertia of the tip mass have been taken into consideration to improve the accuracy on modeling a robot arm as a clamped-free Timoshenko beam with a lumped mass and a lumped moment of inertia at its free end. The same idea of simulating a robot arm by Timoshenko beam theory can be used to model the RCA when the sample is short in length relative to its diameter. Farghaly commented in his published paper that in [16], Liu used Timoshenko beam theory for the system differential equation, while Euler-Bernoulli theory was applied to treat the boundary conditions. Farghaly stressed that, when using Liu’s formula to compute the resonant frequency with proper inputs, inaccurate natural frequencies maybe obtained, particularly for significant values of the slenderness ratio and higher modes of vibration.

The governing differential equations of the boundary conditions were presented as:

file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image248.png, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image250.png file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image252.png

file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image254.png, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image256.png

file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image258.png

The general solutions of the coupled equations of motion are:

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image260.png

| style="width:22.78%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image262.png

|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image264.png

| style="width:22.78%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image266.png

|}

By following the same procedure as in Bruch and Mitchell [3], combining equations (3.53) and (3.54) with the boundary conditions equation (3.50) to (3.52), the system frequency equation in terms of the root rigidity parameters can be written as:

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image268.png

| style="width:22.78%;" |
(3.55)

|}

In which

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image270.png

| style="width:22.78%;" |
(3.56)

|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image272.png

| style="width:22.78%;" |
(3.57)

|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image274.png

| style="width:22.78%;" |
(3.58)

|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image276.png

| style="width:22.78%;" |
(3.59)

|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image278.png

| style="width:22.78%;" |
(3.60)

|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image280.png

| style="width:22.78%;" |
(3.61)

|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image282.png

| style="width:22.78%;" |
(3.62)

|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image284.png

| style="width:22.78%;" |
(3.63)

|}

Where

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image286.png

| style="width:22.78%;" |
(3.64)

|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image288.png

| style="width:22.78%;" |
(3.65)

|}

{| border="0" cellspacing="0" cellpadding="0" width="91%" style="width:91.72%;"
|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image290.png, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image292.png, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image294.png, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image296.png, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image298.png

| style="width:22.78%;" |
(3.66)

|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image300.png, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image236.png, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image302.png, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image304.png, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image306.png

| style="width:22.78%;" |
(3.67)

|-
| style="width:77.22%;" |
file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image308.png, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image310.png, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image312.png, file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image314.png

| style="width:22.78%;" |
(3.68)

|}

In theory, to model the perfectly clamped support condition of the resonant column, the spring constants file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image316.png and K should be made to approach infinity. However, in Matlab, for simplicity, extreme values have been assigned to file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image316.png and K to give significant values of the root rigidity parameters file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image318.png and file:///C:/Users/ACER57~1/AppData/Local/Temp/msohtmlclip1/01/clip_image320.png. As mentioned in Farghaly (1992), when using the matrix determinant equation (3.55) to compute the resonant frequencies, inaccurate results might be obtained for large values of the slenderness ratio and for higher modes of vibration. However, in our case, the length of the specimen is only about more than twice the diameter and vibration was limited to the 1st fundamental mode. According to Liu in Author’s Reply (1992), from his own practical point of view, if one can accept the idea of treating a complicated cantilever structure as a Timoshenko beam, then the discrepancies caused by non-exact boundary conditions might be considered as tolerable.

Off-site prefabrication of buildings: A guide to connection choices

2012-12-14T01:26:17Z

Nicky nguyen 91:

= Introduction =

Off-site prefabrication is a possible solution to many issues regarding the underachievement of the UK’s construction industry including: safety record, public perception, client satisfaction, profitability, delays, skilled workforce and overall contribution to the national economy. Prefabrication is believed to be advantageous over traditional construction methods in the followings:
*Quality – Higher-quality finishes with defects eliminated prior to completion.
*Safety – Safer working environment under factory conditions.
*Cost – Repeated use of moulds through standardisation reduces formwork materials, preliminaries, site storage and on-site facilities.
*Waste – Reduced off-cuts from formwork and the introduction of prefabricated rebars.
*Programme – Increased predictability due to reduced external factors such as weather.
*Local disruption – less environmental impacts such as dust and noise pollution.
*Accuracy – Increased accuracy since templates produced using Computer Aided Design (CAD) systems.
*Timescale – Components built off-site leads to reduced on-site construction time.

Among those mentioned, a driving factor for using prefabrication is to improve both quality and safety, as are rated 4.3 and 3.9 respectively on a five point Likert scale (Pann et al, 2008).

[[File:Laings O Rourke precast factory.JPG|RTENOTITLE]]

Figure 1: Laing O'Rourke's Explore precast manufacturing facility, Steetley, UK (Croxon, 2010) 

= History =

The World War One (1910s) and World War Two (1940s) stimulated research into newer methods than traditional brick construction. At that time, the requirement for rapid construction and the willingness to pay for it were the driving forces, in contrast to the major shortages of skilled labour and building materials. In the 1940s, UK government promoted methods of newer construction from the industries. This eventually led to the construction of hundreds of prefabricated concrete tower blocks and thousands of schools in the 1950s and 1960s which were often poorly designed. These were of low cost and often built without the lifetime of the buildings considered. Volumetric construction, the construction technique involving the production of buildings as a number of boxes connected on site, was used throughout the 1960s and 1970s. The collapse of the Ronan Point tower block in East London in 1968 is well known and attributed as one of the reasons for continued suspicion, fear and decline of prefabrication in this country.

Following the above downturn, there is currently a shift towards prefabrication within the industry. Many UK major construction firms are starting to see the benefits of prefabrication. Kier, Interserve, NG Bailey, Arup, Capita Symonds and Laing O’Rourke (LOR) are some of the market leaders registering their interest. LOR is recognised as having the largest interest, investing £100m in its Design for Manufacture and Assembly facility located at its Explore industrial park, which is “the most advanced facility of its type in Europe”. They aim to take advantage of public and private sector clients including BAA, Premier Inn, the Department of Education and both the Ministry of Justice and the Ministry of Defence (Wright, 2010), who all consider off-site prefabrication solutions.

Although a current shortage of a skilled workforce is said to be the cause, the increased uptake of prefabrication is believed to be a permanent move, as opposed to the short-lived uptake seen in the 60s and the 70s.

[[File:Ronan point tower block.JPG|RTENOTITLE]] 

Figure 2 - Ronan Point tower block failure, East London 1968 (Daily Telegraph, 1968)

= Precast construction method =

As with in-situ reinforced concrete construction, precast construction lends itself to a variety of different construction techniques, layouts and sequences.

== Frame and Deck Construction ==

[[File:Frame and deck construction.JPG|RTENOTITLE]] 

Figure 3 - Frame and deck systems, a) single storey columns, b) multi storey columns (Task Group 6.2 F.I.B., 2008, p. 5).

A precast deck supported by precast beams and columns form the building’s structural system. This form is frequently used in the construction of multi storey car parks with up to 16m spans to reduce columns between car parking spaces. It can also be used where floor to beam soffit height does not need to be minimised. The overall column height within a frame and deck system may correspond to greater than one storey.

The following connections are utilised in frames and deck construction:
*column to column
*column to base
*beam to column
*beam to base

[[File:Diagram.jpg|RTENOTITLE]] 

Figure 4 - Diagram of the precast elements and connections used in frame and deck construction (Irish Precast Concrete Association, 2003)

== Crosswall Construction ==

Crosswall is a modern method where load bearing walls provide the primary vertical support for precast floors and lateral stability. External wall panels, lift cores or staircases are used to provide the required longitudinal stability. Bridging components such as floors, roofs and beams are supported by the load bearing walls or façade wall. The system is ideal for buildings with cellular and orthogonal grids, with rooms of up to 4mx9m as standard. Thus it leads to a structurally efficient building with high levels of sound and fire insulation between adjacent rooms.

Crosswall construction utilises the following connections:
*wall to wall at vertical joints
*wall to wall at horizontal joints
*wall to base/foundation

[[File:Crosswall construction.JPG|RTENOTITLE]] 

Figure 5 - Crosswall systems, a) load bearing crosswall, b) load bearing facade wall (Task Group 6.2 F.I.B., 2008, p. 8). 

The precast elements are brought to site just in time allowing them to be lifted from the transport vehicle and installed in place. Hidden joints and ties, both horizontally and vertically are grouted in place as the work develops, allowing progressive collapse criteria of the Building Regulations to be met. With the possibility of incorporated mechanical and electrical components and minimal finishing needed, following trades can start prior to the completion of precast erection. 

== Volumetric Construction ==

The term volumetric construction is given when concrete modules (constructed in the factory) are installed on site to form a cellular system or used independently as a self-contained cell. The modules can be cast as a room or as panels which are subsequently joined together in the factory prior to site delivery. For a cellular system, the ground floor cells are laid on pre-prepared ground floor slabs with individual modules lowered into place usually forming the roof of the unit below.

[[File:Volumetric construction.JPG|RTENOTITLE]] 

Figure 6 - Examples of volumetric construction used for prison construction with an example of a possible finish (Oldcastle Precast Inc. ). 

Cellular systems are advantageous when being incorporated with a repetitive design (Figure 6). Common uses include hotels, prison cells, student halls and residential buildings. Self-contained cells are used mostly for specialised purposes where services are needed such as wet rooms, bathroom pods (Figure 7) and service utility rooms. Once lifted into place, the modules are secured by a number of methods including bolted and doweled connections. 

[[File:Volumetric bathroom.JPG|RTENOTITLE]]

Figure 7 - Volumetric construction in the form of a bathroom pod lifted into a crosswall frame (The Concrete Centre, 2007a). 

== Hybrid Construction ==

Hybrid construction is not a standalone method of precast construction. It is mentioned for its use in conjunction with the above construction methods. Precast elements can be used to provide permanent formwork for in-situ concrete. The combination of in-situ and precast concrete allows the benefits of both to be utilised. Figure 8 shows how safe working platforms are created by the precast floors, which increases safety on site and omits the need for in-situ concrete formwork, both factors significantly decrease the construction time. Greater spans can be achieved with hybrid construction as composite action is achieved by using different structural materials for the upper and lower areas of the element. The interface between the two materials will have to withstand shear stresses which can be overcome through the use of shear studs or precast reinforcement in the floor slab. 

[[File:Hybrid construction.JPG|RTENOTITLE]]

Figure 8 - Construction site utilising hybrid construction. a) Temporary props in place to support precast lattice floor slabs b) Installed precast lattice floor slabs awaiting reinforcement c) Concrete curing and binding with precast slab beneath. (webbaviation on behalf of Laing O'Rourke, 2011) 

= Precast Connections =

== Classification of connections ==

[[File:Precast connection.jpg|RTENOTITLE]] 

Figure 9 - Generic forms of beam-column connections, A: hidden beam end connection, B: corbel / haunch bearing connection, C: continuous column connection, D: continuous beam connection (Task Group 6.2, Fédération internationale du béton., 2008, p. 289)

For structural design, the connection’s stiffness is important in designing for moment distribution. There are three classes of connections based upon their degree of rigidity:
*Rigid connection – This connection can sustain vertical and horizontal actions as well as bending moment. The relative angle between connected members is maintained due to the stiffness of the connection.
*Pinned connection – This connection can sustain vertical and horizontal actions but not bending moment. The connected members are free to rotate in one direction with the connection having no degree of stiffness.
*Semi-rigid connection – This connection is between rigid or pinned as it is able to sustain vertical and horizontal actions and some amount of moment.

With in-situ reinforced concrete construction, a monolithic rigid connection is usually produced through design and provided on site. Precast connections range in their level of rigidity, from fully rigid to a completely pinned connection. A true pinned connection containing zero moment capacity is rare. In fact, many connections have some degree of rigidity but are conservatively assumed pinned. The steel connection shown in Figure 10 will retain some degree of rigidity, yet is usually modelled in design as a pinned connection. This is a conservative measure as beams spanning pinned connections are subject to the full action moment. Due to the connection having some degree of stiffness and therefore moment capacity, the negative bending moment acting upon the beam will be overestimated. 

[[File:Connectin classification.JPG|RTENOTITLE]] 

Figure 10 - Steel shear plate connection to CHS column (Kurobane et al., 2005)

Within the steel industry, research has shown cost reductions of between 10 to 20% for semi-rigid frames over rigid frames (Kurobane et al., 2005). Therefore the level of rigidity is an important consideration when choosing a method of connecting precast concrete elements.

== Continuous column with Corbel connections ==

[[File:Corbel connection.JPG|RTENOTITLE]]
<div>Figure 11 -Examples of concrete corbel connection with continuity reinforcement. A. (The Concrete Centre, 2007a) B. (Task Group 6.2 F.I.B., 2008, p. 49) </div>
Corbel connections, as shown in Figure 11 are most often used to support long span beams or heavy loads. Due to the visual and physical intrusion caused by the corbel or haunch widening the column, this connection is not widely used in the construction of multi-storey concrete frames. The basic corbel connection is designed as simply supported, dowel bars and/or fixing cleats. This type of connection can be used to prevent lateral movement and provide some joint fixity, although research has proven that the basic dowelled connection is best modelled as pinned. In-situ, structural screed can be used to increase continuity of the connection, thus allowing the tension reinforcement to resist the forces arising from beam movements. This can either be at the end, or across the whole length of the beam or floor slab. It was shown that a corbel / haunch connections with small amounts of cast in place reinforced concrete, although designed as a simply supported pinned connection, can improve strength and stiffness resulting in a semi or often fully rigid connection.

[[File:Corbel connection real example.JPG|RTENOTITLE]] 

Figure 12 - Photo of example beam to column corbel connection (General Precast Concrete Ltd, 2008). 

== Continuous beam connection ==

[[File:Continuous beam connection.JPG|RTENOTITLE]] 

Figure 13 - Discontinuous column with continuous beam, Left: (Elliot, 1992, p.67), Right: (Task Group 6.2 F.I.B., 2008, p. 299)

This type of connection is mainly used in portal frames or in skeletal frames when beams need to be continuous over supports, as is required for a cantilever. The beams are seated on dry pack mortar on top of the vertical members and reinforcing starter bars are projected through sleeves in the beam from the lower column up into the upper column. These sleeves are subsequently grouted to provide vertical continuity. Once the beam is lowered into place, this connection requires no additional formwork providing the grout is poured through vents in the upper column. Therefore, provided the remaining beam end is secured, loads for construction access can be placed upon the beam. This enhances the simplicity of installation and therefore safety on site.

== Wall and Column shoes ==

[[File:Wall and Column shoe connection 1.JPG|RTENOTITLE]] 

Figure 14 - Photos of example hidden corbel systems used with a precast column and precast beam (Peikko concrete connections, 2009)

Investment into modern technologies has resulted in the production of the hidden corbel connection. This is the most popular type of precast connection used in the UK so far. This type produces fireproof connections which are architecturally advantageous as they minimise visual intrusion whilst maximising floor to soffit height.

The connection area is minimal, protecting the reinforcement steel used in the connection. The connection also benefits from superior adjustability with the modern connection utilising a small adjustable plate, allowing fine tuning of the column corbel prior to installation of the beam.

[[File:Wall and Column shoe connection 2.JPG|RTENOTITLE]] 

Figure 15 – (Left) Inside detailed view of the anchorage of the corbel connection into the precast column and beams (Peikko concrete connections, 2009). (Right) Labelled example of an alternative hidden corbel system (JVI inc.)

== Ground foundation column connections ==

There are three main methods of connecting columns to foundations:
*Projecting starter bars – The in-situ foundation houses cast in starter bars which the precast column is later lowered onto and grouted to provide continuity.
*Pocket connection – This is the most rigid connection and is utilised when the moment resisting capacity of the connection is required for the lateral stability of the structure. A pocket is provided within the foundation into which the precast column is lowered. The surrounding area is grouted or filled with in-situ concrete.
*Baseplate connection – The base of the precast column contains steel base plates which cast-in bolts are fed through and bolted into place. The surrounding area to the holding down bolts is then filled with non-shrink grout to complete the connection.

The three types above are conservatively modelled as pinned connections resulting in an underestimate of the moments transferred to the columns and beams above. The foundation column connection is subjected to certain degree of variability such as possible rotations due to ground conditions.

= Connection’s discussion and evaluation =

There exists a variety of precast connection types within each big group above. The connections are assessed against different criteria including: the amount of additional materials; aesthetic/space intrusion of the finished connections; allowable tolerance; amount of wet work formwork required; possibility of future reuse/dismantle; operative involvement on site; level of rigidity; safety; skills required; amount of temporary works; time of assembly; tools required; weather sensitivity and level of wet casting needed.

== Continuous beam connection ==

[[File:Discussion - continuous beam.JPG|RTENOTITLE]] 

Figure 16 - Bolted steel shoe diagram (Halfen GmbH, 2011)

The bolted steel shoe is considered to be the most favourable of the continuous beam connection type as it is simple to produce and quick to assemble on site. This connection requires no structural in-situ works compared to other sub-types. The connection can have a number of different bolt arrangements depending on the size and shape of the column. When used correctly the anchor bolts can be utilised to transfer both tensile and compressive load through to the column below, thus minimising stress on the beam / slab in between. Alternatively the beam / slab can be suitably designed to transfer the load to the column below.

Due to the presence of a continuous beam, the large hogging moments generated at the connection will be transferred to the column. The moment capacity of this connection is high due to the high tensile capacity of the steel holding down bolts resisting the rotation of the column due to buckling, which may result from the hogging moments transferred from the beams.

The amount and positioning of the holding down bolts will determine the connections rigidity. The closer the bolts are to the centre point in the plane of rotation the more the connection will represent a pinned connection between the columns and the beam. As shown in Figure 16, the bolts have been positioned to give the maximum lever arm against any point of pivot and thus maximises rotational resistance.

== Corbel/Billet connection ==

[[File:Discussion-corbel.JPG|RTENOTITLE]] 

Figure 17 - Bearing only connection diagram (Pujol Group)

The bearing only is the favoured connection due to its simple straightforward design which facilitates quick assembly time on site. This fully pinned connection type will therefore transfer vertical and horizontal loads into the column, but all moment will be contained within the beam. For this reason, the beam will be designed to resist greater moment. The connection is therefore less efficient than moment sustaining connections.

The bearing with bolted dowel bar which has been taken as the preferred connection method for this connection type. The bearing with bolted dowel bar allows reduced connection width due to dowel action acting as horizontal restraint. The connection is fixed at the top using a bolt which extends through the column. This is positioned to resist maximum bending moment as the point of pivot will be within the underside of the beam. It is common practice in design to ensure that fixing elements of connections are not the limiting element and therefore the bolt will be able to transfer a considerable amount of moment to the column. The flat landing of the corbel, although unsightly, when combined with the dowel, acts as a torsional restraint. This can be further improved by using a cleat which extends the width of the beam. As the column is continuous, the beam will be required to sustain the majority of bending moment. This connection is semi rigid; therefore it can sustain vertical and horizontal loads with a degree of hogging moment transferred to the column.

[[File:Discussion-corbel 2.JPG|RTENOTITLE]]

Figure 18 - Bearing with bolted dowel, the preferred connection from billet / corbel type as bearing only connection disregarded

== Concealed fixing ==

The concealed bolted steel billet, which is one of the most modern connections, is favoured within this category. It utilises the advantages of other connections with an aesthetically pleasing and simple design, which allows minor adjustments to be made to the plate rather than the beam. There are other connections which show greater rigidity but require greater installing time on site, therefore are less favoured. This connection transfers horizontal and vertical loads to the column through the bolted connection. Its moment resisting capacity is small when compared to the bolted doweled corbel. However, as the connection extends to full height of the beam it is well positioned to sustain some moment. As there are just two bolts per beam to column connection, the level of moment transfer will be limited. For this reason the connection will act as a semi-rigid connection.

[[File:Concealed fixing.JPG|RTENOTITLE]]

Figure 19 - Preferred concealed connection, bolted steel corbel connection (Peikko concrete connections, 2009)

= Conclusion =

== Vertical load resistance ==

All three connections are capable of transferring the vertical load both from the column above and from the beams. As all three of the preferred connections from each group utilise a bolted mechanism to provide fixity, it is feasible that a structure constructed using only these connections would be able to resist against vertical loading without using any in-situ casting.

== Horizontal lateral restraint ==

The connections identified above, unless suitably designed for using excessive sized members and reinforcement, will struggle to resist lateral loading. The lateral loading will need to be taken by shear walls and/or concrete cores such as lift shafts or steel bracing to create a hybrid structure. But as with the initial problem of over-engineered connections through the neglect of their moment capacity, the lateral loading capacity of the preferred connections would need to be assessed and accounted for in order to produce the most efficient design.

Combination of continuous column and continuous beam joints can be used to help transfer moments to stiffer areas of the structure. It thus also allows for a more efficient design with only critical members designed to facilitate the load transfer.

== Frame analysis ==

In-situ frames have fully rigid connections. Should a precast connection be capable of transferring moment to the columns and thus down to the supports, then it can be assed as a complete frame or a series of sub frames. Moments, either hogging or sagging are attracted to stiffer members. Should the connection be capable of transferring these moments, the moments at the columns will then be in hogging and will need to be accounted for. Many published papers (Gorgun, 1997; Aguiar et al., 4 June 2012; Baharuddin et al., 2006) have discovered that some precast connections (including the ones mentioned above) can sustain hogging moment, and are therefore over engineered using the current design process. Therefore the structural frame should be modelled similar to a steel frame, where if almost no moment can be sustained then the connections are designed as pinned.

== Disproportionate Collapse ==

Since 2004, the Building Regulations in England and Wales have been revised to ensure all buildings are designed against disproportionate collapse. The connections above have been analysed with moment capacity as the desired attribute, but they will also provide some tensional resistance which would inherently provide resistance against disproportionate collapse.

= References =

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[[Category:Construction_management]]
[[Category:Health_and_safety_/_CDM]]
[[Category:Products_/_components]]
[[Category:Project_types]]
[[Category:Sustainability]]
[[Category:Student_architect_essay_competition]]
[[Category:Student_engineer_essay_competition]]

Off-site prefabrication of buildings: A guide to connection choices

2012-12-14T01:14:50Z

Nicky nguyen 91:

= Introduction =

Off-site prefabrication is a possible solution to many issues regarding the underachievement of the UK’s construction industry including: safety record, public perception, client satisfaction, profitability, delays, skilled workforce and overall contribution to the national economy. Prefabrication is believed to be advantageous over traditional construction methods in the followings:
*Quality – Higher-quality finishes with defects eliminated prior to completion.
*Safety – Safer working environment under factory conditions.
*Cost – Repeated use of moulds through standardisation reduces formwork materials, preliminaries, site storage and on-site facilities.
*Waste – Reduced off-cuts from formwork and the introduction of prefabricated rebars.
*Programme – Increased predictability due to reduced external factors such as weather.
*Local disruption – less environmental impacts such as dust and noise pollution.
*Accuracy – Increased accuracy since templates produced using Computer Aided Design (CAD) systems.
*Timescale – Components built off-site leads to reduced on-site construction time.

Among those mentioned, a driving factor for using prefabrication is to improve both quality and safety, as are rated 4.3 and 3.9 respectively on a five point Likert scale (Pann et al, 2008).

[[File:Laings O Rourke precast factory.JPG|RTENOTITLE]]

Figure 1: Laing O'Rourke's Explore precast manufacturing facility, Steetley, UK (Croxon, 2010) 

= History =

The World War One (1910s) and World War Two (1940s) stimulated research into newer methods than traditional brick construction. At that time, the requirement for rapid construction and the willingness to pay for it were the driving forces, in contrast to the major shortages of skilled labour and building materials. In the 1940s, UK government promoted methods of newer construction from the industries. This eventually led to the construction of hundreds of prefabricated concrete tower blocks and thousands of schools in the 1950s and 1960s which were often poorly designed. These were of low cost and often built without the lifetime of the buildings considered. Volumetric construction, the construction technique involving the production of buildings as a number of boxes connected on site, was used throughout the 1960s and 1970s. The collapse of the Ronan Point tower block in East London in 1968 is well known and attributed as one of the reasons for continued suspicion, fear and decline of prefabrication in this country.

Following the above downturn, there is currently a shift towards prefabrication within the industry. Many UK major construction firms are starting to see the benefits of prefabrication. Kier, Interserve, NG Bailey, Arup, Capita Symonds and Laing O’Rourke (LOR) are some of the market leaders registering their interest. LOR is recognised as having the largest interest, investing £100m in its Design for Manufacture and Assembly facility located at its Explore industrial park, which is “the most advanced facility of its type in Europe”. They aim to take advantage of public and private sector clients including BAA, Premier Inn, the Department of Education and both the Ministry of Justice and the Ministry of Defence (Wright, 2010), who all consider off-site prefabrication solutions.

Although a current shortage of a skilled workforce is said to be the cause, the increased uptake of prefabrication is believed to be a permanent move, as opposed to the short-lived uptake seen in the 60s and the 70s.

[[File:Ronan point tower block.JPG|RTENOTITLE]] 

Figure 2 - Ronan Point tower block failure, East London 1968 (Daily Telegraph, 1968)

= Precast construction method =

As with in-situ reinforced concrete construction, precast construction lends itself to a variety of different construction techniques, layouts and sequences.

== Frame and Deck Construction ==

[[File:Frame and deck construction.JPG|RTENOTITLE]] 

Figure 3 - Frame and deck systems, a) single storey columns, b) multi storey columns (Task Group 6.2 F.I.B., 2008, p. 5).

A precast deck supported by precast beams and columns form the building’s structural system. This form is frequently used in the construction of multi storey car parks with up to 16m spans to reduce columns between car parking spaces. It can also be used where floor to beam soffit height does not need to be minimised. The overall column height within a frame and deck system may correspond to greater than one storey.

The following connections are utilised in frames and deck construction:
*column to column
*column to base
*beam to column
*beam to base

[[File:Diagram.jpg|RTENOTITLE]] 

Figure 4 - Diagram of the precast elements and connections used in frame and deck construction (Irish Precast Concrete Association, 2003)

== Crosswall Construction ==

Crosswall is a modern method where load bearing walls provide the primary vertical support for precast floors and lateral stability. External wall panels, lift cores or staircases are used to provide the required longitudinal stability. Bridging components such as floors, roofs and beams are supported by the load bearing walls or façade wall. The system is ideal for buildings with cellular and orthogonal grids, with rooms of up to 4mx9m as standard. Thus it leads to a structurally efficient building with high levels of sound and fire insulation between adjacent rooms.

Crosswall construction utilises the following connections:
*wall to wall at vertical joints
*wall to wall at horizontal joints
*wall to base/foundation

[[File:Crosswall construction.JPG|RTENOTITLE]] 

Figure 5 - Crosswall systems, a) load bearing crosswall, b) load bearing facade wall (Task Group 6.2 F.I.B., 2008, p. 8). 

The precast elements are brought to site just in time allowing them to be lifted from the transport vehicle and installed in place. Hidden joints and ties, both horizontally and vertically are grouted in place as the work develops, allowing progressive collapse criteria of the Building Regulations to be met. With the possibility of incorporated mechanical and electrical components and minimal finishing needed, following trades can start prior to the completion of precast erection. 

== Volumetric Construction ==

The term volumetric construction is given when concrete modules (constructed in the factory) are installed on site to form a cellular system or used independently as a self-contained cell. The modules can be cast as a room or as panels which are subsequently joined together in the factory prior to site delivery. For a cellular system, the ground floor cells are laid on pre-prepared ground floor slabs with individual modules lowered into place usually forming the roof of the unit below.

[[File:Volumetric construction.JPG|RTENOTITLE]] 

Figure 6 - Examples of volumetric construction used for prison construction with an example of a possible finish (Oldcastle Precast Inc. ). 

Cellular systems are advantageous when being incorporated with a repetitive design (Figure 6). Common uses include hotels, prison cells, student halls and residential buildings. Self-contained cells are used mostly for specialised purposes where services are needed such as wet rooms, bathroom pods (Figure 7) and service utility rooms. Once lifted into place, the modules are secured by a number of methods including bolted and doweled connections. 

[[File:Volumetric bathroom.JPG|RTENOTITLE]]

Figure 7 - Volumetric construction in the form of a bathroom pod lifted into a crosswall frame (The Concrete Centre, 2007a). 

== Hybrid Construction ==

Hybrid construction is not a standalone method of precast construction. It is mentioned for its use in conjunction with the above construction methods. Precast elements can be used to provide permanent formwork for in-situ concrete. The combination of in-situ and precast concrete allows the benefits of both to be utilised. Figure 8 shows how safe working platforms are created by the precast floors, which increases safety on site and omits the need for in-situ concrete formwork, both factors significantly decrease the construction time. Greater spans can be achieved with hybrid construction as composite action is achieved by using different structural materials for the upper and lower areas of the element. The interface between the two materials will have to withstand shear stresses which can be overcome through the use of shear studs or precast reinforcement in the floor slab. 

[[File:Hybrid construction.JPG|RTENOTITLE]]

Figure 8 - Construction site utilising hybrid construction. a) Temporary props in place to support precast lattice floor slabs b) Installed precast lattice floor slabs awaiting reinforcement c) Concrete curing and binding with precast slab beneath. (webbaviation on behalf of Laing O'Rourke, 2011) 

= Precast Connections =

== Classification of connections ==

[[File:Precast connection.jpg|RTENOTITLE]] 

Figure 9 - Generic forms of beam-column connections, A: hidden beam end connection, B: corbel / haunch bearing connection, C: continuous column connection, D: continuous beam connection (Task Group 6.2, Fédération internationale du béton., 2008, p. 289)

For structural design, the connection’s stiffness is important in designing for moment distribution. There are three classes of connections based upon their degree of rigidity:
*Rigid connection – This connection can sustain vertical and horizontal actions as well as bending moment. The relative angle between connected members is maintained due to the stiffness of the connection.
*Pinned connection – This connection can sustain vertical and horizontal actions but not bending moment. The connected members are free to rotate in one direction with the connection having no degree of stiffness.
*Semi-rigid connection – This connection is between rigid or pinned as it is able to sustain vertical and horizontal actions and some amount of moment.

With in-situ reinforced concrete construction, a monolithic rigid connection is usually produced through design and provided on site. Precast connections range in their level of rigidity, from fully rigid to a completely pinned connection. A true pinned connection containing zero moment capacity is rare. In fact, many connections have some degree of rigidity but are conservatively assumed pinned. The steel connection shown in Figure 10 will retain some degree of rigidity, yet is usually modelled in design as a pinned connection. This is a conservative measure as beams spanning pinned connections are subject to the full action moment. Due to the connection having some degree of stiffness and therefore moment capacity, the negative bending moment acting upon the beam will be overestimated. 

[[File:Connectin classification.JPG|RTENOTITLE]] 

Figure 10 - Steel shear plate connection to CHS column (Kurobane et al., 2005)

Within the steel industry, research has shown cost reductions of between 10 to 20% for semi-rigid frames over rigid frames (Kurobane et al., 2005). Therefore the level of rigidity is an important consideration when choosing a method of connecting precast concrete elements.

== Continuous column with Corbel connections ==

[[File:Corbel connection.JPG|RTENOTITLE]]
<div>Figure 11 -Examples of concrete corbel connection with continuity reinforcement. A. (The Concrete Centre, 2007a) B. (Task Group 6.2 F.I.B., 2008, p. 49) </div>
Corbel connections, as shown in Figure 11 are most often used to support long span beams or heavy loads. Due to the visual and physical intrusion caused by the corbel or haunch widening the column, this connection is not widely used in the construction of multi-storey concrete frames. The basic corbel connection is designed as simply supported, dowel bars and/or fixing cleats. This type of connection can be used to prevent lateral movement and provide some joint fixity, although research has proven that the basic dowelled connection is best modelled as pinned. In-situ, structural screed can be used to increase continuity of the connection, thus allowing the tension reinforcement to resist the forces arising from beam movements. This can either be at the end, or across the whole length of the beam or floor slab. It was shown that a corbel / haunch connections with small amounts of cast in place reinforced concrete, although designed as a simply supported pinned connection, can improve strength and stiffness resulting in a semi or often fully rigid connection.

[[File:Corbel connection real example.JPG|RTENOTITLE]] 

Figure 12 - Photo of example beam to column corbel connection (General Precast Concrete Ltd, 2008). 

== Continuous beam connection ==

[[File:Continuous beam connection.JPG|RTENOTITLE]] 

Figure 13 - Discontinuous column with continuous beam, Left: (Elliot, 1992, p.67), Right: (Task Group 6.2 F.I.B., 2008, p. 299)

This type of connection is mainly used in portal frames or in skeletal frames when beams need to be continuous over supports, as is required for a cantilever. The beams are seated on dry pack mortar on top of the vertical members and reinforcing starter bars are projected through sleeves in the beam from the lower column up into the upper column. These sleeves are subsequently grouted to provide vertical continuity. Once the beam is lowered into place, this connection requires no additional formwork providing the grout is poured through vents in the upper column. Therefore, provided the remaining beam end is secured, loads for construction access can be placed upon the beam. This enhances the simplicity of installation and therefore safety on site.

== Wall and Column shoes ==

[[File:Wall and Column shoe connection 1.JPG|RTENOTITLE]] 

Figure 14 - Photos of example hidden corbel systems used with a precast column and precast beam (Peikko concrete connections, 2009)

Investment into modern technologies has resulted in the production of the hidden corbel connection. This is the most popular type of precast connection used in the UK so far. This type produces fireproof connections which are architecturally advantageous as they minimise visual intrusion whilst maximising floor to soffit height.

The connection area is minimal, protecting the reinforcement steel used in the connection. The connection also benefits from superior adjustability with the modern connection utilising a small adjustable plate, allowing fine tuning of the column corbel prior to installation of the beam.

[[File:Wall and Column shoe connection 2.JPG|RTENOTITLE]] 

Figure 15 – (Left) Inside detailed view of the anchorage of the corbel connection into the precast column and beams (Peikko concrete connections, 2009). (Right) Labelled example of an alternative hidden corbel system (JVI inc.)

== Ground foundation column connections ==

There are three main methods of connecting columns to foundations:
*Projecting starter bars – The in-situ foundation houses cast in starter bars which the precast column is later lowered onto and grouted to provide continuity.
*Pocket connection – This is the most rigid connection and is utilised when the moment resisting capacity of the connection is required for the lateral stability of the structure. A pocket is provided within the foundation into which the precast column is lowered. The surrounding area is grouted or filled with in-situ concrete.
*Baseplate connection – The base of the precast column contains steel base plates which cast-in bolts are fed through and bolted into place. The surrounding area to the holding down bolts is then filled with non-shrink grout to complete the connection.

The three types above are conservatively modelled as pinned connections resulting in an underestimate of the moments transferred to the columns and beams above. The foundation column connection is subjected to certain degree of variability such as possible rotations due to ground conditions.

= Connection’s discussion and evaluation =

There exists a variety of precast connection types within each big group above. The connections are assessed against different criteria including: the amount of additional materials; aesthetic/space intrusion of the finished connections; allowable tolerance; amount of wet work formwork required; possibility of future reuse/dismantle; operative involvement on site; level of rigidity; safety; skills required; amount of temporary works; time of assembly; tools required; weather sensitivity and level of wet casting needed.

== Continuous beam connection ==

[[File:Discussion - continuous beam.JPG|RTENOTITLE]] 

Figure 16 - Bolted steel shoe diagram (Halfen GmbH, 2011)

The bolted steel shoe is considered to be the most favourable of the continuous beam connection type as it is simple to produce and quick to assemble on site. This connection requires no structural in-situ works compared to other sub-types. The connection can have a number of different bolt arrangements depending on the size and shape of the column. When used correctly the anchor bolts can be utilised to transfer both tensile and compressive load through to the column below, thus minimising stress on the beam / slab in between. Alternatively the beam / slab can be suitably designed to transfer the load to the column below.

Due to the presence of a continuous beam, the large hogging moments generated at the connection will be transferred to the column. The moment capacity of this connection is high due to the high tensile capacity of the steel holding down bolts resisting the rotation of the column due to buckling, which may result from the hogging moments transferred from the beams.

The amount and positioning of the holding down bolts will determine the connections rigidity. The closer the bolts are to the centre point in the plane of rotation the more the connection will represent a pinned connection between the columns and the beam. As shown in Figure 16, the bolts have been positioned to give the maximum lever arm against any point of pivot and thus maximises rotational resistance.

== Corbel/Billet connection ==

[[File:Discussion-corbel.JPG|RTENOTITLE]] 

Figure 17 - Bearing only connection diagram (Pujol Group)

The bearing only is the favoured connection due to its simple straightforward design which facilitates quick assembly time on site. This fully pinned connection type will therefore transfer vertical and horizontal loads into the column, but all moment will be contained within the beam. For this reason, the beam will be designed to resist greater moment. The connection is therefore less efficient than moment sustaining connections.

The bearing with bolted dowel bar which has been taken as the preferred connection method for this connection type. The bearing with bolted dowel bar allows reduced connection width due to dowel action acting as horizontal restraint. The connection is fixed at the top using a bolt which extends through the column. This is positioned to resist maximum bending moment as the point of pivot will be within the underside of the beam. It is common practice in design to ensure that fixing elements of connections are not the limiting element and therefore the bolt will be able to transfer a considerable amount of moment to the column. The flat landing of the corbel, although unsightly, when combined with the dowel, acts as a torsional restraint. This can be further improved by using a cleat which extends the width of the beam. As the column is continuous, the beam will be required to sustain the majority of bending moment. This connection is semi rigid; therefore it can sustain vertical and horizontal loads with a degree of hogging moment transferred to the column.

[[File:Discussion-corbel 2.JPG|RTENOTITLE]]

Figure 18 - Bearing with bolted dowel, the preferred connection from billet / corbel type as bearing only connection disregarded

== Concealed fixing ==

The concealed bolted steel billet, which is one of the most modern connections, is favoured within this category. It utilises the advantages of other connections with an aesthetically pleasing and simple design, which allows minor adjustments to be made to the plate rather than the beam. There are other connections which show greater rigidity but require greater installing time on site, therefore are less favoured. This connection transfers horizontal and vertical loads to the column through the bolted connection. Its moment resisting capacity is small when compared to the bolted doweled corbel. However, as the connection extends to full height of the beam it is well positioned to sustain some moment. As there are just two bolts per beam to column connection, the level of moment transfer will be limited. For this reason the connection will act as a semi-rigid connection.

[[File:Concealed fixing.JPG|RTENOTITLE]]

Figure 19 - Preferred concealed connection, bolted steel corbel connection (Peikko concrete connections, 2009)

= Conclusion =

== Vertical load resistance ==

All three connections are capable of transferring the vertical load both from the column above and from the beams. As all three of the preferred connections from each group utilise a bolted mechanism to provide fixity, it is feasible that a structure constructed using only these connections would be able to resist against vertical loading without using any in-situ casting.

== Horizontal lateral restraint ==

The connections identified above, unless suitably designed for using excessive sized members and reinforcement, will struggle to resist lateral loading. The lateral loading will need to be taken by shear walls and/or concrete cores such as lift shafts or steel bracing to create a hybrid structure. But as with the initial problem of over-engineered connections through the neglect of their moment capacity, the lateral loading capacity of the preferred connections would need to be assessed and accounted for in order to produce the most efficient design.

Combination of continuous column and continuous beam joints can be used to help transfer moments to stiffer areas of the structure. It thus also allows for a more efficient design with only critical members designed to facilitate the load transfer.

== Frame analysis ==

In-situ frames have fully rigid connections. Should a precast connection be capable of transferring moment to the columns and thus down to the supports, then it can be assed as a complete frame or a series of sub frames. Moments, either hogging or sagging are attracted to stiffer members. Should the connection be capable of transferring these moments, the moments at the columns will then be in hogging and will need to be accounted for. Many published papers (Gorgun, 1997; Aguiar et al., 4 June 2012; Baharuddin et al., 2006) have discovered that some precast connections (including the ones mentioned above) can sustain hogging moment, and are therefore over engineered using the current design process. Therefore the structural frame should be modelled similar to a steel frame, where if almost no moment can be sustained then the connections are designed as pinned.

== Disproportionate Collapse ==

Since 2004, the Building Regulations in England and Wales have been revised to ensure all buildings are designed against disproportionate collapse. The connections above have been analysed with moment capacity as the desired attribute, but they will also provide some tensional resistance which would inherently provide resistance against disproportionate collapse.

= References =

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=== ===

[[Category:Products_/_components]]
[[Category:Project_types]]
[[Category:Sustainability]]
[[Category:Student_architect_essay_competition]]
[[Category:Student_engineer_essay_competition]]

Off-site prefabrication of buildings: A guide to connection choices

2012-12-14T01:13:56Z

Nicky nguyen 91:

= Introduction =

Off-site prefabrication is a possible solution to many issues regarding the underachievement of the UK’s construction industry including: safety record, public perception, client satisfaction, profitability, delays, skilled workforce and overall contribution to the national economy. Prefabrication is believed to be advantageous over traditional construction methods in the followings:
*Quality – Higher-quality finishes with defects eliminated prior to completion.
*Safety – Safer working environment under factory conditions.
*Cost – Repeated use of moulds through standardisation reduces formwork materials, preliminaries, site storage and on-site facilities.
*Waste – Reduced off-cuts from formwork and the introduction of prefabricated rebars.
*Programme – Increased predictability due to reduced external factors such as weather.
*Local disruption – less environmental impacts such as dust and noise pollution.
*Accuracy – Increased accuracy since templates produced using Computer Aided Design (CAD) systems.
*Timescale – Components built off-site leads to reduced on-site construction time.

Among those mentioned, a driving factor for using prefabrication is to improve both quality and safety, as are rated 4.3 and 3.9 respectively on a five point Likert scale (Pann et al, 2008).

[[File:Laings O Rourke precast factory.JPG|RTENOTITLE]]

Figure 1: Laing O'Rourke's Explore precast manufacturing facility, Steetley, UK (Croxon, 2010) 

= History =

The World War One (1910s) and World War Two (1940s) stimulated research into newer methods than traditional brick construction. At that time, the requirement for rapid construction and the willingness to pay for it were the driving forces, in contrast to the major shortages of skilled labour and building materials. In the 1940s, UK government promoted methods of newer construction from the industries. This eventually led to the construction of hundreds of prefabricated concrete tower blocks and thousands of schools in the 1950s and 1960s which were often poorly designed. These were of low cost and often built without the lifetime of the buildings considered. Volumetric construction, the construction technique involving the production of buildings as a number of boxes connected on site, was used throughout the 1960s and 1970s. The collapse of the Ronan Point tower block in East London in 1968 is well known and attributed as one of the reasons for continued suspicion, fear and decline of prefabrication in this country.

Following the above downturn, there is currently a shift towards prefabrication within the industry. Many UK major construction firms are starting to see the benefits of prefabrication. Kier, Interserve, NG Bailey, Arup, Capita Symonds and Laing O’Rourke (LOR) are some of the market leaders registering their interest. LOR is recognised as having the largest interest, investing £100m in its Design for Manufacture and Assembly facility located at its Explore industrial park, which is “the most advanced facility of its type in Europe”. They aim to take advantage of public and private sector clients including BAA, Premier Inn, the Department of Education and both the Ministry of Justice and the Ministry of Defence (Wright, 2010), who all consider off-site prefabrication solutions.

Although a current shortage of a skilled workforce is said to be the cause, the increased uptake of prefabrication is believed to be a permanent move, as opposed to the short-lived uptake seen in the 60s and the 70s.

[[File:Ronan point tower block.JPG|RTENOTITLE]] 

Figure 2 - Ronan Point tower block failure, East London 1968 (Daily Telegraph, 1968)

= Precast construction method =

As with in-situ reinforced concrete construction, precast construction lends itself to a variety of different construction techniques, layouts and sequences.

== Frame and Deck Construction ==

[[File:Frame and deck construction.JPG|RTENOTITLE]] 

Figure 3 - Frame and deck systems, a) single storey columns, b) multi storey columns (Task Group 6.2 F.I.B., 2008, p. 5).

A precast deck supported by precast beams and columns form the building’s structural system. This form is frequently used in the construction of multi storey car parks with up to 16m spans to reduce columns between car parking spaces. It can also be used where floor to beam soffit height does not need to be minimised. The overall column height within a frame and deck system may correspond to greater than one storey.

The following connections are utilised in frames and deck construction:
*column to column
*column to base
*beam to column
*beam to base

[[File:Diagram.jpg|RTENOTITLE]] 

Figure 4 - Diagram of the precast elements and connections used in frame and deck construction (Irish Precast Concrete Association, 2003)

== Crosswall Construction ==

Crosswall is a modern method where load bearing walls provide the primary vertical support for precast floors and lateral stability. External wall panels, lift cores or staircases are used to provide the required longitudinal stability. Bridging components such as floors, roofs and beams are supported by the load bearing walls or façade wall. The system is ideal for buildings with cellular and orthogonal grids, with rooms of up to 4mx9m as standard. Thus it leads to a structurally efficient building with high levels of sound and fire insulation between adjacent rooms.

Crosswall construction utilises the following connections:
*wall to wall at vertical joints
*wall to wall at horizontal joints
*wall to base/foundation

[[File:Crosswall construction.JPG|RTENOTITLE]] 

Figure 5 - Crosswall systems, a) load bearing crosswall, b) load bearing facade wall (Task Group 6.2 F.I.B., 2008, p. 8). 

The precast elements are brought to site just in time allowing them to be lifted from the transport vehicle and installed in place. Hidden joints and ties, both horizontally and vertically are grouted in place as the work develops, allowing progressive collapse criteria of the Building Regulations to be met. With the possibility of incorporated mechanical and electrical components and minimal finishing needed, following trades can start prior to the completion of precast erection. 

== Volumetric Construction ==

The term volumetric construction is given when concrete modules (constructed in the factory) are installed on site to form a cellular system or used independently as a self-contained cell. The modules can be cast as a room or as panels which are subsequently joined together in the factory prior to site delivery. For a cellular system, the ground floor cells are laid on pre-prepared ground floor slabs with individual modules lowered into place usually forming the roof of the unit below.

[[File:Volumetric construction.JPG|RTENOTITLE]] 

Figure 6 - Examples of volumetric construction used for prison construction with an example of a possible finish (Oldcastle Precast Inc. ). 

Cellular systems are advantageous when being incorporated with a repetitive design (Figure 6). Common uses include hotels, prison cells, student halls and residential buildings. Self-contained cells are used mostly for specialised purposes where services are needed such as wet rooms, bathroom pods (Figure 7) and service utility rooms. Once lifted into place, the modules are secured by a number of methods including bolted and doweled connections. 

[[File:Volumetric bathroom.JPG|RTENOTITLE]]

Figure 7 - Volumetric construction in the form of a bathroom pod lifted into a crosswall frame (The Concrete Centre, 2007a). 

== Hybrid Construction ==

Hybrid construction is not a standalone method of precast construction. It is mentioned for its use in conjunction with the above construction methods. Precast elements can be used to provide permanent formwork for in-situ concrete. The combination of in-situ and precast concrete allows the benefits of both to be utilised. Figure 8 shows how safe working platforms are created by the precast floors, which increases safety on site and omits the need for in-situ concrete formwork, both factors significantly decrease the construction time. Greater spans can be achieved with hybrid construction as composite action is achieved by using different structural materials for the upper and lower areas of the element. The interface between the two materials will have to withstand shear stresses which can be overcome through the use of shear studs or precast reinforcement in the floor slab. 

[[File:Hybrid construction.JPG|RTENOTITLE]]

Figure 8 - Construction site utilising hybrid construction. a) Temporary props in place to support precast lattice floor slabs b) Installed precast lattice floor slabs awaiting reinforcement c) Concrete curing and binding with precast slab beneath. (webbaviation on behalf of Laing O'Rourke, 2011) 

= Precast Connections =

== Classification of connections ==

[[File:Precast connection.jpg|RTENOTITLE]] 

Figure 9 - Generic forms of beam-column connections, A: hidden beam end connection, B: corbel / haunch bearing connection, C: continuous column connection, D: continuous beam connection (Task Group 6.2, Fédération internationale du béton., 2008, p. 289)

For structural design, the connection’s stiffness is important in designing for moment distribution. There are three classes of connections based upon their degree of rigidity:
*Rigid connection – This connection can sustain vertical and horizontal actions as well as bending moment. The relative angle between connected members is maintained due to the stiffness of the connection.
*Pinned connection – This connection can sustain vertical and horizontal actions but not bending moment. The connected members are free to rotate in one direction with the connection having no degree of stiffness.
*Semi-rigid connection – This connection is between rigid or pinned as it is able to sustain vertical and horizontal actions and some amount of moment.

With in-situ reinforced concrete construction, a monolithic rigid connection is usually produced through design and provided on site. Precast connections range in their level of rigidity, from fully rigid to a completely pinned connection. A true pinned connection containing zero moment capacity is rare. In fact, many connections have some degree of rigidity but are conservatively assumed pinned. The steel connection shown in Figure 10 will retain some degree of rigidity, yet is usually modelled in design as a pinned connection. This is a conservative measure as beams spanning pinned connections are subject to the full action moment. Due to the connection having some degree of stiffness and therefore moment capacity, the negative bending moment acting upon the beam will be overestimated. 

[[File:Connectin classification.JPG|RTENOTITLE]] 

Figure 10 - Steel shear plate connection to CHS column (Kurobane et al., 2005)

Within the steel industry, research has shown cost reductions of between 10 to 20% for semi-rigid frames over rigid frames (Kurobane et al., 2005). Therefore the level of rigidity is an important consideration when choosing a method of connecting precast concrete elements.

== Continuous column with Corbel connections ==

[[File:Corbel connection.JPG|RTENOTITLE]]
<div>Figure 11 -Examples of concrete corbel connection with continuity reinforcement. A. (The Concrete Centre, 2007a) B. (Task Group 6.2 F.I.B., 2008, p. 49) </div>
Corbel connections, as shown in Figure 11 are most often used to support long span beams or heavy loads. Due to the visual and physical intrusion caused by the corbel or haunch widening the column, this connection is not widely used in the construction of multi-storey concrete frames. The basic corbel connection is designed as simply supported, dowel bars and/or fixing cleats. This type of connection can be used to prevent lateral movement and provide some joint fixity, although research has proven that the basic dowelled connection is best modelled as pinned. In-situ, structural screed can be used to increase continuity of the connection, thus allowing the tension reinforcement to resist the forces arising from beam movements. This can either be at the end, or across the whole length of the beam or floor slab. It was shown that a corbel / haunch connections with small amounts of cast in place reinforced concrete, although designed as a simply supported pinned connection, can improve strength and stiffness resulting in a semi or often fully rigid connection.

[[File:Corbel connection real example.JPG|RTENOTITLE]] 

Figure 12 - Photo of example beam to column corbel connection (General Precast Concrete Ltd, 2008). 

== Continuous beam connection ==

[[File:Continuous beam connection.JPG|RTENOTITLE]] 

Figure 13 - Discontinuous column with continuous beam, Left: (Elliot, 1992, p.67), Right: (Task Group 6.2 F.I.B., 2008, p. 299)

This type of connection is mainly used in portal frames or in skeletal frames when beams need to be continuous over supports, as is required for a cantilever. The beams are seated on dry pack mortar on top of the vertical members and reinforcing starter bars are projected through sleeves in the beam from the lower column up into the upper column. These sleeves are subsequently grouted to provide vertical continuity. Once the beam is lowered into place, this connection requires no additional formwork providing the grout is poured through vents in the upper column. Therefore, provided the remaining beam end is secured, loads for construction access can be placed upon the beam. This enhances the simplicity of installation and therefore safety on site.

== Wall and Column shoes ==

[[File:Wall and Column shoe connection 1.JPG|RTENOTITLE]] 

Figure 14 - Photos of example hidden corbel systems used with a precast column and precast beam (Peikko concrete connections, 2009)

Investment into modern technologies has resulted in the production of the hidden corbel connection. This is the most popular type of precast connection used in the UK so far. This type produces fireproof connections which are architecturally advantageous as they minimise visual intrusion whilst maximising floor to soffit height.

The connection area is minimal, protecting the reinforcement steel used in the connection. The connection also benefits from superior adjustability with the modern connection utilising a small adjustable plate, allowing fine tuning of the column corbel prior to installation of the beam.

[[File:Wall and Column shoe connection 2.JPG|RTENOTITLE]] 

Figure 15 – (Left) Inside detailed view of the anchorage of the corbel connection into the precast column and beams (Peikko concrete connections, 2009). (Right) Labelled example of an alternative hidden corbel system (JVI inc.)

== Ground foundation column connections ==

There are three main methods of connecting columns to foundations:
*Projecting starter bars – The in-situ foundation houses cast in starter bars which the precast column is later lowered onto and grouted to provide continuity.
*Pocket connection – This is the most rigid connection and is utilised when the moment resisting capacity of the connection is required for the lateral stability of the structure. A pocket is provided within the foundation into which the precast column is lowered. The surrounding area is grouted or filled with in-situ concrete.
*Baseplate connection – The base of the precast column contains steel base plates which cast-in bolts are fed through and bolted into place. The surrounding area to the holding down bolts is then filled with non-shrink grout to complete the connection.

The three types above are conservatively modelled as pinned connections resulting in an underestimate of the moments transferred to the columns and beams above. The foundation column connection is subjected to certain degree of variability such as possible rotations due to ground conditions.

= Connection’s discussion and evaluation =

There exists a variety of precast connection types within each big group above. The connections are assessed against different criteria including: the amount of additional materials; aesthetic/space intrusion of the finished connections; allowable tolerance; amount of wet work formwork required; possibility of future reuse/dismantle; operative involvement on site; level of rigidity; safety; skills required; amount of temporary works; time of assembly; tools required; weather sensitivity and level of wet casting needed.

== Continuous beam connection ==

[[File:Discussion - continuous beam.JPG|RTENOTITLE]] 

Figure 16 - Bolted steel shoe diagram (Halfen GmbH, 2011)

The bolted steel shoe is considered to be the most favourable of the continuous beam connection type as it is simple to produce and quick to assemble on site. This connection requires no structural in-situ works compared to other sub-types. The connection can have a number of different bolt arrangements depending on the size and shape of the column. When used correctly the anchor bolts can be utilised to transfer both tensile and compressive load through to the column below, thus minimising stress on the beam / slab in between. Alternatively the beam / slab can be suitably designed to transfer the load to the column below.

Due to the presence of a continuous beam, the large hogging moments generated at the connection will be transferred to the column. The moment capacity of this connection is high due to the high tensile capacity of the steel holding down bolts resisting the rotation of the column due to buckling, which may result from the hogging moments transferred from the beams.

The amount and positioning of the holding down bolts will determine the connections rigidity. The closer the bolts are to the centre point in the plane of rotation the more the connection will represent a pinned connection between the columns and the beam. As shown in Figure 16, the bolts have been positioned to give the maximum lever arm against any point of pivot and thus maximises rotational resistance.

== Corbel/Billet connection ==

[[File:Discussion-corbel.JPG|RTENOTITLE]] 

Figure 17 - Bearing only connection diagram (Pujol Group)

The bearing only is the favoured connection due to its simple straightforward design which facilitates quick assembly time on site. This fully pinned connection type will therefore transfer vertical and horizontal loads into the column, but all moment will be contained within the beam. For this reason, the beam will be designed to resist greater moment. The connection is therefore less efficient than moment sustaining connections.

The bearing with bolted dowel bar which has been taken as the preferred connection method for this connection type. The bearing with bolted dowel bar allows reduced connection width due to dowel action acting as horizontal restraint. The connection is fixed at the top using a bolt which extends through the column. This is positioned to resist maximum bending moment as the point of pivot will be within the underside of the beam. It is common practice in design to ensure that fixing elements of connections are not the limiting element and therefore the bolt will be able to transfer a considerable amount of moment to the column. The flat landing of the corbel, although unsightly, when combined with the dowel, acts as a torsional restraint. This can be further improved by using a cleat which extends the width of the beam. As the column is continuous, the beam will be required to sustain the majority of bending moment. This connection is semi rigid; therefore it can sustain vertical and horizontal loads with a degree of hogging moment transferred to the column.

[[File:Discussion-corbel 2.JPG|RTENOTITLE]]

Figure 18 - Bearing with bolted dowel, the preferred connection from billet / corbel type as bearing only connection disregarded

== Concealed fixing ==

The concealed bolted steel billet, which is one of the most modern connections, is favoured within this category. It utilises the advantages of other connections with an aesthetically pleasing and simple design, which allows minor adjustments to be made to the plate rather than the beam. There are other connections which show greater rigidity but require greater installing time on site, therefore are less favoured. This connection transfers horizontal and vertical loads to the column through the bolted connection. Its moment resisting capacity is small when compared to the bolted doweled corbel. However, as the connection extends to full height of the beam it is well positioned to sustain some moment. As there are just two bolts per beam to column connection, the level of moment transfer will be limited. For this reason the connection will act as a semi-rigid connection.

[[File:Concealed fixing.JPG|RTENOTITLE]]

Figure 19 - Preferred concealed connection, bolted steel corbel connection (Peikko concrete connections, 2009)

= Conclusion =

== Vertical load resistance ==

All three connections are capable of transferring the vertical load both from the column above and from the beams. As all three of the preferred connections from each group utilise a bolted mechanism to provide fixity, it is feasible that a structure constructed using only these connections would be able to resist against vertical loading without using any in-situ casting.

== Horizontal lateral restraint ==

The connections identified above, unless suitably designed for using excessive sized members and reinforcement, will struggle to resist lateral loading. The lateral loading will need to be taken by shear walls and/or concrete cores such as lift shafts or steel bracing to create a hybrid structure. But as with the initial problem of over-engineered connections through the neglect of their moment capacity, the lateral loading capacity of the preferred connections would need to be assessed and accounted for in order to produce the most efficient design.

Combination of continuous column and continuous beam joints can be used to help transfer moments to stiffer areas of the structure. It thus also allows for a more efficient design with only critical members designed to facilitate the load transfer.

== Frame analysis ==

In-situ frames have fully rigid connections. Should a precast connection be capable of transferring moment to the columns and thus down to the supports, then it can be assed as a complete frame or a series of sub frames. Moments, either hogging or sagging are attracted to stiffer members. Should the connection be capable of transferring these moments, the moments at the columns will then be in hogging and will need to be accounted for. Many published papers (Gorgun, 1997; Aguiar et al., 4 June 2012; Baharuddin et al., 2006) have discovered that some precast connections (including the ones mentioned above) can sustain hogging moment, and are therefore over engineered using the current design process. Therefore the structural frame should be modelled similar to a steel frame, where if almost no moment can be sustained then the connections are designed as pinned.

== Disproportionate Collapse ==

Since 2004, the Building Regulations in England and Wales have been revised to ensure all buildings are designed against disproportionate collapse. The connections above have been analysed with moment capacity as the desired attribute, but they will also provide some tensional resistance which would inherently provide resistance against disproportionate collapse.

= References =

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=== ===

[[Category:Products_/_components]]
[[Category:Project_types]]
[[Category:Sustainability]]
[[Category:Student_architect_essay_competition]]
[[Category:Student_engineer_essay_competition]]

Off-site prefabrication of buildings: A guide to connection choices

2012-12-14T01:03:24Z

Nicky nguyen 91:

= Introduction =

Off-site prefabrication is a possible solution to many issues regarding the underachievement of the UK’s construction industry including: safety record, public perception, client satisfaction, profitability, delays, skilled workforce and overall contribution to the national economy. Prefabrication is believed to be advantageous over traditional construction methods in the followings:
*Quality – Higher-quality finishes with defects eliminated prior to completion.
*Safety – Safer working environment under factory conditions.
*Cost – Repeated use of moulds through standardisation reduces formwork materials, preliminaries, site storage and on-site facilities.
*Waste – Reduced off-cuts from formwork and the introduction of prefabricated rebars.
*Programme – Increased predictability due to reduced external factors such as weather.
*Local disruption – less environmental impacts such as dust and noise pollution.
*Accuracy – Increased accuracy since templates produced using Computer Aided Design (CAD) systems.
*Timescale – Components built off-site leads to reduced on-site construction time.

Among those mentioned, a driving factor for using prefabrication is to improve both quality and safety, as are rated 4.3 and 3.9 respectively on a five point Likert scale (Pann et al, 2008).

[[File:Laings O Rourke precast factory.JPG|RTENOTITLE]]

Figure 1: Laing O'Rourke's Explore precast manufacturing facility, Steetley, UK (Croxon, 2010) 

= History =

The World War One (1910s) and World War Two (1940s) stimulated research into newer methods than traditional brick construction. At that time, the requirement for rapid construction and the willingness to pay for it were the driving forces, in contrast to the major shortages of skilled labour and building materials. In the 1940s, UK government promoted methods of newer construction from the industries. This eventually led to the construction of hundreds of prefabricated concrete tower blocks and thousands of schools in the 1950s and 1960s which were often poorly designed. These were of low cost and often built without the lifetime of the buildings considered. Volumetric construction, the construction technique involving the production of buildings as a number of boxes connected on site, was used throughout the 1960s and 1970s. The collapse of the Ronan Point tower block in East London in 1968 is well known and attributed as one of the reasons for continued suspicion, fear and decline of prefabrication in this country.

Following the above downturn, there is currently a shift towards prefabrication within the industry. Many UK major construction firms are starting to see the benefits of prefabrication. Kier, Interserve, NG Bailey, Arup, Capita Symonds and Laing O’Rourke (LOR) are some of the market leaders registering their interest. LOR is recognised as having the largest interest, investing £100m in its Design for Manufacture and Assembly facility located at its Explore industrial park, which is “the most advanced facility of its type in Europe”. They aim to take advantage of public and private sector clients including BAA, Premier Inn, the Department of Education and both the Ministry of Justice and the Ministry of Defence (Wright, 2010), who all consider off-site prefabrication solutions.

Although a current shortage of a skilled workforce is said to be the cause, the increased uptake of prefabrication is believed to be a permanent move, as opposed to the short-lived uptake seen in the 60s and the 70s.

[[File:Ronan point tower block.JPG|RTENOTITLE]] 

Figure 2 - Ronan Point tower block failure, East London 1968 (Daily Telegraph, 1968)

 

= Precast construction method =

As with in-situ reinforced concrete construction, precast construction lends itself to a variety of different construction techniques, layouts and sequences.

== Frame and Deck Construction ==

[[File:Frame and deck construction.JPG|RTENOTITLE]] Figure 3 - Frame and deck systems, a) single storey columns, b) multi storey columns (Task Group 6.2 F.I.B., 2008, p. 5).

A precast deck supported by precast beams and columns form the building’s structural system. This form is frequently used in the construction of multi storey car parks with up to 16m spans to reduce columns between car parking spaces. It can also be used where floor to beam soffit height does not need to be minimised. The overall column height within a frame and deck system may correspond to greater than one storey.

The following connections are utilised in frames and deck construction:
*column to column
*column to base
*beam to column
*beam to base

[[File:Diagram.jpg|RTENOTITLE]] 

Figure 5 - Diagram of the precast elements and connections used in frame and deck construction (Irish Precast Concrete Association, 2003)

== Crosswall Construction ==

Crosswall is a modern method where load bearing walls provide the primary vertical support for precast floors and lateral stability. External wall panels, lift cores or staircases are used to provide the required longitudinal stability. Bridging components such as floors, roofs and beams are supported by the load bearing walls or façade wall. The system is ideal for buildings with cellular and orthogonal grids, with rooms of up to 4mx9m as standard. Thus it leads to a structurally efficient building with high levels of sound and fire insulation between adjacent rooms.

Crosswall construction utilises the following connections:
*wall to wall at vertical joints
*wall to wall at horizontal joints
*wall to base/foundation

[[File:Crosswall construction.JPG|RTENOTITLE]] 

Figure 6 - Crosswall systems, a) load bearing crosswall, b) load bearing facade wall (Task Group 6.2 F.I.B., 2008, p. 8). 

The precast elements are brought to site just in time allowing them to be lifted from the transport vehicle and installed in place. Hidden joints and ties, both horizontally and vertically are grouted in place as the work develops, allowing progressive collapse criteria of the Building Regulations to be met. With the possibility of incorporated mechanical and electrical components and minimal finishing needed, following trades can start prior to the completion of precast erection. 

== Volumetric Construction ==

The term volumetric construction is given when concrete modules (constructed in the factory) are installed on site to form a cellular system or used independently as a self-contained cell. The modules can be cast as a room or as panels which are subsequently joined together in the factory prior to site delivery. For a cellular system, the ground floor cells are laid on pre-prepared ground floor slabs with individual modules lowered into place usually forming the roof of the unit below.

[[File:Volumetric construction.JPG|RTENOTITLE]] 

Figure 7 - Examples of volumetric construction used for prison construction with an example of a possible finish (Oldcastle Precast Inc. ). 

Cellular systems are advantageous when being incorporated with a repetitive design (Figure 7). Common uses include hotels, prison cells, student halls and residential buildings. Self-contained cells are used mostly for specialised purposes where services are needed such as wet rooms, bathroom pods (Figure 8) and service utility rooms. Once lifted into place, the modules are secured by a number of methods including bolted and doweled connections. 

[[File:Volumetric bathroom.JPG|RTENOTITLE]]

Figure 8 - Volumetric construction in the form of a bathroom pod lifted into a crosswall frame (The Concrete Centre, 2007a). 

== Hybrid Construction ==

Hybrid construction is not a standalone method of precast construction. It is mentioned for its use in conjunction with the above construction methods. Precast elements can be used to provide permanent formwork for in-situ concrete. The combination of in-situ and precast concrete allows the benefits of both to be utilised. Figure 9 shows how safe working platforms are created by the precast floors, which increases safety on site and omits the need for in-situ concrete formwork, both factors significantly decrease the construction time. Greater spans can be achieved with hybrid construction as composite action is achieved by using different structural materials for the upper and lower areas of the element. The interface between the two materials will have to withstand shear stresses which can be overcome through the use of shear studs or precast reinforcement in the floor slab. 

[[File:Hybrid construction.JPG|RTENOTITLE]]

Figure 9 - Construction site utilising hybrid construction. a) Temporary props in place to support precast lattice floor slabs b) Installed precast lattice floor slabs awaiting reinforcement c) Concrete curing and binding with precast slab beneath. (webbaviation on behalf of Laing O'Rourke, 2011) 

= Precast Connections =

== Classification of connections ==

[[File:Precast connection.jpg|RTENOTITLE]] 

Figure 10 - Generic forms of beam-column connections, A: hidden beam end connection, B: corbel / haunch bearing connection, C: continuous column connection, D: continuous beam connection (Task Group 6.2, Fédération internationale du béton., 2008, p. 289)

For structural design, the connection’s stiffness is important in designing for moment distribution. There are three classes of connections based upon their degree of rigidity:
*Rigid connection – This connection can sustain vertical and horizontal actions as well as bending moment. The relative angle between connected members is maintained due to the stiffness of the connection.
*Pinned connection – This connection can sustain vertical and horizontal actions but not bending moment. The connected members are free to rotate in one direction with the connection having no degree of stiffness.
*Semi-rigid connection – This connection is between rigid or pinned as it is able to sustain vertical and horizontal actions and some amount of moment.

With in-situ reinforced concrete construction, a monolithic rigid connection is usually produced through design and provided on site. Precast connections range in their level of rigidity, from fully rigid to a completely pinned connection. A true pinned connection containing zero moment capacity is rare. In fact, many connections have some degree of rigidity but are conservatively assumed pinned. The steel connection shown in Figure 11 will retain some degree of rigidity, yet is usually modelled in design as a pinned connection. This is a conservative measure as beams spanning pinned connections are subject to the full action moment. Due to the connection having some degree of stiffness and therefore moment capacity, the negative bending moment acting upon the beam will be overestimated. 

[[File:Connectin classification.JPG|RTENOTITLE]] 

Figure 11 - Steel shear plate connection to CHS column (Kurobane et al., 2005)

Within the steel industry, research has shown cost reductions of between 10 to 20% for semi-rigid frames over rigid frames (Kurobane et al., 2005). Therefore the level of rigidity is an important consideration when choosing a method of connecting precast concrete elements.

== Continuous column with Corbel connections ==

[[File:Corbel connection.JPG|RTENOTITLE]]
<div>Figure 13 -Examples of concrete corbel connection with continuity reinforcement. A. (The Concrete Centre, 2007a) B. (Task Group 6.2 F.I.B., 2008, p. 49) </div>
Corbel connections, as shown in Figure 13 are most often used to support long span beams or heavy loads. Due to the visual and physical intrusion caused by the corbel or haunch widening the column, this connection is not widely used in the construction of multi-storey concrete frames. The basic corbel connection is designed as simply supported, dowel bars and/or fixing cleats. This type of connection can be used to prevent lateral movement and provide some joint fixity, although research has proven that the basic dowelled connection is best modelled as pinned. In-situ, structural screed can be used to increase continuity of the connection, thus allowing the tension reinforcement to resist the forces arising from beam movements. This can either be at the end, or across the whole length of the beam or floor slab. It was shown that a corbel / haunch connections with small amounts of cast in place reinforced concrete, although designed as a simply supported pinned connection, can improve strength and stiffness resulting in a semi or often fully rigid connection.

[[File:Corbel connection real example.JPG|RTENOTITLE]] Figure 12 - Photo of example beam to column corbel connection (General Precast Concrete Ltd, 2008). 

== Continuous beam connection ==

[[File:Continuous beam connection.JPG|RTENOTITLE]] Figure 14 - Discontinuous column with continuous beam, Left: (Elliot, 1992, p.67), Right: (Task Group 6.2 F.I.B., 2008, p. 299)

This type of connection is mainly used in portal frames or in skeletal frames when beams need to be continuous over supports, as is required for a cantilever. The beams are seated on dry pack mortar on top of the vertical members and reinforcing starter bars are projected through sleeves in the beam from the lower column up into the upper column. These sleeves are subsequently grouted to provide vertical continuity. Once the beam is lowered into place, this connection requires no additional formwork providing the grout is poured through vents in the upper column. Therefore, provided the remaining beam end is secured, loads for construction access can be placed upon the beam. This enhances the simplicity of installation and therefore safety on site.

== Wall and Column shoes ==

[[File:Wall and Column shoe connection 1.JPG|RTENOTITLE]] Figure 16 - Photos of example hidden corbel systems used with a precast column and precast beam (Peikko concrete connections, 2009)

Investment into modern technologies has resulted in the production of the hidden corbel connection. This is the most popular type of precast connection used in the UK so far. This type produces fireproof connections which are architecturally advantageous as they minimise visual intrusion whilst maximising floor to soffit height.

The connection area is minimal, protecting the reinforcement steel used in the connection. The connection also benefits from superior adjustability with the modern connection utilising a small adjustable plate, allowing fine tuning of the column corbel prior to installation of the beam.

[[File:Wall and Column shoe connection 2.JPG|RTENOTITLE]] Figure 17 – (Left) Inside detailed view of the anchorage of the corbel connection into the precast column and beams (Peikko concrete connections, 2009). (Right) Labelled example of an alternative hidden corbel system (JVI inc.)

== Ground foundation column connections ==

There are three main methods of connecting columns to foundations:
*Projecting starter bars – The in-situ foundation houses cast in starter bars which the precast column is later lowered onto and grouted to provide continuity.
*Pocket connection – This is the most rigid connection and is utilised when the moment resisting capacity of the connection is required for the lateral stability of the structure. A pocket is provided within the foundation into which the precast column is lowered. The surrounding area is grouted or filled with in-situ concrete.
*Baseplate connection – The base of the precast column contains steel base plates which cast-in bolts are fed through and bolted into place. The surrounding area to the holding down bolts is then filled with non-shrink grout to complete the connection.

The three types above are conservatively modelled as pinned connections resulting in an underestimate of the moments transferred to the columns and beams above. The foundation column connection is subjected to certain degree of variability such as possible rotations due to ground conditions.

= Connection’s discussion and evaluation =

There exists a variety of precast connection types within each big group above. The connections are assessed against different criteria including: the amount of additional materials; aesthetic/space intrusion of the finished connections; allowable tolerance; amount of wet work formwork required; possibility of future reuse/dismantle; operative involvement on site; level of rigidity; safety; skills required; amount of temporary works; time of assembly; tools required; weather sensitivity and level of wet casting needed.

== Continuous beam connection ==

[[File:Discussion - continuous beam.JPG|RTENOTITLE]] Figure 18 - Bolted steel shoe diagram (Halfen GmbH, 2011)

The bolted steel shoe is considered to be the most favourable of the continuous beam connection type as it is simple to produce and quick to assemble on site. This connection requires no structural in-situ works compared to other sub-types. The connection can have a number of different bolt arrangements depending on the size and shape of the column. When used correctly the anchor bolts can be utilised to transfer both tensile and compressive load through to the column below, thus minimising stress on the beam / slab in between. Alternatively the beam / slab can be suitably designed to transfer the load to the column below.

Due to the presence of a continuous beam, the large hogging moments generated at the connection will be transferred to the column. The moment capacity of this connection is high due to the high tensile capacity of the steel holding down bolts resisting the rotation of the column due to buckling, which may result from the hogging moments transferred from the beams.

The amount and positioning of the holding down bolts will determine the connections rigidity. The closer the bolts are to the centre point in the plane of rotation the more the connection will represent a pinned connection between the columns and the beam. As shown in Figure 18, the bolts have been positioned to give the maximum lever arm against any point of pivot and thus maximises rotational resistance.

== Corbel/Billet connection ==

[[File:Discussion-corbel.JPG|RTENOTITLE]] 

Figure 19 - Bearing only connection diagram (Pujol Group)

The bearing only is the favoured connection due to its simple straightforward design which facilitates quick assembly time on site. This fully pinned connection type will therefore transfer vertical and horizontal loads into the column, but all moment will be contained within the beam. For this reason, the beam will be designed to resist greater moment. The connection is therefore less efficient than moment sustaining connections.

Table 2 also highlights the bearing with bolted dowel bar which has been taken as the preferred connection method for this connection type. The bearing with bolted dowel bar allows reduced connection width due to dowel action acting as horizontal restraint. The connection is fixed at the top using a bolt which extends through the column. This is positioned to resist maximum bending moment as the point of pivot will be within the underside of the beam. It is common practice in design to ensure that fixing elements of connections are not the limiting element and therefore the bolt will be able to transfer a considerable amount of moment to the column. The flat landing of the corbel, although unsightly, when combined with the dowel, acts as a torsional restraint. This can be further improved by using a cleat which extends the width of the beam. As the column is continuous, the beam will be required to sustain the majority of bending moment. This connection is semi rigid; therefore it can sustain vertical and horizontal loads with a degree of hogging moment transferred to the column.

[[File:Discussion-corbel 2.JPG]]

Figure 20 - Bearing with bolted dowel, the preferred connection from billet / corbel type as bearing only connection disregarded

== Concealed fixing ==

The concealed bolted steel billet, which is one of the most modern connections, is favoured within this category. It utilises the advantages of other connections with an aesthetically pleasing and simple design, which allows minor adjustments to be made to the plate rather than the beam. There are other connections which show greater rigidity but require greater installing time on site, therefore are less favoured. This connection transfers horizontal and vertical loads to the column through the bolted connection. Its moment resisting capacity is small when compared to the bolted doweled corbel. However, as the connection extends to full height of the beam it is well positioned to sustain some moment. As there are just two bolts per beam to column connection, the level of moment transfer will be limited. For this reason the connection will act as a semi-rigid connection.

[[File:Concealed fixing.JPG]]

Figure 21 - Preferred concealed connection, bolted steel corbel connection (Peikko concrete connections, 2009)

= Conclusion =

== Vertical load resistance ==

All three connections are capable of transferring the vertical load both from the column above and from the beams. As all three of the preferred connections from each group utilise a bolted mechanism to provide fixity, it is feasible that a structure constructed using only these connections would be able to resist against vertical loading without using any in-situ casting.

== Horizontal lateral restraint ==

The connections identified above, unless suitably designed for using excessive sized members and reinforcement, will struggle to resist lateral loading. The lateral loading will need to be taken by shear walls and/or concrete cores such as lift shafts or steel bracing to create a hybrid structure. But as with the initial problem of over-engineered connections through the neglect of their moment capacity, the lateral loading capacity of the preferred connections would need to be assessed and accounted for in order to produce the most efficient design.

Combination of continuous column and continuous beam joints can be used to help transfer moments to stiffer areas of the structure. It thus also allows for a more efficient design with only critical members designed to facilitate the load transfer.

== Frame analysis ==

In-situ frames have fully rigid connections. Should a precast connection be capable of transferring moment to the columns and thus down to the supports, then it can be assed as a complete frame or a series of sub frames. Moments, either hogging or sagging are attracted to stiffer members. Should the connection be capable of transferring these moments, the moments at the columns will then be in hogging and will need to be accounted for. Many published papers (Gorgun, 1997; Aguiar et al., 4 June 2012; Baharuddin et al., 2006) have discovered that some precast connections (including the ones mentioned above) can sustain hogging moment, and are therefore over engineered using the current design process. Therefore the structural frame should be modelled similar to a steel frame, where if almost no moment can be sustained then the connections are designed as pinned.

== Disproportionate Collapse ==

Since 2004, the Building Regulations in England and Wales have been revised to ensure all buildings are designed against disproportionate collapse. The connections above have been analysed with moment capacity as the desired attribute, but they will also provide some tensional resistance which would inherently provide resistance against disproportionate collapse.

 
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