- Open Access
Experimental and Measurement Methods for the Small-Scale Model Testing of Lateral and Torsional Stability
© The Author(s) 2017
- Received: 15 September 2016
- Accepted: 23 March 2017
- Published: 19 May 2017
Tests of the lateral and torsional stability are quite sensitive to the experimental conditions, such as support conditions and loading system. Controlling all of these conditions in a full-size test is a very challenging task. Therefore, in this paper, an experimental measurement method that can control the experimental conditions using a small-scale model was proposed to evaluate the lateral and torsional stability of beams. For this, a loading system was provided to maintain the vertical direction of the load applied to the beam, and a support frame was produced to satisfy the in-plane and out-of-plane support conditions. The experimental method using a small-scale model was applied successively to the lateral and torsional behavior and stability of I-shaped beams. The proposed experimental methods, which effectively accommodate the changes in the geometry and length of the beam, could contribute to further experimental studies regarding the lateral and torsional stability of flexural members.
- lateral torsional stability
- critical load
- in-plane behavior
- out-of-plane behavior
- lateral deformation
- rotational angle
The lateral and torsional stability is the main issue for the safety of structures and users from the stages of design to construction in the field of architectural and civil engineering (Kalkan 2014; Hurff and Kahn 2012; Lee 2012a, b; Lee et al. 2016; Kim et al. 2016; Hou and Song 2016; Petrone et al. 2016; Ramin and Fereidoonfar 2015; Srikar et al. 2016). Analytical investigations into the lateral and torsional stability have been performed to develop a classical stability theory based on the steel structural members (Horne 1954; Salvadori 1955; Timoshenko 1956; Timoshenko and Gere 1961). Several studies (i.e., Talbot and Dhatt 1987; Rengarajan et al. 1995; Li et al. 2002; Darilmaz 2011; Brsoum and Gallagher 1970; Fafard et al. 1987) developed finite element analysis using shell and plate elements for buckling analysis, which requires large geometric nonlinearity and bifurcation.
For slender and long precast concrete flexural members, most studies (Hansell and Winter 1959; Sant and Bletzacker 1961; Massey 1967; Siev 1960; Revathi and Menon 2007a, b) focused on the classical formulation to propose the weak-axis flexural rigidity and torsional rigidity associated with the lateral and torsional instability of concrete beams. The lateral and torsional stability of the slender and long concrete beams are also strongly affected by initial geometric imperfections. Recently, Kalkan (2014) and Hurff and Kahn (2012) have attempted to evaluate the influence of the initial lateral imperfection in the lateral and torsional buckling of slender, rectangular reinforced and prestressed concrete beams. In addition, Lee (2012a, b) examined the initial lateral deformation caused by environmental thermal effects to evaluate the lateral behavior and stability of bridge I-girders using three-dimensional finite element analysis and experimental data. Moreover, such these imperfections along the length of the beam are considered to be among the main causes of the rollover instability collapse of bridge girders (Oesterle et al. 2007; Zureick et al. 2005), which is the main issue during the construction of concrete and precast beams.
On the other hand, the experiment of lateral and torsional stability requires special caution to minimize geometric imperfections and material irregularities in the manufacturing and testing processes of a specimen, as well as to implementing support and loading conditions. That is, the loading system applied to the beam should retain its vertical direction throughout the testing while allowing longitudinal (in-plane) and transverse (out-of-plane) transitional and flexural movements of the beam. In addition, a lateral support condition should be provided to restrain the transitional and rotational movements but allow flexural behavior in both the longitudinal and lateral directions. Such support and loading conditions are extremely difficult to perform completely, particularly in a full-size beam test. A slight deviation in the support and loading conditions completely changes the experimental results. Moreover, the lateral and torsional buckling of a beam induces lateral and vertical deformation combined with a twisting rotation of the beam.
Therefore, this study presented a small-scale experimental method to implement the support and loading conditions and minimize the experimental errors when assessing the lateral and torsional stability of a beam. An experimental frame was provided to install a specimen and satisfy the loading and support conditions. The frame was also designed to adapt the changes in the loading and support conditions, as well as the size and length of the beam. A loading transfer system was then provided to retain the initially vertical orientation of the load to a beam undergoing the coupled deformation and rotation. The support condition was also designed to satisfy the in-plane and out-of-plane restraint and movement conditions. Finally, the instrumentation was carried out to measure the variations in the deformation and angle and longitudinal strain in the beam. The lateral and vertical deformations and a rotational angle of the beam were determined from the coupled transformations using the measurement method, which was proposed initially by Zhao et al. (1994, 1995) and later modified by Stoddard (1997), and applied successively by Kalkan (2009) and Hurff (2010). In the present study, the measurement method was expanded to allow easier and more useful instrumentation and data processing. As a result, the proposed experimental and measurement methods using a small-scale model can be applied easily and effectively for the lateral and torsional stability testing of flexural members to evaluate the critical load and the influencing factors, such as geometric, support, and loading conditions, in the field of engineering.
Tests of the lateral and torsional stability are quite sensitive to the experimental conditions, such as support condition, loading system, and material nonlinearity. Moreover, the lateral and torsional buckling test using a full-size beam requires considerable time and effort to control all these conditions. Even with careful consideration of these factors, unexpected errors, such as a deviation from the designed support, loading, and material conditions, could completely change the experimental results. Therefore, the appropriate loading, support, and material conditions in a full-size test are extremely difficult to achieve. Therefore, this study presented a small-scale experimental method that can control and maintain the loading and support conditions in the lateral and torsional stability tests of beams. The support system was developed to satisfy both the in-plane and out-of-plane restraints and movement conditions. The loading system was designed to retain its initial vertical direction throughout the course of loading and allow the lateral and torsional movement of the specimen at the application point of the load. The experimental method was applied successively to the lateral and torsional buckling of small-scaled I-shaped beams. The proposed experimental methods, which effectively accommodate the changes in the geometry and length of the beam, could contribute to further experimental studies regarding the lateral and torsional stability of flexural members.
3.1 Beam Specimen
The material selected for the small-scale beam model was polycarbonate, which is strong and durable, as well as linearly elastic. Polycarbonate allows easy specimen preparation and the installation of various sensors, such as strain gauges, displacement meters, and accelerometers, for an experiment. In addition, with the advantage of the linear elasticity of a material, various experiments and explicit interpretation are available to evaluate the influential factors on the lateral and torsional stability. The modulus of elasticity, which plays a decisive role in determining the buckling load, was analyzed using the relationship between the applied load and vertical deflection measured at the middle of the test beam.
The size of the model beam was scaled down using the aspect ratios of the section and length of a typical precast beam, AASHTO Type VI beam.
3.2 Support Conditions and Frame
Therefore, lateral support and support frames were designed using a ball caster and circular rod, shown in Fig. 2. The circular support rod can move laterally to adjust the width of the beam and is fastened to act as a lateral support condition. The ball caster, which is in contact with the side surface of the top and bottom flanges of the beam, contains grease to minimize friction resistance at the contact surface of the ball caster and the circular rod. The top and bottom ball casters, which provide lateral support to the top and bottom flanges of the beam, respectively, is free to move up and down, as well as rotate when the rotations about the strong and weak axes occur in the beam. Therefore, the in-plane and out-of-plane rotations of a beam are unconstrained at the support locations.
3.3 Vertical Loading System
The lateral and torsional stability is strongly dependent on the vertical loading system along with the support condition. The vertical load applied to a beam should retain its vertical direction throughout the course of loading even when lateral and torsional behavior is observed. In addition, the loading transfer mechanism should not have any resistance on the lateral deformation and torsional rotation at the application point of the load. Previous studies using lateral and torsional buckling experiments focused on full-scale size beams. The loading mechanisms used in previous studies were either distributed or point loads. König and Pauli (1990) utilized steel and water weights as the distributed load mechanism to perform the lateral and torsional buckling experiment in reinforced and prestressed concrete beams. The distributed load mechanism has limitations on space, logistics, and safety, which has rarely been applied in the test (Kalkan and Hurff 2012). Therefore, most of the lateral and torsional buckling experiments have attempted to use a point load mechanism to retain the initial vertical and concentrical position of the applied load without its resistance to the lateral and torsional restraints of the test beam at the loading point. Jensen (1978) initially applied a point load for the lateral and torsional buckling experiment but had difficulties in retaining the position of the applied load. Therefore, many studies (i.e., Hansell and Winter 1959; Sant and Bletzacker 1961; Massey 1967; Siev 1960; Revathi and Menon 2007a, b) used roller bearing and ball-and-socket joint mechanisms to minimize the resistance to the lateral and rotational deformations of the test beam at the loading point during the test. Kalkan and Hurff (2012) indicated that the roller bearing mechanism could induce load eccentricity due to some difference between the centroid of the applied load and the beam when the specimen undergoes lateral deformation. Therefore, another load mechanism using a gravity load simulator, which was originally designed by Yarimci et al. (1967) to apply a vertical load to a full-scale frame experiment, was used in the lateral and torsional buckling experiments on full-scale steel I-beams (Yura and Phillips 1992; Helwig et al. 2005), composite I-beams (Stoddard 1997), and reinforced concrete beams (Kalkan 2014; Hurff and Kahn 2012). Nevertheless, the gravity load simulator also inclined the loading unit during the test, which requires an additional control mechanism to maintain the vertical line of the load.
4.1 Instrumentation and Measurements
4.2 Method of Calculating Displacements and Rotational Angle
5.1 Critical Loads and Displacements of the Beam
Therefore, the critical moment of the beam is calculated to be 100 kN mm, and the critical vertical load, P cr , is determined to be 421 N. The modulus of elasticity of polycarbonate ranged from approximately 1100 to 1300 N/mm2. In this study, an average of 1200 N/mm2 was used as the modulus of elasticity for the analytical calculation. Compared with the critical loads obtained from the analytical equations, those obtained from the experiment was 12 and 7% larger than that from the analytical procedure for the ST 1 and ST 2 specimens, respectively. The results from the proposed experimental method were reasonably in agreement with the analytical results. Slight difference might be attributed to some frictional resistance at the contact surface of the lateral support and the side surface of the top and bottom flanges or some errors in the value of the modulus of elasticity of polycarbonate.
5.2 Strains and Neutral Axis of the Beam
The lateral and torsional stability experiment should achieve the in-plane and out-of-plane support conditions. In this study, the simply supported condition in the longitudinal direction of the beam was implemented using a roller that restrains the vertical and lateral translations but allows in-plane flexure of the beam. The lateral support conditions, which should restrain lateral translation and twisting rotation but allow flexural rotations about the strong and weak axes of the cross-section, were provided using a ball caster and circular rod support.
This study also provides a vertical loading system based on the roller mechanism for easy instrumentation and handling. The loading transfer system was designed to free the lateral deformation using a semicircle notch along the width of the beam and the torsional rotation with the help of the ball-and-socket joint. The proposed method has the advantage of maintaining its initially vertical and concentric position throughout the course of loading, even when the beam undergoes coupled deformation and rotation.
The lateral and vertical deflections, combined with a twisting rotation of the beam, cannot be obtained directly from the measurement values of the potentiometers. Thus, a modified measurement method using two vertical potentiometers, which would be easier and more useful for instrumentation and data processing, was presented to determine the lateral and vertical deflections and rotation from the coupled measurements.
For small-scaled I-beams, the critical loads obtained from the proposed experimental method were in good agreement with those calculated using the analytical equations. The strains in the top and bottom flanges, which were under compression and tension, respectively, were induced by the in-plane flexure. On the other hand, the compressive strains in the convex side of the beam decreased due to the out-of-plane flexure. Therefore, the neutral axis in the concave side moves from the centroid of the cross-section up to the top flange. The proposed experimental method, validated in the linear and elastic range of material, could be further applied to the lateral and torsional stability testing of flexural members accounting for various parameters, such as material nonlinearity, shapes and sizes of geometry, and boundary conditions.
This study was supported by Basic Science Research Program through the National Research Foundation (NRF) of Korea funded by the Ministry of Science, ICT & Future Planning (NRF-2014R1A1A1005992).
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
- Brsoum, R. S., & Gallagher, R. H. (1970). Finite element analysis of torsional and torsional flexural stability problems. International Journal for Numerical Methods in Engineering, 2(3), 335–352.View ArticleMATHGoogle Scholar
- Chen, W. F., & Lui, E. M. (1987). Structural Stability: Theory and Implementation (pp. 317–333). New York: Elsevier.Google Scholar
- Darılmaz, K. (2011). An assumed stress hybrid finite element for buckling analysis. Mathematical and Computational Applications, 16(2), 690–701.MathSciNetView ArticleGoogle Scholar
- DIANA. (2016). TNO DIANA-finite element analysis user’s manual release 10.1. Delft, The Netherlands: TNO.Google Scholar
- Fafard, M., Beaulieu, D., & Dhatt, G. (1987). Buckling of thin-walled members by finite elements. Computers & Structures, 25(2), 183–190.View ArticleMATHGoogle Scholar
- Hansell, W., & Winter, G. (1959). Lateral stability of reinforced concrete beams. ACI JouRNAL Proceedings, 56(3), 193–214.Google Scholar
- Helwig, T. A., Wang, L., Deaver, J., & Romero, C. (2005). Cross-frame and diaphragm behavior in bridges with skewed supports: Summary. Project Summary Report 0-1772. Austin, TX: Texas Department of Transportation.Google Scholar
- Horne, M. R. (1954). The flexural torsional buckling of members of symmetric I-section under combined thrust and unequal terminal moments. The Quarterly Journal of Mechanics and Applied Mathematics, 7(4), 410–426.MathSciNetView ArticleMATHGoogle Scholar
- Hou, J., & Song, L. (2016). Progressive collapse resistance of RC frames under a side column removal scenario: the mechanism explained. International Journal of Concrete Structures and Materials, 10(2), 237–242.View ArticleGoogle Scholar
- Hurff, J. (2010). Stability of precast prestressed bridge girders considering imperfections and thermal effects. PhD thesis, Georgia Institute of Technology, Atlanta, GA.Google Scholar
- Hurff, J. B., & Kahn, L. F. (2012). Lateral-torsional buckling of structural concrete beams: experimental and analytical study. ASCE Journal of Structural Engineering, 138, 1138–1148.View ArticleGoogle Scholar
- Jensen, L. M. (1978). Buckling of reinforced concrete beams. Dissertation, Engineering Academy of Denmark, Copenhagen, Denmark.Google Scholar
- Kalkan, I. (2009). Lateral torsional buckling of rectangular reinforced concrete beams. PhD thesis, Georgia Institute of Technology, Atlanta, GA.Google Scholar
- Kalkan, I. (2014). Lateral torsional buckling of rectangular reinforced concrete beams. ACI Structural Journal, 111, 71–81.Google Scholar
- Kalkan, I., & Hurff, J. B. (2012). Experimental techniques for lateral stability testing of beams. Experimental Techniques, 39, 1–6.Google Scholar
- Kim, S. J., Kim, J. H., Yi, S. T., Noor, N. B., & Kim, S. C. (2016). Structural performance evaluation of a precast PSC curved girder bridge constructed using multi-tasking formwork. International Journal of Concrete Structures and Materials, 10(Suppl. 3), 1–17.View ArticleGoogle Scholar
- Kirby, P. A., & Nethercot, D. A. (1997). Design for structural stability. Suffolk: Granada Publishing.Google Scholar
- König, G., & Pauli, W. (1990). Ergebnisse von sechs Kippversuchen an schlanken Fertigteilträgern aus Stahlbeton und Spannbeton (Results of six buckling tests on slender prefabricated girders made of reinforced concrete and prestressed concrete). Beton-und Stahlbetonbau, 85(10), 253–258. (in German).View ArticleGoogle Scholar
- Lee, J. H. (2012a). Behavior of precast prestressed concrete bridge girders involving thermal effects and initial imperfections during construction. Engineering Structures, 42, 1–8.View ArticleGoogle Scholar
- Lee, J. H. (2012b). Investigation of extreme environmental conditions and design thermal gradients during construction for prestressed concrete bridge girders. ASCE Journal of Bridge Engineering, 17(3), 547–556.View ArticleGoogle Scholar
- Lee, J. H., Kalkan, I., Lee, J. J., & Cheung, J. H. (2016). Rollover instability of precast girders subjected to wind load. Mag. Conc. Res., 69(2), 68–83.View ArticleGoogle Scholar
- Li, J. Z., Hung, K. C., & Cen, Z. Z. (2002). Shell element of relative degree of freedom and its application on buckling analysis of thin-walled structures. Thin-Walled Structures, 40, 865–876.View ArticleGoogle Scholar
- Massey, C. (1967). Lateral instability of reinforced concrete beams under uniform bending moments. ACI Journal Proceedings, 64(3), 164–172.Google Scholar
- Nethercot, D. A., & Rockey, K. C. (1971). A unified approach to the elastic lateral buckling of beams. The Structural Engineer, 49(7), 321–330.Google Scholar
- Oesterle, R. G., Sheehan, M. J., Lotfi, H. R., Corley, W. G., & Roller, J. J. (2007). Investigation of red mountain freeway bridge girder collapse. CTL Group project no. 262291 final report. Phoenix: Arizona Department of Transportation.Google Scholar
- Petrone, F., Shan, L., & Kunnath, S. K. (2016). Modeling of RC frame buildings for progressive collapse analysis. International Journal of Concrete Structures and Materials, 10(1), 1–13.View ArticleGoogle Scholar
- Ramin, K., & Fereidoonfar, M. (2015). Finite element modeling and nonlinear analysis for seismic assessment of off-diagonal steel braced RC frame. International Journal of Concrete Structures and Materials, 9(1), 89–118.View ArticleGoogle Scholar
- Rengarajan, G., Aminpour, M. A., & Knight, N. F. (1995). Improved assumed-stress hybrid shell element with drilling degrees of freedom for linear stress, buckling and free vibration analyses. International Journal for Numerical Methods in Engineering, 28(11), 1917–1943.View ArticleMATHGoogle Scholar
- Revathi, P., & Menon, D. (2007a). Estimation of critical buckling moments in slender reinforced concrete beams. ACI Structural Journal, 103(2), 296–303.Google Scholar
- Revathi, P., & Menon, D. (2007b). Slenderness effects in reinforced concrete beams. ACI Structural Journal, 104(4), 412–419.Google Scholar
- Salvadori, M. C. (1955). Lateral buckling of I-beams. ASCE Transactions, 120, 1165–1177.Google Scholar
- Sant, J. K., & Bletzacker, R. W. (1961). Experimental study of lateral stability of reinforced concrete beams. ACI Journal Proceedings, 58(6), 713–736.Google Scholar
- Siev, A. (1960). The lateral buckling of slender reinforced concrete beams. Magazine of Concrete Research, 12(36), 155–164.View ArticleGoogle Scholar
- Srikar, G., Anand, G., & Suriya, S. (2016). Prakash, A study on residual compression behavior of structural fiber reinforced concrete exposed to moderate temperature using digital image correlation. International Journal of Concrete Structures and Materials, 10(1), 75–85.View ArticleGoogle Scholar
- Stoddard, W. P. (1997). Lateral-torsional buckling behavior of polymer composite I-shaped members. PhD thesis, Georgia Institute of Technology, Atlanta, GA.Google Scholar
- Talbot, M., & Dhatt, G. (1987). Three discrete Kirchhoff elements for shell analysis with large geometrical non-linearities and bifurcations. Engineering with Computers, 4, 15–22.View ArticleGoogle Scholar
- Timoshenko, S. P. (1956). Strength of Materials: Part II (3rd ed.). Princeton: Van Nostrand Co.MATHGoogle Scholar
- Timoshenko, S. P., & Gere, J. M. (1961). Theory of elastic stability (2nd ed.). New York: McGraw-Hill.Google Scholar
- Yarimci, E., Yura, J. A., & Lu, L. W. (1967). Techniques for testing structures permitted to sway. Experimental Mechanics, 7(8), 321–331.View ArticleGoogle Scholar
- Yura, J. A., & Phillips, B.A. (1992). Bracing requirements for elastic steel beams. Research Report 1239-1. Austin, TX: Center for Transportation Research, The University of Texas.Google Scholar
- Zhao, X. L., Hancock, G. J., & Trahair, N. S. (1994). Lateral buckling tests of cold-formed RHS beams. Research Report R699. Camperdown: School of Civil and Mining Engineering, The University of Sydney.Google Scholar
- Zhao, X. L., Hancock, G. J., & Trahair, N. S. (1995). Lateral buckling tests of cold-formed RHS beams. ASCE Journal of Structural Engineering, 121(11), 1565–1573.View ArticleGoogle Scholar
- Zureick, A. H., Kahn, L. F., & Will, K. M. (2005). Stability of precast prestressed concrete bridge girders considering sweep and thermal effects. GDOT Project No. RP 05-15 Research Proposal. Atlanta, GA: Georgia Department of Transportation, 2005.Google Scholar