BUCKLING ANALYSIS OF PULTRUDED GFRP HOLLOW BOX BEAM
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1 BUCKLING ANALYSIS OF PULTRUDED GFRP HOLLOW BOX BEAM Donna CHEN Ph.D. Candidate University of Calgary, Department of Civil Engineering 2500 University Drive NW, Calgary, Alberta, T2N 1N4, Canada Raafat EL-HACHA Associate Professor University of Calgary, Department of Civil Engineering 2500 University Drive NW, Calgary, Alberta, T2N 1N4, Canada Abstract Experimental testing, under monotonic four-point flexural load, of a pultruded Glass Fibre Reinforced Polymer (G FRP) hollow box beam section indicated a strong influence of buckling on the ultimate failure mechanism of the beam. Using finite element (FE) methods in the ABAQUS software environment, the experimentally validated model was used to determine the theoretical buckling load and mode shapes. Comparisons between the experimental results and the finite element solution were made, with particular attention towards the incorporation of geometric nonlinearities in the FE analysis and its impact on the accuracy of the solution. It was found that a good approximation of the buckling mechanism can be ascertained using the classic linear elastic eigenvalue approach while including a preceding preload step that takes into consideration large deformation in the initial geometry. Future works will include further investigation into nonlinear buckling analysis techniques, such as Riks method, to eliminate the overestimation of buckling load by the eigenvalue approach. Keywords: Fibre reinforced polymer, finite element, buckling, eigenvalue 1. Introduction The increased implementation of Fibre Reinforced Polymers (FRPs), in its wide spectrum of available structural forms, in bridge applications has resulted in a growing number of research topics focused on the behaviour of FRPs and their failure modes. When FRP structural shapes are used in the design of bridge systems, whether as I-beams [1], sheet piles [2] or hollow sections with square or trapezoidal shapes [3,4], web buckling as a dominant failure mode becomes a major design concern. The finite element (FE) analysis method used in software environments such as ANSYS and ABAQUS can be an effective way of assessing the structural behaviour of actual systems, especially for the purpose of parametric optimization studies. This paper will discuss the development of a FE model representing an experimentally tested rectangular Glass FRP (GFRP) hollow box section. Experimental results had shown significant buckling at ultimate failure and, following validation of the FE model with experimental load-deflection behaviour, the accuracy of the calculated critical buckling load, as compared with the actual ultimate failure load, will be examined. Details regarding buckling mode shapes will also be investigated. Page 1 of 7
2 2. Background Information The GFRP beam under investigation in this paper is a hollow box section beam fabricated using the pultrusion process. The material composition of the GFRP beam consists of internal glass strand rovings in the longitudinal direction for strength purposes as well as either continuous strand glass mats or stitched reinforcements in the transverse strength [5]. The overall cross-sectional dimensions are as follows: outer height and width of mm and mm, respectively, with wall thicknesses of mm all around. Figure 1 shows a schematic of the cross-section as well as a photograph. Figure 1 Dimensions (in mm) and details of the GFRP beam 3. Summary of Results from Experimental Testing The details regarding the experimental results for the pultruded GFRP beams were presented in a separate publication [6] and full discussion of the results is not in the scope of this paper. However, for the purpose of model validation and comparison with the finite element model, a brief summary of the key aspects are presented here. The GFRP beam was tested under monotonic loading using a four-point loading configuration. The span of the beam tested was 2900 mm from center-to-center between the two supports. The point loads were positioned 300 mm away from the center of the beam. The ultimate load reached was 124 kn, with a mid-span deflection equal to 48 mm. At ultimate failure, physical symptoms and failure modes included compression and caving in of the top flange as well as web buckling in the mid-span location between the two point loads. Rupture at the flange-web connections were also noted. Photographs depicting the failure modes are shown in Figure 2. Figure 2 Failure mode for the GFRP beam Page 2 of 7
3 4. Finite Element Model Development 4.1 Model Description The finite element model was developed using ABAQUS 6.9. Due to the symmetry of the loading configuration as well as of the beam s cross-sectional dimensions, only one-quarter of the full beam was modelled. The GFRP beam was modelled using 8-noded continuum shell elements. The steel bearing plates at the support and the point load were modelled using 8- noded solid continuum elements. During experimental testing, Ultra-High Performance Concrete (UHPC) end blocks were cast-in-place within the interior of the GFRP hollow box beam. The intent of the end block is to prevent support crushing at the beam ends. The end block was similarly modelled using 8-noded solid continuum elements. The model was meshed using an overall mesh size of 6.25 mm. In the corner sections of the GFRP box beam, due to the larger stress concentrations expected as well as the cylindrical geometry, higher density meshes in the form of wedges were used. A similarly fine mesh was applied to the corner region of the UHPC end block, which is dimensionally matched to the interior sides of the GFRP hollow box beam. A limited degree of sliding was permitted at the interface between the inner surface of the GFRP hollow box beam and the outer surface of the end block, with a given contact static friction coefficient of The beam was first loaded, at the point load, incrementally up to an applied load level of 100 kn. This was intended to act as the preload prior to the buckling analysis step. Considerations for nonlinear geometry changes in the base state (preload) was made in the analysis, where the intent of the preload was to minimize the occurrence of numerical problems in the subsequent step to determine the buckling load. A linear elastic buckling analysis step was then performed using the linear perturbation procedure. Symmetry boundary conditions were maintained solely during the stress perturbation calculation, allowing for free motion at the symmetry boundaries to occur during the buckling mode analysis and calculation. 4.2 Constitutive Materials Properties The GFRP beam was modelled in three separate portions: a) the top and bottom flange, b) the side webs and c) the four rounded corners connections between the flange and webs. From experimental tension coupon tests, it was found that each section had slightly different material properties. The GFRP material was modelled as a lamina, with negligible strength in the sheet lay-up direction as compared with the in-plane longitudinal and transverse directions. The material properties for the GFRP beam are provided in Table 1. The isotropic material properties for the UHPC end block and the steel support plates are given in Table 2. GFRP portion Table 1 Material Properties for GFRP Beam Modulus of elasticity Longitudinal direction E 1 Transverse direction E 2 In-plane shear modulus G 12 Poisson s Ratio ν Flanges Webs Corners Page 3 of 7
4 Table 2 Material Properties for Steel and UHPC Material Orthotropic modulus of elasticity E Poisson s Ratio ν Steel UHPC Finite Element Model Results The load-deflection behaviour of the GFRP beam during the initial static step, with considerations towards nonlinear geometry during the analysis, was first compared with experimental data for model validation. The comparison between the experimental and finite element solution, shown in Figure 3, illustrates very good approximation of the actual experimental behaviour Load (kn) Experimental Finite Element Mid-span Deflection (mm) Figure 3 Model validation of load-deflection behaviour with experimental data With the model validation completed, the results obtained from the buckling analysis can now be examined. Diagrams representing the buckled mode shape are presented in Figure 4. The deformations are exaggerated to better describe the beam behaviour. The deformed shape obtained from the finite element solution closely resemble the actual failure behaviour exhibited by the GFRP beam, most prominently shown by the following features: a) outwards transverse movement of the GFRP beam away from its axis of longitudinal symmetry, b) significant concavity exhibited in the top flange of the beam in the shear span near the location of the point load and c) large deformations and buckling in the web, particularly below the point load. Page 4 of 7
5 Figure 4 Deformed buckling mode of GFRP beam For further comparison of the finite element solution with experimental results, values for the experimental and predicted buckling loads were assessed. In the linear elastic eigenvalue analysis, the buckling load is calculated using the following relationship shown in Eq. (1): where: P cr P Q (1) P cr is the critical buckling load; P is the applied preload during the static step; Q is the incremental applied load during the buckling step; and, λ i is the eigenvalue for the i th buckling mode. The eigenvalue for the buckling mode shown in Figure 4 was equal to 38.8 and, for convenience, an incremental load, Q, of 1 kn was applied to the beam. Therefore, the critical buckling load determined using the finite element method was equal to kn. 6. Discussion Comparison between the experimental and finite element solution for the critical buckling load of the GFRP hollow box beam shows that the finite element solution overestimated the load by 12%. This overestimation is expected given the nature of the method used to determine the finite element solution, where the linear buckling analysis procedure does not take into account the geometric nonlinearities caused by the coupling of in-plane membrane deformation with bending deformations in the beam. For a system with a near linear-elastic load-deflection behaviour, as in the case for the GFRP beam, the bifurcation point i Page 5 of 7
6 (determined using the linear perturbation procedure) would tend to calculate a critical buckling load higher than the actual limit load. The limit load represents the actual state of a buckled system, with geometric nonlinearities taken into account during the buckling step. Figure 5 is provided below to better depict this trend. Figure 5 Comparison of load-deflection behaviour with and without consideration for geometric nonlinearity during buckling The incorporation of a preload step that includes considerations for geometric nonlinearities, as was performed in this investigation, prior to the buckling step would provide a better approximation, particularly for systems that deviate more from the perfectly linear-elastic behaviour, as compared to a purely linear-elastic analysis. This is due to the use of an updated stiffness matrix based on the final geometry of the system immediately prior to the initiation of the buckling analysis, which would be used in the stress perturbation calculation. Nevertheless, the critical buckling load determined using the linear eigenvalue method should only be taken as an upper bound estimate, to be used a point of reference for further nonlinear buckling analysis methods, such as Riks method. In addition, Riks method, recommended to be used in further studies, has the advantage of providing a continuous load-displacement relationship. Compared with the linear eigenvalue method, a continuous relationship can only be obtained up until the end of the preload step since the time step used in the linear eigenvalue method approximates an infinitely small time frame. The influence of structural imperfections can also play a significant role by decreasing the actual buckling load of the system, as exemplified by the bottom curve in Figure 5. Though the effect does not necessarily need to be at the magnitude shown in the figure, the calculated finite element solution assumes a perfect system, thus leading to an upper bound solution as well. 7. Conclusion Comparison between the FE solution for the critical buckling load and the actual experimental ultimate failure load shows a good approximation, where the FE analysis provided an overestimation of 12% in regards to the buckling load. The use of nonlinear analysis in the preload preceding the buckling load step to a load level close to the ultimate load can provide greater accuracy by including large deformations and a revised stiffness matrix in the determination of the stress perturbation base state. Due to the upper bound nature of the bifurcation point for near linear-elastic structural systems, it is recommended to perform further studies using nonlinear buckling analysis methods, like Riks method, to ascertain the Page 6 of 7
7 limit load in the buckling mechanism. The use of the nonlinear analysis procedure is particularly desirable if a full and continuous load-displacement relationship up until buckling is to be obtained. 8. Acknowledgement The authors would like to thank Lafarge Canada for providing the UHPC, and acknowledge the University of Calgary as well as the Natural Sciences and Engineering Research Council of Canada (NSERC) for their financial support towards this research. 9. References [1] HAI, N.D., MUTSUYOSHI, H., ASAMOTO, S., MATSUI, T., Structural Behaviour of Hybrid FRP Composite I-beam, Construction and Building Materials, 2010, p [2] HONICKMAN, H., FAM, A., Investigation a Structural Form System for Concrete Girders Using Commercially Available GFRP Sheet-Pile Sections, Journal of Composites for Construction, 2009, p [3] HEIJLL, A., TALJSTEN, B., MOTAVALLI, M., Large Scale Hybrid FRP Composite Girders for Use in Bridge Structures Theory, Test and Field Application, Composites: Part B, p [4] ALNAHHAL, W., AREF, A., Structural Performance of Hybrid Fiber Reinforced Polymer-Concrete Bridge Superstructure Systems, Composite Structures, 2008, p [5] STRONGWELL, EXTRENE properties, Strongwell Corporation, [6] EL-HACHA, R., CHEN, D., Behaviour of Hybrid FRP-UHPC Beams Subjected to Static Flexural Loading, Composites: Part B, 2011, 12p. (doi: /j.compositesb ) Page 7 of 7
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