Full length articleBuckling analysis of homogeneous or composite I-beams using a 1D refined beam theory built on Saint Venant's solution
Introduction
Beam-like structures are one of the most used elements in structural engineering. Nowadays, high performance requirements make an increasing use of composite materials and a design of non-conventional cross-sections (shapes and isotropic/anisotropic materials). These optimized structures exhibit a 3D complex mechanical behaviour for which 3D details become essential in order to understand and predict the equilibrium and buckling behaviour of such structures.
In literature, different critical points have been investigated such as the effect of the length-to-thickness ratio, the shear deformation, the material coupling effects when laminated profiles are involved, the boundary conditions and the kind and position of the buckling loads. All these points may have a significant influence on the buckling structural behaviour and one can find a diversity of approaches in the interesting review of Sayyad and Ghugal [1] and the literature review proposed by Sahraei and Mohareb [2]. Most of these works on buckling analysis are available for particular beam configurations developed to find, as much as possible, analytical closed form results intended to be easily used in engineering practice. The most known case is a simply supported beam under an axial load or a pure bending moment. However, finding closed form solution is not always possible, and the use of 3D finite element (3D-FEM) analysis is expensive specially when thin/thick open/closed walled composite sections are involved. This situation calls for the development of general and realistic beam theories accompanied by efficient numerical tools, designed for engineering practice, and suitable for the analysis of beams for which the classical beam theory assumptions are obviously no longer valid. The present work, devoted to the buckling analysis of beams is a contribution to that expectation.
Many beam models have been proposed based on lamination theory to investigate thin-walled members. The main hypotheses used by these models are Kirchhoff-Love thin plate assumption and rigid cross-section contour in its own plane; and when open profiles are investigated, Vlasov null shear at mid-membrane is also assumed. An analytical beam model applicable to thin walled laminated I-section is proposed by Lee and Kim [3] and Lee et al. [4] to study flexural-torsional buckling and lateral buckling respectively. In [5], this model is extended to a geometrically non-linear formulation for an arbitrary open cross-section. In [3], [4], [5], the governing equations are established and solved numerically through 1D-FEM; and the effect of fiber orientation, type and location of load, and boundary conditions are investigated.
Developing a formulation for thin-walled symmetric balanced laminated beams with open or closed cross-section, Cortinez and Piovan [6], [7] showed the importance of the (bending and non-uniform warping) shear flexibility on the stability behaviour. In [6], closed form solution is obtained for a simply supported beam subjected to axial load or bending moment; and the buckling behaviour as well as the influence of initial stress on natural frequencies for different cross-sections and stacking sequences are investigated. In [7], a non linear displacement field based on the rule of semi-tangential rotations is used and the governing equations are solved through 1D-FEM. It is demonstrated that the shear flexibility enhances the critical loads prediction.
Another thin-walled composite beam theory to analyse buckling behaviour is presented in [8], [9] with an exact evaluation of the stiffness matrix. The exact element stiffness matrix is derived from equilibrium equations and force-displacement relations using power series expansion. In [8], flexural-torsional buckling is investigated where the energy functional is consistently obtained corresponding to semitangential moments and rotations. In [10], the numerical method to evaluate the element stiffness matrix is based on the homogeneous form of ordinary differential equations, and in [11] shear deformation is included in the beam model. Critical buckling loads are obtained by FEM using Hermitian interpolation polynomials and the effects of fiber orientation is parametrically studied.
The significant contribution for thin-walled members based on lamination theory is given by the Generalized Beam Theory (GBT) initiated by Schardt [12] and currently developed by Camotin, Silvestre and their colleagues [13]. GBT displacement model includes in and out of plane sectional deformation mode provided by a first cross-section mid-line analysis. Initially developed [14], [15] to analyse open cross-section according to Vlasov's null shear strain assumption, GBT has been extended to deal with arbitrary open/closed cross-sections free from Vlasov's assumption in [16]. Later, Silvestre and Camotim [17] investigated laminated profile where a thin walled member is considered as a ”folded-plate”, Kirchoff's hypothesis is deemed valid for the individual wall (plate) flexural behaviour, and plane stress state is assumed for each plate. GBT's main contribution is its ability to detect both global buckling modes and local, distortional buckling modes as shown in [18] for example. In order to use GBT development, a numerical academic software named GBTUL (acronym for GBT at the University of Lisbon) is available on the website http://www.civil.ist.utl.pt/gbt/.
In Ibrahim et al. [19], Giunta et al. [20] the buckling analysis is performed through a hierarchical N-order (polynomial expansion) 1D-beam theories derived from the Carrera Unified Formulation (CUF) [21] to investigate slender and short columns considering thin-walled beams with open and closed cross-section. CUF in conjunction with Dynamic Stiffness Method is also proposed in Carrera et al. [22] to analyse the buckling of isotropic and composite beam-columns. The numerical results showed that classical and lower-order theories are accurate for flexural buckling modes of slender beams only, but higher-order models are needed to deal with short beams, shear deformable materials (composite), and thin-walled cross-sections. One can find more details on CUF applications and a software using CUF, named MUL2, on the website http://www.mul2.com.
Another interesting work available for buckling of compact or thin-walled beams is proposed by Genoese et al. [23]. The buckling analysis is developed in the framework of large displacements-rotations and small strains and based on a beam model using a mixed-formulation. The beam behaviour is described in terms of generalized static and kinematic quantities. The generalized beam displacement field uses sectional in and out-of plane strain modes deduced from a first cross-section analysis related to the computation of the correspondent 3D Saint Venant's solution [24]. The numerical results presented for homogeneous and composite walled members show that such beam approach is able to describe the 3D buckling mode and even to detect its localisation when pointed load or sectional stiffener are involved.
An elegant and general approach is developed by Hodges and Yu to construct efficient 1D models [25], [26] without invoking the assumptions commonly used for beams. In this formulation, the beam theory is based on the variational asymptotic method (VAM) initiated by Berdichevsky [27]; and the method consists on the asymptotic expansion of the 3D solution using small parameters1 inherent to the beam features. It is shown that the generalized Timoshenko model obtained from VAM can be used to carry out 1D beam analyses such as buckling or vibration [28]. To apply such method for the static/dynamic linear/nonlinear homogeneous/composite beam analysis, a package of two complementary numerical tools is proposed: VABS [29] (Variational Asymptotic Beam Section analysis) and GEBT [30] (Geometrically Exact Beam Theory).
More recently, a refined 1D beam theory built on 3D Saint Venant (SV) solution (RBT/SV) has been proposed by El Fatmi [31]. Free from all the classical beam assumptions, this theory is only based on the choice of the displacement model which includes in and out of plane sectional deformation mode provided by a first cross-section analysis deduced from the computation of 3D SV's solution. The later is not the original one established by SV for homogeneous and isotropic case, but the extended one established by Iesan [32] for an arbitrary composite cross-section (any shape and any arrangement of anisotropic materials). SV's solution (also called central solution) is fundamental because known to describe the exact 3D solution in the interior area of the beam; therefore it reflects the real mechanical behaviour of the section that results from its shape and its materials. This displacement model leads to a beam theory that really fits the cross-section nature. In order to apply RBT/SV, a user friendly numerical tool, named CSB (Cross-Section and Beam analysis) has been also developed. However, the scope of RBT/SV was limited to static beam problem subjected to mechanical loads in [31] and temperature changes in [33].
The objective of this work is to extend the application of RBT/SV to deal henceforth with the buckling analysis of a beam for an arbitrary cross-section (shape and materials) and any direction of loading. This is done according to the usual structural stability condition where the critical case of the structural instability or buckling is identified by the adjacent equilibrium for which the second variation of the potential energy vanishes.
The proposed method being intended to be numerically solved by 1D-FEM, only the theoretical developments to express the 1D weak formulation of the problem will be detailed. To do so, we start from the general 3D expression of the second variation of the potential energy established for a 3D body [34]; then, using the 1D beam displacement model, the correspondent 1D weak formulation is deduced. The essential of the numerical implementation of the proposed beam model will be described, and to show the capabilities of such method, a significant set of thin-to-thick homogeneous/laminated I-beam problems subjected to axial or transversal loads and different boundary conditions is analysed. Most of these I-beams configurations are taken from literature [35], [7], [36]. The results will be confronted to that literature and some of them to 3D-FEM computations, using the finite element commercial code Abaqus.
Section snippets
Kinematic modeling
We consider the composite beam Fig. 1 occupying a prismatic (x, y, z) domain where the beam is along the axis. S denotes the cross-section and L the length of the beam. The materials constituting the beam are linear elastic, anisotropic and perfectly bonded together and the elastic tensor field, denoted by , is z-constant (vectors and tensors are noted in boldface characters).
A reference beam problem
We consider just for convenience, the equilibrium of the cantilever composite beam (Fig. 2) subjected to a body loading and a surface traction acting on its free end ( and are considered as dead loads). The buckling problem to solve is defined by the direction of loading for a positive (multiplicator) scalar . According to the usual structural stability condition, the -equilibrium is stable if the second variation of the potential energy (SVPE) is a minimum; and vanishing of
Comments
It is worth noting that the present 1D beam buckling approach is free from all the classical beam and shell-like assumptions and is available for an arbitrary cross-section (solid or walled, thin or thick, open or closed and any arrangement of isotropic/anisotropic materials) and any direction of load. The only hypothesis is the choice of the displacement model, which is used to solve both the linear problem and the buckling one.
Including displacement modes which reflect the own in and out of
Numerical implementation
The numerical tool CSB which comes with RBT/SV [31], has been upgraded to henceforth deal with buckling problems according to the present developments.
CSB is a package of two complementary numerical tools named CSection and CBeam. For a given beam problem, CSection ensures the cross-section analysis by 2D-FEM to provide the set of sectional modes () which are then used by CBeam to solve the beam problem by 1D-FEM according to the displacement model. Note that:
- •
CSection is a numerical tool
Numerical examples and discussion
Even if the proposed method is available to deal with arbitrary cross-section and loading, the applications in the present paper focus on the buckling of different configurations of homogeneous and laminated I-beams (Fig. 3) subjected to axial or transversal loads. Most of these examples have been already investigated in literature:
- •
Andrade et al. [35] presented the lateral torsional buckling (LTB) of cantilever I-beams, using the isotropic (E= 210 Gpa, = 0.3, = 7800 kg/m3) homogeneous doubly
Conclusion
In this work, the linear buckling of homogeneous/composite beams is performed using a 1D refined beam theory (RBT) available for an arbitrary composite cross-section (CS) (any shape and any materials) and any direction of loading. Free from all the usual beam and shell-like assumptions, RBT displacement model is mainly built on 3D Saint Venant (SV) solution. The kinematic model includes the 6 SV fundamental 3D sectional displacement modes related to the 6 cross-sectional stresses, and
Acknowledgement
This work is a part of a MOBIDOC PhD thesis under the program PASRI-ANPR Tunisia funded by the European Union. This support is acknowledged.
References (38)
- et al.
Bending, buckling and free vibration of laminated composite and sandwich beams: a critical review of literature
Compos. Struct.
(2017) - et al.
Upper and lower bound solutions for lateral-torsional buckling of doubly symmetric members
Thin-Walled Struct.
(2016) - et al.
Flexural-torsional buckling of thin-walled I-section composites
Comput. Struct.
(2001) - et al.
Lateral buckling of I-section composite beams
Eng. Struct.
(2002) Lateral buckling analysis of thin-walled laminated composite beams with monosymmetric sections
Eng. Struct.
(2006)- et al.
Vibration and buckling of composite thin-walled beams with shear deformability
J. Sound Vib.
(2002) - et al.
Stability of composite thin-walled beams with shear deformability
Comput. Struct.
(2006) - et al.
Improved flexural-torsional stability analysis of thin-walled composite beam and exact stiffness matrix
Int. J. Mech. Sci.
(2007) - et al.
Coupled stability analysis of thin-walled composite beams with closed cross-section
Thin-Walled Struct.
(2010) - et al.
Lateral buckling of shear deformable laminated composite I-beams using the finite element method
Int. J. Mech. Sci.
(2013)