Timoshenko Buckling Formula
Marthe Bernskoetter
In this numerical tour, we will compute the critical buckling load of a straight beam under normal compression, the classical Euler buckling problem. Usually, buckling is an important mode of failure for slender beams so that a standard Euler-Bernoulli beam model is sufficient. However, since FEniCS does not support Hermite elements ensuring \(C^1\)-formulation for the transverse deflection, implementing such models is not straightforward and requires using advanced DG formulations forinstance, see the fenics-shell implementation of the Love-Kirchhoff plate model or the FEniCS documented demo on the biharmonic equation.
where \(I=bh^3/12\) is the bending inertia for a rectangular beam of width \(b\) and height \(h\), \(S=bh\) the cross-section area, \(E\) the material Young modulus and \(\mu\) the shear modulus and \(\kappa=5/6\) the shear correction factor. We will use a \(P^2/P^1\) interpolation for the mixed field \((w,\theta)\).
As in the ModalAnalysis tour, a dummy linear form l_form is used to call the assemble_system function which retains the symmetric structure of the associated matrix when imposing boundary conditions. Here, we will consider clamped conditions on the left side \(x=0\) and simple supports on the right side \(x=L\).
Note that we made use of the zero method of DirichletBC making the rows of the matrix associated with the boundary condition zero. If we used instead the apply method, the rows would have been replaced with a row of zeros with a 1 on the diagonal (as for the stiffness matrix K). As a result, we would have obtained an eigenvalue equal to 1 for each row with a boundary condition which can make more troublesome the computation of eigenvalues if they happen to be close to 1.Replacing with a full row of zeros in KG results in infinite eigenvalues for each boundary condition which is more suitable when looking for the lowest eigenvalues of the buckling problem.
Up to the negative sign cancelling from the previous definition of KG, we now formulate the generalized eigenvalue problem \(\mathbfKU=-\lambda\mathbfK_G U\) using the SLEPcEigenSolver. The only difference from what has already been discussed in the dynamic modal analysis numerical tour is that buckling eigenvalue problem may be more difficult to solve than modal analysis in certain cases, it is therefore beneficial to prescribe a value of the spectral shift close to the criticalbuckling load.
The problem of beams resting on an elastic foundation is also a crucial and technical topic in structural and geotechnical engineering. Practical examples of these are railroad structures, highway pavements, the foundations of buildings, and pipelines embedded in the soil. There are various types of foundation models for soil-structure interaction in static and dynamic analysis of structures on an elastic foundation. The Winkler model, which consists of infinitely closed spaced linear translational springs which are independent of each other, is extensively used in the solution of the problems mentioned above related to soil-structure interaction. Several types of research have been conducted on the mechanical behavior of beam elements lying on an elastic foundation.
Also, Zhu and Leung [23] suggested a new finite element formulation for the non-linear free and forced vibration analysis of non-prismatic Timoshenko beams lying on two-parameter foundations. According to the nonlocal Timoshenko beam theory, stability analysis of nanotubes embedded in an elastic matrix was also performed by Wang et al. [24]. Mirzabeigy [25] investigated the free vibration behavior of non-uniform beams resting on an elastic foundation by presenting a semi-analytical technique, based on the DTM. Tsiatas [26] developed a new influential approach to accurately determine stiffness and mass matrices of non-uniform Euler-Bernoulli beam from inhomogeneous linearly elastic material resting on an elastic foundation. Hassan and Nassar [27] assessed the linear buckling and the free vibration analysis of the Timoshenko beam resting on a two-parameter foundation by employing the ADM. Akgoz and Civalek [28] applied higher-order shear deformation microbeams and a modified strain gradient theory to analyze the static bending response of single-walled carbon nanotubes embedded in an elastic medium. The stability analysis of the AFG non-prismatic beam on an elastic foundation was comprehensively examined by Shvartsman and Majak [29]. Based on the modified strain gradient theory and surface stress effects, Mohammadimehr et al. [30] exploited the size-dependent effect on the free vibration behavior of Timoshenko microbeams subjected to pre-stress loading embedded in an elastic medium. Mercan and Civalak [31] analyzed the stability of boron nitride nanotube on the elastic matrix by utilizing a discrete singular convolution technique. By considering the impact of the viscoelastic foundation, Calim [32] studied free and forced vibration of AFG Timoshenko beams. Soltani and Asgarian [33] combined the power series approximation and the Rayleigh-Ritz method to assess the free vibration and stability of AFG tapered beam resting on the Winkler-Pasternak foundation.
Therefore, the main objective of the present paper is to derive highly accurate static and buckling stiffness matrices of axially functionally graded Timoshenko tapered beams subjected to compressive axial concentrated load and supported by a uniform Winkler foundation. For this, a novel numerical methodology based on a general and straightforward procedure presented in [34-37] is established. The superiority of the finite element method over the other semi-analytical and mathematical techniques is its simplicity, excellent precision, and generality. This methodology is applicable to analyzing a vast range of problems subjected to various circumstances. Considering these facts, the majority of the structural engineering simulation software is commonly developed based on the finite element solution.
1- Coupled governing equations for the buckling of AFG tapered Timoshenko beams resting on a uniform Winkler foundation have been derived via the energy principle, and they are analytically solved using the power series method. According to the aforementioned method, the expressions of the vertical deflection and cross-sectional rotation modes are also determined.
3- To assure the precision and practical usefulness of this hybrid formulation, comparisons with existing results in the literature are provided for a particular case. Subsequently, the effects of the taper ratio, the material non-homogeneity index, end restraints, and the Winkler parameter on critical buckling loads are investigated.
A significant point of departure of the present finite element solution from others is in the interpolating shape functions used for derivation the structural stiffness matrices. Unlike the Hermitian interpolation polynomials, the expressions of proposed shape functions are dependent on geometrical properties, Winker elastic foundation coefficient, material characteristics, and the compressive axial load. Besides, the accuracy of this method is improved by contemplating the influence of material gradient and varying cross-sections in the calculation procedure of the terms of structural and buckling stiffness matrices. This numerical methodology is not restricted by any computational operations. It can be easily used for linear stability analysis of non-prismatic Timoshenko beam with axially varying materials subjected to different boundary conditions. It is also believed that the rate of convergence of the present formulation is faster than that of the conventional finite element technique.
d illustrates a virtual variation in the last formulation. In the case of the constant axial load (P) and according to Eq. (2) with respect to u0, w and q, the equilibrium equations for a non-prismatic Timoshenko beam are derived as
The last two equilibrium equations (3b and 3c) are coupled differential equations due to the presence of vertical and rotation displacement components (w and θ) as well as shear rigidity (GA), while the axial stability equation (Eq. (3a)) is uncoupled from the others. It has no incidence on linear stability analysis of the Timoshenko beam.
In the following section, the application of the Power Series Method (PSM) in the linear stability analysis of non-homogeneous Timoshenko beams with non-uniform cross-section is presented. According to this semi-analytical method, all variable geometric and material properties of a beam and the displacement components are developed into the power series form.
where , and are coefficients of power series at order i. In order to facilitate the solution of the equations (3b) and (3c), a non-dimensional variable ( ) is introduced. Furthermore, the general solutions of two displacement parameters ( , ) should be presented by the following power series of the form:
where ai and bi are unknown coefficients and the amplitude of the i component. Substituting Eqs. (4) and (5) and the non-dimensional variable e into the system of stability equations, the following expressions are obtained:
According to the above recurrence formulations and from a mathematical point of view, it is culminated that all the ak and bkcoefficients can be obtained except for the first two ( and ), which can be derived by imposing the natural boundary conditions of Timoshenko beam. Note that the terms of ak+2 and bk+2 converge to zero as .
In equation (12), [B] is a matrix including the fundamental solutions of equilibrium equations for linear stability ( and ), and A represents the column vector of four unknown parameters. All terms of and are derived with the aid of the symbolic software MATLAB [44] and the expressions of displacement functions ( and ) are shown in Appendix A.
It has to be noted that the four undefined coefficients ( ) are functions of the displacements of the degree of freedom (DOF), then all the rest of coefficients are also functions of the displacements of DOF. The expression of the angle of rotation and vertical displacement ( and ) can thus be derived as a function of the displacement of DOF. These mentioned unknown parameters can be obtained by imposing the right and left end boundary conditions of the element (two at each end).
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