A Unified Approach to the Timoshenko Geometric Stiffness Matrix Considering Higher-Order Terms in the Strain Tensor

Abstract

Nonlinear analyses using an updated Lagrangian formulation considering the Euler-Bernoulli beam theory have been developed with consistency in the literature, with different geometric matrices depending on the nonlinear displacement parts considered in the strain tensor. When performing this type of analysis using the Timoshenko beam theory, in general, the stiffness and the geometric matrices present additional degrees of freedom. This work presents a unified approach for the development of a geometric matrix employing the Timoshenko beam theory and considering higher-order terms in the strain tensor. This matrix is obtained using shape functions calculated directly from the solution of the differential equation of the problem. The matrix is implemented in the Ftool software, and its results are compared against several matrices found in the literature, with or without higher-order terms in the strain tensor, as well as the Euler-Bernoulli or Timoshenko beam theories. Examples show that the use of the Timoshenko beam theory has a strong influence, especially when the structure has small slenderness (short members). For high axial load values, the consideration of higher-order terms in the strain tensor results in larger displacements as expected.

An 8-Nodes 3D Hexahedral Finite Element and an 1D 2-Nodes Structural Element for Timoshenko Beams, Both Based on Hermitian Intepolation, in Linear Range

“…Timoshenko's beam theory states that, unlike the Euler-Bernoulli's beam theory, where the cross section does not rotate after loads are applied to it, the shear distortion v is considered as an additional rotation of the cross section. Therefore, transverse rotation and transverse displacement are considered as independent variables [22].…”

“…When beams are being formulated, with a 3D FE or 1D FE, it is necessary a formulation capable to capture the bending and shear effects; for a 3D FE, the bending and shear effects are obtained by the three main displacements of each node, with a quadratic shape function (minimum), due to the complexity of the formulation; for a 1D FE, the formulation is quite easy, with a full 6 DOF each node (axial forces, shear forces, and bending moments). There are two main theories for beam formulation: Bernoulli's formulation, which neglects the shear deformation (valid for small beams in heigth), and Timoshenko's Theory, which considers the shear deformation (valid for all types of beams) [21][22][23][24][25][26].…”

Second-order two-cycle analysis of frames based on interpolation functions from the solution of the beam-column differential equation

“…On the other hand, in the case of a compressive axial load (P<0), μ is a complex number and the expression for v h (x) can be written in terms of trigonometric functions. All those expressions are available in Rodrigues et al (2019). In matrix form, the conditions given by Equations ( 5) and ( 6) are written as:…”

“…From this numerical approach, the continuous structure response is directly influenced by the engineer's experience in choosing the most suitable number of elements to be used in structural analysis (discretization). All these influences occur because the FEM discrete solution approximates the analytical solution, i.e., the interpolation functions that define the structure deformed configuration do not always agree with the problem's exact solution (Burgos & Martha, 2013;Rodrigues et al, 2019).…”

Evaluation of the P-Δ (P-Delta) effect in columns and frames using the two-cycle method based on the solution of the beam-column differential equation

“…A detailed discussion on how to obtain these equations and solutions is given in [4] . Since trigonometric functions are difficult to deal with in numerical schemes due to the possibility of singularity in the matrices involved, the usual way to capture this behavior is to subdivide the domain into finite elements obtained using the cubic solutions of Eq.…”