Abstract
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.
Highlights
A structural geometric nonlinear analysis using the finite element method (FEM) depends on the consideration of four aspects: the bending theory, the kinematic description, the strain-displacement relations and the interpolating functions.The most commonly used bending solution for frame elements is the Euler-Bernoulli beam theory (EBBT), which is the one implemented in most structural analysis software, with a large number of applications.in some cases, this theory cannot predict the correct behaviour of the structure
The results clearly show the influence of the beam theory used and the importance of considering higher-order terms in strain tensors to predict the critical loads of framed structures, especially for small slenderness beam-columns and high axial loads
All stiffness matrices were implemented in the Ftool computer software (Martha 1999), and their results were compared with results developed in the literature and with the Mastan2 v3.5 software developed by McGuire et al (2000), by considering both beam theories, i.e., EBBT_Mastan (Euler-Bernoulli beam theory) and TBT_Mastan (Timoshenko beam theory)
Summary
A structural geometric nonlinear analysis using the finite element method (FEM) depends on the consideration of four aspects: the bending theory, the kinematic description, the strain-displacement relations and the interpolating (shape) functions. A geometric stiffness matrix is developed considering higher-order terms in the strain tensor based on an updated Lagrangian formulation for the equilibrium equations, in which the complete Green strain tensor is employed. In the formulation presented in this work, no additional terms are necessary to develop the geometric stiffness matrix since shape functions are obtained directly from the solution of the differential equations of the problem, considering the Timoshenko beam theory. The main contribution of this research is to provide a unified approach to building the geometric stiffness matrices of two-dimensional Timoshenko beam-column elements, considering higher-order terms in the Green strain tensor. The results clearly show the influence of the beam theory used and the importance of considering higher-order terms in strain tensors to predict the critical loads of framed structures, especially for small slenderness beam-columns and high axial loads. Future work will consider shape functions obtained from the equilibrium of a deformed infinitesimal element, that includes the influence of axial force and 3D elements
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