Abstract
In the static analysis of beam-column systems using matrix methods, polynomials are using as the shape functions. The transverse deflections along the beam axis, including the axial- flexural effects in the beam-column element, are not adequately described by polynomials. As an alternative method, the element stiffness matrix is modeling using stability parameters. The shape functions which are obtaining using the stability parameters are more compatible with the system’s behavior. A mass matrix used in the dynamic analysis is evaluated using the same shape functions as those used for derivations of the stiffness coefficients and is called a consistent mass matrix. In this study, the stiffness and consistent mass matrices for prismatic three-dimensional Bernoulli-Euler and Timoshenko beam-columns are proposed with consideration for the axial-flexural interactions and shear deformations associated with transverse deflections along the beam axis. The second-order effects, critical buckling loads, and eigenvalues are determined. According to the author’s knowledge, this study is the first report of the derivations of consistent mass matrices of Bernoulli-Euler and Timoshenko beam-columns under the effect of axially compressive or tensile force.
Highlights
In linear structures, deformations are proportional to the external loads, the displacements and the internal forces of the system are obtaining via superposition
Analyzing a frame system using the matrix method provides favorable results than the finite element method because, while a mesh arrangement is required in the finite element method, matrix methods enable each element taken into account in the frame system to be considered independently
In cases where the effects of the axial forces are significant, designers are faced with problems that included second-order effects and buckling issues
Summary
Deformations are proportional to the external loads, the displacements and the internal forces of the system are obtaining via superposition. When an element of a structure is subjecting to an axial-flexural effect, non-linear interactions develop This interaction produces second-order moments, which consist of the product of the axial force and flexural displacements. [16] presented a method for evaluating the shape functions and structural matrices derived for non-prismatic Euler-Bernoulli beam elements. [18] used a force-based approach to derive a three-dimensional consistent mass matrix for vibration analysis of beams. The remainder of this paper is organized as follows: in sections 2 and 3 are obtained the stiffness matrices in prismatic three dimensional Bernoulli-Euler and Timoshenko beam elements, respectively. 2. The stiffness matrices in prismatic three dimensional Bernoulli-Euler beam element. Eqs. are the basic parameters to calculate of the element end forces and stiffness matrices. G denotes the torsional elastic modulus and can be assumed to be equal to the shear modulus, while J denoted the torsional constant
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