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| Derivation and application of the geometric stiffness matrix for beam elements based on the rigid body rule |
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Abstract Geometric stiffness is a critical factor influencing the buckling load and nonlinear characteristics of structures. By leveraging the linearized incremental virtual work equation and the spatial torque-rotation characteristics of three-dimensional solid beams, this study analyzes the induced moment matrix, which represents the key attribute enabling equilibrium at beam element nodes. Subsequently, the geometric stiffness matrix for three-dimensional beam elements is constructed. Using the symmetry and rigid-body compliance of the geometric stiffness matrix, a concise explicit expression for the geometric stiffness matrix of three-dimensional beam elements is derived. Additionally, the geometric stiffness matrix for two-dimensional beam elements is obtained by simplifying that of the three-dimensional beam elements. Through linear buckling analysis and nonlinear analysis of typical numerical cases, the results demonstrate that the derived geometric stiffness matrix can be effectively applied to the buckling and post-buckling analysis of beam-type structures. This method for deriving the geometric stiffness matrix of beam elements possesses clear physical significance and a straightforward derivation process, offering a novel approach to deriving the geometric stiffness in finite element analysis.
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Received: 07 May 2025
Published: 27 August 2025
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2025, Vol. 46 
