Calculation of Element Stiffness Matrix for 3D Beam Elements in Finite Element Analysis

This document presents a MATLAB-based computational program designed to calculate element stiffness matrices for three-dimensional beam elements in finite element analysis. Each node possesses six degrees of freedom (three translational and three rotational), enabling comprehensive modeling of complex structural behaviors. The program implements fundamental finite element methodologies including: - Euler-Bernoulli beam theory formulations for bending and torsional stiffness - Coordinate transformation algorithms for local-to-global stiffness matrix conversion - Material property integration through Young's modulus and shear modulus parameters - Geometric property handling via cross-sectional area and moment of inertia calculations This computational tool serves multiple purposes in structural analysis: 1. Educational demonstration of stiffness matrix derivation and assembly processes 2. Parametric studies through modular modification of node coordinates, material properties, and cross-sectional characteristics 3. Validation of theoretical formulations against numerical results 4. Foundation for larger structural system assembly and analysis Key program features include: - Modular architecture allowing independent adjustment of material and geometric parameters - Built-in verification checks for matrix symmetry and positive-definite properties - Clear output formatting for stiffness matrix visualization and further processing - Compatibility with standard finite element pre- and post-processing workflows This implementation provides researchers and students with a practical tool for understanding beam element behavior, verifying theoretical concepts, and conducting numerical experiments in structural mechanics.