Detailed MATLAB Code with Annotations for Finite Element Analysis of a Cantilever Beam (Semi-Original Example)

Finite element analysis of a cantilever beam is a common numerical simulation case in structural mechanics. Implementing it in MATLAB helps understand the fundamental principles and implementation steps of the finite element method. This example typically includes the following key stages: First, establish the geometric model and material parameters of the cantilever beam. With one end fixed and the other free, you need to define parameters such as beam length, cross-sectional properties (area, moment of inertia), and material properties like elastic modulus. These parameters directly influence the generation of stiffness matrices and final calculation results. In MATLAB code, these are typically defined as constants at the beginning using variables like L for length, E for elastic modulus, and I for moment of inertia. Next comes mesh generation. Based on accuracy requirements, the cantilever beam is discretized into several elements connected by nodes. Common cantilever beam problems can be modeled using 1D beam elements (like Euler-Bernoulli beams) or 2D plane elements, with the specific choice depending on analysis objectives. The MATLAB implementation usually involves creating node coordinate arrays and element connectivity matrices, where functions like linspace() can help distribute nodes evenly along the beam length. Then, construct element stiffness matrices and assemble the global stiffness matrix. Each element's stiffness matrix is determined by its geometric and material properties. Through coordinate transformation and superposition, the global stiffness matrix for the entire structure is formed. During this process, boundary condition handling is particularly important, especially the displacement constraints at the fixed end, which must be correctly applied to avoid matrix singularity issues. In code implementation, this often involves using direct stiffness method assembly with for-loops and applying constraints by modifying corresponding rows and columns in the global matrix. Load application and solution form the core steps. Concentrated forces or moments typically applied at the free end of the cantilever beam are converted into equivalent nodal forces. By solving the system of linear equations, nodal displacement responses are obtained, enabling calculation of strain and stress distributions throughout the beam. The MATLAB code typically uses the backslash operator (\) or linsolve() function to solve Ku=F, where K is the global stiffness matrix and F is the force vector. Finally, visualization tools (like MATLAB's plotting functions) display the beam's deformation and stress contours, helping to intuitively understand analysis results. Well-annotated code usually explains the above workflow step by step, including specific implementations of matrix assembly, boundary handling, solvers, and post-processing, making it excellent for learning and secondary development. Common visualization functions include plot() for deformation patterns and contourf() for stress distribution plots. For semi-original examples, improvements might be made in element types, solution optimization, or post-processing visualization, making the code more readable or efficient - this represents common practice in learning finite element programming. Such enhancements could include implementing more efficient sparse matrix storage using sparse() or adding interactive visualization features.