% Apply simply supported boundary conditions: w=0 and Mxx=0 => w,xx=0 on x-edges % We'll set w=0 on all edges and use ghost points to enforce curvature=0 % For simplicity, we set w=0 on boundary nodes and eliminate their equations.
boundary_nodes = []; for i = 1:Nx for j = [1, Ny] boundary_nodes = [boundary_nodes, idx(i,j)]; end end for j = 2:Ny-1 boundary_nodes = [boundary_nodes, idx(1,j), idx(Nx,j)]; end boundary_nodes = unique(boundary_nodes);
[ \left(\frac{\partial^4 w}{\partial x^4}\right) {ij} \approx \frac{w {i-2,j} - 4w_{i-1,j} + 6w_{i,j} - 4w_{i+1,j} + w_{i+2,j}}{\Delta x^4} ] Composite Plate Bending Analysis With Matlab Code
We assemble a sparse linear system ( [K] {w} = {f} ) and solve. Below is the complete code. It computes deflections, curvatures, and then stresses in each ply at Gauss points.
%% Finite Difference Grid Nx = 41; Ny = 25; % odd numbers to include center dx = a/(Nx-1); dy = b/(Ny-1); x = linspace(0, a, Nx); y = linspace(0, b, Ny); % Apply simply supported boundary conditions: w=0 and
% Load (uniform pressure) F(n) = 1000; % Pa end end
%% Geometry a = 0.5; % length (m) b = 0.3; % width ply_thick = 0.125e-3; % m num_plies = 4; h = num_plies * ply_thick; % total thickness It computes deflections, curvatures, and then stresses in
[ \begin{Bmatrix} \mathbf{N} \ \mathbf{M} \end{Bmatrix} = \begin{bmatrix} \mathbf{A} & \mathbf{B} \ \mathbf{B} & \mathbf{D} \end{bmatrix} \begin{Bmatrix} \boldsymbol{\epsilon}^0 \ \boldsymbol{\kappa} \end{Bmatrix} ]