Bending Analysis With Matlab Code - Composite Plate
[NM]=[ABBD][ϵ0κ]the 2 by 1 column matrix; cap N, cap M end-matrix; equals the 2 by 2 matrix; Row 1: cap A, cap B; Row 2: cap B, cap D end-matrix; the 2 by 1 column matrix; epsilon to the 0 power, kappa end-matrix;
Assemble the global stiffness matrix from element matrices derived via FSDT or CLPT.
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The core of composite analysis is the , which relates the in-plane force resultants ( ) and moment resultants ( ) to the mid-plane strains ( ϵ0epsilon sub 0 ) and curvatures ( Composite Plate Bending Analysis With Matlab Code
Compute global strains at any specific layer depth
For a simply supported, rectangular composite plate under uniform transverse load ( ), the maximum deflection ( wmaxw sub m a x end-sub ) occurs at the center. The governing equation is:
The following MATLAB script automates this analysis. It calculates the ABD matrix, solves for the center deflection of a simply supported cross-ply laminate under a uniform load, and plots the deflected surface shape. Use code with caution. Interpreting the Analysis [NM]=[ABBD][ϵ0κ]the 2 by 1 column matrix; cap N,
% Initialize laminate stiffness matrices A = zeros(3,3); B = zeros(3,3); D = zeros(3,3); As = zeros(2,2); % Transverse shear stiffness
%% Composite Plate Bending Analysis using CLPT (Navier's Method) clear; clc; close all; %% 1. Material Properties & Geometry E1 = 175e9; % Longitudinal Elastic Modulus (Pa) E2 = 7e9; % Transverse Elastic Modulus (Pa) G12 = 3.5e9; % Shear Modulus (Pa) nu12 = 0.25; % Poisson's Ratio nu21 = (E2 / E1) * nu12; a = 0.5; % Length of plate (m) b = 0.5; % Width of plate (m) q0 = 10e3; % Uniformly Distributed Load (Pa) % Layup sequence (angles in degrees) and thickness per ply angles = [0, 90, 90, 0]; t_ply = 0.00125; % Thickness of each ply (m) N_plies = length(angles); h = N_plies * t_ply; % Total thickness %% 2. Coordinate System for Plies % Total height spans from -h/2 to h/2 z = zeros(N_plies + 1, 1); for k = 1:(N_plies + 1) z(k) = -h/2 + (k-1) * t_ply; end %% 3. Reduced Stiffness Matrix (Q) and Transformed Q (Qbar) % Standard reduced stiffness components Q11 = E1 / (1 - nu12 * nu21); Q12 = (nu12 * E2) / (1 - nu12 * nu21); Q22 = E2 / (1 - nu12 * nu21); Q66 = G12; Q = [Q11, Q12, 0; Q12, Q22, 0; 0, 0, Q66]; % Initialize A, B, D matrices A = zeros(3,3); B = zeros(3,3); D = zeros(3,3); for k = 1:N_plies theta = deg2rad(angles(k)); m = cos(theta); n = sin(theta); % Transformation matrix T T = [m^2, n^2, 2*m*n; n^2, m^2, -2*m*n; -m*n, m*n, m^2-n^2]; % Reuter's matrix R R = [1, 0, 0; 0, 1, 0; 0, 0, 2]; % Transformed reduced stiffness matrix Qbar Qbar = inv(T) * Q * R * T * inv(R); % Integrate through thickness to populate ABD A = A + Qbar * (z(k+1) - z(k)); B = B + 0.5 * Qbar * (z(k+1)^2 - z(k)^2); D = D + (1/3) * Qbar * (z(k+1)^3 - z(k)^3); end disp('--- Calculated Bending Stiffness Matrix (D) ---'); disp(D); %% 4. Navier's Solution for Deflection Nx = 50; Ny = 50; % Grid resolution for plotting x = linspace(0, a, Nx); y = linspace(0, b, Ny); [X, Y] = meshgrid(x, y); W = zeros(Nx, Ny); % Initialize deflection matrix MaxTerms = 49; % Number of Fourier terms (odd numbers only) for m = 1:2:MaxTerms for n = 1:2:MaxTerms % Fourier load coefficient for UDL Qmn = (16 * q0) / (pi^2 * m * n); % Denominator for cross-ply laminates (assuming D16 = D26 = 0) denom = pi^4 * (D(1,1)*(m/a)^4 + 2*(D(1,2) + 2*D(3,3))*(m/a)^2*(n/b)^2 + D(2,2)*(n/b)^4); % Deflection amplitude Wmn = Qmn / denom; % Accumulate spatial contribution W = W + Wmn * sin(m * pi * X / a) .* sin(n * pi * Y / b); end end % Extract Max Center Deflection center_idx_x = round(Nx/2); center_idx_y = round(Ny/2); fprintf('Max Center Deflection: %.6f mm\n', W(center_idx_y, center_idx_x) * 1000); %% 5. Visualization figure('Color', 'w'); surf(X, Y, W * 1000, 'EdgeColor', 'interp'); colorbar; colormap(jet); title(sprintf('Bending Deflection Profile of [%s] Laminate', num2str(angles))); xlabel('X-axis (m)'); ylabel('Y-axis (m)'); zlabel('Deflection (mm)'); view(-37.5, 30); grid on; Use code with caution. Result Interpretation and Post-Processing Deflection Profile
matrix for a symmetric laminate and determines the maximum center deflection for a simply supported plate. If you share with third parties, their policies apply
[NM]=[ABBD][ϵ0κ]the 2 by 1 column matrix; cap N, cap M end-matrix; equals the 2 by 2 matrix; Row 1: cap A, cap B; Row 2: cap B, cap D end-matrix; the 2 by 1 column matrix; epsilon to the 0 power, kappa end-matrix; Extensional Stiffness
Straight lines normal to the mid-surface remain normal to the mid-surface after deformation. The thickness of the plate does not change. 1.1. Laminate Constitutive Relations The relationship between forces , mid-plane strains , and curvatures is given by the ABD matrix:
Substituting into the governing equation yields: