To analyze composite plates, engineers treat the material as a stack of distinct layers (laminae), each having unique fiber orientations. Classical Laminate Plate Theory (CLPT) extends standard Kirchhoff-Love plate theory to thin, laminated composite structures. Constitutive Equations
end
function [N, dN_dxi, detJ] = shape_functions(xy, xi, eta) % xy: nodal coordinates, 4x2 N = zeros(4,1); dN_dxi = zeros(2,4); % dN/dxi and dN/deta for i = 1:4 xi_i = [-1, 1, 1, -1]; eta_i = [-1, -1, 1, 1]; N(i) = 0.25 * (1 + xi_i(i)*xi) * (1 + eta_i(i)*eta); dN_dxi(1,i) = 0.25 * xi_i(i) * (1 + eta_i(i)*eta); dN_dxi(2,i) = 0.25 * eta_i(i) * (1 + xi_i(i)*xi); end % Jacobian J = zeros(2,2); for i = 1:4 J(1,1) = J(1,1) + dN_dxi(1,i) * xy(i,1); J(1,2) = J(1,2) + dN_dxi(1,i) * xy(i,2); J(2,1) = J(2,1) + dN_dxi(2,i) * xy(i,1); J(2,2) = J(2,2) + dN_dxi(2,i) * xy(i,2); end detJ = det(J); % Derivatives w.r.t. physical coordinates dN_dxi = J \ dN_dxi; end
% Display stiffness matrices disp('Extensional stiffness A (N/m):'); disp(A); disp('Coupling stiffness B (N):'); disp(B); disp('Bending stiffness D (N*m):'); disp(D);
, Navier's solution solves this equation using double Fourier series expansions. Navier's Analytical Solution Navier's method assumes the lateral deflection and the distributed load can be expressed as: Composite Plate Bending Analysis With Matlab Code
%% 6. Boundary Conditions (Simply supported: w=0 at edges, theta_tangential free) % Simply supported: w = 0 on all edges, but rotations free. % For simplicity, fix w on all boundary nodes boundary_tol = 1e-6; fixedDOFs = []; for i = 1:nNodes x = nodeCoords(i,1); y = nodeCoords(i,2); if abs(x) < boundary_tol || abs(x - a) < boundary_tol || ... abs(y) < boundary_tol || abs(y - b) < boundary_tol % Fix w (DOF 1) fixedDOFs = [fixedDOFs, (i-1)*ndof + 1]; end end freeDOFs = setdiff(1:nDofs, fixedDOFs);
For moderately thick plates (side‑to‑thickness ratio ( a/h < 20 )), transverse shear effects become important. The FSDT introduces two additional rotation variables ( \phi_x, \phi_y ) and requires a shear correction factor. The Navier solution for FSDT is similar but leads to a different set of algebraic equations. Extending the code to FSDT is a valuable next step.
While CLPT works for thin plates, it overestimates the stiffness of thick composite plates because it neglects transverse shear deformation. For thicker plates, , or Mindlin-Reissner plate theory, is necessary, which requires calculating the shear stiffness terms.
Composite materials, particularly fibre-reinforced polymer laminates, are widely used in aerospace, automotive, civil and marine engineering due to their high stiffness-to-weight and strength-to-weight ratios. A critical aspect of designing composite structures is the analysis of thin to moderately thick laminated plates under transverse loading. This article presents a complete, step‑by‑step approach to bending analysis of composite plates using Classical Lamination Theory (CLPT) and the Navier solution technique. A fully commented Matlab code is provided to compute deflections, strains and stresses for simply supported rectangular laminates subjected to arbitrary transverse loads expanded in double Fourier series. To analyze composite plates, engineers treat the material
% Pre-allocate A = zeros(3,3); B = zeros(3,3); D = zeros(3,3);
for different loading types (e.g., concentrated load). Add stress calculations ( ) at specific ply layers.
Different plate theories are used based on the thickness of the plate and the desired accuracy:
clear; clc; close all;
Composite Plate Bending Analysis With Matlab Code: A Comprehensive Guide
Change the theta array to represent your desired layup (e.g., [0 90 0 90] ). Update geometry: Change a, b, and ply_thickness .
D11𝜕4w𝜕x4+2(D12+2D66)𝜕4w𝜕x2𝜕y2+D22𝜕4w𝜕y4=q0cap D sub 11 partial to the fourth power w over partial x to the fourth power end-fraction plus 2 open paren cap D sub 12 plus 2 cap D sub 66 close paren the fraction with numerator partial to the fourth power w and denominator partial x squared partial y squared end-fraction plus cap D sub 22 partial to the fourth power w over partial y to the fourth power end-fraction equals q sub 0 Using Navier’s solution, the deflection at any point
[ A_s_ij = k_s \sum_k=1^N \int_z_k-1^z_k \barQ_ij^(k) , dz, \quad i,j=4,5. ] physical coordinates dN_dxi = J \ dN_dxi; end
For each element, the nodal displacement vector is: