function Q = orthotropic_Q(E1, E2, nu12, G12) nu21 = nu12 * E2 / E1; denom = 1 - nu12 nu21; Q11 = E1/denom; Q12 = nu12 E2/denom; Q22 = E2/denom; Q66 = G12; Q = [Q11, Q12, 0; Q12, Q22, 0; 0, 0, Q66]; end
A = A + Q_bar * (z_top - z_bot); B = B + 0.5 * Q_bar * (z_top^2 - z_bot^2); D = D + (1/3) * Q_bar * (z_top^3 - z_bot^3); As = As + Q_s_bar * (z_top - z_bot);
Substituting the series into the governing equation yields: Composite Plate Bending Analysis With Matlab Code
The coefficients ( Q_mn ) are determined from the type of loading. For a uniformly distributed load ( q_0 ): [ Q_mn = \frac16 q_0mn\pi^2 \quad (m,n \text odd); \quad Q_mn=0 \text otherwise. ] For a sinusoidal load ( q(x,y) = q_0 \sin(\pi x/a)\sin(\pi y/b) ), only the term ( m=n=1 ) is non‑zero with ( Q_11=q_0 ).
% Denominator for specially orthotropic plate (D16=D26=0) denom = pi^4 * ( D11*(m/a)^4 + 2*(D12+2*D66)*(m/a)^2*(n/b)^2 + D22*(n/b)^4 ); Wmn(m,n) = Qmn(m,n) / denom; end function Q = orthotropic_Q(E1, E2, nu12, G12) nu21
% Full displacement vector U = zeros(nDof,1); U(freeDOF) = Uf;
Aij=∑k=1n(Q̄ij)k(zk−zk−1)cap A sub i j end-sub equals sum from k equals 1 to n of open paren cap Q bar sub i j end-sub close paren sub k open paren z sub k minus z sub k minus 1 end-sub close paren Coupling Stiffness Matrix ( n) = Qmn(m
D11𝜕4w𝜕x4+2(D12+2D66)𝜕4w𝜕x2𝜕y2+D22𝜕4w𝜕y4=q(x,y)cap 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 open paren x comma y close paren For a simply supported rectangular plate of dimensions
The following MATLAB script automates the calculation of the ABDcap A cap B cap D