Unlike isotropic materials (like steel or aluminum), composite laminates have directional properties that vary based on fiber orientation and stacking sequence. Analyzing the bending of these plates requires calculating the ABD matrices (stiffness) and solving for curvatures and stresses.
We will focus on Classical Lamination Theory (CLT), which is the standard engineering approach for thin to moderately thick plates.
For a laminate without in-plane forces (( N_x = N_y = N_xy = 0 )), the equilibrium equation for transverse load ( q(x,y) ) is:
[ D_11 \frac\partial^4 w\partial x^4 + 4 D_16 \frac\partial^4 w\partial x^3 \partial y + 2(D_12 + 2 D_66) \frac\partial^4 w\partial x^2 \partial y^2 + 4 D_26 \frac\partial^4 w\partial x \partial y^3 + D_22 \frac\partial^4 w\partial y^4 = q(x,y) ] Composite Plate Bending Analysis With Matlab Code
For symmetric laminates (( B_ij = 0 )), only bending occurs. For unsymmetric laminates, solving directly is difficult; instead we solve the coupled equations:
[ A_11 \frac\partial u_0\partial x + \dots - B_11 \frac\partial^2 w\partial x^2 - \dots = 0 ] [ D_11 \frac\partial^4 w\partial x^4 + \dots - B_11 \frac\partial^3 u_0\partial x^3 - \dots = q ]
But in practice, we use the reduced bending stiffness approach or solve the system numerically. For a plate subjected to moments ($M$) and
For simplicity, we first analyze symmetric cross-ply or quasi-isotropic plates where ( B_ij = 0 ) and ( D_16 = D_26 = 0 ). Then the equation simplifies to:
[ D_11 \frac\partial^4 w\partial x^4 + 2(D_12 + 2D_66) \frac\partial^4 w\partial x^2 \partial y^2 + D_22 \frac\partial^4 w\partial y^4 = q(x,y) ]
For a plate subjected to moments ($M$) and forces ($N$): $$ \beginBmatrix N \ M \endBmatrix = \beginbmatrix A & B \ B & D \endbmatrix \beginBmatrix \epsilon^0 \ \kappa \endBmatrix $$ Where $\epsilon^0$ are mid-plane strains and $\kappa$ are curvatures. You can extend the code to:
You can extend the code to:
u(x,y,z) = u0(x,y) - z * ∂w/∂x
v(x,y,z) = v0(x,y) - z * ∂w/∂y
w(x,y,z) = w0(x,y)
Since we consider pure bending (no in-plane forces), u0 = v0 = 0. The strains become:
ε = -z * κ, where κ = ∂²w/∂x² , ∂²w/∂y² , 2∂²w/∂x∂y ^T
If you run the code with the provided [0/90/0] stack: