Shell element, laminate and orthotropic section

Results.

Problem. In laminated and orthotropic shells the section stiffness matrices are given directly as input. In an isotropic shell these matrices are computed from the modulus of elasticity, Poisson's ratio and the thickness. The cases tested are an isotropic thick plate given through the laminate input, a specially orthotropic plate, a two-way CLT plate and an unsymmetric laminate. For the CLT plate the transverse shear force near the support is compared as well as the deflection.

Reference document. For the specially orthotropic plate the Navier series in Whitney's Structural Analysis of Laminated Anisotropic Plates is used. For the CLT plate the first-order shear deformation solution in Reddy's Mechanics of Laminated Composite Plates is used. The reference for the unsymmetric laminate is classical lamination theory.

The deflection of a specially orthotropic simply supported plate is computed with the Navier series.

Classical lamination theory relates the section forces and moments to strain and curvature through this matrix. The matrix $B$ carries the coupling between membrane action and bending and is not zero for an unsymmetric lay-up.

Numerical model. On all four edges of the plates the vertical translation and the rotation about the axis normal to the edge are held. In the unsymmetric laminate example the strip is supported at one end. The rotation is held at the support and the other end is left free.

Two-way CLT plate, L/t = 40#

Parameters. a = 4 · t = 0.1 · mesh = 24×24 · Ds₁₁/Ds₂₂ = 1.635

Published source. The Navier series of first-order shear deformation theory for a specially orthotropic laminate

Note. The remaining deviation comes from the mesh density and shrinks as the mesh is refined.

Two-way CLT plate, transverse shear force at the middle of a supported edge#

Parameters. a = 4 · mesh = 24×24 · x = 0.0833

Published source. The Navier series of first-order shear deformation theory (Mindlin)

Note. The transverse shear force is read at the middle of a supported edge, at the centre of the element next to the support ($x = a/48$, $y = a/2$). The reference value is computed at the same point. The shear force reaches its largest value at the middle of the supported edge, and this value is the direct input to the rolling shear check.

Two-way CLT plate, hand check of the layer stresses#

Parameters. element count = 256 · q = 10

Closed form. Hand calculation with the transformed section

Note. The moment and the shear force are read from the most heavily loaded elements of the model. The element next to the plate centre carries $M_{xx} = 15.23$ kNm/m and the element at the middle of a supported edge carries $Q_x = 16.81$ kN/m. The lay-up gives the bending stiffness $D_{11} = 765.4$ kNm²/m, the outer fibre at $z = 0.05$ m and, for the inner face of the upper cross layer, the weighted first moment $S = E_0 t \cdot 0.04 + E_{90} t \cdot 0.02 = 9348$. The stresses are in kPa. With Poisson's ratio zero, $\kappa_x = M_{xx}/D_{11}$ is exact and the hand calculation reproduces the engine's result exactly.

Isotropic thick plate given through the laminate input#

Parameters. t/a = 0.05 · ν = 0.2 · mesh = 24×24

Published source. The Navier series of first-order shear deformation theory (Mindlin)

Note. The section matrices are built from an isotropic material and given directly as input. Before it is used, the reference series is checked in two cases. With a very large shear stiffness it returns to the Kirchhoff series, and for an isotropic thick plate it gives the Mindlin correction (1.01133).

Specially orthotropic simply supported plate#

Parameters. a = 5 · t = 0.05 · E₁/E₂ = 30 · mesh = 16×16

Published source. Navier series, specially orthotropic plate (Whitney)

Note. Part of the remaining deviation comes from the mesh density and part from the reference solution leaving out shear deformation.

Unsymmetric laminate, axial tension#

Parameters. t₁ = 0.06 · t₂ = 0.04 · E₁/E₂ = 3

Published source. In classical lamination theory the strain and curvature are computed with the inverse of the stiffness matrix.

Note. The lower layer is thicker and stiffer, so the stiffness centre of the section lies $e = 10.9$ mm below the mid-plane. The tension applied at the mid-plane is eccentric to this centre and puts a bending moment $M = N\,e = 10.9$ kNm/m on the strip. The strip therefore bends downward as it is pulled. The membrane-bending coupling is therefore measured by the vertical displacement. In a symmetric laminate this value is zero. A hand check with the beam analogy gives $EI_c = 1412$ kNm²/m about the stiffness centre and $\kappa = M/EI_c = 7.73\cdot10^{-3}$ 1/m. At the tip of a cantilever held against rotation at the support the vertical displacement is $w = \kappa L^2/2 = 0.0618$ m. With Poisson's ratio taken as zero this result is the same as classical lamination theory.