(14), [N.sub.x] and [N.sub.y] are the normal membrane forces in the x and y directions, respectively; [N.sub.xy] is the shear membrane force; [M.sub.x] and [M.sub.y] are the bending moments corresponding to the normal stresses in the x and y directions, respectively, [M.sub.xy] is the twisting moment; [Q.sub.x] and [Q.sub.y] are the transverse shear forces on planes normal to the x- and y-axes, respectively.
The resulting membrane forces in each case were exact.
As with Pro, the optimization uses all of the FEA computed shell element in-plane membrane forces
, out of plane transverse shear forces, and bending moments.
For any plate problem described by a fourth-order partial differential equation or coupled with two second-order equations in the presence of membrane forces
, the classical plate equation and two Poisson equations serve as substitute equations.
The constitutive relations between the membrane forces
[N.sub.x], [N.sub.y], and [N.sub.xy] and the strains [[epsilon].sub.x], [[epsilon].sub.y], and [[gamma].sub.xy] are
[W.sub.i]--volume; [V.sub.i]--volume fraction; [E.sub.i]--Young's modulus; [v.sub.i]--Poisson's ratio; [G.sub.i]--shear modulus; X-longitudinal strength; Y-transverse strength; S-shear strength; [[sigma].sub.n], [[tau].sub.n],--stress in direction; N-matrix of membrane forces
; M-matrix of bending moments; A-mid-plain strains to membrane forces
relation; B-coupling between bending and membrane load; D-curvatures to bending moments relation; [epsilon]--mid-plane; [kappa]--curvatures; [Q.sub.n]-stiffness in direction; i--index: m--matrix; f--fibres; n--index: 1, 2, 6--appropriate direction
The membrane forces
molten material forward, instead of the polymer forcing the water to the side.