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II.1.6.8 Ply out-of-plane shear stresses

II.1.6.8.1 With (μx,μy) approach

When the (μx,μy) approach is adopted, Y k lam(z) matrices allow the calculation of out-of-plane shear stress from the global out-of-plane shear force:

Y k lam(z) = Y 0k lam + Y 1k lamz + Y 2k lamz2,
τs(z) lam = Y lam(z) Qs lam.

Actually, one is interested in stresses in ply axes, rather than in laminate axes. If ξ is the orientation of the ply in laminate axes, then:

τs(z) ply = S+ (ξ) τs(z) lam.

Then, one has simply:

τs(z) ply = S+ (ξ) Y lam(z) Qs lam, = Y plylam(z) Qs lam.

One stores a 2 × 2 matrix Y plylam for each station through laminate thickness where out-of-plane shear stress might be requested. Actually it is done at top, mid and bottom surfaces in each ply. This means that 3N 2 × 2 matrices Y j plylam are stored in the ClaLam object.

II.1.6.8.2 With “resolution in shear force axes” approach

One first estimates the components of bending moments gradient in laminate axes with equation (II.1.49):

Mlam = Mxx,x Myy,x Mxy,x Mxx,y Myy,y Mxy,y lam = 1 Qxzlam Qxzlam + Qyzlam Qyzlam Qxzlam Q xzlam Q xzlam Qyzlam Q yzlam Q xzlam Qxzlam Q yzlam Q xzlam Qxzlam Q xzlam Q yzlam Qyzlam Q yzlam Q yzlam Qxzlam Q yzlam Q yzlam .

Then, the laminate out-of-plane shear stresses can be estimated by an expression like (II.1.43):

τxz(z) τyz(z) lam = V lam(z)Mlam.

In this expression, V lam(z) is given by (II.1.44):

V lam(z) = V 0k lam + V 1k lamz + V 2k lamz2.

The three 2 × 6 matrices V 0k lam, V 1k lam and V 2k lam are calculated recursively using the expressions (II.1.46), (II.1.45) and (II.1.45):

V 0k lam = V 0k-1 lam + V 1k-1 lam -J0k lam zk-1 + V 2k-1 lam -J1k lam 2 zk-12.
V 1k lam = J0k lam,
V 2k lam = J1k lam 2 .

To resolve the recursion for V 0k lam one need an estimate for the first ply. One uses (II.1.47):

V 01 lam = -V 11 lamz0 -V 21 lamz02.

Again, one is more interested in the out-of-plane shear stresses in ply axes than in laminate axes.

τs(z) ply = S+ (ξ) V lam(z) Mlam, = V plylam(z) Mlam.

One stores a 2 × 6 matrix V plylam for each station through laminate thickness where out-of-plane shear stress might be requested. Actually it is done at top, mid and bottom surfaces in each ply. This means that 3N 2 × 6 matrices V j plylam are stored in the ClaLam object.