Calculation of plies stresses and strains

II.1.9.2 Calculation of plies stresses and strains

The calculation of ply stresses and strains from the laminate loads is easily accelerated. Indeed, let us consider a laminate loading corresponding to:

Laminate in-plane average strain and curvature tensors ${\{ϵ^{0}\}}_{lam}$ and ${\{κ\}}_{lam}$ (in laminate axes),
Laminate bending gradient ${\{\nabla M\}}_{lam}$ (in laminate axes),
Laminate temperature loading characterized by the two real values $(T_{0} - T_{ref})$ and $T_{, z}$ ,
Laminate moisture loading characterized by the two real values $(H_{0} - H_{ref})$ and $H_{, z}$ .

These quantities correspond to 16 real values that characterize entirely the laminate loading. For a given ply, the stresses can be calculated at a given height ( $z$ value) from these 16 real values. All the calculations are linear.

If one defines a vector with 16 components as follows:

{\{L L d\}}_{lam} = \{\begin{matrix} {\{ϵ^{0}\}}_{lam} \\ {\{κ\}}_{lam} \\ {\{\nabla M\}}_{lam} \\ T_{0} - T_{ref} \\ T_{, z} \\ H_{0} - H_{ref} \\ H_{, z} \end{matrix}\} .

That contains all the laminate loading, there must be a matrix ${[K_{σ}]}_{ply} (z_{i})$ that allows to calculate ply stresses as follows:

{\{σ\}}_{ply} (z_{i}) = {[K_{σ}]}_{ply} (z_{i}) \cdot {\{L L d\}}_{lam} .

Matrix ${[K_{σ}]}_{ply} (z_{i})$ is a $6 \times 16$ matrix that depends only on laminate definition. This means that this matrix can be calculated once and for all when laminate is created in the database.

Similarly one can also define matrices ${[K_{ϵ}]}_{ply} (z_{i})$ and ${[K_{ϵ}^{M e c h}]}_{ply} (z_{i})$ for the calculations of ${\{ϵ\}}_{ply} (z_{i})$ and ${\{ϵ^{Mech}\}}_{ply} (z_{i})$ respectively. We explain here, how these three matrices can be constructed.

II.1.9.2.1 Loading in ply axes

The first step of ply stresses or strain calculations consists in expressing the laminate loading in ply axes. The following operations are performed:

{\{ϵ^{0}\}}_{ply} = {[T_{+}^{'}]}_{(ξ)} \cdot {\{ϵ^{0}\}}_{lam},

{\{κ\}}_{ply} = {[T_{+}^{'}]}_{(ξ)} \cdot {\{κ\}}_{lam},

{\{τ_{s}^{j} (z)\}}_{ply} = {[V^{j} (z)]}_{ply}^{lam} \cdot {\{\nabla M\}}_{lam} .

The four real values corresponding to laminate temperature and moisture loading are not affected by the modification of coordinate system. The three relations above allow us to write the following equation:

\{\begin{matrix} {\{ϵ^{0}\}}_{ply} \\ {\{κ\}}_{ply} \\ {\{τ_{s}^{j} (z)\}}_{ply} \\ T_{0} - T_{ref} \\ T_{, z} \\ H_{0} - H_{ref} \\ H_{, z} \end{matrix}\} = [\begin{matrix} {[T_{+}^{'}]}_{(ξ)} & [0] & [0] & 0 & 0 & 0 & 0 \\ [0] & {[T_{+}^{'}]}_{(ξ)} & [0] & 0 & 0 & 0 & 0 \\ [0] & [0] & {[V^{j} (z)]}_{ply}^{lam} & 0 & 0 & 0 & 0 \\ [0] & [0] & [0] & 1 & 0 & 0 & 0 \\ [0] & [0] & [0] & 0 & 1 & 0 & 0 \\ [0] & [0] & [0] & 0 & 0 & 1 & 0 \\ [0] & [0] & [0] & 0 & 0 & 0 & 1 \end{matrix}] \cdot \{\begin{matrix} {\{ϵ^{0}\}}_{lam} \\ {\{κ\}}_{lam} \\ {\{\nabla M\}}_{lam} \\ T_{0} - T_{ref} \\ T_{, z} \\ H_{0} - H_{ref} \\ H_{, z} \end{matrix}\} .

(II.1.103)

Note that the out-of-plane shear response is now expressed as out-of-plane shear stresses at the specified height in selected ply and no longer as laminate out-of-plane shear forces.

The strains, temperatures and moisture at height $z$ in selected ply is easily obtained with the following expressions:

{\{ϵ\}}_{ply} (z) = {\{ϵ^{0}\}}_{ply} + z {\{κ\}}_{ply},

T (z) - T_{ref} = T_{0} - T_{ref} + z T_{, z},

H (z) - H_{ref} = H_{0} - H_{ref} + z H_{, z} .

The combination of these three expressions in a single matricial expression gives:

\{\begin{matrix} {\{ϵ\}}_{ply} (z) \\ {\{τ_{s}^{j} (z)\}}_{ply} \\ T (z) - T_{ref} \\ H (z) - H_{ref} \end{matrix}\} = [\begin{matrix} [δ_{3}] & [δ_{3}] z & [0] & 0 & 0 & 0 & 0 \\ [0] & [0] & [δ_{2}] & 0 & 0 & 0 & 0 \\ [0] & [0] & [0] & 1 & z & 0 & 0 \\ [0] & [0] & [0] & 0 & 0 & 1 & z \end{matrix}] \cdot \{\begin{matrix} {\{ϵ^{0}\}}_{ply} \\ {\{κ\}}_{ply} \\ {\{τ_{s}^{j} (z)\}}_{ply} \\ T_{0} - T_{ref} \\ T_{, z} \\ H_{0} - H_{ref} \\ H_{, z} \end{matrix}\} .

(II.1.104)

( $[δ_{3}]$ and $[δ_{2}]$ are $3 \times 3$ and $2 \times 2$ unit matrices respectively.) Then, considering equation (92), one writes:

\{\begin{matrix} {\{σ\}}_{ply} (z) \\ {\{τ_{s}^{j} (z)\}}_{ply} \\ T (z) - T_{ref} \\ H (z) - H_{ref} \end{matrix}\} = [\begin{matrix} {[C]}_{ply} & [0] & - {[C]}_{ply} \cdot {\{α\}}_{ply} & - {[C]}_{ply} \cdot {\{β\}}_{ply} \\ [0] & [δ_{2}] & {0} & {0} \\ [0] & [0] & 1 & 0 \\ [0] & [0] & 0 & 1 \end{matrix}] \cdot \{\begin{matrix} {\{ϵ\}}_{ply} (z) \\ {\{τ_{s}^{j} (z)\}}_{ply} \\ T (z) - T_{ref} \\ H (z) - H_{ref} \end{matrix}\} .

(II.1.105)

The vector at left hand side of expression (II.1.105) contains all the ply stress components. One can remove the two lower lines of the equation as follows:

\{\begin{matrix} {\{σ\}}_{ply} (z) \\ {\{τ_{s}^{j} (z)\}}_{ply} \end{matrix}\} = [\begin{matrix} {[C]}_{ply} & [0] & - {[C]}_{ply} \cdot {\{α\}}_{ply} & - {[C]}_{ply} \cdot {\{β\}}_{ply} \\ [0] & [δ_{2}] & {0} & {0} \end{matrix}] \cdot \{\begin{matrix} {\{ϵ\}}_{ply} (z) \\ {\{τ_{s}^{j} (z)\}}_{ply} \\ T (z) - T_{ref} \\ H (z) - H_{ref} \end{matrix}\} .

(II.1.106)

Then, the components can be reordered as follows:

{\{σ\}}_{mat} (z) = (\begin{matrix} σ_{11} \\ σ_{22} \\ σ_{33} \\ τ_{23} \\ τ_{13} \\ τ_{12} \end{matrix}) (z) = [\begin{matrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 \end{matrix}] \cdot \{\begin{matrix} {\{σ\}}_{ply} (z) \\ {\{τ_{s}^{j} (z)\}}_{ply} \end{matrix}\} .

(II.1.107)

One will also use:

(\begin{matrix} {\{σ\}}_{mat} (z) \\ T (z) - T_{ref} \\ H (z) - H_{ref} \end{matrix}) = (\begin{matrix} σ_{11} \\ σ_{22} \\ σ_{33} \\ τ_{23} \\ τ_{13} \\ τ_{12} \\ T - T_{ref} \\ H - H_{ref} \end{matrix}) (z) = [\begin{matrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}] \cdot \{\begin{matrix} {\{ϵ\}}_{ply} (z) \\ {\{τ_{s}^{j} (z)\}}_{ply} \\ T (z) - T_{ref} \\ H (z) - H_{ref} \end{matrix}\} .

(II.1.108)

One uses (12) to estimate the strain tensor:

{\{ϵ\}}_{mat} (z) = (\begin{matrix} ϵ_{11} \\ ϵ_{22} \\ ϵ_{33} \\ γ_{23} \\ γ_{31} \\ γ_{12} \end{matrix}) (z) = (\begin{matrix} [c_{i j k l}] & {\{α\}}_{mat} & {\{β\}}_{mat} \end{matrix}) \cdot (\begin{matrix} {\{σ\}}_{mat} (z) \\ T (z) - T_{ref} \\ H (z) - H_{ref} \end{matrix}) .

(II.1.109)

The vector that appears in right-hand-side of the previous expression The so-called “mechanical strain tensor” is given by:

{\{ϵ\}}_{mat}^{Mech} (z) = (\begin{matrix} ϵ_{11}^{Mech} \\ ϵ_{22}^{Mech} \\ ϵ_{33}^{Mech} \\ γ_{23}^{Mech} \\ γ_{31}^{Mech} \\ γ_{12}^{Mech} \end{matrix}) (z) = [c_{i j k l}] \cdot {\{σ\}}_{mat} (z) .

(II.1.110)

All the operations (II.1.103) to (II.1.110) reduce to matricial products. The characteristics of the matrices used in these operations are summarized in Table II.1.1. Two of these matrices are “re-ordering” metrices, and do not depend on the ply material or position accross laminate thickness. The other matrices depend on the ply material. Only two of the matrices depends on height $z$ .

Table II.1.1: Summary of the matrices that have been defined for the acceleration of ply stresses and strains calculation.


Matrix	Reference	Size	Dependence on


$[K_{1}]$	(II.1.103)	$12 \times 16$	Ply angle, material and height $z$

$[K_{2}]$	(II.1.104)	$7 \times 12$	Height $z$

$[K_{3}]$	(II.1.105)	$7 \times 7$	Ply material

$[K_{4}]$	(II.1.106)	$5 \times 7$	Ply material

$[K_{5}]$	(II.1.107)	$6 \times 5$	—

$[K_{6}]$	(II.1.108)	$8 \times 7$	—

$[K_{7}]$	(II.1.109)	$6 \times 8$	Ply material

$[K_{8}]$	(II.1.110)	$6 \times 6$	Ply material

In the end, one writes:

\begin{align} {\{σ\}}_{mat} (z) & = [K_{5}] \cdot [K_{4}] \cdot [K_{2}] (z) \cdot [K_{1}] (z) \cdot {\{L L d\}}_{lam}, \\ = [K_{σ}] (z) \cdot {\{L L d\}}_{lam} . & (II.1.111) \end{align}

\begin{align} {\{ϵ\}}_{mat} (z) & = [K_{7}] \cdot [K_{6}] \cdot [K_{3}] \cdot [K_{2}] (z) \cdot [K_{1}] (z) \cdot {\{L L d\}}_{lam}, \\ = [K_{ϵ}] (z) \cdot {\{L L d\}}_{lam} . & (II.1.112) \end{align}

\begin{align} {\{ϵ\}}_{mat}^{Mech} (z) & = [K_{8}] \cdot {\{σ\}}_{mat} (z), \\ = [K_{8}] \cdot [K_{σ}] (z) \cdot {\{L L d\}}_{lam}, \\ = [K_{ϵ}^{Mech}] (z) \cdot {\{L L d\}}_{lam} . & (II.1.113) \end{align}

We have shown that if one wishes to calculate stresses and strains in plies from the laminate loading ${\{L L d\}}_{lam}$ , at the three “bot” “mid” and “sup” heights of laminate plies, one needs to calculate $3 \times 3 = 9$ matrices per ply. Each of the matrices has $6 \times 16$ size and depends only on the laminate definition. This means that these matrices can be calculated once and for all when laminate is defined.

Note that the acceleration matrices for the ply stresses and strains calculation must be re-estimated each time the laminate or one of its materials is modified. This is the reason why method “reInitAllPliesAccelMatrices” has been added to the “ClaLam” and “ClaDb” classes.

Finally, as usual, the out-of-plane shear calculation approach will influence the results, because the gradient of bending tensor can be calculated different ways:

With

(μ_{x}, μ_{y})

approach, one calculates:

{\{\nabla M\}}_{lam} = [\begin{matrix} μ_{x} & 0 \\ 0 & 0 \\ 0 & (1 - μ_{x}) \\ 0 & 0 \\ 0 & μ_{y} \\ (1 - μ_{y}) & 0 \end{matrix}] \cdot {\{\begin{matrix} Q_{x z} \\ Q_{y z} \end{matrix}\}}_{lam} .

(II.1.114)

If the “resolution in shear force axes” approach is adopted, this vector is given by:

{\{\begin{matrix} M_{x x, x} \\ M_{y y, x} \\ M_{x y, x} \\ M_{x x, y} \\ M_{y y, y} \\ M_{x y, y} \end{matrix}\}}_{lam} = \frac{1}{Q_{x z}^{lam} \cdot Q_{x z}^{lam} + Q_{y z}^{lam} \cdot Q_{y z}^{lam}} \{\begin{matrix} Q_{x z}^{lam} \cdot Q_{x z}^{lam} \cdot Q_{x z}^{lam} \\ Q_{y z}^{lam} \cdot Q_{y z}^{lam} \cdot Q_{x z}^{lam} \\ Q_{x z}^{lam} \cdot Q_{y z}^{lam} \cdot Q_{x z}^{lam} \\ Q_{x z}^{lam} \cdot Q_{x z}^{lam} \cdot Q_{y z}^{lam} \\ Q_{y z}^{lam} \cdot Q_{y z}^{lam} \cdot Q_{y z}^{lam} \\ Q_{x z}^{lam} \cdot Q_{y z}^{lam} \cdot Q_{y z}^{lam} \end{matrix}\} .

The adoption of the first or second approach affects only the corresponding components of ${\{L L d\}}_{lam}$ vector. Matrices $[K_{σ}] (z)$ , $[K_{ϵ}] (z)$ and $[K_{ϵ}^{Mech}] (z)$ remain the same.

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