Integration of out-of-plane shear stress equation

II.1.6.4 Integration of out-of-plane shear stress equation

The out-of-plane shear stress components are obtained by integration of expression (II.1.42) along the thickness. This leads to:

{\begin{matrix} τ_{x z} (z) \\ τ_{y z} (z) \end{matrix}}_{lam} = (\int_{z_{0}}^{z} {[J]}_{lam} (z) d z) \cdot {\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} + {\begin{matrix} τ_{x z} (z_{0}) \\ τ_{y z} (z_{0}) \end{matrix}}_{lam} .

Zero shear stress along the bottom surface of the laminate is assumed. This corresponds to a free surface, or at least to a surface that receives no contact forces in direction $x$ and $y$ . This assumption leads to the following expression:

\begin{align} {\begin{matrix} τ_{x z} (z) \\ τ_{y z} (z) \end{matrix}}_{lam} & = (\int_{z_{0}}^{z} {[J]}_{lam} (z) d z) \cdot {\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}, \\ = {[V]}_{lam} (z) \cdot {\nabla M}_{lam}, & (II.1.43) \end{align}

in which a new matrix notation is introduced:

{[V]}_{lam} (z) = \int_{z_{0}}^{z} {[J]}_{lam} (z) d z,

and a new vector notation for the gradient of bending moments:

{\nabla M}_{lam} = {\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} .

The new matrix ${[V]}_{lam} (z)$ is of course a $2 \times 6$ matrix.

An explicit expression of the integrated matrix is calculated ply-by-ply, from bottom layer to top layer. If $z_{k - 1} \leq z \leq z_{k}$ :

\begin{align} {[V]}_{lam} (z) & = {[V^{k}]}_{lam} (z), \\ = \int_{z_{0}}^{z} {[J^{k}]}_{lam} (z) d z, \\ = {[V^{k}]}_{lam} (z_{k - 1}) + \int_{z_{k - 1}}^{z} {[J^{k}]}_{lam} (z) d z, \\ = {[V^{k}]}_{lam} (z_{k - 1}) + \int_{z_{k - 1}}^{z} {[J_{0}^{k}]}_{lam} + z {[J_{1}^{k}]}_{lam} d z, \\ = {[V^{k}]}_{lam} (z_{k - 1}) + {[J_{0}^{k}]}_{lam} (z - z_{k - 1}) + {[J_{1}^{k}]}_{lam} \frac{z^{2} - z_{k - 1}^{2}}{2}, \\ = {[V_{0}^{k}]}_{lam} + {[V_{1}^{k}]}_{lam} z + {[V_{2}^{k}]}_{lam} z^{2} . & (II.1.44) \end{align}

In expression (II.1.44), three new matrices are introduced, calculated as follows:

{[V_{0}^{k}]}_{lam} = {[V^{k}]}_{lam} (z_{k - 1}) - {[J_{0}^{k}]}_{lam} z_{k - 1} - {[J_{1}^{k}]}_{lam} \frac{z_{k - 1}^{2}}{2},

{[V_{1}^{k}]}_{lam} = {[J_{0}^{k}]}_{lam},

(II.1.45)

{[V_{2}^{k}]}_{lam} = \frac{{[J_{1}^{k}]}_{lam}}{2} .

Note that the expression above involve the a priori unknown quantity ${[V^{k}]}_{lam} (z_{k - 1})$ . To calculate this expression, the continuity of ${[V^{k}]}_{lam} (z)$ across ply interfaces is used:

{[V^{k}]}_{lam} (z_{k - 1}) = {[V^{k - 1}]}_{lam} (z_{k - 1}) .

This relation corresponds to the continuity of out-of-plane shear stress at each interface between two consecutive plies, and is developed as follows.

\begin{align} {[V_{0}^{k}]}_{lam} & = {[V^{k - 1}]}_{lam} (z_{k - 1}) - {[J_{0}^{k}]}_{lam} z_{k - 1} - {[J_{1}^{k}]}_{lam} \frac{z_{k - 1}^{2}}{2}, \\ = {[V_{0}^{k - 1}]}_{lam} + {[V_{1}^{k - 1}]}_{lam} z_{k - 1} + {[V_{2}^{k - 1}]}_{lam} z_{k - 1}^{2} - {[J_{0}^{k}]}_{lam} z_{k - 1} - {[J_{1}^{k}]}_{lam} \frac{z_{k - 1}^{2}}{2}, \\ = {[V_{0}^{k - 1}]}_{lam} + ({[V_{1}^{k - 1}]}_{lam} - {[J_{0}^{k}]}_{lam}) z_{k - 1} + ({[V_{2}^{k - 1}]}_{lam} - \frac{{[J_{1}^{k}]}_{lam}}{2}) z_{k - 1}^{2} . & (II.1.46) \end{align}

The last line of this development enables recursive calculation of the components of ${[V_{0}^{k}]}_{lam}$ from the bottom ply to the top ply. For bottom ply, the condition $τ_{x z} (z_{0}) = τ_{y z} (z_{0}) = 0$ leads to the following expressions:

{[V^{1}]}_{lam} (z_{0}) = {[V_{0}^{1}]}_{lam} + {[V_{1}^{1}]}_{lam} z_{0} + {[V_{2}^{1}]}_{lam} z_{0}^{2} = 0 .

\begin{align} {[V_{0}^{1}]}_{lam} & = - {[V_{1}^{1}]}_{lam} z_{0} - {[V_{2}^{1}]}_{lam} z_{0}^{2}, \\ = - {[J_{0}^{1}]}_{lam} z_{0} - \frac{{[J_{1}^{1}]}_{lam}}{2} z_{0}^{2} . & (II.1.47) \end{align}

Then, it becomes possible to calculate recursively the

{[V_{0}^{i}]}_{lam}

matrices.

It is easy to verify that the condition $τ_{x z} (z_{0}) = τ_{y z} (z_{0}) = 0$ also ensures that $τ_{x z} (z_{N}) = τ_{y z} (z_{N}) = 0$ . Indeed:

\begin{align} {[V]}_{lam} (z_{N}) & = \int_{z_{0}}^{z_{N}} {[J]}_{lam} (z) d z, \\ = - [P_{1}] \cdot \int_{z_{0}}^{z_{N}} [\begin{matrix} {[F]}_{lam} (z) & \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} \\ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} & {[F]}_{lam} (z) \end{matrix}] d z, \\ = - [P_{1}] \cdot [\begin{matrix} \int_{z_{0}}^{z_{N}} {[F]}_{lam} (z) d z & \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} \\ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} & \int_{z_{0}}^{z_{N}} {[F]}_{lam} (z) d z \end{matrix}] . \end{align}

The last line of previous equation contains twice the integral of ${[F]}_{lam} (z)$ along the laminate thickness. This integral is expanded as follows:

\begin{align} \int_{z_{0}}^{z_{N}} {[F]}_{lam} (z) d z & = \int_{z_{0}}^{z_{N}} {[C]}_{lam} (z) \cdot ({[b]}_{lam} + z {[d]}_{lam}) d z, \\ = (\int_{z_{0}}^{z_{N}} {[C]}_{lam} (z) d z) \cdot {[b]}_{lam} + (\int_{z_{0}}^{z_{N}} z {[C]}_{lam} (z) d z) \cdot {[d]}_{lam}, \\ = {[A]}_{lam} \cdot {[b]}_{lam} + {[B]}_{lam} \cdot {[d]}_{lam} . \end{align}

On the other hand, equation (II.1.33) allows one to write:

\begin{align} [\begin{matrix} [A] & [B] \\ [B] & [D] \end{matrix}] \cdot [\begin{matrix} [a] & [b] \\ [b] T & [d] \end{matrix}] & = [\begin{matrix} [A] \cdot [a] + [B] \cdot [b] T & [A] \cdot [b] + [B] \cdot [d] \\ [B] \cdot [a] + [D] \cdot [b] T & [B] \cdot [b] + [D] \cdot [d] \end{matrix}], \\ = [\begin{matrix} [I] & [0] \\ [0] & [I] \end{matrix}] . \end{align}

(The “lam” subscript has been omitted for conciseness.) The identification of the right upper corner of the last expression with the integration of ${[F]}_{lam} (z)$ along the laminate thickness shows that this integral must be zero. Consequently:

\begin{align} {[V]}_{lam} (z_{N}) & = 0, \\ τ_{x z} (z_{N}) & = 0, \\ τ_{y z} (z_{N}) & = 0 . \end{align}

It is worth noting that the ply out-of-plane shear moduli have not been used in the calculations to obtain (II.1.43). The out-of-plane shear stresses depend only on laminate in-plane bending moments and ply in-plane material properties. As shown in section II.1.6.6, the calculation of out-of-plane shear strains caused by out-of-plane shear forces requires the knowledge of ply out-of-plane material constants.

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