User talk:Egm6936.s09/Test Word macros

=Classical Laminate Theory=

The strain-displacement (geometry) relations remain those of equations (2.9), (2.13), the membrane forces, bending moments and twists and vertical shear forces remain those of equations (1.6), (1.7), (2.15). The P.D.E's equations (2.19), (2.21) involving the vertical and membrane forces, respectively, remain valid. The relationship between transverse shear forces and the moments or also accounting for the effect of membrane forces as expressed by equation (2.20) are valid irrespective of whether the response is isotropic or anisotropic. For general orthotropy: $$\displaystyle \left\{ \sigma \right\}=\left[ Q \right]\left\{ \varepsilon  \right\}$$

Thus, the $$\displaystyle \sigma -\varepsilon $$ in the $$\displaystyle k$$ orthotropic ply or lamina of a stacked laminate is (assuming $$\displaystyle \sigma_{z}=\gamma_{xz}=\gamma_{yz}=0$$)
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$$ \displaystyle {{\left\{ \begin{align} & {{\sigma }_{x}} \\ & {{\sigma }_{y}} \\ & {{\tau }_{xy}} \\ \end{align} \right\}}_{k}}={{\left[ \begin{matrix} {{Q}_{xx}} & {{Q}_{xy}} & {{Q}_{xs}} \\ {{Q}_{yx}} & {{Q}_{yy}} & {{Q}_{ys}} \\ {{Q}_{sx}} & {{Q}_{sy}} & {{Q}_{ss}} \\ \end{matrix} \right]}_{k}}{{\left\{ \begin{align} & {{\varepsilon }_{x}} \\ & {{\varepsilon }_{y}} \\ & {{\gamma }_{xy}} \\ \end{align} \right\}}_{k}}$$ (N)
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Where $$\displaystyle x, y$$ are general orthotropic coordinates, $$\displaystyle {{\left[ {{Q}_{ij}} \right]}_{k}}$$ are the transformed reduced stiffness of the $$\displaystyle k$$ ply which can be related to $$\displaystyle E_{11}^{k},E_{22}^{k},\nu _{12}^{k}$$ and $$\displaystyle G_{12}^{k}$$ of this $$\displaystyle k$$ ply and the angle between the 1-direction (fiber direction) of this lamina and the $$\displaystyle x$$-axis Since the strains in the $$\displaystyle k$$ ply are given by equation (2.13):
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$$ \displaystyle {{\left[ \begin{matrix} {{\varepsilon }_{x}} \\ {{\varepsilon }_{x}} \\ {{\gamma }_{xy}} \\ \end{matrix} \right]}_{k}}=\left[ \begin{matrix} \varepsilon _{x}^{0} \\ \varepsilon _{y}^{0} \\ \gamma _{xy}^{0} \\ \end{matrix} \right]+z\left[ \begin{matrix} {{\kappa }_{x}} \\ {{\kappa }_{y}} \\ {{\kappa }_{xy}} \\ \end{matrix} \right]$$ (N)
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$$ \displaystyle {{\left\{ \begin{align} & {{\sigma }_{x}} \\ & {{\sigma }_{y}} \\ & {{\tau }_{xy}} \\ \end{align} \right\}}_{k}}={{\left[ \begin{matrix} {{Q}_{xx}} & {{Q}_{xy}} & {{Q}_{xs}} \\ {{Q}_{yx}} & {{Q}_{yy}} & {{Q}_{ys}} \\ {{Q}_{sx}} & {{Q}_{sy}} & {{Q}_{ss}} \\ \end{matrix} \right]}_{k}}\left( \left[ \begin{matrix}  \varepsilon _{x}^{0}  \\   \varepsilon _{y}^{0}  \\   \gamma _{xy}^{0}  \\ \end{matrix} \right]+z\left[ \begin{matrix}   {{\kappa }_{x}}  \\   {{\kappa }_{y}}  \\   {{\kappa }_{xy}}  \\ \end{matrix} \right] \right)$$ (N)
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$$ \displaystyle \left\{ \begin{align} & {{N}_{x}} \\ & {{N}_{y}} \\ & {{N}_{xy}} \\ \end{align} \right\}=\int\limits_{\begin{smallmatrix} -h/2 \\ \left( z=-{{z}_{0}} \right) \end{smallmatrix}}^{\begin{smallmatrix} \left( z={{z}_{N}} \right) \\ h/2 \end{smallmatrix}}{\left\{ \begin{align} & {{\sigma }_{x}} \\ & {{\sigma }_{y}} \\ & {{\tau }_{xy}} \\ \end{align} \right\}}dz=\sum\limits_{k=1}^{N}{{{\int\limits_^{\left\{ \begin{align} & {{\sigma }_{x}} \\ & {{\sigma }_{y}} \\ & {{\tau }_{xy}} \\ \end{align} \right\}}}_{k}}dz}=\sum\limits_{k=1}^{N}\left( \int\limits_^{\left[ \begin{matrix}  \varepsilon _{x}^{0}  \\   \varepsilon _{y}^{0}  \\   \gamma _{xy}^{0}  \\ \end{matrix} \right]}dz+\int\limits_^{\left[ \begin{matrix}   {{\kappa }_{x}}  \\   {{\kappa }_{y}}  \\   {{\kappa }_{xy}}  \\ \end{matrix} \right]}zdz \right)$$ (N)
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$$ \displaystyle \left\{ \begin{align} & {{N}_{x}} \\ & {{N}_{y}} \\ & {{N}_{xy}} \\ \end{align} \right\}=\left[ \begin{matrix} {{A}_{xx}} & {{A}_{xy}} & {{A}_{xs}} \\ {{A}_{yx}} & {{A}_{yy}} & {{A}_{ys}} \\ {{A}_{sx}} & {{A}_{sy}} & {{A}_{ss}} \\ \end{matrix} \right]\left\{ \begin{align} & \varepsilon _{x}^{0} \\ & \varepsilon _{y}^{0} \\ & \gamma _{xy}^{0} \\ \end{align} \right\}+\left[ \begin{matrix} {{B}_{xx}} & {{B}_{xy}} & {{B}_{xs}} \\ {{B}_{yx}} & {{B}_{yy}} & {{B}_{ys}} \\ {{B}_{sx}} & {{B}_{sy}} & {{B}_{ss}} \\ \end{matrix} \right]\left\{ \begin{align} & {{\kappa }_{x}} \\ & {{\kappa }_{y}} \\ & {{\kappa }_{xy}} \\ \end{align} \right\}$$ (N)
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Where
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$$ \displaystyle {{A}_{ij}}=\sum\limits_{k=1}^{N}\int\limits_^{dz}=\sum\limits_{k=1}^{N}\underbrace{\left( {{z}_{k}}-{{z}_{k-1}} \displaystyle \right)}_=\sum\limits_{k=1}^{N}{{t}_{k}}$$ (N)
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Also
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$$ \displaystyle \left\{ \begin{align} & {{M}_{x}} \\ & {{M}_{y}} \\ & {{M}_{xy}} \\ \end{align} \right\}=\int\limits_{\begin{smallmatrix} -h/2 \\ \left( z=-{{z}_{0}} \right) \end{smallmatrix}}^{\begin{smallmatrix} \left( z={{z}_{N}} \right) \\ h/2 \end{smallmatrix}}{\left\{ \begin{align} & {{\sigma }_{x}} \\ & {{\sigma }_{y}} \\ & {{\tau }_{xy}} \\ \end{align} \right\}z}dz=\sum\limits_{k=1}^{N}{{{\int\limits_^{\left\{ \begin{align} & {{\sigma }_{x}} \\ & {{\sigma }_{y}} \\ & {{\tau }_{xy}} \\ \end{align} \right\}}}_{k}}zdz}=\sum\limits_{k=1}^{N}\left( \int\limits_^{\left[ \begin{matrix}  \varepsilon _{x}^{0}  \\   \varepsilon _{y}^{0}  \\   \gamma _{xy}^{0}  \\ \end{matrix} \right]z}dz+\int\limits_^{\left[ \begin{matrix}   {{\kappa }_{x}}  \\   {{\kappa }_{y}}  \\   {{\kappa }_{xy}}  \\ \end{matrix} \right]}{{z}^{2}}dz \right)$$ (N)
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$$ \displaystyle \left\{ \begin{align} & {{M}_{x}} \\ & {{M}_{y}} \\ & {{M}_{xy}} \\ \end{align} \right\}=\left[ \begin{matrix} {{B}_{xx}} & {{B}_{xy}} & {{B}_{xs}} \\ {{B}_{yx}} & {{B}_{yy}} & {{B}_{ys}} \\ {{B}_{sx}} & {{B}_{sy}} & {{B}_{ss}} \\ \end{matrix} \right]\left\{ \begin{align} & \varepsilon _{x}^{0} \\ & \varepsilon _{y}^{0} \\ & \gamma _{xy}^{0} \\ \end{align} \right\}+\left[ \begin{matrix} {{D}_{xx}} & {{D}_{xy}} & {{D}_{xs}} \\ {{D}_{yx}} & {{D}_{yy}} & {{D}_{ys}} \\ {{D}_{sx}} & {{D}_{sy}} & {{D}_{ss}} \\ \end{matrix} \right]\left\{ \begin{align} & {{\kappa }_{x}} \\ & {{\kappa }_{y}} \\ & {{\kappa }_{xy}} \\ \end{align} \right\}$$ (N)
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$$ \displaystyle \begin{align} & {{B}_{ij}}=\sum\limits_{k=1}^{N}\int\limits_^{zdz}=\frac{1}{2}\sum\limits_{k=1}^{N}{{{\left[ {{Q}_{ij}} \right]}_{k}}\left( z_{k}^{2}-z_{k-1}^{2} \right)} \\ & {{D}_{ij}}=\sum\limits_{k=1}^{N}\int\limits_^{{{z}^{2}}dz}=\frac{1}{3}\sum\limits_{k=1}^{N}{{{\left[ {{Q}_{ij}} \right]}_{k}}\left( z_{k}^{3}-z_{k-1}^{3} \right)} \\ \end{align}$$ (N)
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Note that unlike the isotropic case, both the in-plane forces and moments depend on membrane displacements $$\displaystyle u_{0},v_{0}$$ and transverse flexure $$\displaystyle w=w_{0}$$, i.e. there is coupling between in-plane and out-of-plane activity. The coupling vanishes if and only if $$\displaystyle B_{ij}=0$$

By determining $$\displaystyle \tau_{xz} $$ and $$\displaystyle \tau_{yz}$$ from the equilibrium equations:


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$$ \displaystyle \begin{align} & \frac{\partial {{\sigma }_{x}}}{\partial x}+\frac{\partial {{\tau }_{yx}}}{\partial y}+\frac{\partial {{\tau }_{xz}}}{\partial z}=0 \\ & \frac{\partial {{\tau }_{xy}}}{\partial x}+\frac{\partial {{\sigma }_{y}}}{\partial y}+\frac{\partial {{\tau }_{yz}}}{\partial z}=0 \\ \end{align}$$ (N)
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and boundary conditions $$\displaystyle \tau_{xz} = \tau_{yz} = 0 $$ at $$\displaystyle z=\pm {}^{t}\!\!\diagup\!\!{}_{2}\;$$ or from (for small deflections)
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$$ \displaystyle \begin{align} & {{Q}_{x}}=\frac{\partial {{M}_{x}}}{\partial x}+\frac{\partial {{M}_{xy}}}{\partial y} \\ & {{Q}_{y}}=\frac{\partial {{M}_{xy}}}{\partial x}+\frac{\partial {{M}_{y}}}{\partial y} \\ \end{align}$$ (N)
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One obtains:
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$$ \displaystyle \begin{align} & {{Q}_{x}}=\int\limits_{-h/2}^{h/2}{{{\tau }_{xz}}dz=\left[ {{B}_{11}}\frac{\partial {{x}^{2}}}+2{{B}_{16}}\frac{\partial x\partial y}+{{B}_{66}}\frac{\partial {{y}^{2}}}+{{B}_{16}}\frac{\partial {{x}^{2}}}+\left( {{B}_{12}}+{{B}_{66}} \right)\frac{\partial x\partial y}+{{B}_{26}}\frac{\partial {{x}^{2}}} \right]} \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-\left[ {{D}_{11}}\frac{{{\partial }^{3}}w}{\partial {{x}^{3}}}+3{{D}_{16}}\frac{{{\partial }^{3}}w}{{{\partial }^{2}}x\partial y}+\left( {{D}_{12}}+2{{D}_{66}} \right)\frac{{{\partial }^{3}}w}{\partial x\partial {{y}^{2}}}+{{D}_{26}}\frac{{{\partial }^{3}}w}{\partial {{y}^{3}}} \right] \\ & {{Q}_{y}}=\int\limits_{-h/2}^{h/2}{{{\tau }_{yz}}dz=\left[ {{B}_{16}}\frac{\partial {{x}^{2}}}+\left( {{B}_{12}}+{{B}_{66}} \right)\frac{\partial x\partial y}+{{B}_{26}}\frac{\partial {{y}^{2}}}+{{B}_{66}}\frac{\partial {{x}^{2}}}+2{{B}_{26}}\frac{\partial x\partial y}+{{B}_{22}}\frac{\partial {{x}^{2}}} \right]} \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-\left[ {{D}_{16}}\frac{{{\partial }^{3}}w}{\partial {{x}^{3}}}+\left( {{D}_{12}}+2{{D}_{66}} \right)\frac{{{\partial }^{3}}w}{{{\partial }^{2}}x\partial y}+3{{D}_{26}}\frac{{{\partial }^{3}}w}{\partial x\partial {{y}^{2}}}+{{D}_{22}}\frac{{{\partial }^{3}}w}{\partial {{y}^{3}}} \right] \\ \end{align}$$ (N)
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Expressions for membrane forces and moments can be written in a matrix form as follows:


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$$ \displaystyle \left[ \begin{align} & N \\ & M \\ \end{align} \right]=\left[ \begin{matrix} A & B \\ B & D \\ \end{matrix} \right]\left[ \begin{align} & {{\varepsilon }^{0}} \\ & \kappa \\ \end{align} \right]$$ (N)
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