The strains at a point in a generally orthotropic ply are εx = 0.005, εy = 0.002, andγxy = 0.0002 referred to the xy reference axis system. If the ply angle is 45°, calculate the strains in the directions of the material axes. Answer: el = 3.6 × 10-3, et = 3.4 × 10-3, glt = -3.0 × 10-3.

Question

The strains at a point in a generally orthotropic ply are [latex]\varepsilon_{x}=0.005,\,\varepsilon_{y}=0.002,\,\mathrm{and}\gamma_{x y}=0.0002[/latex] referred to the xy reference axis system. If the ply angle is 45°, calculate the strains in the directions of the material axes.

Accepted Answer

As in S.26.4, m²=0.5, n²=0.5, and mn=0.5. Eq. (26.25) then becomes

[latex]\left\{\begin{matrix} \varepsilon _{l} \ \varepsilon _{t} \ \gamma _{lt} \end{matrix} ight\} =\left[\begin{matrix} m^{2} & n^{2} & mn \ n^{2} & m^{2} & -mn \ -2mn & 2mn & m^{2}-n^{2} \end{matrix} ight] \left\{\begin{matrix} \varepsilon _{x} \ \varepsilon _{y} \ \gamma _{xy} \end{matrix} ight\} [/latex] (26.25)

[latex]\left\{\begin{matrix} \varepsilon _{l} \ \varepsilon _{t} \ \gamma _{lt} \end{matrix} ight\} =\left[\begin{matrix} 0.5 & 0.5 & 0.5 \ 0.5 & 0.5 & -0.5 \ -1 & 1 & 0 \end{matrix} ight] \left\{\begin{matrix} 0.005 \ 0.002 \ 0.0002 \end{matrix} ight\} [/latex]

Then

[latex]\begin{array}{l}{{\varepsilon_{l}=0.5 imes0.005+0.5 imes0.002+0.5 imes0.0002=3.6 imes10^{-3}}}\ {{\varepsilon_{t}=0.5 imes0.005+0.5 imes0.002-0.5 imes 0.0002=3.4 imes10^{-3}}}\ {{\gamma_{\mathrm{lt}}=-1 imes0.005+1 imes0.002+0=-3.0 imes10^{-3}}}\end{array}[/latex]

Question 26.5: The strains at a point in a generally orthotropic ply are εx......

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