A generally orthotropic ply is subjected to direct stresses of 60 N/mm^2 parallel to the x reference axis and 40 N/mm^2 perpendicular to the x reference axis. If the longitudinal plies are inclined at an angle of 45° to the x axis and the elastic constants are El=150,000 N/mm^2, Et=90,000 N/mm^2,

Question

A generally orthotropic ply is subjected to direct stresses of $60 N / mm ^{2}$ parallel to the x reference axis and $40 N / mm ^{2}$ perpendicular to the x reference axis. If the longitudinal plies are inclined at an angle of $45^{\circ}$ to the x axis and the elastic constants are $E_{l}=150,000 N / mm ^{2}, E_{t}=90,000 N / mm ^{2}, G_{l t}=5,000 N / mm ^{2}$ and $ν_{l t}=0.3$ , calculate the direct strains parallel to the x and y directions and the shear strain referred to the xy axes.

Use MATLAB to repeat Example 24.6.

Accepted Answer

We note that there is no applied shear stress, so it is unnecessary to calculate the terms in the third column of the matrix of Eqs. (24.33). Then,

[latex]\left\{\begin{array}{c}\varepsilon_{x}\\varepsilon_{y}\\gamma_{xy}\end{array} ight\}=\left[\begin{array}{ccc}m^{4}s_{11}+n^{4} s_{22} & m^{2} n^{2} s_{11}+m^{2}n^{2}s_{22}&2m^{3}ns_{11}-2mn^{3}s_{22}\+2m^{2}n^{2}s_{12}+m^{2}n^{2}s_{33}&+\left(m^{4}+n^{4} ight) s_{12} & +2\left(m n^{3}-m^{3} n ight) s_{12} \m^{2} n^{2} s_{11}+m^{2} n^{2} s_{22} & -m^{2} n^{2} s_{33} & +\left(m n^{3}-m^{3} n ight) s_{33} \+\left(m^{4}+n^{4} ight) s_{12} & n^{4} s_{11}+m^{4} s_{22} & 2 m n^{3} s_{11}-2 m^{3} n s_{22} \-m^{2} n^{2}s_{33}&+2m^{2}n^{2}s_{12}+m^{2}n^{2}s_{33}&+2\left(m^{3} n-m n^{3} ight) s_{12} \ & & +(m^3n-mn^3)s_{33} \2 m^{3} n s_{11}-2 m n^{3} s_{22} & 2 m n^{3} s_{11}-2 m^{3} n s_{22} & 4 m^{2} n^{2} s_{11}+4 m^{2} n^{2} s_{22} \+2\left(m n^{3}-m^{3} n ight)s_{12}&+2\left(m^{3} n-m n^{3} ight) s_{12} & -8 m^{2} n^{2} s_{12} \+\left(m n^{3}-m^{3} n ight) s_{33} & +\left(m^{3} n-n m^{3} ight) s_{33} &+\left(m^{2}-n^{2} ight)^{2}s_{33}\end{array} ight]\left\{\begin{array}{c}\sigma_{x}\\sigma_{y}\ au_{xy}\end{array} ight\}[/latex] (24.33)

[latex]s_{11}=\frac{1}{E_{l}}=\frac{1}{150,000}=6.7 imes 10^{-6}[/latex]

[latex]s_{22}=\frac{1}{E_{t}}=\frac{1}{90,000}=11.1 imes 10^{-6}[/latex]

[latex]s_{12}=-\frac{ν_{l t}}{E_{l}}=-\frac{0.3}{150,000}=-2.0 imes 10^{-6}[/latex]

[latex]s_{33}=\frac{1}{G_{l t}}=\frac{1}{5,000}=200 imes 10^{-6}[/latex]

Also,

[latex]\cos heta=\sin heta=\cos 45^{\circ}=1 / \sqrt{2}[/latex]

so that

[latex]m^{2}=0.5=n^{2}, \quad m^{4}=n^{4}=0.25, \quad m^{2} n^{2}=0.25[/latex], and so forth Substituting these values in Eqs. (24.33), we have

[latex]\left\{\begin{array}{c}\varepsilon_{x}\\varepsilon_{y}\\gamma_{xy}\end{array} ight\}=\left[\begin{array}{ccc}53.45 & -46.55 & - \-46.55 & 53.45 & - \-2.2 & -2.2 & -\end{array} ight]\left\{\begin{array}{c}60 \40 \0\end{array} ight\}[/latex]

which gives

[latex]\varepsilon_{x}=1345 imes 10^{-6}[/latex]

[latex]\varepsilon_{y}=-655 imes 10^{-6}[/latex]

[latex]\gamma_{x y}=-220 imes 10^{-6}[/latex]

The direct and shear strains are obtained through the following MATLAB file:

Question 24.6: A generally orthotropic ply is subjected to direct stresses ...

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