Using the material properties from Problem 4.3 and assuming that the stiffnesses are the same in tension and compression, determine the allowable off-axis shear stress, \(\tau_{x y}\) at \(\theta=45^{\circ}\) (refer to Figure 4.7) according to:

(a) the maximum stress criterion,

(b) the maximum strain criterion, and

(c) the TsaiHill criterion. Compare and discuss the results and check both positive and negative values of \(\tau_{x y}\).

Problem 4.3

An orthotropic lamina has the following properties:

\[E_{1}=160 \mathrm{GPa} \quad s_{\mathrm{L}}^{(+)}=1800 \mathrm{MPa}\]

\[\begin{aligned} & E_{2}=10 \mathrm{GPa} \quad s_{\mathrm{L}}^{(-)}=1400 \mathrm{MPa} \\ & v_{12}=0.3 \quad s_{\mathrm{T}}^{(+)}=40 \mathrm{MPa} \\ & G_{12}=7 \mathrm{GPa} \quad s_{\mathrm{T}}^{(-)}=230 \mathrm{MPa} \\ & s_{\mathrm{LT}}=100 \mathrm{MPa} \end{aligned}\]

Construct the failure surfaces in the \(\sigma_{1}-\sigma_{2}\) stress space for this material according to:

• the maximum stress criterion,

• the maximum strain criterion, and

• the Tsai-Hill criterion.