Strains and stresses in a CST element

(a) Area and $[B]$

$$2A = x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) = 0 + 40(30) + 0 = 1200 \Rightarrow A = 600\ \text{mm}^2$$ $$[B] = \frac{1}{1200}\begin{bmatrix} -30 & 0 & 30 & 0 & 0 & 0 \\ 0 & -40 & 0 & 0 & 0 & 40 \\ -40 & -30 & 0 & 30 & 40 & 0 \end{bmatrix}$$

(b) Strains

$$\begin{aligned} \varepsilon_x &= \frac{30(0.02)}{1200} = 5.0\times10^{-4} \\ \varepsilon_y &= \frac{40(-0.01) - 40(0)}{1200} = -3.33\times10^{-4} \\ \gamma_{xy} &= \frac{30(0.005) + 40(0.004)}{1200} = \frac{0.31}{1200} = 2.58\times10^{-4} \end{aligned}$$

(c) Stresses (plane stress)

$\dfrac{E}{1 - \nu^2} = \dfrac{210\,000}{0.9375} = 224\,000$ MPa and $G = \dfrac{E}{2(1 + \nu)} = 84\,000$ MPa.

$$\begin{aligned} \sigma_x &= 224\,000\,(5.0\times10^{-4} + 0.25(-3.33\times10^{-4})) = 93.3\ \text{MPa} \\ \sigma_y &= 224\,000\,(-3.33\times10^{-4} + 0.25(5.0\times10^{-4})) = -46.7\ \text{MPa} \\ \tau_{xy} &= 84\,000\,(2.58\times10^{-4}) = 21.7\ \text{MPa} \end{aligned}$$

(d) Principal stresses

$$\sigma_{1,2} = \frac{\sigma_x + \sigma_y}{2} \pm \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2} = 23.3 \pm \sqrt{70.0^2 + 21.7^2} = 23.3 \pm 73.3$$ $$\sigma_1 = 96.6\ \text{MPa},\quad \sigma_2 = -50.0\ \text{MPa}$$ $$\tan 2\theta_p = \frac{2\tau_{xy}}{\sigma_x - \sigma_y} = \frac{43.4}{140} = 0.310 \Rightarrow \theta_p = 8.6^\circ$$

Answer: $\varepsilon = (5.0\times10^{-4},\ -3.33\times10^{-4},\ 2.58\times10^{-4})$; $\sigma_x = 93.3$ MPa, $\sigma_y = -46.7$ MPa, $\tau_{xy} = 21.7$ MPa; $\sigma_1 = 96.6$ MPa, $\sigma_2 = -50.0$ MPa at $8.6^\circ$ from the x-axis.