Question

for the given state of stress, (1) plot the stress circle, (2) determine the principal stresses. (unit: mpa)

Explanation:

Step1: Identify Stress Components

From the stress element, we have:

• Normal stress in x - direction, $\sigma_x = 20\ MPa$
• Normal stress in y - direction, $\sigma_y = 100\ MPa$
• Shear stress, $\tau_{xy}=- 30\ MPa$ (sign convention: positive if it causes counter - clockwise rotation on the positive face)

Step2: Calculate Center and Radius of Stress Circle

The center of the stress circle (average normal stress), $\sigma_{avg}=\frac{\sigma_x+\sigma_y}{2}=\frac{20 + 100}{2}=60\ MPa$
The radius of the stress circle, $R=\sqrt{(\frac{\sigma_x-\sigma_y}{2})^2+\tau_{xy}^2}=\sqrt{(\frac{20 - 100}{2})^2+(-30)^2}=\sqrt{(-40)^2+(-30)^2}=\sqrt{1600 + 900}=\sqrt{2500}=50\ MPa$

Step3: Determine Principal Stresses

The principal stresses are given by $\sigma_{1,2}=\sigma_{avg}\pm R$
$\sigma_1=\sigma_{avg}+R=60 + 50 = 110\ MPa$
$\sigma_2=\sigma_{avg}-R=60 - 50 = 10\ MPa$

(For plotting the stress circle: Plot a circle with center at $(\sigma_{avg},0)=(60,0)$ on the $\sigma-\tau$ plane and radius $R = 50$. The points corresponding to the x - face is $(\sigma_x,\tau_{xy})=(20,-30)$ and the y - face is $(\sigma_y,-\tau_{xy})=(100,30)$ (since $\tau_{yx}=-\tau_{xy}$) can be plotted on the circle)

Answer:

The principal stresses are $\sigma_1 = 110\ MPa$ and $\sigma_2 = 10\ MPa$ (To plot the stress circle, use center $(60,0)$ and radius $50$ on the $\sigma-\tau$ coordinate system)