The state of plane-stress at a point is given by σₓ = −200 MPa, σ${}_y$ = 100 MPa and τ${}_{xy}$ = 100 MPa. The maximum shear stress in MPa is

Plane Stress Analysis: Determining Maximum Shear Stress

Understanding the state of plane stress at a specific point is crucial in mechanics of materials. The problem provides the normal stress components, ${\sigma }_x$ and ${\sigma }_y$, along with the shear stress component, ${\tau }_{xy}$. Our goal is to calculate the maximum shear stress, often denoted as ${\tau }_{max}$, which is a key parameter for design and failure analysis.

Stress Components Given

The problem provides the following stress components at a point:

- Normal stress in the x-direction, ${\sigma }_x=-200\text{MPa}$
- Normal stress in the y-direction, ${\sigma }_y=100\text{MPa}$
- Shear stress in the xy-plane, ${\tau }_{xy}=100\text{MPa}$

We need to find the maximum shear stress, ${\tau }_{max}$.

Maximum Shear Stress Formula

For a state of plane stress defined by ${\sigma }_x$, ${\sigma }_y$, and ${\tau }_{xy}$, the maximum shear stress, ${\tau }_{max}$, can be calculated using the following formula:

$${\tau }_{max}=\sqrt{{\left(\frac{{\sigma }_x-{\sigma }_y}{2}\right)}^2+{\tau }_{xy}^2}$$

This formula is derived from Mohr's Circle for plane stress, where ${\tau }_{max}$ represents the radius of the circle.

Shear Stress Calculation Steps

Let's substitute the given values into the formula and perform the calculation step-by-step:

Step 1: Calculate the average normal stress difference.

$$\frac{{\sigma }_x-{\sigma }_y}{2}=\frac{-200\text{MPa}-100\text{MPa}}{2}$$

$$=\frac{-300\text{MPa}}{2}$$

$$=-150\text{MPa}$$

Step 2: Square the calculated difference and the shear stress component.

$${\left(\frac{{\sigma }_x-{\sigma }_y}{2}\right)}^2=(-150\text{MPa}{)}^2=22500{\text{MPa}}^2$$

$${\tau }_{xy}^2=(100\text{MPa}{)}^2=10000{\text{MPa}}^2$$

Step 3: Sum the squared values.

$${\left(\frac{{\sigma }_x-{\sigma }_y}{2}\right)}^2+{\tau }_{xy}^2=22500{\text{MPa}}^2+10000{\text{MPa}}^2=32500{\text{MPa}}^2$$

Step 4: Take the square root to find the maximum shear stress.

$${\tau }_{max}=\sqrt{32500{\text{MPa}}^2}$$

$${\tau }_{max}\approx 180.2776\text{MPa}$$

Rounding to one decimal place, the maximum shear stress is approximately $180.3\text{MPa}$.

Result Comparison

Comparing our calculated value with the given options:

- Option 1: $111.8\text{MPa}$
- Option 2: $150.1\text{MPa}$
- Option 3: $180.3\text{MPa}$
- Option 4: $223.6\text{MPa}$

The calculated maximum shear stress of $180.3\text{MPa}$ matches Option 3.