What is Measured?

The experiment begins with the given state of plane stress:

• Normal stress along the $x$-direction, $\sigma_x$
• Normal stress along the $y$-direction, $\sigma_y$
• Shear stress, $\tau_{xy}$

Using these values, the experiment determines:

• Principal stresses
• Principal plane angle
• Maximum shear stress
• Tresca equivalent stress
• Von Mises equivalent stress

Why are the Calculations Required?

These calculations help engineers:

• Identify the maximum and minimum normal stresses acting on a component.
• Determine the orientation of the critical planes.
• Evaluate the likelihood of yielding.
• Compare different failure theories for ductile materials.
• Design structural and machine components safely.

Observation Table

Parameter	Symbol	Unit
Normal stress in x-direction	$\sigma_x$	MPa
Normal stress in y-direction	$\sigma_y$	MPa
Shear stress	$\tau_{xy}$	MPa
Principal stress	$\sigma_1$	MPa
Principal stress	$\sigma_2$	MPa
Principal plane angle	$\theta_p$	degree
Maximum shear stress	$\tau_{max}$	MPa
Von Mises stress	$\sigma_v$	MPa
Tresca stress	$\sigma_T$	MPa

Sequential Calculations

Compute the average normal stress

$$\sigma_{avg} = \frac{\sigma_x+\sigma_y}{2}$$

Compute the radius

$$R = \sqrt{ \left( \frac{\sigma_x-\sigma_y}{2} \right)^2 + \tau_{xy}^{\,2} }$$

Principal stresses

$$\sigma_1=\sigma_{avg}+R$$

$$\sigma_2=\sigma_{avg}-R$$

Principal plane angle

$$\tan2\theta_p = \frac{2\tau_{xy}} {\sigma_x-\sigma_y}$$

Maximum shear stress

$$\tau_{max}=R$$

Von Mises stress

$$\sigma_v = \sqrt{ \sigma_x^2+\sigma_y^2-\sigma_x\sigma_y+3\tau_{xy}^{\,2} }$$

Tresca stress

$$\sigma_T=\sigma_1-\sigma_2$$

Solved Numerical Example

Given

• $\sigma_x=100$ MPa
• $\sigma_y=40$ MPa
• $\tau_{xy}=30$ MPa

Average stress

$$\sigma_{avg} = \frac{100+40}{2} = 70\text{ MPa}$$

Radius

$$R = \sqrt{30^2+30^2} = 42.43\text{ MPa}$$

Principal stresses

$$\sigma_1=70+42.43=112.43\text{ MPa}$$

$$\sigma_2=70-42.43=27.57\text{ MPa}$$

Maximum shear stress

$$\tau_{max}=42.43\text{ MPa}$$

Von Mises stress

$$\sigma_v = \sqrt{100^2+40^2-100(40)+3(30)^2} = 105.36\text{ MPa}$$

Interpretation of Results

• Larger principal stress indicates the most critical tensile stress.
• Maximum shear stress is used in the Tresca criterion.
• Von Mises stress predicts yielding based on distortion energy.
• If the equivalent stress exceeds the material yield strength, yielding is expected.

Result

The experiment determines the principal stresses, principal plane orientation, maximum shear stress, and compares the Tresca and Von Mises failure criteria to evaluate the safety of a component subjected to plane stress.