Question

According to maximum shear stress theory, the yield strength in shear \((\tau_s)\) and yield strength in tension \((\sigma_t)\) is related as:

• A. \(\tau_s = \sigma_t\)
• B. \(\tau_s = \frac{\sigma_t}{2}\)
• C. \(\sigma_t = \frac{\tau_s}{2}\)
• D. \(\sigma_t = \frac{\tau_s}{\sqrt{2}}\)

Hint:

For ductile materials, always remember: \(\tau_{yield} = \frac{\sigma_{yield}}{2}\) (Tresca theory).

Answer 1

The Correct Option is B

Concept: The Maximum Shear Stress Theory (also known as Tresca's Criterion) is one of the most fundamental failure theories used in Strength of Materials. It states that yielding of a ductile material begins when the maximum shear stress in the material reaches the maximum shear stress at the yield point in a simple tension test. Mathematically, the theory states: \[ \tau_{\max} = \frac{\sigma_1 - \sigma_3}{2} \] where \(\sigma_1\) and \(\sigma_3\) are the maximum and minimum principal stresses respectively.

Step 1: Consider a uniaxial tensile test condition.
In a standard tensile test: \[ \sigma_1 = \sigma_t, \quad \sigma_2 = 0, \quad \sigma_3 = 0 \] This means that the material is only subjected to stress in one direction, and the other principal stresses are zero.

Step 2: Determine maximum shear stress under this condition.
Using the formula: \[ \tau_{\max} = \frac{\sigma_1 - \sigma_3}{2} \] Substituting: \[ \tau_{\max} = \frac{\sigma_t - 0}{2} = \frac{\sigma_t}{2} \]

Step 3: Apply failure condition.
According to Tresca theory, yielding occurs when: \[ \tau_s = \tau_{\max} \] Thus: \[ \tau_s = \frac{\sigma_t}{2} \]

Step 4: Physical interpretation.
This means that the shear stress required to cause yielding is exactly half of the tensile stress required to cause yielding in the same material. Final Answer: \[ \boxed{\tau_s = \frac{\sigma_t}{2}} \]