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A can has tensile yield strength of 380 N/mm2. It has internal diameter of 20 ...

Jun 8, 2024

A can has tensile yield strength of 380 N/mm2. It has internal diameter of 20 cm and thickness of 0.4 cm. Considering a factor of safety of 3, what is the maximum internal pressure that can be applied to the can to cause yield. Use the two theories. (Max 200 words) i. Tresca failure theory ii. Von-mises failure theory

Solution by Steps

step 1

Determine the allowable stress using the factor of safety. The tensile yield strength is 380 N/mm², and the factor of safety is 3. Therefore, the allowable stress is:

\sigma_{allowable} = \frac{380}{3} = 126.67 \, \text{N/mm}^2

step 2

Convert the internal diameter and thickness to consistent units (mm):

\text{Internal diameter} = 20 \, \text{cm} = 200 \, \text{mm}

\text{Thickness} = 0.4 \, \text{cm} = 4 \, \text{mm}

step 3

Calculate the internal radius:

r = \frac{200}{2} = 100 \, \text{mm}

step 4

Apply Tresca failure theory. According to Tresca, the maximum shear stress is equal to half the difference between the maximum and minimum principal stresses. For a thin-walled cylinder, the maximum shear stress is:

\tau_{max} = \frac{p \cdot r}{2t}

Setting this equal to the allowable stress:

\frac{p \cdot 100}{2 \cdot 4} = 126.67

p = \frac{126.67 \cdot 8}{100} = 10.13 \, \text{N/mm}^2

step 5

Apply Von-Mises failure theory. According to Von-Mises, the equivalent stress is given by:

\sigma_{eq} = \sqrt{\sigma_1^2 - \sigma_1 \sigma_2 + \sigma_2^2}

For a thin-walled cylinder, the principal stresses are:

\sigma_1 = \frac{p \cdot r}{t}

\sigma_2 = 0

Thus,

\sigma_{eq} = \sqrt{\left( \frac{p \cdot 100}{4} \right)^2} = \frac{p \cdot 100}{4}

Setting this equal to the allowable stress:

\frac{p \cdot 100}{4} = 126.67

p = \frac{126.67 \cdot 4}{100} = 5.07 \, \text{N/mm}^2

Answer

The maximum internal pressure that can be applied to the can to cause yield is 10.13 N/mm² according to Tresca failure theory and 5.07 N/mm² according to Von-Mises failure theory.

Key Concept

Failure Theories in Thin-Walled Cylinders

Explanation

Tresca and Von-Mises theories provide different criteria for yielding, with Tresca focusing on maximum shear stress and Von-Mises on equivalent stress.