If the major (σ1) and minor (σ3) principal stresses for a rock element have a relationship as σ3 = -1/2 σ1, what is the expression for the maximum shear stress?
Understand the Problem
The question is asking for the expression of maximum shear stress based on the given relationship between the principal stresses (σ1 and σ3) for a rock element. We need to analyze the relationship provided to derive the maximum shear stress formula.
Answer
The maximum shear stress is $τ_{max} = \frac{3}{4}σ_1$.
Answer for screen readers
The maximum shear stress is expressed by $τ_{max} = \frac{3}{4}σ_1$.
Steps to Solve
- Understanding Principal Stresses The two principal stresses for the problem are given as:
- Major principal stress: $σ_1$
- Minor principal stress: $σ_3 = -\frac{1}{2}σ_1$
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Formula for Maximum Shear Stress The maximum shear stress ($τ_{max}$) can be calculated using the formula: $$ τ_{max} = \frac{σ_1 - σ_3}{2} $$
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Substituting for Minor Principal Stress Substituting $σ_3$ in the maximum shear stress formula: $$ τ_{max} = \frac{σ_1 - (-\frac{1}{2}σ_1)}{2} $$
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Simplifying the Expression Simplifying the expression gives: $$ τ_{max} = \frac{σ_1 + \frac{1}{2}σ_1}{2} $$
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Combining Terms Combining the terms in the numerator: $$ τ_{max} = \frac{\frac{3}{2}σ_1}{2} $$
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Final Simplification This can be simplified further: $$ τ_{max} = \frac{3}{4}σ_1 $$
The maximum shear stress is expressed by $τ_{max} = \frac{3}{4}σ_1$.
More Information
In the context of rock mechanics, understanding the relationship between principal stresses and maximum shear stress is crucial for predicting failure conditions in geological materials. The factor of $\frac{3}{4}$ reflects how these stresses interact under given conditions.
Tips
- Incorrect Substitution: Sometimes, students may misinterpret which stress should be used in the formula.
- Simplification Errors: Errors when combining terms or simplifying fractions are common. Always double-check each step.
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