A Mohr-Coulomb envelope between shear stress, τ and normal stress, σn of a sandstone rock is given as τ = 7.5 + 0.84 σn (unit of stresses is MPa). A sandstone sample is tested in t... A Mohr-Coulomb envelope between shear stress, τ and normal stress, σn of a sandstone rock is given as τ = 7.5 + 0.84 σn (unit of stresses is MPa). A sandstone sample is tested in triaxial mode with confining pressure of 5.0 MPa. The value of the shear stress, τ in MPa at the failure, is _____ (round off up to 2 decimals).
Understand the Problem
The question requires us to determine the shear stress in a sandstone sample using a Mohr-Coulomb envelope equation given a specific confining pressure. We will substitute the confining pressure into the equation and calculate the shear stress.
Answer
The value of the shear stress, $\tau$, at failure is $11.7$ MPa.
Answer for screen readers
The value of the shear stress, $\tau$, at failure is $11.7$ MPa.
Steps to Solve
- Identify the equation of Mohr-Coulomb envelope
The Mohr-Coulomb envelope equation is given as:
$$ \tau = 7.5 + 0.84 \sigma_n $$
where $\tau$ is the shear stress and $\sigma_n$ is the normal stress.
- Substitute the confining pressure into the equation
In triaxial testing, the confining pressure is equal to the normal stress ($\sigma_n$). Here, the confining pressure is given as 5.0 MPa.
Substituting $\sigma_n = 5.0$ MPa into the equation:
$$ \tau = 7.5 + 0.84 \times 5.0 $$
- Calculate the shear stress
Now, calculate $\tau$:
$$ \tau = 7.5 + 0.84 \times 5.0 = 7.5 + 4.2 $$
- Final calculation
Add the values together:
$$ \tau = 7.5 + 4.2 = 11.7 \text{ MPa} $$
The value of the shear stress, $\tau$, at failure is $11.7$ MPa.
More Information
The shear stress determined here is critical for understanding the mechanical behavior of sandstone under load, particularly in geological and engineering applications.
Tips
- Forgetting to substitute the confining pressure correctly into the normal stress variable $\sigma_n$.
- Miscalculating the multiplication when substituting values; ensure accurate arithmetic.
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