At a Point in a stressed body the principal stresses are 100 MN/m2 (tensile) and 60 MN/m2 (Compressive). The maximum shear stress at the point is
80 MN/m2
Principal Stresses Analysis
This problem involves finding the maximum shear stress at a point within a stressed body, given the principal stresses acting on it. Principal stresses are the maximum and minimum normal stresses at a point. In this case, we are given:
- Major Principal Stress ($\sigma_1$) = 100 MN/m2 (Tensile)
- Minor Principal Stress ($\sigma_2$) = 60 MN/m2 (Compressive)
Tensile stresses are typically represented as positive values, and compressive stresses are represented as negative values in stress calculations.
- $\sigma_1 = +100$ MN/m2
- $\sigma_2 = -60$ MN/m2
Calculating Maximum Shear Stress
The maximum shear stress ($\tau_{max}$) at a point can be calculated using the principal stresses ($\sigma_1$ and $\sigma_2$) with the following formula:
$$ \tau_{max} = \frac{\sigma_1 - \sigma_2}{2} $$
Let's substitute the given values into the formula:
$$ \tau_{max} = \frac{(+100 \text{ MN/m}^2) - (-60 \text{ MN/m}^2)}{2} $$
First, handle the subtraction of the negative value, which is equivalent to addition:
$$ \tau_{max} = \frac{100 \text{ MN/m}^2 + 60 \text{ MN/m}^2}{2} $$
Next, sum the values in the numerator:
$$ \tau_{max} = \frac{160 \text{ MN/m}^2}{2} $$
Finally, perform the division:
$$ \tau_{max} = 80 \text{ MN/m}^2 $$
Result Summary
The calculation shows that the maximum shear stress at the point is 80 MN/m2. This value represents the peak shear stress experienced on a plane within the material at that specific point under the given loading conditions.
| Parameter | Value | Unit |
|---|---|---|
| Major Principal Stress ($\sigma_1$) | 100 | MN/m2 |
| Minor Principal Stress ($\sigma_2$) | -60 | MN/m2 |
| Maximum Shear Stress ($\tau_{max}$) | 80 | MN/m2 |
Therefore, the maximum shear stress at the point is 80 MN/m2.
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