Calculate the max normal stress if the axial tensile load in the x direction is given as 200 kN, shear stress is given as 100 N/mm2 and cross sectional area is given as 2000 mm2.
161.8 N/mm2
Calculate Maximum Normal Stress
To calculate the maximum normal stress, we first need to determine the normal stress component acting on the material due to the axial load and then combine it with the given shear stress to find the principal stresses.
Calculating Normal Stress from Axial Load
The normal stress ($\sigma_x$) in the x-direction is caused by the axial tensile load. It can be calculated using the formula:
\(\sigma_x = \frac{\text{Load (P)}}{\text{Area (A)}}\)
- Given Load (P) = 200 kN = \(200 \times 10^3\) N
- Given Area (A) = 2000 mm\(^2\)
Substituting the values:
\(\sigma_x = \frac{200 \times 10^3 \text{ N}}{2000 \text{ mm}^2} = \frac{200000}{2000} \text{ N/mm}^2\)
\(\sigma_x = 100 \text{ N/mm}^2\)
Understanding the Stress State
We are given a normal stress component in the x-direction (\(\sigma_x = 100 \text{ N/mm}^2\)) and a shear stress (\(\tau_{xy} = 100 \text{ N/mm}^2\)). Since there is no mention of load or stress in the y-direction, we can assume \(\sigma_y = 0\). This represents a plane stress state.
The stress components are:
- \(\sigma_x = 100 \text{ N/mm}^2\)
- \(\sigma_y = 0 \text{ N/mm}^2\)
- \(\tau_{xy} = 100 \text{ N/mm}^2\)
Calculating Principal Stresses
The maximum and minimum normal stresses on a body are known as principal stresses. These occur on planes where the shear stress is zero. For a plane stress state, the principal stresses (\(\sigma_{1,2}\)) are calculated using the formula:
\(\sigma_{1,2} = \frac{\sigma_x + \sigma_y}{2} \pm \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2}\)
Substituting the stress components:
\(\sigma_{1,2} = \frac{100 + 0}{2} \pm \sqrt{\left(\frac{100 - 0}{2}\right)^2 + 100^2}\)
\(\sigma_{1,2} = \frac{100}{2} \pm \sqrt{\left(\frac{100}{2}\right)^2 + 100^2}\)
\(\sigma_{1,2} = 50 \pm \sqrt{50^2 + 100^2}\)
\(\sigma_{1,2} = 50 \pm \sqrt{2500 + 10000}\)
\(\sigma_{1,2} = 50 \pm \sqrt{12500}\)
Now, calculate the value of \(\sqrt{12500}\):
\(\sqrt{12500} = \sqrt{2500 \times 5} = 50\sqrt{5}\)
Using the approximate value \(\sqrt{5} \approx 2.236\):
\(50\sqrt{5} \approx 50 \times 2.236 = 111.8\)
So, the principal stresses are:
\(\sigma_1 = 50 + 111.8 = 161.8 \text{ N/mm}^2\)
\(\sigma_2 = 50 - 111.8 = -61.8 \text{ N/mm}^2\)
Identifying the Maximum Normal Stress
The maximum normal stress is the algebraically larger of the two principal stresses.
Maximum normal stress = \(\sigma_1 = 161.8 \text{ N/mm}^2\)
The minimum normal stress is \(\sigma_2 = -61.8 \text{ N/mm}^2\).
Therefore, the maximum normal stress is 161.8 N/mm\(^2\).
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