Calculate maximum and minimum stresses at the base of a hollow circular column having external diameter as 300 mm and internal diameter as 200 mm, subjected to 100 mm from the neut... Calculate maximum and minimum stresses at the base of a hollow circular column having external diameter as 300 mm and internal diameter as 200 mm, subjected to 100 mm from the neutral axis of the column.
Understand the Problem
The question is asking to calculate the maximum and minimum stresses experienced at the base of a hollow circular column with specified diameters and a given distance from the neutral axis. This involves applying principles from mechanics of materials to determine the stress distribution.
Answer
Maximum and minimum stresses in a hollow circular column can be calculated using the formulas: $$ \sigma_{\max} = \frac{M y_{\max}}{I} $$ and $$ \sigma_{\min} = \frac{M y_{\min}}{I} $$
Answer for screen readers
The maximum and minimum stresses at the base of the hollow circular column depend on the specific values of the outer and inner diameters, the distance from the neutral axis, and the applied bending moment.
$$ \sigma_{\max} = \frac{M y_{\max}}{I}, \quad \sigma_{\min} = \frac{M y_{\min}}{I} $$
Steps to Solve
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Identify the Parameters
We need to define the parameters for the hollow circular column:
- Outer diameter, $D_o$
- Inner diameter, $D_i$
- Distance from the neutral axis, $y$
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Calculate the Centroid of the Hollow Section
For a hollow circular section, the centroid can be found using the formula:
$$ y_c = \frac{D_o^4 - D_i^4}{D_o^4 - D_i^4} $$ -
Calculate the Moment of Inertia
The moment of inertia, $I$, for a hollow circular section is given by:
$$ I = \frac{\pi}{64} (D_o^4 - D_i^4) $$ -
Determine the Bending Stress
Bending stress at a distance $y$ from the neutral axis is calculated using the formula:
$$ \sigma = \frac{M y}{I} $$
where $M$ is the bending moment acting on the column. -
Calculate Maximum and Minimum Stresses
Substituting the values into the bending stress formula, we need to calculate:
For maximum stress when $y$ is positive and minimum stress when $y$ is negative. -
Summarize the Results
State the maximum and minimum stresses based on the calculations.
The maximum and minimum stresses at the base of the hollow circular column depend on the specific values of the outer and inner diameters, the distance from the neutral axis, and the applied bending moment.
$$ \sigma_{\max} = \frac{M y_{\max}}{I}, \quad \sigma_{\min} = \frac{M y_{\min}}{I} $$
More Information
Understanding how to compute stresses in hollow circular columns is important in structural engineering, especially in applications like bridges, buildings, and various mechanical systems. As structures are subjected to loads, knowing the stress distribution helps in ensuring structural safety and integrity.
Tips
- Miscalculating the moment of inertia, which can significantly affect stress calculations.
- Forgetting to correctly identify whether $y$ is measured from the centroid or the neutral axis.
- Using incorrect units or failing to keep track of them throughout the calculations.
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