A horizontal cantilever 2 m long is constructed from the Z-section shown in Fig. A load of 10 kN is applied to the end of the cantilever at an angle of 60° to the horizontal. Assum... A horizontal cantilever 2 m long is constructed from the Z-section shown in Fig. A load of 10 kN is applied to the end of the cantilever at an angle of 60° to the horizontal. Assuming that no twisting moment is applied to the section: i. Determine the principal second moment of areas of the sections and the angle of inclination it's making with the X-axis. ii. Determine the stresses at point A and B.
Understand the Problem
The question is asking to calculate the principal second moment of areas of a horizontal cantilever Z-section subjected to a load and to find the stresses at two specific points, A and B. It requires using principles of mechanics and material strength to determine geometric properties and stress distributions.
Answer
$$ \sigma_A = \frac{M \cdot c_A}{I}, \ \sigma_B = \frac{M \cdot c_B}{I} $$
Answer for screen readers
The stresses at points A and B can be expressed as: $$ \sigma_A = \frac{M \cdot c_A}{I} $$ $$ \sigma_B = \frac{M \cdot c_B}{I} $$
Steps to Solve
- Identify the dimensions of the Z-section
We need to know the dimensions of the Z-section to calculate the first and second moments of area. Identify the height (h), width (b), thickness (t), and the distances of centroid from the base of the Z-section.
- Calculate the area of each section
The Z-section can be broken down into its constituent rectangles. To find the area $A_i$ of each rectangle (for example, two flanges and a web), use: $$ A_i = b_i \times h_i $$
- Find the centroid of the Z-section
Use the formula for the centroid (y_{\text{centroid}}): $$ y_{\text{centroid}} = \frac{\sum (A_i \cdot y_i)}{\sum A_i}$$
Where (y_i) is the distance of each area from a reference line (bottom of the Z-section).
- Calculate the first moment of area
Calculate the first moment of the area (Q) about the neutral axis: $$ Q = \sum (A_i \cdot d_i) $$ Where (d_i) is the distance from the centroid to the area.
- Calculate the second moment of area (I)
Using the formula: $$ I = \sum \left( I_i + A_i \cdot d_i^2 \right) $$ Where (I_i) is the second moment of area of each individual rectangle about its own centroid.
- Determine the maximum bending moment (M)
Using the loading conditions, calculate the maximum bending moment at the fixed end, which often is given by: $$ M = P \cdot L $$ Where (P) is the load and (L) is the length of the cantilever.
- Calculate the bending stress at points A and B
Using the bending stress formula: $$ \sigma = \frac{M \cdot c}{I} $$ Where (c) is the distance from the neutral axis to the point of interest.
- Find the stresses at Points A and B
Substitute the respective distances from the neutral axis for points A and B into the bending stress formula to find the stresses $\sigma_A$ and $\sigma_B$.
The stresses at points A and B can be expressed as: $$ \sigma_A = \frac{M \cdot c_A}{I} $$ $$ \sigma_B = \frac{M \cdot c_B}{I} $$
More Information
The second moment of area, also known as the area moment of inertia, helps describe how a beam will bend under a load. The farther the area is from the neutral axis, the larger the stress will be, which is why we need to find distances (c_A) and (c_B) carefully.
Tips
- Not correctly calculating the centroid of the Z-section, which leads to incorrect distances for (c).
- Forgetting to convert all dimensions to the same unit before performing calculations.
- Not including all components of the Z-section when calculating the area and its moments.
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