23. Simply supported beam 8 m length
has a 2 000 kg load at the center.
The flexural stress is 25 000 kPa.
Find the sectionmodulus.
A. 6 434.56 in3 B. 1 846.
56 in3 C. 95.78 in3
D. 79.82 in3

To find the section modulus (\(Z\)) for the simply supported beam, we can use the flexural stress formula:

\[
\sigma = \frac{M}{Z}
\]

Where:
- \(\sigma\) is the flexural stress (given as 25,000 kPa).
- \(M\) is the maximum bending moment.
- \(Z\) is the section modulus.

### Step 1: Calculate the Maximum Bending Moment
For a simply supported beam with a point load at the center, the maximum bending moment (\(M\)) is given by:

\[
M = \frac{W \cdot L}{4}
\]

Where:
- \(W\) is the load (2,000 kg = 19,620 N using \(1 \text{kg} = 9.81 \, \text{m/s}^2\)).
- \(L\) is the length of the beam (8 m).

Substitute the values into the formula:

\[
M = \frac{19,620 \, \text{N} \cdot 8 \, \text{m}}{4} = 39,240 \, \text{Nm}
\]

### Step 2: Use the Flexural Stress Formula to Find Section Modulus

Rearranging the flexural stress formula to solve for \(Z\):

\[
Z = \frac{M}{\sigma}
\]

Convert the flexural stress to the same units (N/m²):
\[
\sigma = 25,000 \, \text{kPa} = 25,000,000 \, \text{N/m}^2
\]

Now, substitute the values of \(M\) and \(\sigma\):

\[
Z = \frac{39,240 \, \text{Nm}}{25,000,000 \, \text{N/m}^2} = 0.0015696 \, \text{m}^3
\]

### Step 3: Convert to Inches
Since the answers are given in cubic inches, convert \(Z\) from cubic meters to cubic inches (\(1 \, \text{m}^3 = 61,023.7 \, \text{in}^3\)):

\[
Z = 0.0015696 \, \text{m}^3 \times 61,023.7 \, \text{in}^3/\text{m}^3 = 95.78 \, \text{in}^3
\]

### Answer:
The section modulus is **95.78 in³**, which corresponds to option **C**.

Would you like more details or have any questions?

Here are 5 related questions:
1. How is the flexural stress in a beam derived?
2. What happens if the load is not applied at the center of the beam?
3. How do you calculate the moment of inertia for a beam?
4. What are the differences between flexural stress and shear stress in beams?
5. How does the material of the beam affect the section modulus?

**Tip**: Always ensure units are consistent throughout calculations to avoid errors!

23. Simply supported beam 8 m length has a 2 000 kg load at the center. The flexural stress is 25 000 kPa. Find the section modulus. A. 6 434.56 in3 B. 1 846.56 in3 C. 95.78 in3 D. 79.82 in3

Find the section modulus for a simply supported beam with an 8-meter length and a 2,000 kg load at the center. Given the flexural stress of 25,000 kPa, use formulas for flexural stress and bending moment to calculate the section modulus. This detailed guide walks through the necessary steps, suitable for undergraduate engineering students.

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