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!