To solve the problem, we need to calculate the normal compressive stress induced at a distance of 50 mm above the neutral axis of the section due to a bending moment of 100 kNm about the x-axis.

Step 1: Determine the Second Moment of Area (Iₓ)

From the image provided, we can see the section is a composite of rectangular areas. The second moment of area about the x-axis ($I_x$) can be calculated using the parallel axis theorem:

$I_x = I_1 + I_2 + I_3$

Where:

• $I_1$, $I_2$, $I_3$ are the second moments of area of each rectangular segment about the x-axis.

Step 2: Calculate Individual Moments of Inertia

For each rectangular segment, $I = \frac{b \cdot h^3}{12} + A \cdot d^2$, where:

• $b$ is the width.
• $h$ is the height.
• $A = b \cdot h$ is the area.
• $d$ is the distance from the centroid of the rectangle to the neutral axis.

Let's break down the calculation.

1. Top Flange (horizontal segment at the top):

• $b_1 = 100 \, \text{mm}$
• $h_1 = 10 \, \text{mm}$
• Distance from the centroid of this section to the neutral axis ($d_1$) = $380 \, \text{mm} - \frac{h_1}{2} = 380 - 5 = 375 \, \text{mm}$

$I_{1x} = \frac{100 \times 10^3}{12} + 100 \times 10 \times (375)^2$

2. Vertical Web:

• $b_2 = 10 \, \text{mm}$
• $h_2 = 380 \, \text{mm}$
• Distance from the centroid of this section to the neutral axis ($d_2$) = $0$ (since it is at the neutral axis).

$I_{2x} = \frac{10 \times 380^3}{12}$

3. Bottom Flange (horizontal segment at the bottom):

• $b_3 = 120 \, \text{mm}$
• $h_3 = 20 \, \text{mm}$
• Distance from the centroid of this section to the neutral axis ($d_3$) = $\frac{h_2}{2} + \frac{h_3}{2} = 190 + 10 = 200 \, \text{mm}$

$I_{3x} = \frac{120 \times 20^3}{12} + 120 \times 20 \times (200)^2$

Step 3: Sum the Moments of Inertia

$I_x = I_{1x} + I_{2x} + I_{3x}$

Step 4: Calculate the Bending Stress

Once we have $I_x$, we can calculate the stress using:

$\sigma = \frac{M \cdot y}{I_x}$

Where:

• $M = 100 \times 10^3 \, \text{N·m}$
• $y = 50 \, \text{mm} = 0.05 \, \text{m}$

Let's calculate

I'll compute these step-by-step.The normal compressive stress induced at a distance of 50 mm above the neutral axis due to a moment of 100 kNm about the x-axis is 17.7 MPa.

---

Relative Questions:

1. How would the stress change if the moment was applied about the y-axis instead of the x-axis?
2. What is the significance of the second moment of area in structural analysis?
3. How does the distance from the neutral axis affect the induced stress in a beam?
4. What would be the effect on stress if the beam span was reduced to 5 meters?
5. How does the shape and size of the cross-section influence the beam's bending strength?

Tip: Always ensure to calculate the correct centroid and moment of inertia when dealing with composite sections to obtain accurate stress results.