The cross-section of a steel T-beam is shown in the figure where all dimensions are in mm.

The plastic section modulus of the given cross-section is ______×104 mm3 (in integer).

The correct option is 12.

1. Analysis of Cross-Section Dimensions from the Image:
Based on the provided T-beam cross-section diagram:

• The width of the top flange is 100 mm.
• The thickness of the top flange is 20 mm.
• The height (depth) of the vertical web is 100 mm.
• The thickness (width) of the vertical web is 20 mm.

2. Calculation of Cross-Sectional Areas:
Let us calculate the area of the flange and the web separately:

$$A_{\mathrm{flange}}=100\text{mm}\times 20\text{mm}=2000{\text{mm}}^2$$

$$A_{\mathrm{web}}=100\text{mm}\times 20\text{mm}=2000{\text{mm}}^2$$

The total cross-sectional area is:

$$A_{\mathrm{total}}=A_{\mathrm{flange}}+A_{\mathrm{web}}=2000+2000=4000{\text{mm}}^2$$

3. Location of the Plastic Neutral Axis (PNA):
The plastic neutral axis (PNA) divides the total cross-section area into two equal halves. Since the tension and compression zones under fully plastic conditions must have equal areas:

$$A_1=A_2=\frac{A_{\mathrm{total}}}{2}=2000{\text{mm}}^2$$

Because the flange area is exactly equal to 2000 mm² and the web area is also 2000 mm², the plastic neutral axis lies precisely at the junction of the flange and the web.

4. Centroidal Distances of the Halves from the PNA:
The area above the PNA (the flange) is a rectangle of depth 20 mm. Its centroid is located at a distance y₁ from the PNA:

$$y_1=\frac{20}{2}=10\text{mm}$$

The area below the PNA (the web) is a rectangle of depth 100 mm. Its centroid is located at a distance y₂ from the PNA:

$$y_2=\frac{100}{2}=50\text{mm}$$

5. Calculation of the Plastic Section Modulus (Z_p):
The plastic section modulus is calculated as:

$$Z_p=A_1{y}_1+A_2{y}_2$$

Substituting the calculated values:

$$Z_p=(2000\times 10)+(2000\times 50)$$

$$Z_p=20000+100000=120000{\text{mm}}^3$$

Expressing this value in the required units:

$$Z_p=12\times {10}^4{\text{mm}}^3$$

Therefore, the plastic section modulus factor is 12.