Bending stress diagram; moment in flanges

Section (I-section): overall depth 20 cm, flange width 10 cm, flange thickness 3 cm, web 2 cm thick and 14 cm deep. $M = 10$ kNm.

Moment of inertia

$$I = \frac{100\times200^3 - 80\times140^3}{12} = 48.373\times10^6\ \text{mm}^4$$

Bending stresses ($\sigma = My/I$)

• Extreme fibre ($y = 100$ mm):

$$\sigma_{max} = \frac{10\times10^6\times100}{48.373\times10^6} = 20.67\ \text{N/mm}^2$$

• At the flange-web junction ($y = 70$ mm):

$$\sigma_j = \frac{10\times10^6\times70}{48.373\times10^6} = 14.47\ \text{N/mm}^2$$

Bending stress diagram

  y (mm)   stress (N/mm^2)
  +100     20.67 (C)   *-----------+   top fibre
                        \          |
  +70      14.47 (C)     *---------+   flange-web junction
                           \       |
     0     0 (NA)            *-----+   neutral axis
                           /       |
  -70      14.47 (T)     *---------+   flange-web junction
                        /          |
  -100     20.67 (T)   *-----------+   bottom fibre

Stress is zero at the neutral axis, varies linearly with distance, and is compressive on one side and tensile on the other (for a sagging moment, top compressive).

Moment carried by the flanges

Flanges (both, from $y = 70$ to $100$ mm):

$$M_f = 2\times\frac{\sigma_{max}}{100}\times100\times\frac{100^3 - 70^3}{3} = 9.05\ \text{kNm}$$

Web (from $-70$ to $70$):

$$M_w = \frac{\sigma_{max}}{100}\times20\times\frac{2\times70^3}{3} = 0.95\ \text{kNm}$$

Check: $9.05 + 0.95 = 10$ kNm.

Answer: maximum stress = 20.67 N/mm² (tension and compression); moment carried by the flanges = 9.05 kNm (90.5% of the total).