Simply supported I-beam: stress at flange-web junction

Data: flanges 40 cm $\times$ 6 cm each (width $\times$ thickness), web 60 cm deep $\times$ 6 cm thick, so total depth $= 6 + 60 + 6 = 72$ cm. Span $L = 6$ m, UDL $w = 25$ kN/m.

Maximum bending moment

$$M_{max} = \frac{wL^2}{8} = \frac{25(6)^2}{8} = 112.5\ \text{kN·m} = 1.1250\times10^{8}\ \text{N·mm}$$

Moment of inertia about the neutral axis (symmetrical section, in mm)

$$I = 2\left[\frac{400(60)^3}{12} + (400\times60)(330)^2\right] + \frac{60(600)^3}{12}$$

where the flange centroid is $300 + 30 = 330$ mm from the neutral axis.

$$I = 2\left[7.2000\times10^{6} + 2.6136\times10^{9}\right] + 1.0800\times10^{9} = 6.3216\times10^{9}\ \text{mm}^4$$

Stress at the flange-web junction

The junction is at $y = 300$ mm from the neutral axis:

$$\sigma = \frac{My}{I} = \frac{1.1250\times10^{8}\times300}{6.3216\times10^{9}} = 5.339\ \text{N/mm}^2$$

The stress is compressive at the top junction and tensile at the bottom junction. For comparison, the extreme fibre stress is $\dfrac{M(360)}{I} = 6.407\ \text{N/mm}^2$.

Answer: bending stress at the flange-web junction $= 5.34\ \text{N/mm}^2$ (about 5.34 MPa).