T-beam: maximum tensile and compressive stress

Section: flange 18 cm × 4 cm on top, web 3 cm × 30 cm; overall depth 34 cm.

Maximum bending moment (mid-span, sagging)

$$M = \frac{wL^2}{8} + \frac{PL}{4} = \frac{30\times4^2}{8} + \frac{40\times4}{4} = 60 + 40 = 100\ \text{kNm}$$

Neutral axis (from base, cm)

$$\bar y = \frac{3\times30\times15 + 18\times4\times32}{90+72} = \frac{1350+2304}{162} = 22.56\ \text{cm}$$

Top fibre distance $y_t = 34 - 22.56 = 11.44$ cm; bottom $y_b = 22.56$ cm.

Moment of inertia

$$I = \frac{3\times30^3}{12} + 90(15-22.56)^2 + \frac{18\times4^3}{12} + 72(32-22.56)^2 = 18406\ \text{cm}^4 = 1.8406\times10^8\ \text{mm}^4$$

Stresses

With the flange at the top and a sagging beam, top = compression, bottom = tension.

$$\sigma_c = \frac{100\times10^6\times114.4}{1.8406\times10^8} = 62.2\ \text{N/mm}^2$$ $$\sigma_t = \frac{100\times10^6\times225.6}{1.8406\times10^8} = 122.5\ \text{N/mm}^2$$

Answer: maximum tensile stress = 122.5 N/mm² (bottom of web); maximum compressive stress = 62.2 N/mm² (top of flange).