Cantilever with self-weight: P and moment of resistance

Section: flange 18 cm × 6 cm on top, stem 4 cm × 18 cm below; overall depth 24 cm. $\sigma_t = \sigma_c = 150\ \text{N/mm}^2$.

 P
 |      w = 1.5 kN/m
 v  vvvvvvvvvvvvvvvvv
 |##################  free end
 fixed    3 m

Section properties (from base, cm)

$$\bar y = \frac{18\times6\times21 + 4\times18\times9}{108+72} = 16.2\ \text{cm}$$ $$I = \frac{18\times6^3}{12} + 108(21-16.2)^2 + \frac{4\times18^3}{12} + 72(9-16.2)^2 = 8488.8\ \text{cm}^4$$

Top fibre distance $y_t = 24 - 16.2 = 7.8$ cm; bottom $y_b = 16.2$ cm.

$$Z_t = \frac{I}{y_t} = 1088.3\ \text{cm}^3,\qquad Z_b = \frac{I}{y_b} = 524.0\ \text{cm}^3$$

Moment of resistance

A downward load causes hogging: top in tension, bottom in compression. Both have the same permissible stress, so the smaller $Z$ governs:

$$M_r = 150\times 524.0\times10^3 = 78.6\times10^6\ \text{Nmm} = 78.6\ \text{kNm}$$

Value of P

$$M_{max} = P\times 3 + \frac{1.5\times 3^2}{2} = 3P + 6.75$$ $$3P + 6.75 = 78.6 \Rightarrow P = 23.95\ \text{kN}$$

Answer: $P = 23.95$ kN; moment of resistance $= 78.6$ kNm.