Wind pressure on pitched roof on hill

Assumptions (not all data are given). $V_b$ is not stated; 47 m/s is assumed. A dense forest with closely spaced trees is terrain Category 3. Greatest dimension $= 30$ m, so Class B. Design life is taken as 50 years ($k_1 = 1.0$). Roof: width $w = 10$ m, rise 3.5 m, so $\tan\alpha = 3.5/5 = 0.7$, $\alpha = 35.0^\circ$; $h/w = 10/10 = 1.0$. Height for $k_2$ = mean roof height $= 10 + 3.5/2 = 11.75$ m. Normal openings: $C_{pi} = \pm0.2$.

Topography factor $k_3$ (Appendix C). Hill slope 1:4, so $\theta = \tan^{-1}(1/4) = 14.0^\circ$, which lies between 3° and 17°. Height of hill $Z = 450$ m, so the upwind slope length is $L = Z/\tan\theta = 1800$ m and $L_e = L = 1800$ m. $C = 1.2Z/L = 1.2\times450/1800 = 0.30$. The building is at mid-height on the upwind slope, i.e. $X = -L/2 = -900$ m (upwind of the crest), so $X/L_e = -0.5$, and $H/L_e \approx 12/1800 \approx 0.007$. From Fig. 15 (hill and ridge, upwind section), $s \approx 0.27$ (read from the chart, approximate). So $k_3 = 1 + Cs = 1 + 0.30\times0.27 = 1.081$.

Step 1: Design wind speed (IS 875 Part 3:1987, cl. 5.3)

$$V_z = V_b\,k_1k_2k_3$$

• Basic wind speed $V_b = 47$ m/s.
• Risk coefficient $k_1 = 1.00$ (50-year life; Table 1).
• Terrain, height and size factor $k_2$: Category 3, Class B, at height 11.75 m, interpolating between 10 m (0.88) and 15 m (0.94): $k_2 = 0.901$ (Table 2).
• Topography factor $k_3 = 1.081$ (hill, Appendix C).

$$\begin{aligned} V_z &= 47\times1.00\times0.901\times1.081 = 45.78\ \text{m/s} \\ p_z &= 0.6V_z^2 = 0.6\times45.78^2 = 1257\ \text{N/m}^2 = 1.257\ \text{kN/m}^2 \end{aligned}$$

Step 2: Pressure coefficients (Table 5, pitched roof of rectangular clad building)

Height ratio $h/w = 1.00$ (band $1/2 < h/w \le 3/2$), roof angle $\alpha = 35.0^\circ$. Values are interpolated between $\alpha = 30^\circ$ and $45^\circ$.

Internal pressure coefficient $C_{pi} = \pm0.2$ (normal openings), cl. 6.2.3.

Design pressure normal to the roof: $p_d = (C_{pe} - C_{pi})\,p_z$ (positive = towards the surface, negative = suction). The most severe case is slope EG (wind 90°), with $C_{pe} - C_{pi} = -1.00$, so $p_d = -1257$ N/m² $= -1.257$ kN/m².

Step 3: Wind force on the roof

$F = (C_{pe} - C_{pi})\,A\,p_z$, acting normal to the roof surface. The most severe net coefficient, $-1.00$, is used for the maximum force.

• One roof slope (slope length $\sqrt{5^2 + 3.5^2} = 6.10$ m × 30 m): $A = 183.1$ m², $F = 1.00\times183.1\times1.257 = 230.2$ kN (suction, i.e. acting away from the roof).

Answer

• Design wind speed $V_z = 45.8$ m/s; design wind pressure $p_z = 1257$ N/m² (1.26 kN/m²).
• Net design pressure on the roof: up to $1257$ N/m² (1.26 kN/m²), suction.
• Force on one roof slope (6.10 m × 30 m): 230.2 kN.