# 014 Review Problem - Weight of concrete surge chamber when full of water

**Problem 14**

A concrete surge chamber with circular cross section and vertical inner walls has an inside diameter of 100 ft. The outer walls taper uniformly. The outer walls taper uniformly ¼ in. to 1 ft. of rise, and at the base the thickness is 5 ft. The height of the surge chamber is 150 ft. above the pressure tunnel, and the material used in its construction weighs 150 lb. per cu. ft. Find the total weight of the chamber when full of water.

**Solution 14**

$a = 37.5 ~ \text{in.} = 3.125 ~ \text{ft.}$

$r = 55 - a = 55 - 3.125$

$r = 51.875 ~ \text{ft.}$

Total volume (water + concrete)

$V_t = \text{Frustum of a cone}$

$V_t = \frac{1}{3}\pi h(R^2 + r^2 + Rr)$

$V_t = \frac{1}{3}\pi (150)[ \, 55^2 + 51.875^2 + 55(51.875) \, ]$

$V_t = 1,346,037.462 ~ \text{ft.}^3$

Volume of water

$V_w = \text{Cylinder}$

$V_w = \pi (50^2)(150)$

$V_w = 1,178,097.245 ~ \text{ft.}^3$

Volume of concrete

$V_c = V_t - V_w$

$V_c = 1,346,037.462 - 1,178,097.245$

$V_c = 167,940.217 ~ \text{ft.}^3$

Weight of water

$W_w = \gamma_w V_w = 62.4(1,178,097.245)$

$W_w = 73,513,268.09 ~ \text{lb}$

Weight of concrete

$W_c = \gamma_c V_c = 150(167,940.217)$

$W_c = 25,191,032.55 ~ \text{lb}$

Total weight when full of water

$W_t = W_w + W_c = 73,513,268.09 + 25,191,032.55$

$W_t = 98,704,300.64 ~ \text{lb} \times \dfrac{1 ~ \text{short ton}}{2000 ~ \text{lb}}$

$W_t = 49,352.15 ~ \text{short tons}$ *answer*