2m 15m 10m Sectional View A concrete dam is 10m high and spans 15m as shown in t

Question

2m 15m 10m Sectional View A concrete dam is 10m high and spans 15m as shown in the figure above. The friction coefficient between the ground and the dam is 0.1. Find the max height of water before the dam starts to either over-flow or move. (Hint. Max horizontal force the dam can provide total vertical force *friction co- efficient.) (Density of concrete-2400kg/m^3. Density of water = 1000kg/m*3) ist of formulae P=gh P Force Area Ic bd3/12 (for a rectangle) A of trapezium = ½(top-base)"h A of triangle-½(bh) A of circle- r 2 SG = fluid density / water density Weight = mass * gravity Density = mass * volume

Explanation / Answer

given, desnity of concrete, rho = 24000 kg/m^3
volume of concrete = area of cross section * span
area of cross section = 0.5*(2 + 10)*h = 0.5*12*10 = 60m^2
span = 15 m
volume, V = 15*60 = 900 m^3
so mass of dam, M = rho*V = 24000*900 = 21.6*10^6 kg

so total friction force = Mg*mu ( where mu is the coefficent of friction )
f = 21.6*10^6 * 9.81 * 0.1 = 21.1896*10^6 N

now, consider the 60 degree wall of the dam to be inner side of the dam, with the water
so, if height of water in dam is d
Pressure ar depth y = rho*gy
comsider small deph dy
length along the dam wall = dy/sin(60)
span, l = 15 m
force, dF = rho*gy*ldy/sin(60)
integrating from y = 0 to y = d
F = rho*g*ld^2/2*sin(60)
foerce parallel top the ground = Fsin(60)
for fully filled water, d = 10 m
so force Fsin(60) = 1000*9.81*15*10^2/2 = 7357500 N = 7.357*10^6 N

this fore is lesser than the maximum friction that the dam can support
hence, the maximum height of water can be 15 m after which the dam begins to overflow

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