Question

A retaining wall against a mud slide is to be constructed by placing 1.2 m-high rectangular concrete blocks (ρconcrete = 2700 kg/m3 ) side by side, as shown in Figure 1. The friction coefficient between the ground and the concrete blocks is f = 0.3, and the density of the mud is about 1800 kg/m3 . There is concern that the concrete blocks may slide or tip over the lower left edge as the mud level rises.

a) Determine the minimum width w of the concrete blocks at which the blocks will overcome friction and start sliding. Plot the results over the mud height ranging from zero to the top of the retaining wall in 0.2 m-increments.

b) Determine the minimum width w of the concrete blocks at which the blocks will tip over. Plot the results over the mud height ranging from zero to the top of the retaining wall in 0.2 m-increments in the same graph as a).

c) Briefly comment on the results.

Answer 1

Assumptions   :   Atmospheric pressure acts on both sides of the wall, and thus it can be ignored in calculations for convenience.
Properties   :   The density is given to be 1800 kg/m3 for the mud, and 2700 kg/m3 for concrete blocks.
Analysis   :   (a) The weight of the concrete wall per unit length (L = 1 m) and the friction force between the wall and the ground are

Wblock = rho*g*v= 2700*9.81*0.2*0.8*1 = 4238 N

The hydrostatic force exerted by the mud to the wall is

FH = Fx = Pavg*A = rho*g*(h/2)*A = 1800*9.81*(h/2)*h = 8829*h^2 N

   

Setting the hydrostatic and friction forces equal to each other gives

FH = Ffriction

8829*h^2 = 1271

h = 0.38 m

(b) The line of action of the hydrostatic force passes through the pressure center, which is 2h/3 from the free surface. The line of action of the weight of the wall passes through the midplane of the wall. Taking the moment about point A and setting it equal to zero gives

Sigma MA = 0

Wblock(t/2) = FH(h/3)

Wblock(t/2) = 8829h^3/3

Solving for h and substituting, the mud height for tip over is determined to be

h = (3Wblock*t/(2*8829))^(1/3) = (3*4238*0.2/(2*8829))^(1/3 = 0.52 m

Discussion   The concrete wall will slide before tipping. Therefore, sliding is more critical than tipping in this case.