Elementary analysis for width/height ratio of retaining masonary structure

As we know that masonry structure cannot resist tension, hence all the lateral force should be resisted by its weight component only. In this article we will derive expression for width/height ratio for a simple rectangular masonry structure.

Let,

γ=density of water

S_m=specific gravity of masonry material

S_L=specific gravity of material giving lateral force (e.g. water, soil etc)

h=height of the structure

x=width of the structure

W=weight of structure=h*x*s* γ

μ= coefficient of friction

Case1: For overturning

Moment dut to lateral force=moment due to weight of the material

½* γ*s_L*h²*h/3 =W*x/2

Or, ½* γ*h²*h/3*s_L = γ *s_m*h*x*x/2

Or, (x/h)²=(s_L/(3*s_m))

Or, x/h=sqrt(s_L/s_m/3)

For water s_L=1;for stone masonry s_m=22/9.81=2.14

Therefore, x/h=0.394≈0.4

Case2: For Sliding

Horizontal force=friction factor*weight of the structure

½* γ*s_L*h²= μ (γ*s_m*h*x)

Or, x/h=s_L/(2*s_m* μ)

For water and stone masonry and using μ=0.65

x/h=0.359≈0.4

Thus from above derivation we see that the minimum base width/height ratio of the rectangular block to resist the lateral (triangular here) force with safety factor 1 is 0.4.

With safety factor of 1.5 the minimum width/height ratio becomes 1.5*0.4=0.6.