## Check for Bearing Capacity of the retaining wall

The vertical pressures transmitted to the soil by the base slab of the retaining wall should be checked against the ultimate bearing capacity of the soil.The nature of variation of the vertical pressure transmitted by the base slab into the soil is shown in fig.8.9. Note that qtoe and qheel are the maximum and the minimum pressures occurring at the ends of the toe and heel sections ,respectively .The magnitudes of qtoe and qheel can be determined in the following manner:

-the sum of the vertical forces acting on the base slab is ƩV,and the horizontal force Ph=Pacosα

R=ƩV+PA

-the net moment of these forces about point C  is     Mnet=ƩMR-ƩMO

-let the line of action of the resultant R intersect the base slab at E,then the distance  CE=X=Mnet/ƩV

Hence, the eccentricity of the resultant R may be expressed as:                   e=B/2-CE .

The pressure distribution under the base slab may be determined by using simple principles from the mechanics of materials.First we have:

q=ƩV/A+/-Mnety/I     where

Mnet-moment=(ƩV)e;

I-moment of inertia per unit length of the base section=1/12(1)B2.

For maximum and minimum pressures ,the value of y in the main equation equals B/2 resulting:

qmax=ƩV/B(1+6e/B)  and

qmin= ƩV/B(1-6e/B)  .

Note that ƩV includes the weight of the soil ,and that when the value of the eccentricity e becomes greater than B/6,qmin becomes negative thus will result some tensile strength at the end of the heel section.This stress is not desirable,because the tensile strength of the soil is very small.If the analysis of a  design shows e>B/6,the design should be reproportioned and calculations redone.

The ultimate bearing capacity of a shallow foundation is defined by the following equation:

qu=c’2NcFcdFØl+qNqFØdFqi+1/2y2B’NyFydFyi    where:

q=y2D

B’=B-2e

Fcd=1+0.4D/B’

FØd=1+2tanØ’2(1-sinØ’2)2D/B’

Fyd=1

Fqi=(1-ψ⁰/90⁰)2

ψ⁰=tan-1(Pacosα/ƩV)

Fyl=(1- ψ⁰/Ø’2⁰)2

Once the ultimate bearing capacity of the soil has been calculated ,the factor of safety can be determinated (generally a factor of 2..3) is required   FS(bearing capacity)=qu/qmax.