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Earth pressures for a cohesive-frictional soil

These equations appear in chapter 5 of the coursebook:

a a a v a

p p p v p

p ' 2c' K qK ' K

p ' 2c' K qK ' K

They are based on considering a cohesive-frictional soil at shear failure and find the major (passive

pressure) and minor (active pressure) principal stresses at failure. Figure 5.13 gives the Mohr-Coulomb

failure envelope for a cohesionless material; it shows how pa and pp are related to v. If we have a

failure envelope for a c - material and draw Mohr circles at failure for a given vertical stress, v+ q, we

have the diagram from which we can obtain equations 5.21. The interface is assumed to be smooth so

that Ka and Kp are the values obtained from equations 5.2 and 5.5. When the interface is not smooth we

use the appropriate Ka and Kp values, but the form of the equations is as in 5.21. Note the notation: lower

case pa and pp indicate the pressure distribution against the wall. The vertical effective stress, v,

increases linearly with depth and so too does the earth pressure.

If it is the active or passive thrust against a wall of height H which is required, the integrated forms of the

equations in 5.21 become:

2a a a a

2p p p p

1P 2c K H qK H K H

2

1P 2c K H qK H K H

2

Note the notation in this case: upper case Pp and Pa indicate the total thrust against a wall of height H. We

could also obtain these by following the Coulomb wedge method used in the first part of the chapter, in

other words these are generalisations of equation 5.3 and 5.6.

Note also the similarity of equations 5.21 to:

BNqNcNq

21

qcu For bearing strength the q is the vertical stress in the soil at the side of the foundation (the surcharge on

the ground surface if the foundation is not embedded). For the earth pressure case q is the surcharge

pressure on the surface of the backfill adjacent to the retaining wall (refer to equation 5.8).

These equations are used in the worked example for the gravity retaining wall in chapter 6; refer to the

sixth ultimate limit state on page 182 where the passive resistance of the embedded toe of the wall is

calculated (the q term does not appear as there is no surcharge on the ground surface adjacent to the toe).

The equations are also used for the embedded wall calculations in chapter 5 in Figs. 5.20b and 5.22 and

example 5.8in these cases there is no cohesion (as the material is sand) and no surcharge on the ground

surface in front of or behind the wall.

The above equation for the active and passive pressure distributions, indicate a linear with depth. The

situation for passive conditions is shown diagrammatically in Fig. 1 over the page. I leave it to you to

figure out how the diagram would look for the active pressure and thrust situations.

5.21

5.22

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Figure 1: Components of the passive pressure distribution and components of the passivethrust against a vertical wall given in equations 5.21 and 5.22.

The above equations deal with the effective stress case. What conclusion do you reach from the above

diagram which has vKp on the left hand side and HKp on the right hand side?

Equations 5.21 and 5.22 in the coursebook are for a c - soil. How would they be re-written for the =

0 material?

backfill surcharge pressure, qno surcharge

H

pressures

back

fill

qK p2c' KpHK p

qK pH

2c ' KpH

'vKp 2c' Kp

KpH2