Lateral Earth Pressure

 

The magnitude of lateral earth pressure depends on:

1.     Shear strength characteristics of soil

2.     Lateral strain condition

3.     Pore water pressure

4.     State of Equilibrium of soil

5.     Wall and ground surface shape  

        Previous conditions depends mainly on:

a)     Drainage conditions

b)    Interaction between soil and wall

 

Soil state of Equilibrium  

The state of Equilibrium of soil can be divided into two states :

a)     State of Elastic Equilibrium

        When a small change in stress produces a corresponding small change in strain.

b)    State of Plastic Equilibrium

          When irreversible strain takes place at a constant stress.

 

 The strain state relating to earth pressure calculation fall into three categories:

a)     At Rest State:

           It is the case when state of elastic Equilibrium with no lateral displacement occurs (Figure 1 ).

b)    Active state:

          It is the case when plastic equilibrium with lateral expansion takes place (wall moves outward from the soil). ( Figure 2 ).

c)     Passive state:

          Plastic equilibrium with lateral contraction takes place (wall moves toward the soil). ( Figure 2 )

Figure 1

 

 

 Figure 2

(1) Earth Pressure at rest (ko condition )

 

         lateral strain , εh = 0

        

 ko  = coefficient of lateral earth presser at rest.

 ko  depends on :

§        Soil type ( sand, silt, clay)

§        Loading –Unloading history

§        Relative density of soil

  For normally consolidated soil (N.C.) and sand

                        

  For over consolidated soil (O.C.)

              

where OCR  is the over consolidated ratio

                  

    and in general                 

                      where   is Poisson ratio 

In this case ,the mohre circle is below the shear envelope of this soil   (Figure 3).

Figure 3

 

 

(2) Active Earth pressure  (Rankine’s Active state )

 

        Rankine’s theory  neglects friction

        between soil and wall .

 

   when the wall moves outward  from the

   soil , the lateral earth pressure starts to

   be reduced until it reaches its , minimum                                                                      Figure 3

   value ,                                                       

   where ,  

   where ,  ka = coefficient of active earth pressure. (Figure 3)

And in this case , the failure plane makes ( 45 – Ø/2 )  with the direction of  the major

principal plane ( usually horizontal axes ) . 

 

(3) Rankin Passive State 

    If the wall moved toward the soil , the passive condition will take place.                                  

    The passive pressure will increase until it reaches its maximum value of:

                   

     

 

and in this case , the failure place makes ( 45 + Ø/2 ) with the direction of the major principal plane.

 

Earth pressure Distribution for Sand 

   In general, the active earth pressure distribution in sand will be:

            

Figure (4)

             where γ = soil unit weight

                       H = wall height

  

      However the distribution and value of passive earth pressure in sand is,  Figure(5)

                   

Figure 5

 

(b) For clay 

          The value of active earth pressure is (Figure 6)

 

 

                               Figure(6)

 

                            

    

                where ,   c = soil cohesion

    The passive pressure values are  (Figure 7)

        

        

Figure(7)

 

      Effect of Ground Water Table 

a)     Static water table

                     For static water table the water pressure should be treated     

                     separately from soil lateral pressure as shown in Figure(8)

 

                                                                      Figure(8)

 

                   

         

          

         where  γsub = submerged unit weight of soil

                       = γsat  - γw

b)     Moving Water Table 

                   In this case , the flow net and equipotential lines should be   

                     drawn for the wall , Then the pore pressure against the wall can  

                     be determined, ( Figure 9) 

        Figure (9 )

 

Effect of Stratified  Soil 

      As an example of stratified soil shown in(Figure10)

                                                (Figure 10)             

 

Effect of Sloping Ground Surface

   In this case the active coefficient of lateral earth pressure is

 

 and the resultant in this case will be parallel to the ground surface,           

 (Figure 11)

Coulomb's Theory for Rough Walls

                                                                    Figure(12)

    δ = angle of wall friction

     for concrete wall

 

            

and

           

and   

        

Strutted Trench sheeting

Figure(13)

The location of struts affects the values and distributions of lateral earth pressures

Figure(14)