This theory was originally developed by Coulomb [5]. Fig. (2) shows the retaining wall under passive earth pressure conditions. At the point of failure, a wedge of soil ABC is under the equilibrium of three forces:

Fig. (2). Long retaining wall under passive earth pressure conditions.

1. The resultant passive force, P
2. The weight of the wedge, W
3. The resultant reaction, R between the wedge and the rest of the soil along the surface BC.

The reaction force R is the resultant of the shear force T and the normal force N. In Fig. (3), force polygon of the above system is shown. It can be shown that the failure wedge BC makes an angle of (45- ϕ/2) with the horizontal and hence the angle α is equal to (45 + ϕ/2). From the figure, the value of the P is calculated as:

(2)

Fig. (3). Force polygon.

where:

W= weight of the wedge per unit length of the wall (see Equation 3 below)

ϕ = effective angle of internal friction of granular soil

(3)

Hence, the total passive force, P is given by,

(4)

This force, P is thus equivalent to triangular earth pressure distribution on the wall as shown on the right-hand side of Fig. (2). The passive pressure on the wall at any depth, Z, is given by:

(5)

where:

K_p = passive earth pressure coefficent =

As indicated earlier, the ultimate lateral resistance per unit width of a rigid pile is greater than that of a corresponding wall, due to the shearing resistance on the vertical sides of the failure wedges in the soil. Using the same distribution of earth pressure at failure of the pile as was assumed for the wall, the three-dimensional effect for a pile can approximately be taken into account by multiplying the net earth pressure on a wall by shape factor (S_f).

The pressure distribution on a rigid pile at any depth z can be calculated by multiplying Equation 5 with the shape factor.

(6)

Typical shape factors adopted by a few investigators for a rigid pile, with depth of L and width of B, are summarized in Table 1.

In Fig. (4), a rigid long retaining wall with two different soil media is shown. The height of a retaining wall is L and the thickness of the annular space around the pole is ‘w’. The following assumptions are made:

Fig. (4). Long retaining wall under passive earth pressure conditions with backfill and in-situ native soil conditions.

• The vertical face of the wall is smooth
• The backfill soil is stronger than the in-situ native soil
• The failure wedges formed in the two soil media makes an angle of 45- ϕ₁/2 and 45- ϕ₂/2 (ϕ₁ = effective angle of internal friction of backfill soil & ϕ₂ effective angle of internal friction of native in-situ soil)
• The total passive force can be equated to the passive force developed due to the individual failure wedges of backfill and in-situ native soil

(7)

where:

(8)

(9)

In the above equations, the weights of wedges in backfill and in native in-situ soil are W₁ and W₂ respectively and they can be expressed as follows,

(10)

(11)

Substituting W₁ and W₂ values in eq (7) above, the total passive force is given by,

(12)

Simplifying Equation (12) by substituting w = nL (n = ratio between thickness of backfill behind the wall and the height of the retaining wall) and γ₁ = γ₂ = γ:

(13)

where,

(14)

In the above equation, K_pm is the modified passive earth pressure coefficient. Equation (13) for P is very similar to the earlier Equation (4). Assuming the triangular soil pressure distribution, the passive earth pressure at any depth, z is given by,

(15)

In the above equation, γ is assumed to be the average of unit weights of backfill and native in-situ soil.

As noted above, the theory and associated equations developed in Step 4 are applicable to direct embedment pole foundations. The following are the assumptions made:

• The theory proposed in Step 3 above can be extended to the rigid pile using appropriate shape factor
• The shape factor is of constant value 3.7 as used by Petrasovits and Award [7] for rigid piles.
• The type of distribution assumed was similar to the one presented by Petrasovits and Award [7] for rigid piles.
• The active earth pressure effect behind the back of the pile is neglected.

By extending Equation (15), the pressure distribution on a rigid pile at any depth z is given by (see Fig. 5),

(16)

Fig. (5). Passive pressure distribution on direct embedment pole foundations.

From the horizontal force equilibrium and moment equilibrium of forces shown in Fig. (5), the ultimate lateral resistance H_u and ultimate moment resistance M_u are calculated using the equations shown below.

(17)

(18)

L = embedment depth of pole (m)

γ = average of unit weights of backfill and native in-situ soils (kN/m³)

R = ratio of depth of point rotation (x) to total embedment depth of pole (L) = x/L

K_pm = combined passive coefficient for backfill and native in-situ soils

e = eccentricity = M_u/H_u

In the calculation of K_pm for pole foundations, ‘n’ is the ratio between the thickness of annular space around the pole and embedment depth of pole.

The value R is estimated from the definition of ‘e’ to be 0.707 for pure bending moment and 0.794 for pure horizontal load. Note that pure bending moment corresponds to the case where e = ∞ and pure horizontal load when e = 0.

For a given combined moment and horizontal load, R can be calculated using the equation below.

(19)

where:

e = eccentricity or elevation of H_u above ground level (m).