2.1.1 Coulomb Earth Pressure Theory
Coulomb (1776) proposed a method of analysis that determines the resultant horizontal force on a retaining system for any slope of wall, wall friction, and slope of backfill. The Coulomb theory is based on the assumption that soil shear resistance develops along the wall and the failure plane. Detailed assumptions are made as the followings:
1. The backfill is isotropic and homogeneous.
2. The rupture surface is plane, as plane BC in Fig. 2.1(a). The backfill surface AC is a plane surface as well.
3. The frictional resistance is distributed uniformly along the rupture surface BC.
4. Failure wedge is a rigid body.
5. There is a friction force between soil and wall when the failure wedge moves toward the wall.
6. Failure is a plane strain condition.
In order to develop an active state, the wall is designed to move away from the soil mass. If the wedge ABC in Fig. 2.1(a) moves down relative to the wall, the wall friction angle δ will develop at the interface between the soil and wall. Let the weight of wedge ABC be W and the force on BC be F. With the given value θ and the summation of vertical forces and horizontal forces, the resultant soil thrust P can be calculated as shown in Fig. 2.1(b).
Similarly, the active forces of other trial wedges, such as ABC2, ABC3 in Fig 2.2 can be determined. The maximum value of Pa thus determined is the Coulomb's active force. Ka = coefficient of active earth pressure
γ = unit weight of soil
β = slope of back of the wall to horizontal i = slope of ground surface behind wall
2.1.2 Rankine Earth Pressure Theory
Rankine (1857) considered the soil in a state of plastic equilibrium and used essentially the same assumptions as Coulomb. The Rankine theory further assumes that there is no wall friction and failure surfaces are straight planes, and that the resultant force acts parallel to the backfill slope. Detailed assumptions are made as the followings:
1. The backfill is isotropic and homogeneous.
2. The retaining wall is a rigid body. The wall surface is vertical and the friction force between the wall and the soil is neglected.
Rankine assumed no friction between wall surface and backfill, and the backfill is cohesionless. The earth pressure on plane AB of Fig. 2.3(a) is the same as that on plane AB inside a semi-infinite soil mass in Fig. 2.3(b). For active condition, the active earth pressure σa at a given depth z can be expressed as:
σa =γzKa (2.3)
The total active force Pa per unit length of the wall is equal to
Pa H2Ka 2
1γ
= (2.4)
The direction of resultant force Pa is parallel to the ground surface as Fig. 2.3(b), where
2.1.3 Terzaghi General Wedge Theory
The assumption of plane failure surface made by Coulomb and Rankine, however, does not apply in practice. Terzaghi (1941) suggested that part of the failure surface in the backfill under an active condition was a log spiral curve, like the curve bd in Fig. 2.4. But the failure surface dc is still assumed a plane.
Fig. 2.5 illustrates the procedure to elevate the active resistance by trial wedge method (Terzaghi and Peck, 1967). The line d1c1 makes an angle of 45o +φ 2 with the surface of the backfill. The arc bd1 of trial wedge abd1c1 is a logarithmic spiral formulated as the following equation
r1 =r0eθtanφ (2.6)
O1 is the center of the log spiral curve in Fig. 2.5, where O1b = r1, O1d1 = r0, and ∠bO1d1 = θ . For the equilibrium and the stability of the soil mass abd1f1 in Fig. 2.6, the following forces per unit width of the wall are considered:
1. Soil weight per unit width in abd1f1: W1 = γ × (area of abd1f1)
2. The vertical face d1f1 is in the zone of Rankine’s active state; hence, the force
Pd1 acts horizontally at a distance of Hd1/3 measured vertically upward from d1.
γ is the unit weight of soil
3. The resultant force of the shear and normal forces dF, acting along the surface of sliding bd1. At any point of the curve, according to the property of the logarithmic spiral, a radial line makes an angle φ with the normal.
Since the resultant dF makes an angle φ with the normal to the spiral at its point of application, its line of application will coincide with a radial line and will pass through the point O1.
4. The active force per unit width of the wall P1 acts at a distance of H/3 measured vertically from the bottom of the wall. The direction of the force P1 is inclined at an angle δ with the normal drawn to the back face of the
The trial active forces per unit width in various trial wedges are shown in Fig.
2.7. Let P1, P2, P3, …, and Pn be the force that respectively correspond to the trial wedges 1, 2, 3, …, and n. The forces are plotted to the same scale as shown in the upper part of the figure. A smooth curve is plotted through the points 1, 2, 3, …, n.
The maximum P3 of the smooth curve defines the active force Pa per unit width of the wall.
2.1.4 Spangler and Handy’s Theory
Spangler and Handy (1984) have applied Janssen’s (1895) theory to design
problem of fascia retaining walls. Fig. 2.8 defines the soils with a width B bounded by two unyielding frictional boundaries (the rock face and wall face). The vertical force equilibrium of the thin horizontal soil element in Fig. 2.9 requires
dh V Bdh
This is a linear differential equation, the solution for which is
( )
μ = tan δ, the coefficient of friction between the soil and the wall
γ = unit weight of the soil B = backfill width
h = backfill depth (i.e. z)
K = the coefficient of lateral earth pressure V = the vertical force
From the solution of eq.(2.11), an equation for lateral earth pressure σh can be calculated
Some solutions for different values of B are shown in Fig. 2.10. The soil pressure, instead of continuing to increase with increasing values of h, levels off at a maximum value σh,max defined as follows.