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Methods of determining passive earth pressure
Passive earth pressure occurs only when deformation of the wall towards the soil is large enough to mobilize the internal resistance of the soil against the external force. Franzius experimentally established that mobilizing passive earth pressure requires wall movement of s=30_h_1,5 in wet sand and s=23h2,5 in submerged sand, where the movement s is in mm and h is the wall height in m. Therefore, if the wall height is h=3,0 m, mobilizing passive pressure in wet sand requires wall movement towards the soil of s=156 mm. Because such a large wall movement is not permitted in most cases, one of the following approaches is used:
a) If the wall height is small, as in the case of the foundation footing CD (fig. 1a), passive earth pressure Ep is disregarded, particularly because the soil layer in front of the footing CD may later be excavated or eroded; only active earth pressure Ea over the full height of the wall AB is considered.
b) For a greater wall height (fig. 1b), Ea is adopted as the resistance on the wall segment CD instead of passive earth pressure Ep, opposing the forces Ea1 and Ea2.
c) For abutments that must resist large horizontal forces, full passive pressure Ep is required over the entire wall height AB to oppose the applied force R (fig. 1c). In this case, the soil is first deformed using hydraulic jacks. If it consists of soft clay or similar material that deforms greatly under load, the material is replaced over the required width b with gravel having a high angle of internal friction and capable of being compacted easily, so that later soil deformation under the structural load is prevented.
Fig. 1. Cases of passive earth pressure
There are several methods of determining passive earth pressure, among which the best known are the graphic methods of Coulomb and Rebhann-Poncelet.
Coulomb’s graphical method
This method assumes a straight slip surface and that the internal resistance of the soil consists only of friction, without cohesion (c=0). Since passive earth pressure acts as a resisting force, the soil’s angle of internal friction and the angle by which force Ep deviates from the wall normal have the opposite sign, namely -δ and -ϕ (fig. 2a).
If the inner face of the wall is AB and an arbitrary slip surface AC is adopted, the external force Ep is opposed by the weight W of the sliding prism ABCA and the frictional resistance Q along the slip surface AC. The magnitude, line of action and sense of weight W are known, while the lines of action and senses of forces Q and Ep are known, so a force diagram can be constructed (fig. 2b) to determine the magnitude of force Ep.
Fig. 2. Passive earth pressure according to Coulomb
Because the required value is the smallest passive pressure Ep,min, corresponding to the critical slip surface AC along which the sliding prism is most likely to form and be displaced by the external force, the following procedure is used (fig. 3).
Fig. 3. Determination of passive earth pressure EP according to Coulomb
Several arbitrary slip surfaces AC1, AC2, AC3 are drawn. . . and the weight of each corresponding sliding prism is determined: W1 = AΔABC1 x 1,00 x γ, W2 = AΔABC2 x 1,00 x γ, W3 = AΔABC3 x 1,00 x γ. A corresponding force diagram is then constructed for each prism and the forces EP1, EP2, EP3 are determined… These forces are plotted at a suitable scale on the EP force diagram. A tangent to this diagram parallel to the abscissa gives the smallest value Ep, corresponding to the critical slip surface AC. The disadvantage is that this graphical procedure may produce a force Ep that deviates significantly from reality, so the method is rarely used.
Culmann’s graphical method
This method assumes that passive earth pressure Ep acts at an angle -δ to the normal of the retaining wall, that the soil’s natural-slope line forms an angle -ϕ with the horizontal, and that the reference line forms an angle -(δ+ϕ) with the wall line (fig. 4a). At limit equilibrium, the three acting forces W, Q and Ep intersect at one point. Since the magnitude, line of action and sense of force W are known, and the lines of action and senses of forces Q and Ep are known, a force diagram can be drawn (fig. 4b). If this force triangle is rotated by 90°- ϕ counterclockwise, force W falls along the soil’s natural-slope line AN, force Q along the slip surface AC, and force Ea along the reference line BH. The angle α+ϕ lies between forces W and Q, while angle δ lies between Ep and the wall normal, so the line of action of Ep coincides with the reference line BH (fig. 4a).
Fig. 4. Determination of passive earth pressure according to Coulomb
The procedure for determining passive earth pressure is as follows (Fig. 4c). From the base of the wall A, several arbitrary slip surfaces AC1, AC2, AC3… are drawn. Each corresponds to a sliding prism ABC1, ABC2, ABC3…, whose length perpendicular to the drawing is taken as unity. When the weights of these soil prisms W1 = AΔABC1 x 1,00 x γ, W2 = AΔABC2 x 1,00 x γ, W3 = AΔABC3 x 1,00 x γ, where γ is the unit weight of the soil, are plotted on the soil’s natural-slope line AN, points I, II, III… are obtained. Lines drawn from these points parallel to the reference line give the intersection points R1, R2, R3… The triangles A_I_R1, A_II_R2, A_III_R3 are force diagrams for W, Q and Ep for the corresponding slip surfaces. Connecting points R1, R2, R3… gives the Culmann line of passive earth pressures, the so-called Culmann Ep line. A tangent to this line parallel to the soil’s natural-slope line gives the smallest length KK’, which, at the scale used for forces W, gives passive earth pressure Ep.
Extending line AK to its intersection with the ground surface gives the slip surface of least resistance AC.
Rebhann-Poncelet method
According to this method, the procedure is similar to the one for determining active earth pressure with the difference that the force Q acts at an angle -ϕ with the normal to the sliding surface, and the force Ep acts at an angle -δ with the normal to the wall line. According to Rebhann, AΔABC = AΔACD (fig. 5a). The procedure for determining the passive earth pressure is as follows:
Fig. 5. Determination of passive earth pressure according to Rebhann-Poncelet
From the foot of the wall A, draw the soil’s natural-slope line at an angle -ϕ to the horizontal.
From the top of the wall B, draw the reference line BH at the angle -(δ+ϕ). At point M, where the reference line intersects the natural-slope line, raise a normal to intersection L with the semicircle constructed over AN as its diameter.
Using compass radius AL, transfer point L to D (AL=AD).
Draw from point D parallel to the reference line until it intersects the ground surface at C, obtaining one side of the earth-pressure triangle DC = e. Transfer point C to F (DC= DF) and drop the normal CK to the soil’s natural-slope line to obtain the earth-pressure triangle DCF, from which Ep=1/2_γ_ ef, where γ is the unit weight of the soil.
Line AC is the slip surface. The passive earth-pressure diagram is ba (fig. 5b). It follows from the expression
Ep = xh/2, whence x = 2_Ep_/h
The point of application of force Ep is at the height of the centroid of triangle abc. Force Ep acts at an angle -δ to the normal of the retaining wall.
The stated methods, units, formulas and values are taken from a historical source and are not a substitute for a geotechnical report, structural calculation, retaining-wall design or construction plan. Mobilizing passive resistance depends on permitted deformation, actual soil parameters and stratification, groundwater and surface water, drainage, surcharge loads, seismic effects, erosion, excavation, construction stages and neighboring structures. A solution for a specific site must be established through appropriate investigation by authorized geotechnical and structural professionals in accordance with current regulations.