Earthquake - Design of Retaining Walls: Pseudostatic Method.

The pseudostatic method has been previously discussed and it is applicable for slope stability and retaining wall analyses. The advantages of this method are that it is easy to understand and apply. Similar to earthquake slope stability analyses, this method ignores the cyclic nature of the earthquake and treats it as if it applies an additional static force upon the retaining wall. In particular, the pseudostatic approach is to apply a lateral earthquake force upon the retaining wall. To derive the lateral force, it can be assumed that the force acts through the centroid of the active wedge. The pseudostatic lateral force PE is calculated by using Eq. 13.12 (i.e., PE = kh W) and it has units of pounds per linear foot of wall length or kilonewtons per linear meter of wall length.

Note that an earthquake could subject the active wedge to both vertical and horizontal pseudostatic forces. However, the vertical force is usually ignored in the standard pseudostatic analysis.

This is because the vertical pseudostatic force acting on the active wedge usually has much less effect on the design of the retaining wall. In addition, most earthquakes produce a peak vertical acceleration that is less than the peak horizontal acceleration, and hence kv is smaller than kh.

The only unknowns in the pseudostatic method are the weight of the active wedge (W) and the seismic coefficient kh. Because of the usual relatively small size of the active wedge, the seismic coefficient kh can be assumed to be equal to amax /g. Using Fig. 11.3, the weight of the active wedge can be calculated as follows:

FIGURE 11.3 Active wedge behind retaining wall.


Using Eq. 14.1, the final result is as follows:


Note that since the pseudostatic force is applied to the centroid of the active wedge, the location of the force PE is at a distance of 2/3H above the base of the retaining wall. Seed and Whitman (1970) developed a similar equation that can be used to determine the horizontal pseudostatic force acting on the retaining wall, as follows:


Note that the terms in Eq 14.3 have the same definitions as the terms in Eq. 14.2. In comparing
Eqs. 14.2 and 14.3, the two equations are identical for the case where

According to
Seed and Whitman (1970), the location of the pseudostatic force from Eq. 14.3 can be assumed to act at a distance of 0.6 H above the base of the wall.

Mononobe and Matsu (1929) and Okabe (1926) also developed an equation that can be used to determine the horizontal pseudostatic force acting on the retaining wall. This method is often referred to as the Mononobe-Okabe method. The equation is an extension of the Coulomb approach and is as follows:

where PAE is the sum of the static (PA) and the pseudostatic earthquake force (PE). The equation for kAE is shown in Fig. 11.4. Note that in Fig. 11.4, the term y is defined as follows:

 The original approach by Mononobe and Okabe was to assume that the force PAE from Eq. 14.4 acts at a distance of 1/3 H above the base of the wall.

For the analysis of sliding and overturning of the retaining wall, it is common to accept a lower factor of safety (1.1 to 1.2) under the combined static and earthquake loads. Thus the retaining wall would be considered marginally stable for the earthquake sliding and overturning conditions. Note in the above table that the factor of safety for overturning is equal to 2.35 based on the Mononobe-Okabe method. This factor of safety is much larger than the other two methods. This is because the force PAE is assumed to be located at a distance of 1/3 H above the base of the wall. Kramer (1996) suggests that it is more appropriate to assume that PE is located at a distance of 0.6 H above the base of the wall (i.e., PE = PAE – PA, see Eq. 14.4).

Although the calculations are not shown, it can be demonstrated that the resultant location of N for the earthquake condition is outside the middle third of the footing. Depending on the type of material beneath the footing, this condition could cause a bearing capacity failure or excess settle- ment at the toe of the footing during the earthquake.

 FIGURE 11.4 Coulomb’s earth pressure kA equation. Part A) presents Coulomb’s equation for static conditions and Part B) presents the modified Coulomb’s equation for earthquake conditions.

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