In the present state of knowledge, the recommended method of obtaining seismic soil forces is that using equivalent-static analysis. Only for exceptional structures would dynamic analyses using finite elements seem warranted.
In the equivalent-static method, a horizontal earthquake force equal to the weight of the soil wedge multiplied by a seismic coefficient is assumed to act at the centre of gravity of the soil mass. This earthquake force is additional to the static forces on the wall.
In general, the total soil pressure on a wall during an earthquake equals the sum of three possible components:
(1) static pressure due to gravity loads;
(2) dynamic pressure due to the earthquake;
(3) pressure due to the wall being displaced into the backfill by an external force, e.g. by the horizontal sway of a bridge deck at a monolithic abutment - design recommendations for this condition are given by Matthewson et al. (1980).
The soil pressures may be estimated by the following methods:
• approximate plasticity theory, e.g. Coulomb and Mononobe-Okabe; or
• numerical methods, modelling the soil as Winkler springs (Section 5.3.3) or as finite elements.
The results of dynamic analyses using finite elements are difficult to interpret because of the inability of such analyses to represent actual modes of failure. It should be noted that it is not appropriate to design all soil-retaining structures for earthquake soil pressures. For example, external retaining walls of modest height with no significant consequences of failure are generally not designed for earthquakes in many countries. Using the Mononobe-Okabe methods, it can be readily seen (Seed and Whitman, 1970) that it requires an effective ground acceleration of about 0.3g to produce an earthquake force increment equal to the static earth pressure for cohesionless soils. So, clearly a safety factor of 2.0 on a non-seismic design should permit walls to survive moderate earthquakes, with acceptable displacements. This rationale is applied to bridge design in New Zealand, where it has been recommended (Matthewson et al., 1980) that higher risks should be accepted for lesser structures, by designing the walls (as wells as the decks) for earthquakes of lower return period.
Active pressure on AB
Figure 9.7 Active pressure due to unsaturated cohesionless soil on a flexible retaining wall during an earthquake, for use with Mononobe-Okabe equations (Coulomb conditions)
Active pressure on AB
Figure 9.7 Active pressure due to unsaturated cohesionless soil on a flexible retaining wall during an earthquake, for use with Mononobe-Okabe equations (Coulomb conditions)
Active seismic pressures in unsaturated cohesionless soils
The most commonly used solution is that derived by Mononobe (1929) and Okabe (1926), based on Coulomb's theory. The effect of an earthquake is represented by a static horizontal force equal to the weight of the wedge of soil multiplied by the seismic coefficient. Referring to Figure 9.7, the Mononobe-Okabe equations are as follows.
The total force on a wall due to the static and earthquake active soil pressures due to unsaturated cohensionless soils is
where
sin(y' + S) sin(y' — i — 0) cos(5 + p +6)cos(P -i)
and cot(aAE — i) = — tan(y ' + s + p — i) + sec(y' + 5 + p — i)
{cos(P+8 + 6)smUp> + 8) X V I cos(P -i)sm((p< -6 - i)
where aAE is the slope angle of the failure plane in an earthquake (Figure 9.7); P the angle of the back face of the wall to the vertical; yd the unit weight of the soil; S the angle of wall friction; the effective angle of shearing resistance; i the slope angle of the backfill; 9 = tan-1[ah/(1 — av)]; ah is a seismic coefficient given by 1/g x (horizontal ground acceleration); and av is a seismic coefficient given 1/g x (vertical ground acceleration).
The ground accelerations ahg and avg would normally correspond to those for the design earthquake ground motions (Chapter 4), except as modified to allow for wall inertia effects, as discussed below.
The effect of vertical acceleration on the wall pressures has been shown to be small (Seed and Whitman, 1970), except in the case of gravity walls (Richards and Elms, 1979), as discussed below. Therefore, for non-gravity walls, the term av disappears from equation (9.23), which thus reduces to
Pae = ^aekdtf2
and the expression for 9 becomes 9 = tan-1 ah.
In the conditions assumed in Coulomb's theory, where the shearing resistance is mobilized between the back of the wall and the soil, the earthquake soil pressure is calculated directly (Figure 9.7). For concrete walls against formwork, the wall friction S may be taken as 1/2^. The static active force PA (Figure 9.7) may be found from the Coulomb equation
where
Mononobe and Okabe apparently considered that the earthquake force APae = PAE — PA (Figure 9.7), calculated by their analysis, would act on the wall at the same position as the initial static force PA, that is, at a height of H/3 above the base. This assumption is reasonable for flexible walls (Matthewson et al., 1980), which rotate as required in the active state. Suggestions that the earthquake force from the Mononobe-Okabe analysis acts at a higher level appear to have been based on tests of more rigid construction which, of course, are not applicable to the active state. Equation (9.26) describes the general case, and becomes considerably simplified in the case of a wall with a vertical back face and horizontal fill, so that P = m = 0. In this case, equation (9.26) reduces to kae =
cos2 9
cos 9
A simple way of obtaining KAE from KA (for which design charts are available) has been derived by Arango, and is described by Seed and Whitman (1970).
While the Mononobe-Okabe analysis is widely accepted as the basis for seismic design of retaining walls in Coulomb conditions, Richards and Elms (1979) have shown that for gravity walls it needs modification to allow for the effect of wall inertia, which causes pressures of the same size as the dynamic pressure derived from the Mononobe-Okabe analysis given above. Their design method, based on a deflection criterion rather than stresses or stability, involves the following steps:
(1) Select values of peak ground acceleration Ag and the ground velocity V.
(2) Select the maximum allowable permanent horizontal displacement, dL.
(3) Find the resistance factor N (where Ng is the acceleration at which the wall begins to slide), such that the actual permanent displacement will just equal dL. For finding N, Richards and Elms recommend using an expression for the dimensionless parameter N/A given by
(4) However, N is equivalent to the limiting value of the seismic coefficient acting on the wall, ah. Hence, we may find the active pressure coefficient KAE from equation (9.26). The horizontal force due to the wall (weight Ww) is ahWw, and the effective design weight of the wall for sliding is (1 — av)Ww. The resistance to sliding is
where 6b is the friction angle for the base of the wall.
Equating F to ah Ww plus the horizontal components of PAE, it is found that
To allow for uncertainties in their design method, Richards and Elms (1979) suggest using a safety factor of 1.5 on the above wall weight, so that the weight of the wall as built should be 1.5Ww.
As an improvement on the above design procedure for gravity walls, in an enlightening study of the uncertainties involved, Whitman and Liao (1984) propose replacing equation (9.32) with
where Fc is a safety factor on the allowable displacement dL. An appropriate value of Fc may be found from the probability distribution of dR(= dL/Fc), which appears to be lognormal. Thus, if the 95th percentile value is required, a value of Fc = 3.8 should be used. Then, using the value of N obtained from equation (9.34) would lead directly to the value of Ww from equation (9.35), which would be used directly in the design without applying the factor 1.5, as in step 4 above.
Active seismic pressures in cohesionless soils containing water
For cohesionless soils containing water, the above solution using the Mononobe-Okabe equations is not realistic, and attempts to use them by applying factors to the densities and using the apparent angle of internal friction $u and the Mononobe-Okabe equations would lead to solving the wrong problem.
The undrained situation is not only undesirable physically, but also difficult to analyse, hence it is recommended that good drainage should be provided to obviate the problem. Such drainage should be effective to well below the potential failure zone behind the wall, and also in front of the wall if cohesionless soils exist there so that the required passive resistance is available.
Active seismic pressures in cohesive soils or with irregular ground surface
The trial wedge method (Figure 9.8) offers the easiest derivation of seismic soil pressure when the material is cohesive or the surface of the ground is irregular. Figure 9.8 is drawn for Rankine conditions, and where the ground surface is very irregular the direction of pae may be taken as approximately parallel to a line drawn between points a and C. For Coulomb conditions the principles of the trial wedge method are similar and the direction of PAE will be at an angle s to the surface on which the pressure is calculated, similar to Figure 9.7.
Note that in seismic conditions, tension cracks may be ignored on the assumption that this introduces relatively small errors compared with others involved in the analysis. For saturated soils the appropriate density will have to be taken in determining w in Figure 9.8.
Where soil is retained by a rigid wall, pressures greater than active develop. In this situation, the static and earthquake soil pressures may be taken as pe = p0 + pqe = \koyh2 +ahyh\ (9.36)
where y is the total unit weight of the soil and K0 is the coefficient of at-rest soil pressure.
As with the active pressure case discussed above, this equation should not be applied to saturated sands. For a vertical wall and horizontal ground surface, and for all normally consolidated materials, K0 may be taken as
where is the effective angle of shearing resistance. For other wall angles and ground slopes, K0 may be assumed to vary proportionately to KA. The at-rest soil pressure force p0 = 1/2K0yH2 may be assumed to act at a height h/3 above the base of the wall, while the dynamic pressure AP0E = ahyh2 may be assumed (Matthewson et al., 1980) to act at a height of 0.58H.
For gravity retaining walls, the at-rest force should be taken as acting normal to the back of the wall, while for cantilever and counterfort walls it should be calculated on the vertical plane through the rear of the heel, and taken as acting parallel to the ground surface.
Seismic displacements of retaining walls
The serviceability of retaining walls after earthquakes is often dictated by whether their displacements are acceptable, rather than the forces that they can withstand. The design methods of Richards and Elms (1979) and Whitman and Liao (1984), referred to above, include methods of estimating displacements. In addition, such estimates may be made by finite-element analysis. However, the prediction of displacements is much less reliable than is desirable.
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