# Seismic active and passive earth pressures on rigid retaining structures by a kinematical approach

Abdul-Hamid Soubra, Borut Macuh

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Abdul-Hamid Soubra, Borut Macuh. Seismic active and passive earth pressures on rigid retaining structures by a kinematical approach. International Conference on Recent Advances in Geotechnical Earthquake, Mar 2001, San Diego, United States. hal-01008866

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Distributed under a Creative Commons Attribution 4.0 International License
SEISMIC ACTIVE AND PASSIVE EARTH PRESSURES ON RIGID RETAINING
STRUCTURES BY A KINEMATICAL APPROACH

Abdul-Hamid Soubra

IUP Génie Civil et Infrastructures, L.G.C.N.S.N.
44603 Saint-Nazaire cedex, France

Borut Macuh

University of Maribor, Faculty of Civil Engineering
SI-2000 Maribor, Slovenia

ABSTRACT

The seismic active and passive earth pressure problems are investigated by means of the kinematical method of the limit analysis theory. Two rotational kinematically admissible failure mechanisms *M1* and *M2* are proposed. Quasi-static representation of earthquake effects using the seismic coefficient concept is adopted. The solutions obtained are rigorous upper-bound ones in the framework of the limit analysis theory. The numerical results of the seismic active and passive earth pressure coefficients are presented and compared with other authors' results. These results improve the best available upper-bound solutions given by Chen and Liu (1990) in the active case and Soubra (2000) in the passive case. The best upper-bound solutions as given by the different upper-bound approaches are presented for practical use in geotechnical engineering.

INTRODUCTION

The well-known Mononobe-Okabe analysis of seismic lateral earth pressures (see Mononobe and Matsuo 1929 and Okabe 1924) is a direct modification of the Coulomb wedge method where the earthquake effects are replaced by a quasi-static inertia force whose magnitude is computed on the basis of the seismic coefficient concept. As in the Coulomb analysis, the failure surface is assumed planar in the Mononobe-Okabe method, regardless of the fact that the most critical sliding surface may be curved. Similar to Coulomb's, the Mononobe-Okabe analysis may underestimate the active earth pressure and overestimate the passive earth pressure. In this paper, the seismic active and passive earth pressure problems are investigated by the upper-bound method of limit analysis using rotational log-spiral failure mechanisms.

THE UPPER-BOUND THEOREM OF LIMIT ANALYSIS

The upper-bound theorem which assumes a perfectly plastic
soil model with an associated flow rule, states that the rate of
energy dissipation in any kinematically admissible velocity
field can be equated to the rate of work done by the external
forces, and so enables a strict upper-bound on the true limit
load to be deduced (see Drucker et al. 1952, Chen 1975 and
Salençon 1990). A kinematically admissible velocity field is
one, which satisfies compatibility, the flow rule and the
velocity boundary conditions.

In this paper, the upper-bound theorem of limit analysis is
applied to the seismic active and passive earth pressure

problems using kinematically admissible velocity fields. It
should be noted here, that the upper-bound theorem gives an
unsafe estimate of the active and passive failure load.

THEORETICAL ANALYSIS OF THE SEISMIC ACTIVE
AND PASSIVE EARTH PRESSURE PROBLEMS

An earthquake has two possible effects on a soil-wall system.
One is to increase the driving forces. The other is to decrease
the shearing resistance of the soil. The assumptions made in
the analysis can be summarized as follows:

* Only the decrease (respectively increase) of the passive (respectively active) pressures due to the increase in driving forces is investigated under seismic loading conditions. The shear strength of the soil is assumed unaffected as the result of the seismic loading.

* A constant seismic coefficient is assumed for the entire soil mass involved. Only the horizontal seismic coefficient $K_H$ is considered, the vertical seismic coefficient being often disregarded.

* An adhesive force $P_a$ is assumed to act at the soil-structure interface. The intensity of this force is $c \cdot l \cdot tg \delta/tg\phi$.

Failure mechanisms

Two rotational log-spiral failure mechanisms are considered in
the present analysis. One for the active state *M1* (cf. Fig. 1a)
and the other for the passive state *M2* (cf. Fig. 1b).
<figure><img src="image_2.png" /><figcaption>Fig. 1. Log-spiral failure mechanisms (a) M1 for active; and (b) M2 for passive earth pressure analyses</figcaption></figure>
For both M1 and M2 mechanisms, the region ABC rotates as a rigid body about the as yet undefined center of rotation O relative to the materials below the logarithmic failure surface BC. Thus, the surface BC is a surface of velocity discontinuity. These failure mechanisms can be specified completely by two variables θ₀ and θ₁.

As shown in Fig. 1, the external forces contributing to the total rate of external work consist of the weight of the soil mass in motion W, the active or passive earth force ($P_{aE}$ or $P_{pE}$), the adhesive force $P_a$ ($P_a=c \cdot l \cdot tg\delta/tg\phi$), the surcharge $q \cdot L$ acting on the ground surface and the different inertia forces. These inertia forces concern the soil mass and the surcharge loading. Since no general plastic deformation of the soil is permitted to occur, the energy is dissipated solely at the velocity discontinuity surface BC between the material at rest and the material in motion. The rate of energy dissipation per unit length along this velocity discontinuity can be expressed as [cf. Chen (1975)]

$$ \dot{D} = cV \cos\phi \quad (1) $$

where V is the velocity which makes an angle φ with the velocity discontinuity.

By equating the total rate of external work to the total rate of internal energy dissipation, one can see that the work equation of the rotational log-spiral mechanism is identical to the moment equilibrium equation around the center of the log-spiral. It should be emphasized that the moment equilibrium equation of the soil mass in motion can be written around the center of the log-spiral slip surface without specifying the normal stress distribution along the slip surface since the log-spiral function has a particular property, that the resultant of the forces ($\sigma \cdot dl$) and ($\tan\phi \cdot \sigma \cdot dl$) passes through the pole of the spiral. Hence, the seismic active and passive forces can be expressed respectively as follows

$$ P_{aE} = K_{a\gamma E} \frac{\gamma l^2}{2} + K_{aqE} q l - K_{ucE} c l \quad (2) $$

$$ P_{pE} = K_{p\gamma E} \frac{\gamma l^2}{2} + K_{pqE} q l + K_{pcE} c l \quad (3) $$

where $K_{a\gamma E}$, $K_{p\gamma E}$, $K_{aqE}$, $K_{pqE}$, $K_{ucE}$ and $K_{pcE}$ are the seismic earth pressure coefficients. The coefficients $K_γ$, $K_q$ and $K_c$ represent respectively the effect of soil weight, surcharge loading and cohesion and the subscripts a and p represent the active and passive cases respectively.

## NUMERICAL RESULTS

The most critical earth pressure coefficients can be obtained by numerical maximization (respectively minimization) of the active (respectively passive) coefficients with regard to the parameters θ₀ and θ₁. The optimization procedure can be performed using the optimization tool available in most

spreadsheet software packages. In this paper, one uses the Solver optimization tool of Microsoft Excel. Two computer programs have been developed using Visual Basic 'VBA' to define the active and passive earth pressure coefficients as function of the two angular parameters θ₀ and θ₁ defined in Fig. 1. In the following sections, the active and passive earth pressure coefficients obtained from the present analysis are presented and compared to those given by other authors.

## Comparison of Results with Existing Solutions

In order to confirm the validity of the present upper-bound solution, the seismic active and passive earth pressure coefficients are calculated and compared with solutions given by other authors.

Chang and Chen (1982) [cf. Chen and Liu 1990] considered a log-sandwich failure mechanism and gave rigorous upper-bound solutions of the $K_{p\gamma E}$ and $K_{a\gamma E}$ coefficients. On the other hand, Soubra (2000) has recently considered a translational multi-block failure mechanism for the study of the seismic passive earth pressure problem. His mechanism improves significantly the available upper-bound solutions given by Chen and Liu (1990). The comparisons between the present upper-bound solutions and the best available upper-bound solutions are presented in Figs. 2 and 3.

<figure><img src="image_3.png" /><figcaption>Fig. 2. Comparison of present $K_{a\gamma E}$ coefficient with that of Chen and Liu (1990) for $\phi=45^\circ$, $\delta/\phi=1$, $\lambda/\phi=0$ and $K_H=0.3$</figcaption></figure>

For the active case, the results given by the present approach are better than those presented by Chen and Liu (1990), the improvement equals 11.5% for $\beta/\phi=1/2$ when are $\phi=45^\circ$, $\delta/\phi=1$, $\lambda/\phi=0$ and $K_H=0.3$. For the passive case, the present upper-bound solutions are better than those of Soubra (2000) for $\beta>15^\circ$, the improvement attains 20.1% for $\beta/\phi=1$ when $\phi=45^\circ$, $\delta/\phi=1$, $\lambda/\phi=0$ and $K_H=0.3$. Therefore, the present failure mechanisms give interesting solutions of the seismic active and passive earth pressure coefficients.
<figure><img src="image_4.png" /><figcaption>Fig. 3. Comparison of present $K_{pγE}$ coefficient with that of Soubra (2000) for $φ=45°$, $δ/φ=1$, $λ/φ=0$ and $K_H=0.3$</figcaption></figure>

## Seismic Active and Passive Earth Pressure Coefficients

Earthquake have the unfavorable effect of decreasing (respectively increasing) the passive (respectively active) earth pressures. It is to be mentioned that the $K_{pcE}$ and $K_{acE}$ coefficients are unaffected by the seismic loading. To investigate how the passive and active earth pressure coefficients $K_{pγE}$, $K_{pqE}$, $K_{aγE}$ and $K_{aqE}$ are affected by earthquake, extensive numerical results based on the present failure mechanisms are presented in Tables 1 and 2.

It can be observed that for $β/φ=0$, the passive earth pressure coefficients $K_{pγE}$ and $K_{pqE}$ (respectively $K_{aγE}$ and $K_{aqE}$) decrease (respectively increase) with the $K_H$ increase. For the passive case, the decrease is more significant for looser soils with lower $φ$ values than for denser soils with higher $φ$ values. For example, for $δ/φ=1$ and $λ/φ=β/φ=0$, the reduction of the $K_{pγE}$ coefficient is equal to 10% for $φ=45°$ and attains 32% for $φ=20°$ when $K_H$ increases from 0 to 0.3.

From Table 1, one can see that for some configurations (see for instance $φ=40°$, $δ/φ=2/3$, $β/φ=1/3$, $λ/φ=0$), the passive earth pressure coefficient $K_{pqE}$ increases with the $K_H$ increase. This seems to be surprising. However, this result is logical for the present failure mechanism since in this case, the center of the log-spiral lies below the line of action of the surcharge inertia force (see Fig. 4) and thus, this force has the favorable effect of increasing the passive earth force.

<table><caption>Table 1. Seismic passive earth pressure coefficients $K_{pγE}$ and $K_{pqE}$ ($λ/φ=0$)</caption><thead><tr><th rowspan="2">β/φ</th><th rowspan="2">δ/φ</th><th rowspan="2">K<sub>H</sub></th><th colspan="4">K<sub>pγE</sub></th><th colspan="4">K<sub>pqE</sub></th></tr><tr><th colspan="4">φ (°)</th><th colspan="4">φ (°)</th></tr><tr><th></th><th></th><th></th><th>20</th><th>25</th><th>30</th><th>35</th><th>40</th><th>20</th><th>25</th><th>30</th><th>35</th><th>40</th></tr></thead><tbody><tr><td rowspan="3">0</td><td rowspan="3">0</td><td>0</td><td>2.04</td><td>2.46</td><td>3.00</td><td>3.69</td><td>4.60</td><td>1.88</td><td>2.34</td><td>2.86</td><td>3.55</td><td>4.44</td></tr><tr><td>0.15</td><td>1.86</td><td>2.28</td><td>2.81</td><td>3.48</td><td>4.37</td><td>1.78</td><td>2.18</td><td>2.69</td><td>3.34</td><td>4.20</td></tr><tr><td>0.3</td><td>1.56</td><td>2.02</td><td>2.55</td><td>3.22</td><td>4.10</td><td>1.48</td><td>1.91</td><td>2.42</td><td>3.06</td><td>3.90</td></tr><tr><td rowspan="3">1/3</td><td rowspan="3">0</td><td>0</td><td>2.39</td><td>3.07</td><td>4.03</td><td>5.44</td><td>7.62</td><td>2.11</td><td>2.74</td><td>3.59</td><td>4.81</td><td>6.64</td></tr><tr><td>0.15</td><td>2.13</td><td>2.78</td><td>3.69</td><td>5.04</td><td>7.09</td><td>1.99</td><td>2.57</td><td>3.38</td><td>4.55</td><td>6.30</td></tr><tr><td>0.3</td><td>1.72</td><td>2.38</td><td>3.27</td><td>4.54</td><td>6.50</td><td>1.64</td><td>2.25</td><td>3.04</td><td>4.17</td><td>5.87</td></tr><tr><td rowspan="3">2/3</td><td rowspan="3">0</td><td>0</td><td>2.75</td><td>4.08</td><td>5.34</td><td>7.95</td><td>12.60</td><td>2.31</td><td>3.15</td><td>4.39</td><td>6.34</td><td>9.62</td></tr><tr><td>0.15</td><td>2.42</td><td>3.37</td><td>4.84</td><td>7.29</td><td>11.68</td><td>2.20</td><td>2.99</td><td>4.18</td><td>6.07</td><td>9.26</td></tr><tr><td>0.3</td><td>1.91</td><td>2.84</td><td>4.24</td><td>6.53</td><td>10.67</td><td>1.82</td><td>2.64</td><td>3.80</td><td>5.64</td><td>8.76</td></tr><tr><td rowspan="3">1/3</td><td rowspan="3">0</td><td>0</td><td>2.37</td><td>3.04</td><td>3.97</td><td>5.34</td><td>7.43</td><td>1.96</td><td>2.60</td><td>3.43</td><td>4.62</td><td>6.39</td></tr><tr><td>0.15</td><td>2.51</td><td>2.92</td><td>3.85</td><td>5.21</td><td>7.30</td><td>2.01</td><td>2.60</td><td>3.42</td><td>4.61</td><td>6.39</td></tr><tr><td>0.3</td><td>2.06</td><td>2.74</td><td>3.67</td><td>5.04</td><td>7.13</td><td>1.91</td><td>2.50</td><td>3.33</td><td>4.52</td><td>6.33</td></tr><tr><td rowspan="3">1/3</td><td rowspan="3">0</td><td>0</td><td>2.82</td><td>3.87</td><td>5.53</td><td>8.28</td><td>13.23</td><td>2.21</td><td>3.08</td><td>4.36</td><td>6.36</td><td>9.75</td></tr><tr><td>0.15</td><td>2.63</td><td>3.66</td><td>5.27</td><td>7.97</td><td>12.86</td><td>2.27</td><td>3.10</td><td>4.36</td><td>6.38</td><td>9.82</td></tr><tr><td>0.3</td><td>2.36</td><td>3.38</td><td>4.96</td><td>7.62</td><td>12.44</td><td>2.15</td><td>2.99</td><td>4.26</td><td>6.30</td><td>9.79</td></tr><tr><td rowspan="3">2/3</td><td rowspan="3">0</td><td>0</td><td>3.28</td><td>4.82</td><td>7.47</td><td>12.42</td><td>22.63</td><td>2.42</td><td>3.53</td><td>5.30</td><td>8.34</td><td>14.04</td></tr><tr><td>0.15</td><td>3.04</td><td>4.52</td><td>7.10</td><td>11.96</td><td>22.07</td><td>2.50</td><td>3.59</td><td>5.37</td><td>8.48</td><td>14.34</td></tr><tr><td>0.3</td><td>2.69</td><td>4.15</td><td>6.66</td><td>11.42</td><td>21.43</td><td>2.40</td><td>3.51</td><td>5.32</td><td>8.49</td><td>14.51</td></tr></tbody></table>
<table><caption>Table 2. Seismic active earth pressure coefficients $K_{u\gamma E}$ and $K_{aqE}$ ($\lambda/\phi=0$)</caption><thead><tr><th rowspan="2">β/φ</th><th rowspan="2">δ/φ</th><th rowspan="2">K<sub>H</sub></th><th colspan="5">K<sub>uγE</sub></th><th colspan="5">K<sub>aqE</sub></th></tr><tr><th colspan="5">φ (°)</th><th colspan="5">φ (°)</th></tr><tr><th></th><th></th><th></th><th>20</th><th>25</th><th>30</th><th>35</th><th>40</th><th>20</th><th>25</th><th>30</th><th>35</th><th>40</th></tr></thead><tbody><tr><td rowspan="3">0</td><td rowspan="3">0</td><td>0</td><td>0.49</td><td>0.41</td><td>0.33</td><td>0.27</td><td>0.22</td><td>0.49</td><td>0.41</td><td>0.33</td><td>0.27</td><td>0.22</td></tr><tr><td>0.15</td><td>0.60</td><td>0.50</td><td>0.41</td><td>0.34</td><td>0.28</td><td>0.62</td><td>0.52</td><td>0.43</td><td>0.36</td><td>0.30</td></tr><tr><td>0.3</td><td>0.82</td><td>0.66</td><td>0.55</td><td>0.46</td><td>0.38</td><td>0.83</td><td>0.68</td><td>0.57</td><td>0.48</td><td>0.40</td></tr><tr><td rowspan="3">2/3</td><td rowspan="3">0</td><td>0</td><td>0.44</td><td>0.36</td><td>0.30</td><td>0.25</td><td>0.20</td><td>0.44</td><td>0.36</td><td>0.30</td><td>0.24</td><td>0.20</td></tr><tr><td>0.15</td><td>0.57</td><td>0.47</td><td>0.39</td><td>0.33</td><td>0.28</td><td>0.58</td><td>0.49</td><td>0.41</td><td>0.34</td><td>0.29</td></tr><tr><td>0.3</td><td>0.85</td><td>0.68</td><td>0.56</td><td>0.48</td><td>0.41</td><td>0.85</td><td>0.68</td><td>0.57</td><td>0.49</td><td>0.42</td></tr><tr><td rowspan="3">1/3</td><td rowspan="3">0</td><td>0</td><td>0.54</td><td>0.45</td><td>0.37</td><td>0.31</td><td>0.25</td><td>0.54</td><td>0.46</td><td>0.38</td><td>0.31</td><td>0.25</td></tr><tr><td>0.15</td><td>0.70</td><td>0.59</td><td>0.49</td><td>0.41</td><td>0.34</td><td>0.72</td><td>0.61</td><td>0.52</td><td>0.43</td><td>0.36</td></tr><tr><td>0.3</td><td>-</td><td>0.97</td><td>0.75</td><td>0.61</td><td>0.50</td><td>-</td><td>0.95</td><td>0.76</td><td>0.63</td><td>0.52</td></tr><tr><td rowspan="3">2/3</td><td rowspan="3">0</td><td>0</td><td>0.49</td><td>0.41</td><td>0.34</td><td>0.28</td><td>0.23</td><td>0.49</td><td>0.41</td><td>0.35</td><td>0.29</td><td>0.24</td></tr><tr><td>0.15</td><td>0.69</td><td>0.58</td><td>0.48</td><td>0.41</td><td>0.34</td><td>0.70</td><td>0.59</td><td>0.50</td><td>0.43</td><td>0.36</td></tr><tr><td>0.3</td><td>-</td><td>1.01</td><td>0.81</td><td>0.67</td><td>0.56</td><td>-</td><td>1.03</td><td>0.81</td><td>0.68</td><td>0.57</td></tr></tbody></table>

<figure><img src="image_5.png" /><figcaption>Fig. 4. Critical slip surfaces for $\phi=40°$, $\delta/\phi=2/3$, $\beta/\phi=1/3$, $\lambda/\phi=0$ and $K_H=0, 0.15, 0.3$ (Passive Case)</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 5. Critical slip surfaces for $\phi=20°$, $\delta/\phi=1$, $\beta/\phi=\lambda/\phi=0$ and $K_H=0, 0.2$ (Passive Case)</figcaption></figure>

Finally, it should be mentioned that the seismic acceleration generated by earthquakes not only imposes extra loading to a soil mass but also shift the sliding surface to less favorable positions.

<figure><img src="image_7.png" /><figcaption>Fig. 6. Critical slip surfaces for $\phi=20°$, $\delta/\phi=1$, $\beta/\phi=\lambda/\phi=0$ and $K_H=0, 0.2$ (Active Case)</figcaption></figure>

Figures 5 and 6 show that the critical slip surface becomes more extended as the acceleration intensity increases.

Since there is no unique mechanism giving the best upper-bound solutions, one presents in Tables 3 and 4 the optimal upper-bound solutions. Table 3 gives the lowest upper-bound solutions given by both Soubra (2000) and the present failure mechanism in the passive case and Table 4 gives the greatest solutions between the present upper-bounds and the solutions by Chen and Liu (1990) in the active case. These tables give the best upper-bound solutions available in the literature and can be used by practicing geotechnical engineers.

CONCLUSIONS

Two rotational kinematically admissible failure mechanisms have been considered for the calculation of the seismic active and passive earth pressures using the upper-bound method of the limit analysis theory. The solutions presented are rigorous upper-bound ones in the framework of the limit analysis theory. The numerical results obtained lead to the following conclusions:
<table><caption>Table 3. Lowest upper-bound solutions for seismic passive earth pressure coefficients $K_{p\gamma\epsilon}$ and $K_{p\gamma E}$ ($\beta/\phi=\lambda/\phi=0$)</caption><thead><tr><th rowspan="2">δ/φ</th><th rowspan="2">K<sub>H</sub></th><th colspan="5">K<sub>pγε</sub></th></tr><tr><th colspan="5">φ (°)</th></tr><tr><th></th><th></th><th>20</th><th>25</th><th>30</th><th>35</th><th>40</th></tr></thead><tbody><tr><td rowspan="3">0</td><td>0</td><td>2.04</td><td>2.46</td><td>3.00</td><td>3.69</td><td>4.60</td></tr><tr><td>0.15</td><td>1.80</td><td>2.21</td><td>2.73</td><td>3.39</td><td>4.27</td></tr><tr><td>0.3</td><td>1.48</td><td>1.91</td><td>2.42</td><td>3.06</td><td>3.91</td></tr><tr><td rowspan="3">1/3</td><td>0</td><td>2.39</td><td>3.07</td><td>4.03</td><td>5.44</td><td>7.62</td></tr><tr><td>0.15</td><td>2.08</td><td>2.72</td><td>3.62</td><td>4.96</td><td>7.04</td></tr><tr><td>0.3</td><td>1.64</td><td>2.28</td><td>3.13</td><td>4.39</td><td>6.33</td></tr><tr><td rowspan="3">2/3</td><td>0</td><td>2.75</td><td>3.79</td><td>5.34</td><td>7.95</td><td>12.60</td></tr><tr><td>0.15</td><td>2.37</td><td>3.32</td><td>4.79</td><td>7.25</td><td>11.68</td></tr><tr><td>0.3</td><td>1.83</td><td>2.74</td><td>4.10</td><td>6.35</td><td>10.44</td></tr></tbody></table>

<table><caption>Table 4. Greatest upper-bound solutions for seismic active earth pressure coefficients $K_{a\gamma\epsilon}$ ($\lambda/\phi=0$)</caption><thead><tr><th rowspan="2">δ/φ</th><th rowspan="2">K<sub>H</sub></th><th colspan="5">K<sub>pγε</sub></th></tr><tr><th colspan="5">φ (°)</th></tr><tr><th></th><th></th><th>20</th><th>25</th><th>30</th><th>35</th><th>40</th></tr></thead><tbody><tr><td rowspan="3">0</td><td>0</td><td>1.88</td><td>2.34</td><td>2.86</td><td>3.55</td><td>4.44</td></tr><tr><td>0.15</td><td>1.78</td><td>2.18</td><td>2.69</td><td>3.34</td><td>4.20</td></tr><tr><td>0.3</td><td>1.48</td><td>1.91</td><td>2.42</td><td>3.06</td><td>3.90</td></tr><tr><td rowspan="3">1/3</td><td>0</td><td>2.11</td><td>2.74</td><td>3.59</td><td>4.81</td><td>6.64</td></tr><tr><td>0.15</td><td>1.99</td><td>2.57</td><td>3.38</td><td>4.55</td><td>6.30</td></tr><tr><td>0.3</td><td>1.64</td><td>2.25</td><td>3.04</td><td>4.17</td><td>5.87</td></tr><tr><td rowspan="3">2/3</td><td>0</td><td>2.31</td><td>3.15</td><td>4.39</td><td>6.34</td><td>9.62</td></tr><tr><td>0.15</td><td>2.20</td><td>2.99</td><td>4.18</td><td>6.07</td><td>9.26</td></tr><tr><td>0.3</td><td>1.82</td><td>2.64</td><td>3.80</td><td>5.64</td><td>8.76</td></tr></tbody></table>

<table><thead><tr><th rowspan="2">β/φ</th><th rowspan="2">δ/φ</th><th rowspan="2">K<sub>H</sub></th><th colspan="5">K<sub>aγε</sub></th></tr><tr><th colspan="5">φ (°)</th></tr><tr><th></th><th></th><th></th><th>20</th><th>25</th><th>30</th><th>35</th><th>40</th></tr></thead><tbody><tr><td rowspan="3">0</td><td>0</td><td>0</td><td>0.49</td><td>0.41</td><td>0.33</td><td>0.27</td><td>0.22</td></tr><tr><td></td><td>0.15</td><td>0.62</td><td>0.52</td><td>0.43</td><td>0.36</td><td>0.30</td></tr><tr><td></td><td>0.3</td><td>0.83</td><td>0.66</td><td>0.57</td><td>0.48</td><td>0.40</td></tr><tr><td rowspan="3">1/3</td><td>0</td><td>0</td><td>0.44</td><td>0.36</td><td>0.30</td><td>0.25</td><td>0.20</td></tr><tr><td></td><td>0.15</td><td>0.58</td><td>0.48</td><td>0.41</td><td>0.34</td><td>0.28</td></tr><tr><td></td><td>0.3</td><td>0.85</td><td>0.68</td><td>0.57</td><td>0.49</td><td>0.42</td></tr><tr><td rowspan="3">2/3</td><td>0</td><td>0</td><td>0.54</td><td>0.45</td><td>0.37</td><td>0.31</td><td>0.25</td></tr><tr><td></td><td>0.15</td><td>0.72</td><td>0.60</td><td>0.51</td><td>0.42</td><td>0.35</td></tr><tr><td></td><td>0.3</td><td>1.21</td><td>0.97</td><td>0.75</td><td>0.61</td><td>0.51</td></tr><tr><td rowspan="3">1/3</td><td>0</td><td>0</td><td>0.49</td><td>0.41</td><td>0.34</td><td>0.28</td><td>0.23</td></tr><tr><td></td><td>0.15</td><td>0.70</td><td>0.59</td><td>0.49</td><td>0.42</td><td>0.35</td></tr><tr><td></td><td>0.3</td><td>1.25</td><td>1.01</td><td>0.81</td><td>0.67</td><td>0.56</td></tr></tbody></table>

For the passive case, the present rotational failure mechanism gives better upper-bound solutions of the $K_{p\gamma\epsilon}$ coefficient than the multi-block mechanism, the improvement attains 20.1% for $\phi=45°$, $\delta/\phi=1$, $\beta/\phi=1$, $\lambda/\phi=0$ and $K_H=0.3$. For the active case, the present results are also better than the ones given by Chen and Liu (1990), the improvement equals 11.5% for $\beta/\phi=1/2$ when $\phi=45°$, $\delta/\phi=1$, $\lambda/\phi=0$ and $K_H=0.3$. Finally, extensive numerical results of the best seismic active and passive earth pressure coefficients as given by the different upper-bound approaches are presented for practical use in geotechnical engineering.

ACKNOWLEDGEMENT

The Ministry of Foreign Affairs in France and the Ministry of Science and Technology in Republic of Slovenia are gratefully acknowledged for their financial support.

REFERENCES

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