2.3.1 Bearing capacity method
As shown in Figure 2.2，the soil weight above the level of the excavation surface（plane abc）can be seen as the load to cause excavation failure.Supposing a trial failure surface caused by the soil weight within the width of B1 acts on plane abc as is shown in Figure 2.2（a），we can find the ultimate load for the width of B1 following Terzaghi’s bearing capacity method with the shear strength along bd considered.The ratio of the ultimate load to the weight of soil within the width of B1is the factor of safety for the trial failure surface.Then increase the value of B1（which denotes increasing of the size of trial failure surfaces）and find the corresponding factor of safety accordingly until the trial failure surface covers the whole excavation（i.e.B1＝B
），as shown in Figures 2.2（b）and 2.2（c）.Since the weight of B1⁃wide soil on each side of the excavation zone may produce failures，the schematic diagram to calculate the factor of safety is illustrated in Figure 2.2（d）.Following the principle of virtual work，the factor of safety induced from Figure 2.2（c）and that from Figure 2.2（d）would be identical.The factor of safety against basal heave（Fb）for the excavation is the smallest one among the safety factors corresponding to the trial failure surfaces.
Terzaghi（1943）did not adopt the above method，where the smallest factor of safety is taken to be the factor of safety against basal heave.Instead，he directly assumed the trial failure surface where B1＝B
（i.e.X＝B
）is the critical failure surface and its corresponding factor of safety is the factor of safety against basal heave，as shown in Figure 2.3.According to Terzaghi’s bearing capacity theory，the bearing capacity of saturated clay under plane ab can be denoted as Pmax＝5.7su.When the soil weight above plane ab is greater than the soil bearing capacity，the excavation will fail.Besides，the failure surface will be restrained by stiff soils.Let D represent the distance between the excavation surface and the stiff soil，and we can discuss Terzaghi’s method in two parts：D≥B
and D＜B
.
When D≥B
，as shown in Figure 2.3（a），the formation of a failure surface is not restrained by the stiff soil.Suppose the unit weight of the soil isγ，then the soil weight（containing the surcharge qs）ranges B1 on plane ab will be
the ultimate load，Qu，of the saturated clay below plane ab will be
When a basal heave failure occurs，vertical plane can offer shear resistance（su1He）and the factor of safety against basal heave（Fb）will be
where su1 and su2 represent respectively the undrained shear strengths of the soils above and below the excavation surface；qs denotes surcharge on the ground surface.
When D＜B
，under such a condition，the failure surface will be restrained by the stiff soil，as shown in Figure 2.3（b），and its factor of safety（Fb）will be
For most excavation cases，Terzaghi’s factor of safety（Fb）should be greater than or equal to 1.5（Mana and Clough，1981；JSA，1988）.
Assuming that the penetration depth of the retaining wall is deep enough，the failure surface may be formed as illustrated in Figure 2.4（a），which is one of the possible failure modes.According to the analysis on the basis of the principle of virtual work，the factor of safety for a failure surface as illustrated in Figure 2.4（a）is close to that of Eqs 2.7 and 2.8.The only difference is that the failure surface in Figure 2.4（a）ranges wider（with the extra failure surface be）and the average soil strength on the failure surface is higher than that in Figure 2.4（b），assuming the undrained shear strength of clay increases with the increase of depth.
Figure 2.4 Relation between the embedded part of retaining wall and failure surface
As shown in Figure 2.4（b），assuming the penetration depth of the retaining wall is not deep enough，the calculation of the factor of safety will still follow Eqs 2.7 or 2.8.That is to say，the value of the factor of safety against basal heave has nothing to do with the existence of the retaining wall according to the equations.However，theoretically speaking，the retaining wall with high stiffness may be capable of restraining basal heave failures.Thus，the actual factor of safety should be greater than the result from Eqs.2.7 or 2.8 though there does not exist a suitable way to estimate it.
The bearing capacity method or Terzaghi’s method is suitable for shallow excavations，where the excavation width（B）is larger than the excavation depth（He）.For deep excavations，B＜He，the bearing capacity method or Terzaghi’s method may not yield reasonable results because the method assumes that the failure surface extends up to the ground surface and that the shear strength of clay is fully mobilized all the way to the ground surface，neither of which is necessarily true for deep excavations.