2.3.2 Negative bearing capacity method
The negative bearing capacity method assumes that the unloading behavior caused by excavation is analogous to the building foundation subject to an upward loading and that the shape of the failure surface is similar to the failure mode of the deep foundation.Then，using the bearing capacity equation for the deep foundation，we can obtain the ultimate unloading pressure.The factor of safety is the ratio of the ultimate unloading pressure to the unloading pressure.As shown in Figure 2.5，assuming there are various failure surfaces for analysis（representing different B1⁃values）with their separate corresponding factors of safety，the smallest factor of safety among them is the factor of safety against basal heave for the excavation.
Figure 2.5 Analysis of basal heave failure by the negative bearing capacity method
Like Terzaghi’s method，Bjerrum and Eide（1956）did not yield the factor of safety against basal heave by finding the smallest，as just mentioned.Instead，they assumed the failure surface where the radius of the circular arc is equal to B
is the critical failure surface and the corresponding factor of safety is the one against basal heave［see Figure 2.5（c）］.The factor of safety can be expressed as follows：
where qs is the surcharge on the ground surface and Nc is Skempton’s bearing capacity factor as shown in Figure 2.6.
Since Nc has taken into account the effects of the embedment depth of foundations and excavation size，Eq.2.9 is equally valid for shallow and deep excavations，as well as rectangular excavations.
Figure 2.6 Skempton’s bearing capacity factor（Skempton，1951）
Like Terzaghi’s method，when there exists stiff soil below the excavation surface，the failure surfaces assumed by Bjerrum and Eide’s method would also be restrained by the stiff soil.According to Reddy and Srinivasan’s study（1967），NAVFAC DM 7.2（1982）modified Bjerrum and Eide’s method to apply the method to the excavations where there are stiff soils below the excavation surfaces or there are two layers of soils.As shown in Figure 2.7，the Extended Bjerrum and Eide’s method can be expressed as follows：
where
γ＝unit weight of the soil；
He＝excavation depth；
su1＝undrained shear strength of the upper clay；
su2＝undrained shear strength of the lower clay；
Nc，s＝bearing capacity factor that does not consider the excavation depth.This can be determined according to Figure 2.7（a）or 2.7（b）；
fd＝depth correction factor，which can be found in Figure 2.7；
fs＝shape correction factor，which can be estimated by the following equation：fs＝1＋0.2B／L.
If the penetration depth of the retaining wall is deep enough，Bjerrum and Eide’s method computes the factor of safety in a way similar to Terzaghi’s method.That is to say，the failure surface will be formed in a deeper level，similar to what is illustrated in Figure 2.4（a）.Under such conditions，Eq.2.9 is still workable to estimate the factor of safety with the slight differences of average soil strengths on failure surfaces.When Hp is not large enough，the calculation of the factor of safety will still follow Eq.2.9.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.
The negative bearing capacity method or Bjerrum and Eide’s method take into account the effects of excavation shape，width，and depth.Therefore，the methods are applicable to various shapes of excavations，shallow excavations as well as deep excavations.For most excavations，the factor of safety obtained according to Bjerrum and Eide’s method（Fb）should be larger than or equal to 1.2（JSA，1988）.
Figure 2.7 Extended bjerrum and eide’s method