2.3.4 Finite⁃element method with strength reductio...
Besides the limit equilibrium methods discussed above，finite element method（FEM）combined with strength reduction（SR）technique is also commonly used to calculate the factor of safety against basal heave.More information about theory and application of FEM can be found in Chapter 6.The factor of safety is defined here as the number by which the original shear strength parameters must be divided in order to bring the excavation to the point of failure，as given in Eqs.2.12a and 2.12b.
This definition of the factor of safety is exactly the same as that used in limit equilibrium methods，namely the ratio of restoring to driving moments.
SR technique allows to apply different factors of safety to the c′andϕ′terms.In common practice，however，the same factor is applied to both terms.To find the true Fb，it is necessary to initiate a systematic search for the value of Fb that will just cause the excavation to fail.This is achieved by the program solving the problem repeatedly using a predefined sequence of Fb values.Moreover，the criteria to define the failure should be chosen properly.There are three commonly utilized criteria as follows：
①A plastic zone going through the slope from the toe to the top，as shown in Figure 2.11.At failure，a band is formed within the slope，in which all elements are in the plastic state and the band would extend through the slope from the toe to the top.
②A significant increase in the nodal displacement within the mesh.
③Non⁃convergence of the solution.
Figure 2.11 Plastic zone inner a slope under limit state
Pei（2010）suggested these three criteria are consistent and would result in similar Fb.Most of commercial FE codes，such as Plaxis（2017），adopt the third as the criteria to define the failure.
Compared to traditional limit equilibrium methods，the advantages of FEM with SR technique can be summarized as follows：
①No assumption needs to be made in advance about the shape or location of the failure surface.Failure occurs naturally through the zones within the soil mass in which the soil shear strength is unable to sustain the applied shear stresses.
②Since there is no concept of slices in the FEM，there is no need for assumptions about slice side forces.The FEM preserves global equilibrium until failure is reached.
③As soil constitutive models are employed in FEM，the stress⁃strain relation is considered.
④It is capable to simulate the soil⁃structure interaction.