2.3.3 Slip circle method
Let the trial failure surfaces of the basal heave failure be assumed to be basically circular arcs，and separately compute the ratios of the resistant moments to the driving moments for the trial circular arc failure surfaces.The smallest factor of safety among them is then the factor of safety against basal heave for the excavation.The method is designated as the slip circle method.The center of the circle with the slip circle method can be set at the lowest level of strut，at the excavation surface，or not be set at any specific position，for example，points A，B，O as shown in Figure 2.8.The slip circle method，without setting the center at a specific position，is to try out the circles for various positions and sizes and to find the corresponding factors of safety.The circle with the smallest factor of safety is the critical circle.
Figure 2.8 Location of the center of a failure circle for slip circle method
Theoretically，the critical circle has the smallest factor of safety though few adopt it for analysis，considering it causes a lot of complication in computation.According to the analysis results，the factor of safety corresponding to the failure circle whose center is set at the lowest level of strut（point O）is smaller than that at the excavation surface（point B）and is close to the factor of safety of the critical circle（Liu et al.，1997）.Thus，the circular arc failure surface whose center is set at the lowest level of strut is often adopted for analysis.
Suppose the failure surface of the basal heave failure is a combination of a circular arc which centers at the lowest level of strut and a vertical plane above the lowest level of struts，as shown in Figure 2.9（a）.Let the shear strength on the vertical failure plane［line bc in Figure 2.9（a）］be ignored，and take the retaining wall and soil below the lowest level of struts as well as above the circular arc as a free body，as shown in Figure 2.9（b）.If we regard the soil weight above the excavation surface in back of the retaining wall as the driving force and the shear strength along the failure surface as the resistant force，the factor of safety against basal heave，the ratio of the resistant moment to the driving moment with regard to the point at the lowest level of strut will be
where
Mr＝resistant moment；
Md＝driving moment；
Ms＝allowable bending moment of the retaining wall；
su＝undrained shear strength of clay；
X＝radius of the failure circle；
W＝total weight of the soil in front of the vertical failure plane and above the excavation surface，including the surcharge on the ground surface.
Equation 2.11 is a commonly used slip circle method.The original source of the slip circle method is untraceable.Nevertheless，TGS（2001）and JSA（1988）adopted the method in their building codes.Both assume that Ms＝0 and recommend that the factor of safety against basal heave（Fb）should be greater than or equal to 1.2.According to design experience in some countries，the value is quite reasonable.In fact，the allowable bending moment value of the retaining wall Ms is far less than the resistance created by shear strength.Thus，to simplify computation，it is reasonable to assume that Ms＝0.
Figure 2.9 Analysis of basal heave by the slip circle method
In analysis，we should try out different radii of circles and find the one with the smallest factor of safety as the critical circle，which represents the factor of safety against basal heave.In fact，a failure circle cannot pass through the embedded part of a retaining wall.Thus，for soils with constant strength or strength increasing with depth，the circle passing through the bottom of a retaining wall is the critical circle with the smallest factor of safety.Therefore，it is rational to let X＝s＋Hp，where s is the distance between the lowest level of strut and the excavation surface and Hp，the penetration depth of a retaining wall.Eq.2.11，therefore，can be used to compute the penetration depth of a retaining wall.Nevertheless，if there exist soft soils below the bottom of a retaining wall，the circle passing through the bottom of a retaining wall is not necessarily the critical failure circle，with the smallest factor of safety.Thus，we have to try out different values of X to find the failure circle with the smallest factor of safety，as shown in Figure 2.10.
Figure 2.10 Analysis of basal heave in layered soft soils