**4.3 Case study of when Coulomb's active earth pressure of retaining walls is not a maximum**

In **Figure 10**, ΔABC is the potential active sliding failure block for a retaining wall, as proposed by Coulomb [9]. Here, *H* is the height of the retaining wall, γ is the unit weight of the soil, *W* is the weight per unit length of ΔABC, β is the inclination angle of AC, θ is the inclination angle of AB, ρ is the inclination angle of the potential active shear failure band BC, ρβ is the angle between CA and CB, *R* is the resultant shear

#### **Figure 10.**

*Various forces acting on a retaining wall as proposed by Coulomb [9]: (a) Coulomb's potential sliding failure block under active conditions; (b) the closed force polygon of* W*,* R*, and Pa.*

resisting force acting on BC, internal friction angle ϕ is the intersection angle of *R* and the normal to BC, *Pa* is the active earth pressure acting on AB, and wall friction angle δ is the angle of intersection between *Pa* and the normal to AB. From **Figure 10**, it can be seen that the Coulomb's active earth pressure *Pa* of retaining walls does not consider the effects of strain softening and shear banding.

We used a retaining wall height of *<sup>H</sup>* = 6 m, soil unit weight *<sup>γ</sup>* <sup>¼</sup> 22 kN*=*m3, internal friction angle <sup>¼</sup> <sup>50</sup>° , cohesion *c* = 0 kPa, wall friction angle *δ* = 33*:*3° , AC inclination angle *β* = 0° , and AB inclination angle *θ* = 105° . When adopting the elastic-perfectly plastic model, the result of the analysis shows that the sliding failure plane inclination angle was *<sup>ρ</sup>* <sup>¼</sup> <sup>72</sup>*:*83° , the failure block weight was *W* = 228.46 kN, and the traditional Coulomb active earth pressure was *Pa* ¼ 98.19 kN.
