**4. The active Coulomb's earth pressure of retaining wall**

### **4.1 The adopted perfectly plastic model**

In **Figure 1a**, the red solid line represents the stress–strain curve of real soil obtained from laboratory triaxial compression tests. It shows the transition from peak strength to residual strength due to plastic strain softening. However, the plastic strain softening curve of soil has generally been simplified as a perfectly plastic curve in the retaining wall literature, as shown in **Figure 1a**.

#### **4.2 Numerical simulation for the formation of shear failure bands**

Coulomb [9] proposed the theory of active earth pressure as applied to retaining walls using both a shear failure band and a perfectly plastic model. At that time Coulomb did not know whether the use of the perfectly plastic model could induce shear failure bands.

For perfectly plastic and strain softening models, the numerical simulation analysis results using the finite element method are presented in **Figures 6** and **7**, respectively. The results show that in the perfectly plastic model, the plate does not develop a shear failure band when the strain of the lateral compression of the plate goes deep into the plastic range. Only in the plastic strain softening model does the plate show the induced shear failure bands.

**Figure 7** shows that the formation of the shear failure bands has two requirements: strain deep into the plastic range and strain softening. The strain softening originates from the volume expansion of the sliding failure block. In other words, the negative incremental strain energy in strain softening is the motivating force for inducing the shear failure band.
