*4.3.1 Increase in the Coulomb's active earth pressure of retaining walls from the strain softening effect*

Coulomb's active earth pressure of retaining walls is the maximum lateral earth pressure. **Figure 11a** shows that during the 921 Jiji earthquake, the shear-banding zone on the left bank of the downstream Shigang Weir caused the sandy gravel layer on the back of the retaining wall to change from its original dense state to a loose state after shear banding. **Figure 11b** shows that after shear banding, the sand and gravel separated and the retaining wall slid into the riverbed.

As the failure block continued to slide along the same shear failure band, it continued to undergo brittle fracture and strain softening during the sliding process. Further, the angle of internal friction ϕ decreased from a peak value of 50° to a residual value of 33° , and the wall friction angle δ decreased from 33*:*3° to 22° . When the inclination angle β of AC , the inclination angle θ of AB, the internal friction angle ϕ, the wall friction angle δ, and the angle *α* shown in **Figure 10b** are all known, the inclination angle ρ of the potential active shear failure band BC can be calculated using *Plasticity Model Required to Prevent Geotechnical Failures in Tectonic Earthquakes DOI: http://dx.doi.org/10.5772/intechopen.107223*

#### **Figure 11.**

*Collapse of the retaining wall induced by shear banding during the 921 Jiji earthquake [10]: (a) different sandy gravel conditions on both sides of the failure plan; (b) the separated sandy gravel and retaining wall slipped to the riverbed.*


**Table 1.**

*Analysis results of the increase of the Coulomb's active earth pressure of the retaining wall induced by soil strain softening [10].*

Eq. 15. Then the Coulomb's active earth pressure *Pa* of the retaining wall shown in **Table 1** can be calculated using Eq. 16 [10]:

$$\cot\left(\theta - \rho\right) + \cot\left(\rho - \beta\right) = \cot\left(\rho - \phi\right) - \cot\left(\rho + a - \phi\right) \tag{15}$$

$$P\_a = W \frac{\sin\left(\rho - \phi\right)}{\sin\left(180^\circ - \rho - a + \phi\right)} = \frac{1}{2}\gamma H^2 \frac{\sin\left(\theta - \beta\right)}{\sin^2\theta} \cdot \frac{\sin\left(\theta - \rho\right)}{\sin\left(\rho - \beta\right)} \cdot \frac{\sin\left(\rho - \phi\right)}{\sin\left(\rho + a - \phi\right)}\tag{16}$$

According to **Table 1**, the Coulomb's active earth pressure of the retaining wall increased from 98*:*19 kN to 146*:*51 kN due to soil strain softening; therefore, the increment of the Coulomb's active earth pressure of the retaining wall, Δ*Pa*, was 48*:*32 kN.

**Figure 12.**

*The mechanism of the shear-band tilting effect inducing an increment in the Coulomb's active earth pressure of the retaining wall [10]: (a) shear-band tilting effect; (b) the action positions of Ps, Rs, and Pas on the retaining wall and the soil behind the wall; (c) the closed force polygon of* P*s,* R*s, and Pas.*
