**3. The ultimate bearing capacity of the strip foundation**

#### **3.1 Equation derivation**

Before deriving the ultimate bearing capacity of a strip foundation, Terzaghi [8] assumed symmetric general shear failure bands in the foundation soil under the ultimate load (see **Figure 8**). The required foundation soil properties include the cohesion c, angle of internal friction ϕ, and unit weight γ. The cohesion *c* ¼ *cult* and internal friction angle *ϕ* ¼ *ϕult* were obtained from the perfectly plastic curve shown in **Figure 1b**. **Figure 8** shows that Terzaghi [8] divided the whole area enclosed by the general shear failure bands into active zone I, radial shear zones II and II1, and passive zones III and III1. The ultimate bearing capacity *Qult* of the foundation was equal to twice the passive earth pressures *Pp* that act on the shear failure bands ad and bd. The passive earth pressure *Pp* included the passive earth pressure *Pp*<sup>1</sup> generated by the soil cohesion *c*, the passive earth pressure *Pp*<sup>2</sup> was generated by the overburden pressure *γDf* (*i.e.*, *q*), and the passive earth pressure *Pp*<sup>3</sup> was generated by the weight of the soil

**Figure 8.**

*The general shear failure bands set before deriving the ultimate bearing capacity equation of the strip foundation (reproduced from Terzaghi [8]).*

enclosed by the shear failure band adef. Based on the balance of the vertical components of all forces, Terzaghi obtained the ultimate bearing capacity of the foundation as:

$$q\_{\rm ult} = 2\left(P\_{p1} + P\_{p2} + P\_{p3} + \frac{1}{2}Bc\tan\phi\right)/B = cN\_c + qN\_q + \frac{1}{2}B\gamma N\_\gamma \tag{14}$$

where *Nc*, *Nq*, and *<sup>N</sup><sup>γ</sup>* are bearing-capacity factors: *Nq* <sup>¼</sup> *<sup>a</sup>*<sup>2</sup> 2 cos <sup>2</sup> <sup>45</sup>° ð Þ <sup>þ</sup>*ϕ=*<sup>2</sup> ; *Nc* <sup>¼</sup> *Nq* � <sup>1</sup> cot *<sup>ϕ</sup>*; *<sup>N</sup><sup>γ</sup>* <sup>¼</sup> tan *<sup>ϕ</sup>* 2 *Kp<sup>γ</sup>* cos <sup>2</sup>*<sup>ϕ</sup>* � 1 ; *<sup>a</sup>* <sup>¼</sup> exp 0*:*75*<sup>π</sup>* � *<sup>ϕ</sup>* 2 tan *ϕ* ; *Kp<sup>γ</sup>* =10.8, 14.7, 25.0, 52.0, and 141.0 when ϕ is equal to 0° , 10° , 20° , 30° , and 40° , respectively.

## **3.2 The overestimation problems**

## *3.2.1 Problems arising from adopting symmetric general shear failure band*

When plastic strain softening induces structural instability, the asymmetric structural matrix results in asymmetric general shear failure bands (detailed in **Figure 9**). Thus, the adoption of symmetric general failure bands by Terzaghi before deriving the ultimate bearing capacity equation of the strip foundation should be improved by the asymmetric shear failure pattern. In this case, when the ultimate bearing capacity of the strip foundation is evaluated by Eq. 14, the evaluation result will be overestimated twice.

*Schematic diagram of the asymmetric general shear failure band of a strip foundation (reproduced from [3]).*

*Plasticity Model Required to Prevent Geotechnical Failures in Tectonic Earthquakes DOI: http://dx.doi.org/10.5772/intechopen.107223*

### *3.2.2 Problems arising from adopting perfectly plastic model*

When the perfectly plastic model is adopted, the shear strength parameters used are obtained from the experimental results of the ultimate shear strength shown in **Figure 1**, but the asymmetric general shear failure band is induced by the plastic strain softening effect; therefore, the shear strength will decrease from the peak value to the residual value as the strain softens. In other words, the cohesion *c* will be reduced from the peak value *cp* to the residual value *cr*, *cr* ≈0, and the angle of internal friction ϕ will also be reduced from the peak value *ϕ<sup>p</sup>* to the residual value *ϕr*. Therefore, when calculating the ultimate bearing capacity *Qult* using the ultimate shear strength parameter, there will also be an overestimation problem [3].
