The Applicability of Correspondence Rule with Inclined Load

Appendix: Iteration Procedure for Slip-Line Analysis

In this appendix, the iteration procedure is given to obtain the dimensionless effective contact stress of a strip footing with an inclined load through slip-line analysis. The cohesive–frictional rock is characterized by cohesion c and internal friction angle $\varphi$. The interstitial fluid pressure $p_{\rm{f}}$ at the footing–rock interface is equal to the confining pressure $\sigma _3$, and the inclination angle $\alpha$ of the effective stress is given.

The iteration procedure is as follows:

• 1. Initialize the inclination angle $\delta$ of the total stress at the interface. As the value of $\delta$ is between 0 and the value of $\alpha$, let $\delta =\alpha /2$.

• 2. Calculate the trial effective contact stress $\sigma _{n1}{^\prime}$ based on the geometrical relationship in Fig. 3:

  $$\sigma _{n1}{^\prime}=\frac{\tan \delta }{\tan \alpha -\tan \delta }\sigma _3$$

  (22)

• 3. Calculate the trial effective contact stress $\sigma _{n2}{^\prime}$ based on slip-line analysis following these steps: (1) solve the fan angle $\Delta \varPsi$ from Eq. (7); (2) calculate the inclination angle $\varPsi _a$ of the major principal stress in the active zone , the coefficient $N_c$ and the force inclination factor $i_c$; and (3) calculate the trial effective contact stress $\sigma _{n2}{^\prime}$ from Eq. (12).

• 4. Calculate the effective contact stress as the average of the two trial effective contact stresses:

  $$\sigma _n{^\prime}=\frac{1}{2}(\sigma _{n1}{^\prime}+\sigma _{n2}{^\prime})$$

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• 5. Compare the two trial effective contact stresses. If $|\sigma _{n1}{^\prime}-\sigma _{n2}{^\prime}|/\sigma _n{^\prime}>10^{-6}$, update the inclination angle $\delta$ of the total stress from Eq. (2), and then repeat steps 2, 3, and 4. Otherwise, go to the next step.

• 6. Calculate the dimensionless effective contact stress $\varSigma$ from Eq. (13).