Seismic Reliability Investigation of Bearing Capacity of Foundations Based on Limit Analysis and Limit Equilibrium Methods

Abstract

In this paper, by considering the upper-bound limit analysis method in calculating the seismic bearing capacity of foundations, the comparison between a limit equilibrium method such as Hansen and limit analysis method in terms of reliability is investigated. Limit equilibrium has been the oldest method for performing bearing capacity of foundations. However, it seems that theoretical equations proposed by the majority of old limit equilibrium methods to calculate the bearing capacity of foundations (such as Hansen) do not consider the effects of inertial forces of the earthquake in the soil mass. These methods only take into account the horizontal load applied to the foundation and disregard the earthquake forces applied to the soil. By comparing the results of this study, it can be inferred that the reliability of the Hansen approach is very close to the limit analysis method that ignores the impact of horizontal earthquake forces applied to rigid soil blocks. Since the upper-bound limit analysis takes into account the seismic effect of inertial forces in an earthquake, its results may be considered more accurate than Hansen. This study shows that in both granular and cohesive soils in high seismicity-prone areas, usage of a more realistic method as the limit analysis is more reasonable. Considering the facility of RBD by Hansen method using EXCEL spreadsheet in comparison to the seismic reliability-based design using MATLAB programming following the upper-bound limit analysis method, graphs that offer coefficients for obtaining the equivalent value of target reliability in the limit analysis by using the Hansen formulation are presented. Additionally, this study shows that the horizontal seismic coefficient (kh) is an impressive parameter that affects the differences between the two mentioned approaches.

Appendix

In this appendix, exact formulations for computation of the parametric velocity related to rigid blocks, the geometry parameters of triangular blocks, and the procedure of calculating the bearing capacity is presented. Equations 4 and 5 represent the parametric velocity related to rigid block i, and Eq 6 shows the relative velocity between adjacent blocks(i and i + 1).

$$V_{1} = \frac{1}{{{ \sin }\left( {\beta_{1} - \varphi } \right)}}$$

(4)

$$V_{i + 1} = V_{i} \frac{{{ \sin }\left( {\pi - \alpha_{i} - \beta_{i} + 2\varphi } \right)}}{{{ \sin }\left( {\beta_{i + 1} - 2\varphi } \right)}}$$

(5)

$$V_{i.i + 1} = V_{i} \frac{{{ \sin }\left( {\alpha_{i} - \beta_{i} - \beta_{i + 1} } \right)}}{{{ \sin }\left( {\beta_{i + 1} - 2\varphi } \right)}}$$

(6)

Equations 7–9 show the geometry parameters of triangular block i.

$$l_{i} = B_{0} \frac{{\sin \beta_{1} }}{{\sin \left( {\alpha_{1} + \beta_{1} } \right)}}\mathop \prod \limits_{j = 2}^{i} \frac{{\sin \beta_{j} }}{{\sin \left( {\alpha_{j} + \beta_{j} } \right)}}$$

(7)

$$d_{i} = B_{0} \frac{{\sin \beta_{1} }}{{\sin \left( {\alpha_{1} + \beta_{1} } \right)}}\frac{{\sin \alpha_{i} }}{{\sin \beta_{i} }}\mathop \prod \limits_{j = 2}^{i} \frac{{\sin \beta_{j} }}{{\sin \left( {\alpha_{j} + \beta_{j} } \right)}}$$

(8)

$$S_{i} = \frac{{B_{0}^{2} }}{2}\frac{{{ \sin }^{2} \beta_{1} }}{{\sin^{2} \left( {\alpha_{1} + \beta_{1} } \right)}}\frac{{\sin \alpha_{i} \sin \left( {\alpha_{i} + \beta_{i} } \right)}}{{\sin \beta_{i} }}\mathop \prod \limits_{j = 2}^{i} \frac{{{ \sin }^{2} \beta_{j} }}{{\sin^{2} \left( {\alpha_{j} + \beta_{j} } \right)}}$$

(9)

Now, according to the upper bound limit analysis theorem, the amount of bearing capacity can be determined by the equalization of internal and external works (Eqs. 14–16) introduced in Eqs. 10–13. The external work consists of the work of imposed load to the foundation ($W_{P} )$, work of soil weight of the block i plus the work due to the inertial force acting on the block ($W_{{w_{i} }}$) and the surcharge work ($W_{q}$). Internal work includes the work dissipated in velocity discontinuities di and li ($W_{{d_{i} }}$ and $W_{{l_{i} }}$). Energy is dissipated along the lines $l_{i}$ (i = 1,…, n-1) and $d_{i}$ (i = 1,…, n). Assuming the velocity of the first block, V₁, equal to unity (δ = 1), the calculated load is the bearing capacity of the foundation.

$$W_{P} = P\left( {1 + K_{h} V_{1} \cos \left( {\lambda_{1} - \varphi } \right)} \right)$$

(10)

$$W_{{w_{i} }} = \left( {\gamma S_{i} V_{i} \sin \left( {\lambda_{i} - \varphi } \right)} \right) + \left( {K_{h} \gamma S_{i} V_{i} \cos \left( {\lambda_{i} - \varphi } \right)} \right)$$

(11)

$$W_{{d_{i} }} = cd_{i} V_{i} \cos \left( \varphi \right)$$

(12)

$$W_{{l_{i} }} = cl_{i} V_{i.i + 1} \cos \left( \varphi \right)$$

(13)

$W_{P}$: Work of imposed load to the foundation, $W_{{w_{i} }}$: Work of soil weight of block I plus work due to inertial force acting on the block, $W_{{d_{i} }}$: Work dissipated on velocity discontinuity $d_{i}$, $W_{{l_{i} }}$: Work dissipated on velocity discontinuity $l_{i}$, C: Soil cohesion, $\varphi$: Internal friction angle of the soil, $\lambda_{i}$ :The angle between $l_{i}$ and horizon, $W_{q}$ :Surcharge work.

$$\overbrace {{W_{P} + W_{{w_{i} }} + W_{q} }}^{External work} = \overbrace {{W_{{d_{i} }} + W_{{l_{i} }} }}^{Internal work}$$

(14)

$$P\left( {1 + K_{h} V_{1} \cos \left( {\lambda_{1} - \varphi } \right)} \right) + \mathop \sum \limits_{i = 1}^{n} \left( {\gamma S_{i} V_{i} \sin \left( {\lambda_{i} - \varphi } \right)} \right) + \left( {K_{h} \gamma S_{i} V_{i} \cos \left( {\lambda_{i} - \varphi } \right)} \right) + W_{q} = \mathop \sum \limits_{i = 1}^{n} cd_{i} V_{i} \cos \left( \varphi \right) + \mathop \sum \limits_{i = 1}^{n - 1} l_{i} V_{i.i + 1} \cos \left( \varphi \right)$$

(15)

$$q_{u} = P = \frac{1}{{\left( {1 + K_{h} V_{1} \cos \left( {\lambda_{1} - \varphi } \right)} \right)}}\left( {\mathop \sum \limits_{i = 1}^{n} cd_{i} V_{i} \cos \left( \varphi \right) + \mathop \sum \limits_{i = 1}^{n - 1} l_{i} V_{i.i + 1} \cos \left( \varphi \right) - \mathop \sum \limits_{i = 1}^{n} \left( {\gamma S_{i} V_{i} \sin \left( {\lambda_{i} - \varphi } \right)} \right) + \left( {K_{h} \gamma S_{i} V_{i} \cos \left( {\lambda_{i} - \varphi } \right)} \right) + W_{q} } \right)$$

(16)