Ultimate Bearing Capacity of Rock Mass Foundations Subjected to Seepage Forces Using Modified Hoek–Brown Criterion

Abstract

The experimental data shows that most rocks behave nonlinearly in nature. The modified nonlinear Hoek–Brown failure criterion was considered to investigate the bearing capacity problem of shallow rigid foundations on rock masses subjected to horizontal seepage forces. Two multi-wedge translational failure mechanisms, including symmetrical and non-symmetrical mechanisms were used in the closed-form of the upper bound method of the limit analysis theory. The symmetrical failure mechanism was used in the case of no seepage, while the seepage effect was considered in the non-symmetrical mechanism. The variation of seepage forces was obtained as a function of gradient ratio i(γ_w/γ_sub) in the developed formulation. The bearing capacity coefficients N_γ, N_q and N_σ are introduced for the case of seepage flow condition. The results show that the magnitude of the bearing capacity coefficients reduces continuously with an increase in the value of gradient ratio i(γ_w/γ_sub). The obtained results were compared and offered for functional use in foundation engineering.

Appendices

Appendix 1: M1 Mechanism (Dry Rock Masses)

1.1 Geometry

For the triangular block i, the lengths l_i and d_i and the area S_i are given as follows:

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

(23)

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

(24)

$$S_{i} = \frac{{B_{0}^{2} }}{2}\frac{{\sin \alpha_{i} \sin \beta_{i} }}{{4\cos^{2} \theta \sin (\alpha_{i} + \beta_{i} )}}\mathop \prod \limits_{j = 1}^{i - 1} \frac{{\sin^{2} \beta_{j} }}{{\sin^{2} (\alpha_{j} + \beta_{j} )}}$$

(25)

1.2 Internal Energy Dissipation

1. Along BC:

$$D_{\text{BC}} = c_{0,1} B_{0} f_{1} \left( {\alpha_{i} , \;\beta_{i} , \;\phi_{i} , \;\phi_{i,i + 1} , \;\theta } \right)V_{0} ,$$

(26)

where

$$f_{1} = \frac{{\cos \phi_{0,1} \cos (\beta_{1} - \theta - \phi_{1} )}}{{2\cos \theta \sin (\beta_{1} - \varphi_{1} - \phi_{0,1} )}} \times c_{0,1}$$

(27)

2. Along lines d_i (i = 1, …, k):

$$D_{{d_{i (i = 1, \ldots , k)} }} = c_{i} B_{0} f_{2} \left( {\alpha_{i} , \;\beta_{i} , \;\phi_{i} ,\; \phi_{i,i + 1} , \;\theta } \right)V_{0} ,$$

(28)

where

$$f_{2} = \frac{{ \cos (\theta - \phi_{0,1} ) }}{{2\cos \theta \sin (\beta_{1} - \phi_{1} - \phi_{0,1} )}} \mathop \sum \limits_{i = 1}^{k} \left[ {c_{i} \cos \phi_{i} \frac{{\sin \alpha_{i} }}{{\sin (\alpha_{i} + \beta_{i} )}}\mathop \prod \limits_{j = 1}^{i - 1} \frac{{\sin \beta_{j} \sin (\alpha_{j} + \beta_{j} - \phi_{j} - \phi_{j,j + 1} )}}{{\sin (\alpha_{j} + \beta_{j} )\sin (\beta_{j + 1} - \phi_{j + 1} - \phi_{j,j + 1} )}}} \right].$$

(29)

3. Along lines l_i (i = 2, …, k):

$$D_{{l_{i (i = 1, \ldots , k)} }} = c_{i,i + 1} B_{0} f_{3} \left( {\alpha_{i} , \;\beta_{i} ,\; \phi_{i} ,\; \phi_{i,i + 1} , \;\theta } \right)V_{0} ,$$

(30)

where

$$f_{3} = \frac{{ \cos (\theta - \phi_{0,1} ) }}{{2\cos \theta \sin (\beta_{1} - \phi_{1} - \phi_{0,1} )}}\sum\limits_{i = 2}^{k} {\left[ {c_{i,i + 1} \cos \phi_{i,i + 1} \frac{{\sin (\alpha_{i - 1} + \beta_{i - 1} + \phi_{i} - \phi_{i - 1} - \beta_{i} )}}{{\sin (\alpha_{i - 1} + \beta_{i - 1} - \phi_{i - 1} - \phi_{i,i - 1} )}} \times \mathop \prod \limits_{j = 1}^{i - 1} \frac{{\sin \beta_{j} }}{{\sin (\alpha_{j} + \beta_{j} )}} \times \mathop \prod \limits_{j = 1}^{i - 2} \frac{{\sin (\alpha_{j} + \beta_{j} - \phi_{j} - \phi_{j,j + 1} )}}{{\sin (\beta_{j + 1} - \phi_{j + 1} - \phi_{j,j + 1} )}}} \right]} .$$

(31)

Because of the symmetry of the M1 mechanism, the total energy dissipation in the whole mechanism is twice the summation of these three parts, i.e., Eqs. (24), (28), and (30):

$$\sum D = 2\left( {D_{\text{BC}} + D_{{d_{i (i = 1, \ldots , k)} }} + D_{{l_{i (i = 2, \ldots , k)} }} } \right)$$

(32)

1.3 External Work

1. External work due to the surcharge loading:

$$W_{{q_{0} }} = q_{0} B_{0} f_{4} \left( {\alpha_{i} , \;\beta_{i} , \;\phi_{i} , \;\phi_{i,i + 1} , \;\theta } \right)V_{0}$$

(33)

$$\begin{aligned} f_{4} = \frac{{ \cos \left( {\theta - \phi_{0,1} } \right) }}{{\cos \theta \sin \left( {\beta_{1} - \phi_{1} - \phi_{0,1} } \right)}}\frac{{\sin \beta_{k} }}{{\sin \left( {\alpha_{k} + \beta_{k} } \right)}}\sin \left( {\beta_{k} - \theta - \mathop \sum \limits_{j = 1}^{k - 1} \alpha_{j} - \phi_{k} } \right) \hfill \\ \times \mathop \prod \limits_{j = 1}^{k - 1} \frac{{\sin \beta_{j} \sin \left( {\alpha_{j} + \beta_{j} - \phi_{j} - \phi_{j,j + 1} } \right)}}{{\sin \left( {\alpha_{j} + \beta_{j} } \right)\sin \left( {\beta_{j + 1} - \phi_{j + 1} - \phi_{j,j + 1} } \right)}} \hfill \\ \end{aligned}$$

(34)

2. External work due to self-weight of the central triangular wedge, ABC:

$$W_{\text{ABC}} = \frac{{\gamma B_{0}^{2} }}{2}\left[ {f_{5} \left( {\alpha_{i} ,\; \beta_{i} , \;\phi_{i} , \;\phi_{i,i + 1} , \;\theta } \right)} \right]V_{0}$$

(35)

where

$$f_{5} = \frac{\tan \theta }{2}$$

(36)

3. External work due to self-weights of the remaining 2k triangular wedges:

$$\mathop \sum \limits_{i = 1}^{2k} W_{i} = \frac{{\gamma B_{0}^{2} }}{2}\left[ {f_{6} \left( {\alpha_{i} , \;\beta_{i} ,\; \phi_{i} ,\; \phi_{i,i + 1} , \;\theta } \right)} \right]V_{0}$$

(37)

where

$$f_{6} = \frac{{\cos (\theta - \phi_{0,1} )}}{{2\cos^{2} \theta \sin (\beta_{1} - \phi_{1} - \phi_{0,1} ) }}\sum\limits_{i = 1}^{k} {\left[ { \frac{{\sin \alpha_{i} \sin \beta_{i} }}{{\sin (\alpha_{i} + \beta_{i} )}}\sin \left( {\beta_{i} - \theta - \mathop \sum \limits_{j = 1}^{i - 1} \alpha_{j} - \phi_{i} } \right) \times \mathop \prod \limits_{j = 1}^{i - 1} \frac{{\sin^{2} \beta_{j} \sin (\alpha_{j} + \beta_{j} - \phi_{j} - \phi_{j,j + 1} )}}{{\sin^{2} (\alpha_{j} + \beta_{j} )\sin (\beta_{j + 1} - \phi_{j + 1} - \phi_{j,j + 1} )}}} \right].}$$

(38)

4. External work due to the footing load:

$$W_{{q_{\text{uD}} }} = q_{\text{uD}} V_{0} .$$

(39)

The total external work is the summation of the four contributions, i.e., Eqs. (33), (35), (37), and (39):

$$\sum W_{\text{ext}} = W_{{q_{0} }} + W_{\text{ABC}} + \mathop \sum \limits_{i = 1}^{2k} W_{i} + W_{{q_{\text{uD}} }} .$$

(40)

Appendix 2: M2 Mechanism (Rock Masses Subjected to Seepage)

2.1 Geometry

For the triangular block i, the lengths l_i and d_i, and the area S_i are given as follows:

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

(41)

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

(42)

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

(43)

2.2 Internal Energy Dissipation

1. Along lines d_i (i = 1, …, k):

$$D_{{d_{i (i = 1, \ldots , k)} }} = c_{i} B_{0} g_{1} \left( {\alpha_{i} , \;\beta_{i} , \;\phi_{i} , \;\phi_{i,i + 1} , \;\theta } \right)V_{0}$$

(44)

where

$$g_{1} = \frac{{\sin \beta_{1} }}{{\sin \left( {\alpha_{1} + \beta_{1} } \right)}}\sum\limits_{i = 1}^{k} {\left[ {c_{i} \cos \phi_{i} \frac{{\sin \alpha_{i} }}{{\sin \beta_{i} }} \times \mathop \prod \limits_{j = 2}^{i} \frac{{\sin \beta_{j} }}{{\sin \left( {\alpha_{j} + \beta_{j} } \right)}}\mathop \prod \limits_{j = 1}^{i - 1} \frac{{\sin \left( {\alpha_{j} + \beta_{j} - \phi_{j} - \phi_{j,j + 1} } \right)}}{{\sin \left( {\beta_{j + 1} - \phi_{j + 1} - \phi_{j,j + 1} } \right)}}} \right]}$$

(45)

2. Along lines l_i (i = 1, …, k − 1):

$$D_{{l_{i (i = 1, \ldots , k - 1)} }} = c_{i,i + 1} B_{0} g_{2} \left( {\alpha_{i} , \;\beta_{i} , \;\phi_{i} ,\; \phi_{i,i + 1} ,\; \theta } \right)V_{0}$$

(46)

where

$$g_{2} = \frac{{\sin \beta_{1} }}{{\sin (\alpha_{1} + \beta_{1} ) }}\sum\limits_{i = 1}^{k = 1} {\left[ {c_{i,i + 1} \cos \phi_{i,i + 1} \frac{{\sin (\alpha_{i} + \beta_{i} + \phi_{i + 1} - \phi_{i} - \beta_{i + 1} )}}{{\sin (\beta_{i + 1} - \phi_{i + 1} - \phi_{i,i + 1} )}}\mathop \prod \limits_{j = 2}^{i} \frac{{\sin \beta_{j} }}{{\sin (\alpha_{j} + \beta_{j} )}}\mathop \prod \limits_{j = 1}^{i - 1} \frac{{\sin (\alpha_{j} + \beta_{j} - \phi_{j} - \phi_{j,j + 1} )}}{{\sin (\beta_{j + 1} - \phi_{j + 1} - \phi_{j,j + 1} )}}} \right]} .$$

(47)

The total energy dissipation in the whole mechanism is equal to the summation of these two parts, i.e., Eqs. (44) and (46):

$$\sum {D = \left( {D_{{d_{i (i = 1, \ldots , k)} }} + D_{{l_{i (i = 2, \ldots , k)} }} } \right)} .$$

(48)

2.3 External Work

1. External work due to self-weights and seepage forces of the rock mass in motion of the k triangular rigid blocks:

$$W_{\text{rockmass}} = \frac{{\gamma B_{0}^{2} }}{2}\left[ {g_{3} + i\left( {\frac{{\gamma_{\text{w}} }}{{\gamma_{\text{sub}} }}} \right) g_{4} } \right]V_{1} ,$$

(49)

where

$$g_{3} = \frac{{\sin \beta_{1} }}{{\sin (\alpha_{1} + \beta_{1} )}}\sin \left( {\beta_{k} - \mathop \sum \limits_{j = 1}^{k - 1} \alpha_{j} - \phi_{k} } \right) \times \mathop \prod \limits_{j = 2}^{k} \frac{{\sin \beta_{j} }}{{\sin (\alpha_{j} + \beta_{j} )}}\mathop \prod \limits_{j = 1}^{k - 1} \frac{{\sin (\alpha_{j} + \beta_{j} - \phi_{j} - \phi_{j,j + 1} )}}{{\sin (\beta_{j + 1} - \phi_{j + 1} - \phi_{j,j + 1} )}}$$

(50)

$$g_{4} = \frac{{\sin \beta_{1} }}{{\sin (\alpha_{1} + \beta_{1} )}}\cos \left( {\beta_{k} - \mathop \sum \limits_{j = 1}^{k - 1} \alpha_{j} - \phi_{k} } \right) \times \mathop \prod \limits_{j = 2}^{k} \frac{{\sin \beta_{j} }}{{\sin (\alpha_{j} + \beta_{j} )}}\mathop \prod \limits_{j = 1}^{k - 1} \frac{{\sin (\alpha_{j} + \beta_{j} - \phi_{j} - \phi_{j,j + 1} )}}{{\sin (\beta_{j + 1} - \phi_{j + 1} - \phi_{j,j + 1} )}}$$

(51)

2. External work due to the surcharge loading and the corresponding seepage forces:

$$W_{{q_{0} }} = qB_{0} \left[ {g_{5} + i\left( {\frac{{\gamma_{\text{w}} }}{{\gamma_{\text{sub}} }}} \right) g_{6} } \right]V_{1} ,$$

(52)

where

$$g_{5} = \frac{{\sin^{2} \beta_{1} }}{{\sin^{2} (\alpha_{1} + \beta_{1} )}}\sum\limits_{i = 1}^{k} {\left[ {\frac{{\sin \alpha_{i} \sin (\alpha_{i} + \beta_{i} )}}{{\sin \beta_{i} }}\sin \left( {\beta_{i} - \mathop \sum \limits_{j = 1}^{i - 1} \alpha_{j} - \phi_{i} } \right) \times \mathop \prod \limits_{j = 2}^{i} \frac{{\sin^{2} \beta_{j} }}{{\sin^{2} (\alpha_{j} + \beta_{j} )}} \times \mathop \prod \limits_{j = 1}^{i - 1} \frac{{\sin (\alpha_{j} + \beta_{j} - \phi_{j} - \phi_{j,j + 1} )}}{{\sin (\beta_{j + 1} - \phi_{j + 1} - \phi_{j,j + 1} )}} } \right]}$$

(53)

$$g_{6} = \frac{{\sin^{2} \beta_{1} }}{{\sin^{2} (\alpha_{1} + \beta_{1} )}}\sum\limits_{i = 1}^{k} {\left[ {\frac{{\sin \alpha_{i} \sin (\alpha_{i} + \beta_{i} )}}{{\sin \beta_{i} }}\cos \left( {\beta_{i} - \mathop \sum \limits_{j = 1}^{i - 1} \alpha_{j} - \phi_{i} } \right) \times \mathop \prod \limits_{j = 2}^{i} \frac{{\sin^{2} \beta_{j} }}{{\sin^{2} (\alpha_{j} + \beta_{j} )}} \times \mathop \prod \limits_{j = 1}^{i - 1} \frac{{\sin (\alpha_{j} + \beta_{j} - \phi_{j} - \phi_{j,j + 1} )}}{{\sin (\beta_{j + 1} - \phi_{j + 1} - \phi_{j,j + 1} )}} } \right]}$$

(54)

3. External work due to the footing load and the corresponding seepage forces:

$$W_{{q_{\text{uS}} }} = q_{\text{uS}} \left[ {\sin (\beta_{1} - \phi_{1} ) + i\left( {\frac{{\gamma_{\text{w}} }}{{\gamma_{\text{sub}} }}} \right) \cos (\beta_{1} - \phi_{1} )} \right]V_{1} .$$

(55)

The total external work is the summation of the three contributions, Eqs. (49), (52), (55):

$$\sum W_{\text{ext}} = W_{\text{rockmass}} + W_{{q_{0} }} + W_{{q_{\text{uS}} }} .$$

(56)