Upper Bound Solution for the Bearing Capacity of Submerged Jointed Rock Foundations

1 Introduction

Despite the fact that important studies such as those by Sutcliffe et al. (2004), Yang and Yin (2005), Merifield et al. (2006), and Saada et al. (2008) have been performed in the field of rock foundations, significant topics such as the effect of groundwater and joint spacing on the bearing capacity have not been investigated. In this regard, the upper bound theorem of limit analysis was used to investigate the bearing capacity of submerged jointed rock foundations.

In the calculations, a rock mass containing two orthogonal tight joint sets was considered and the Mohr–Coulomb failure criterion was used for both the intact rock and the joint sets. Orientation angles (α) equal to 15°, 30°, and 45° were considered for one of the joint sets. The concept of “spacing ratio” (SR), which was initially proposed by Serrano and Olalla (1996), was used to account the joint spacing. The SR can be computed as follows:

$$ {\text{SR}} = B\sum\limits_{i = 1}^{n} {\frac{1}{{S_{i} }}} $$

(1)

where B is the footing width, S _i is the spacing of the ith joint set, and n is the number of joint sets.

Different failure mechanisms were assumed for the rock foundation and analytical relationships were derived for the bearing capacity of dry rocks based on each mechanism. The mechanism by which the minimum bearing capacity is predicted is named “the most appropriate failure mechanism”. Assuming that the mechanical properties of the intact rock and the joint sets are not affected by the groundwater, the rock buoyancy effect is incorporated into the upper bound equations obtained based upon the most appropriate failure mechanism. The proposed upper bound solution is able to take into account different depths of the water table beneath the footing.

2 Bearing Capacity of Shallow Foundations on Rocks

2.1 Selection of the Failure Mechanism

In the foundations with centric and vertical loadings, the failure mechanism is usually considered to be symmetrical, i.e., a two-sided mechanism. However, in jointed rock masses, the failure mechanism may be affected by the joint sets, and, thus, converted to an asymmetrical shape. For determining the shape of the failure mechanism, numerical analyses were performed using the Universal Distinct Element Code, UDEC (Itasca Consulting Group, Inc. 2000). A large number of jointed rock models with various joint spacings and mechanical properties for the intact rock and the joint sets were simulated, and the foundation load was applied vertically at the top boundary of the models. As an example, for the intact rock cohesion (c _i) equal to 5 MPa, the joint sets cohesion (c _j) equal to 50 kPa and friction angles for both intact rock and joint sets (ϕ _i and ϕ _j, respectively) equal to 35°, the displacement vectors of the rock blocks are shown in the left hand side of Fig. 1. For α = 45°, the mechanism is symmetrical (a two-sided mechanism), while for α = 30° and 15°, the mechanism is asymmetrical (a one-sided mechanism). Moreover, using different values of SR, the effect of joint spacing on the failure mechanism configuration was also investigated and no remarkable effect of joint spacing on the shape of the failure mechanism was observed. Hence, a two-sided symmetrical failure mechanism (TS failure mechanism) was used for the case of α = 45° and a one-sided asymmetrical failure mechanism (OS failure mechanism) was used for α = 15° and 30°.

Fig. 1

Displacement vectors obtained using the distinct element method for the case of SR = 10 and the overall shape of the failure mechanisms

For obtaining the least magnitude of the upper bound bearing capacity, two different TS mechanisms (TS1 and TS2) and, also, two different OS mechanisms (OS1 and OS2) were considered, as shown in the right hand side of Fig. 1.

2.2 Bearing Capacity of Foundations on Dry Jointed Rocks

The mechanisms TS1 and OS1, along with the corresponding hodographs, are shown in Fig. 2. In order to obtain minimum values for the upper bound bearing capacity, the greatest possible length of velocity discontinuity lines have been passed along the joints. It is assumed that both the beginning and the end points of line CD are located at the junction of the two joint sets, not within the intact rock block. To satisfy this assumption, CD was considered to be a straight line. It will be shown in this study that the shape of CD (either a straight line or a log-spiral curve) does not have any significant effect on the submerged bearing capacity. Since the displacements in the mechanism TS1 are symmetrical about the footing vertical axis, half of the mechanism is only considered for the analysis.

Fig. 2

Failure mechanisms and the corresponding hodographs: a TS1 and b OS1

According to the normality rule, the velocity on every discontinuity line must be inclined at an angle ϕ with that line, where ϕ is the friction angle of the medium in which the discontinuity line lays (either ϕ _i or ϕ _j). All the velocities of the mechanism determined in this way constitute a kinematically admissible velocity field.

In Fig. 2, the foundation width (B), S ₁, α, ϕ _i, and ϕ _j are known. The angle of line CD with the horizontal direction (θ) is:

$$ \theta = \tan^{ - 1} \left( {\frac{{n\;_{0} S_{1} }}{B\cos \alpha }} \right) - \alpha $$

(2)

where n ₀ is an integer number. The length of the velocity discontinuity line BD can be obtained by multiplying n ₀ by S ₁. So, the only unknown parameter of the failure mechanisms is n ₀.

The rate of energy dissipation (ΔD _L) along each velocity discontinuity is:

$$ \Updelta D_{\text{L}} = c\;\Updelta V\cos \phi $$

(3)

where c is the cohesion (either c _i or c _j) and ΔV is the incremental velocity that makes the angle ϕ _i or ϕ _j with a velocity discontinuity line. Thus, the total energy dissipation (D) in the mechanism TS1 is:

$$ D = 2\left( {D_{\text{BC}} + D_{\text{CD}} + D_{\text{BD}} + D_{\text{DE}} } \right) $$

(4)

In the mechanism OS1, this changes to:

$$ D = D_{\text{AC}} + D_{\text{BC}} + D_{\text{CD}} + D_{\text{BD}} + D_{\text{DE}} $$

(5)

where D _XY is the energy dissipation along the discontinuity line XY.

The total external work (W) in the mechanism TS1 is:

$$ W = 2\left( {W_{\text{ABC}} + W_{\text{BCD}} + W_{\text{BDE}} + W_{q} } \right) + W_{{q_{\text{u}} }} $$

(6)

and, in the mechanism OS1, this changes to:

$$ W = W_{\text{ABC}} + W_{\text{BCD}} + W_{\text{BDE}} + W_{q} + W_{{q_{\text{u}} }} $$

(7)

where W _XYZ is the external work of wedge XYZ, W _q is the external work of surcharge q, and $W_{{q_{\text{u}} }}$ is the external work of the foundation load.

Equating the total external work to the total energy dissipation, and after some rearrangements, the general equation of the ultimate bearing capacity of a shallow foundation on the assumed dry rock (q _u) is obtained as:

$$ q_{\text{u}} = c_{\text{j}} N_{\text{cj}} + c_{\text{i}} N_{\text{ci}} + qN_{\text{q}} + \frac{1}{2}\gamma \;BN_{\gamma } $$

(8)

where γ is the total unit weight of the rock mass and N _cj, N _ci, N _q, and N _γ are the bearing capacity coefficients that are obtained as follows:

Assuming:

$$ \xi_{1} = \alpha + \theta - \phi_{i} - \phi_{j} $$

(9)

$$ \xi_{2} = \alpha + \theta - \phi_{i} + \phi_{j} $$

(10)

$$ \xi_{3} = \alpha + \phi_{j} $$

(11)

$$ \xi_{4} = \alpha + \theta $$

(12)

$$ \xi_{5} = \phi_{i} - \theta $$

(13)

$$ \xi_{6} = \alpha - \phi_{j} $$

(14)

For the mechanism TS1:

$$ N_{\text{cj}} = \frac{{2\cos \phi_{j} \cos \alpha }}{{f_{1} }}\left[ {\cos^{2} \xi_{5} + \frac{{f_{2} }}{{f_{3} }} \cdot \tan \xi_{4} \left( {\sin^{2} \xi_{2} + \frac{{f_{4} }}{\tan \alpha }} \right)} \right] $$

(15)

$$ N_{\text{ci}} =\,\frac{{f_{2} }}{{f_{1} }} \cdot \frac{{2\cos \phi_{i} \cos \alpha }}{{\cos \xi_{4} }} $$

(16)

$$ N_{\text{q}} =\,\frac{{f_{2} f_{4} }}{{f_{1} f_{3} }} \cdot \frac{{2\sin \xi_{3} \tan \xi_{4} }}{\tan \alpha } $$

(17)

$$ N_{\gamma } = \cos \alpha \left[ {\frac{{2f_{2} }}{{f_{1} }} \cdot \cos \alpha \tan \xi_{4} \left( {\sin \xi_{5} + \frac{{f_{4} }}{{f_{3} }} \cdot \frac{{\sin \xi_{3} \tan \xi_{4} }}{\tan \alpha }} \right) - \sin \alpha } \right] $$

(18)

where:

$$ f_{1} = \sin \xi_{6} - \sin \xi_{5} \cos \xi_{1} $$

(19)

$$ f_{2} = - \sin \xi_{6} \sin \xi_{5} + \cos \xi_{1} $$

(20)

$$ f_{3} = \sin 2\phi_{j} \cos \xi_{2} + \sin \xi_{1} $$

(21)

$$ f_{4} = \sin 2\phi_{j} + \cos \xi_{2} \sin \xi_{1} $$

(22)

For the mechanism OS1:

$$ N_{\text{cj}} = \frac{{\cos \phi_{j} \cos \alpha }}{{\cos \xi_{3} }}\left[ {\tan \alpha + \frac{{\cos^{2} \xi_{2} }}{{g_{1} }} + \frac{{g_{2} }}{{g_{1} g_{3} }} \times \tan \xi_{4} \left( {\sin^{2} \xi_{2} + \frac{{g_{4} }}{\tan \alpha }} \right)} \right] $$

(23)

$$ N_{\text{ci}} = \frac{{g_{2} }}{{g_{1} }} \cdot \frac{{\cos \phi_{i} \cos \alpha }}{{\cos \xi_{3} \cos \xi_{4} }} $$

(24)

$$ N_{\text{q}} = \frac{{g_{2} g_{4} }}{{g_{1} g_{3} }} \cdot \frac{{\tan \xi_{3} \tan \xi_{4} }}{\tan \alpha } $$

(25)

$$ N_{\gamma } = \cos \alpha \left[ {\frac{{g_{2} }}{{g_{1} }} \cdot \frac{{\cos \alpha \tan \xi_{4} }}{{\cos \xi_{3} }}\left( {\sin \xi_{5} + \frac{{g_{4} }}{{g_{3} }} \cdot \frac{{\sin \xi_{3} \tan \xi_{4} }}{\tan \alpha }} \right) - \sin \alpha } \right] $$

(26)

where:

$$ g_{1} = \cos \xi_{1} \sin \xi_{2} - \sin 2\phi_{j} $$

(27)

$$ g_{2} = \cos \xi_{1} - \sin \xi_{2} \sin 2\phi_{j} $$

(28)

$$ g_{3} = \sin 2\phi_{j} \cos \xi_{2} + \sin \xi_{1} $$

(29)

$$ g_{4} = \sin 2\phi_{j} + \cos \xi_{2} \sin \xi_{1} $$

(30)

The minimum upper bound bearing capacity is obtained by minimizing Eq. 8 with respect to the only unknown parameter, n ₀. Two computer programs were prepared in MATLAB code to solve the bearing capacity equations for TS1 and OS1. The genetic algorithm provided in the code was used for the minimization. The minimization procedure was performed subject to the following constraints for OS1:

$$ \alpha + \theta < \frac{\pi }{2},\quad 0 < \alpha + \theta - \phi_{i} - \phi_{j} < \frac{\pi }{2},\quad n_{0} > 0 $$

(31)

And for TS1, the constraint 0 < α − θ + ϕ _i + ϕ _j < π/2 was applied in addition to the above constraints.

It was not clear prior to calculations which mechanism would predict the minimum bearing capacity. So, in addition to TS1 and OS1, the mechanisms TS2 and OS2 were considered in order not to predefine the failure mechanism. For all cases analyzed, failure mechanisms TS1 and OS1 ensured the minimum limit load. For example, for the case of α = 45°, c _i = 5 MPa, c _j = 50 kPa, ϕ _i = ϕ _j = 35°, q = 20 kPa, γ = 27 kN/m³, and S ₁ = S ₂ = 3.9 cm, the mechanism TS1 yields n ₀ = 100 and q _u = 210.8 MPa, while the mechanism TS2 yields n ₀ = 101 and q _u = 572.3 MPa. Therefore, the mechanisms TS2 and OS2 are not presented here and the effect of submergency was only applied to TS1 and OS1.

It should be noted that, for different mechanical properties of the intact rock and the joint sets, the SR ratio did not show a remarkable effect on the bearing capacities obtained from the upper bound solution proposed in this study. This is due to the fact that the variation of SR imposes small changes in the location of discontinuity line CD and, hence, does not change the overall configuration of the failure mechanisms. The same result was obtained by Halakatevakis and Sofianos (2010) for rock models with one to three joint sets. Therefore, a SR value equal to 50 was taken into account in the following sections.

2.3 Bearing Capacity of Foundations on Submerged Rocks

In practice, intact rock permeability is small and negligible in comparison to the permeability of joints. However, in the field, the whole rock mass below the water table is saturated after a specific time lag and leads to buoyancy of the rock mass.

The water table was considered to be located at the depth d _w from the foundation base, which can vary to any depth of the rock mass. For various combinations of mechanical properties of intact rock and joint set, there is a depth beyond which the groundwater has no effect on the bearing capacity. This depth is called the “critical depth” and is shown here by d _cr, which is equal to the depth of point D. According to Fig. 2:

$$ d_{\text{cr}} = B\cos^{2} \alpha \tan \left( {\alpha + \theta } \right) = n_{0} S_{1} \cos \alpha $$

(32)

For a submerged rock, the forces contributing to the external work are similar to the case of a dry rock, except for the work due to the rock weight, which will be the sum of the work due to the total weight of the rock above the water table and the work due to the submerged weight of the rock below the water table. Also, the work due to the seepage forces must be considered as an external work. As the water does not flow within the rock mass, the work due to seepage forces will be equal to zero. Ausilio and Conte (2005) used this approach to obtain the bearing capacity of foundations on submerged soils.

The total energy dissipation is similar to the dry case. Finally, the general equation of the ultimate bearing capacity of shallow foundations on the assumed submerged rocks (q _uw) is obtained as:

$$ q_{\text{uw}} = c_{\text{j}} N_{\text{cj}} + c_{\text{i}} N_{\text{ci}} + qN_{\text{q}} + \frac{1}{2}\gamma \;BN_{\gamma }^{\text{sub}} $$

(33)

where N _cj, N _ci, and N _q are the same as in the dry case and can be obtained from Eqs. 15–17 or Eqs. 23–25, depending upon the failure mechanism. N ^sub _γ is the submerged bearing capacity coefficient and is expressed as:

$$ N_{\gamma }^{\text{sub}} = \frac{{\gamma^{\prime } }}{\gamma }N_{\gamma } + \frac{{d_{\text{w}} }}{B}\left( {1 - \frac{{\gamma^{\prime } }}{\gamma }} \right)N_{{\gamma {\text{w}} }} $$

(34)

where γ′ is the submerged unit weight of the rock mass, N _γ is the bearing capacity coefficient for foundations on dry rock that can be obtained from Eq. 18 or Eq. 26, and N _γw is an additional factor that depends on the groundwater depth and is obtained as follows:

For the mechanism TS1:

• (a) For 0 ≤ d _w ≤ h _C:

  $$ N_{{\gamma {\text{w}} }} = \frac{{d_{\text{w}} }}{B} \cdot \frac{2}{\sin 2\alpha }\left( {1 + \frac{{2f_{2} }}{{f_{1} }} \cdot \sin \xi_{5} - \frac{{2f_{2} f_{4} }}{{f_{1} f_{3} }} \cdot \sin \xi_{3} } \right) + \frac{{4f_{2} f_{4} }}{{f_{1} f_{3} }} \cdot \frac{{\sin \xi_{3} \tan \xi_{4} }}{\tan \alpha } - 2 $$

  (35)

  where h _C is the depth of point C given by: h _C = Bsinα.cosα.

• (b) For h _C ≤ d _w ≤ h _D:

  $$ \begin{aligned} N_{\gamma {\text{w}} } = & \frac{{4f_{2} }}{{f_{1} }} \cdot \tan \xi_{4} \cos \alpha \left( {\frac{{\cos \xi_{4} \sin \xi_{5} }}{\sin \theta } + \frac{{f_{4} }}{{f_{3} }} \cdot \frac{{\sin \xi_{3} }}{\sin \alpha }} \right) \\ - \frac{{2d_{\text{w}} }}{B} \cdot \frac{{f_{2} }}{{f_{1} }} \cdot \frac{1}{\cos \alpha }\left( {\frac{{\cos \xi_{4} \sin \xi_{5} }}{\sin \theta } + \frac{{f_{4} }}{{f_{3} }} \cdot \frac{{\sin \xi_{3} }}{\sin \alpha }} \right) \\ + \frac{B}{{d_{\text{w}} }} \cdot \cos \alpha \left[ {\frac{{2f_{2} }}{{f_{1} }} \cdot \cos \alpha \tan \xi_{4} \sin \xi_{5} \times \left( {1 - \frac{{\cos \alpha \sin \xi_{4} }}{\sin \theta }} \right) - \sin \alpha } \right] \\ \end{aligned} $$

  (36)

  where h _D is the depth of point D given by: h _D = Bcos² α.tan(α + θ).

• (c) For 0 ≤ d _w ≤ h _C, the N ^sub _γ in Eq. 33 is equal to N _γ in Eq. 8.

For the mechanism OS1:

• (a) For 0 ≤ d _w ≤ h _C:

  $$ N_{{\gamma {\text{w}} }} = \frac{{d_{\text{w}} }}{B} \cdot \frac{2}{\sin 2\alpha }\left( {1 + \frac{{g_{2} }}{{g_{1} }} \cdot \frac{{\sin \xi_{5} }}{{\cos \xi_{3} }} - \frac{{g_{2} g_{4} }}{{g_{1} g_{3} }} \cdot \tan \xi_{3} } \right) + \frac{{2g_{2} g_{4} }}{{g_{1} g_{3} }} \cdot \frac{{\tan \xi_{3} \tan \xi_{4} }}{\tan \alpha } - 2 $$

  (37)

  where h _C is the depth of point C given by: h _C = Bsinα.cosα.

• (b) For h _C ≤ d _w ≤ h _D:

  $$ \begin{aligned} N_{{\gamma {\text{w}} }} = &\,\frac{{2g_{2} }}{{g_{1} }} \cdot \frac{{\sin \xi_{4} \cos \alpha }}{{\cos \xi_{3} }}\left( {\frac{{\sin \xi_{5} }}{\sin \theta } + \frac{{g_{4} }}{{g_{3} }} \cdot \frac{{\sin \xi_{3} }}{{\sin \alpha \cos \xi_{4} }}} \right) \\ - \frac{{d_{\text{w}} }}{B} \cdot \frac{{g_{2} }}{{g_{1} }} \cdot \frac{1}{{\cos \alpha \cos \xi_{3} }}\left( {\frac{{\cos \xi_{4} \sin \xi_{5} }}{\sin \theta } + \frac{{g_{4} }}{{g_{3} }} \cdot \frac{{\sin \xi_{3} }}{\sin \alpha }} \right) \\ + \frac{B}{{d_{\text{w}} }} \cdot \cos \alpha \left[ {\frac{{g_{2} }}{{g_{1} }} \cdot \frac{{\cos \alpha \tan \xi_{4} \sin \xi_{5} }}{{\cos \xi_{3} }}\left( {1 - \frac{{\cos \alpha \sin \xi_{4} }}{\sin \theta }} \right) - \sin \alpha } \right] \\ \end{aligned} $$

  (38)

  where h _D is the depth of point D given by: h _D = Bcos² α.tan(α + θ).

• (c) For d _w > h _D, the N ^sub _γ in Eq. 33 is equal to N _γ in Eq. 8.

The lowest upper bound bearing capacity is obtained by minimizing Eq. 33. The unknown parameter, the minimization method, and the corresponding constraints are similar to those considered for the dry condition.

3 Discussion of the Results

3.1 Comparison with the Existing Solutions

Since there is no published document for determining the effect of groundwater on the bearing capacity of rock masses, the results obtained using the upper bound solution were compared to the methods of Hansen et al. (1987) and Ausilio and Conte (2005) for determining the bearing capacity of foundations on submerged soils. For greater compatibility in the results, the same c and ϕ for the intact rock and the joint sets were used and the symmetrical mechanism (TS1) was selected for comparison.

For a footing resting on the surface of a rock mass (q = 0) with γ = 27 kN/m³ and assuming that ϕ _i = ϕ _j = 30°, c _i = c _j = 0, and the water unit weight (γ _w) is equal to 10 kN/m³, the results obtained are presented in Fig. 3. It is clear that there is a good agreement between the upper bound solution and the results of others. The figure also shows that the shape of line CD [either as a straight line assumed in this study or a log-spiral curve assumed by Ausilio and Conte (2005)] does not significantly affect the reduction in the bearing capacity due to the submergence of rocks.

Fig. 3

Comparison of q _uw/q _u versus d _w/B obtained using the upper bound solution with that obtained by the methods of Ausilio and Conte (2005) and Hansen et al. (1987)

3.2 Effect of Shear Strength Parameters

For investigating the effect of shear strength parameters on the bearing capacity of submerged rocks, analyses were carried out using Eqs. 8 and 33, assuming that q = 0, γ = 27 kN/m³, and γ _w = 10 kN/m³. Figure 4 shows the q _uw/q _u versus d _w/B for different c and ϕ values of intact rock and joint sets.

Fig. 4

Variation of q _uw/q _u versus d _w/B for: a ϕ _i = ϕ _j = 35° and c _i = 5 MPa and b ϕ _i = 35°, c _i = 0.1 MPa, and c _j = 0

According to Fig. 4a, the maximum reduction in the bearing capacity due to the rock submergence occurred for the case of α = 15° and the minimum occurred for the case of α = 45°. Moreover, the effect of both the α angle and the rock submergence on the bearing capacity was decreased by increasing the c _j/c _i ratio. Also, the critical depths can be compared as: d _{cr (α=45°)} < d _{cr (α=30°)} < d _{cr (α=15°)}.

Figure 4b reveals that, by increasing ϕ _j, a further reduction in the bearing capacity occurs due to the rock submergence. This is because of the fact that an increase in the friction angle leads to a deeper failure mechanism, which will result in higher d _cr and a further reduction in the bearing capacity. This result is more pronounced for α = 15° than for other orientations. The same result was obtained by increasing ϕ _i, but is not shown here.

3.3 Bearing Capacity Coefficients

Table 1 presents the N _cj, N _ci, N _q, and N _γ coefficients required for determining the bearing capacity of dry rocks using the upper bound solution. For the case of a submerged rock, N ^sub _γ can be obtained using Fig. 5, in which the N _γ factor is presented in Table 1.

Table 1 Bearing capacity coefficients for dry rocks

Fig. 5

Variation of N ^sub _γ /N _γ versus d _w/B for: a α = 15°, b α = 30°, c α = 45°

4 Conclusions

Analytical expressions derived for the bearing capacity of dry and submerged rocks indicate that submergence of the rock below the footing base reduces the contribution of the rock weight in the bearing capacity. The amount of this reduction and also the critical depth in the case of α = 15° is larger than for α = 30° and, for the latter, is larger than for α = 45°.

Because of the inhomogeneity and discontinuum nature of rock masses, application of the upper bound theorem was used for relatively simple configurations. Further investigations are required in order to deal with more complicated geometries.