1. Introduction
Nowadays, more land resources are being converted to accommodate urban construction, including rocky foundations that were once difficult to develop upon [
1,
2,
3,
4,
5,
6,
7,
8]. Numerous residential buildings, office buildings, and transportation facilities that create large loads have been built on rocky foundations, and these facilities may be damaged or even destroyed during an earthquake, resulting in large economic and human losses. To ensure the safety of these facilities, engineers have focused their attention on the study of seismic bearing capacity of foundations [
1]. The problem of concern in this study was the ultimate bearing capacity of strip foundations under the action of seismic load.
The methods used for shallow foundation bearing capacity assessment can be roughly divided into the following four categories: (1) the limit equilibrium method, (2) the slip-line method, (3) the limit analysis method, and (4) the numerical analysis method, based on finite element technique or finite difference technique [
2,
3,
4,
5,
6,
7,
8]. The first three methods are all traditional soil mechanics analysis methods: the limit equilibrium method establishes a simplified failure mode that makes it possible to use simple statics methods to solve various problems. The slip-line method is used to derive the basic differential equation and then obtain the solutions of various problems by determining the slip-line network. Unlike the traditional limit equilibrium and slip-line methods, the limit analysis method selects a certain flow law to consider the stress-strain relationship of the soil in an idealized way. As for the numerical analysis methods, they are modern calculation means that are rapidly emerging following the development of computers. During the calculation process, the entire problem area is decomposed, and each sub-area becomes a simple part. For each unit, a suitable (simpler) approximate solution is assumed, and then the total satisfying conditions of this domain (such as structural equilibrium conditions) can be inferred to obtain the solution of the problem. The kinematics method selected for this study belongs to the limit analysis upper method, in which both the failure mechanism and energy consumption are considered. The upper limit of the actual failure load is obtained by simply selecting the appropriate stress field and velocity field, according to the principle that the external power must be equal to the internal power consumed by the mechanism. Since the foundation bearing capacity studied in this paper is an actual engineering problem, the safety conditions require that the bearing capacity should be greater than the maximum value of the calculated ultimate load, which means the foundation bearing capacity should be at the upper limit of the ultimate load.
The calculation of the ultimate bearing capacity of a foundation placed on a rock mass is a classic subject in the field of civil engineering [
2,
9,
10,
11]. Most analytical work is based on the general assumption that the strength of rock is dominated by the Mohr-Coulomb (MC) criterion as soil [
12], under which the classical textbook of Terzaghi describes the bearing capacity of strip loads as [
13]:
where
denotes soil cohesion;
is equivalent uniform load;
is unit weight of soil or rock mass;
is foundation width; and
are the coefficients of Terzaghi bearing capacity, which are only determined by the friction angle
.
The stress-strain relationship of most rock masses is nonlinear; this has been confirmed by numerous experiments [
14,
15,
16]. Among the nonlinear failure criteria proposed in numerous studies, the Hoek-Brown (HB) criterion shows a preferable ability to simulate the strength properties of isotropic rock masses. Hoek et al. [
17,
18,
19,
20] proposed a method that can transform the various parameters (
, etc.) describing the properties of rock into the commonly used parameters
and
of the MC criterion describing the properties of soil. After the transformation, the stress-strain relationship of the rock can be approximated by the MC criterion, and then the bearing capacity of the strip foundation placed on the rock mass can be calculated by the above-mentioned formula for the bearing capacity proposed by Terzaghi [
13].
Based on the method used by Hoek et al. [
17,
19,
21,
22], Yang and Yin [
12] assumed that the rock mass followed the modified HB criterion, while searching for the process of the bearing capacity for strip foundations on rock mass, and in which a certain optimal MC criterion tangent on the original HB curve is selected for substitution. This substitution transforms the stress-strain relationship of the foundation from the original nonlinear relationship described by the rock parameters (
and
) into a linear relationship described by the soil parameters (
and
). Although this method has been adopted by some researchers, it overestimates the strength of the rock foundation, which leads to a subsequent overestimation of the ultimate bearing capacity of the foundation.
In addition, reductions in foundation bearing capacity caused by seismic motion is also a key concern in civil engineering. In many studies and engineering stability analyses, the effect of seismic motion on the foundation is usually modeled as a pseudo-static force, which has been shown to yield a relatively reliable estimate across a wide range of applications. Since the stability calculation is not considered in this paper, to simplify the calculation process, the seismic action is simplified to the load imposed on the structure by the proposed pseudo-static approach.
Most of the existing studies on the bearing capacity of rock foundations [
12,
13,
14,
23,
24] have adopted the optimization method recommended by Drescher and Christopoulos [
25] and Collins et al. [
26]. The tangent angle (i.e., equivalent friction angle,
) is determined by moving the position of the tangent point on the HB nonlinear damage envelope, and then the equivalent cohesion,
, is inferred from the equivalent friction angle,
. The bearing capacity can be calculated according to the equivalent friction angle,
, and cohesion,
. Finally, the process is repeated until the minimum value of the calculated bearing capacity is obtained.
However, due to the nonlinearity of the rock strength envelope, the slope of the tangent line varies greatly when the tangent point is located at different positions on the envelope, which explains why the slope of the equivalent MC strength line at the completion of the iteration does not correctly represent the trend of the nonlinear HB strength envelope. Referring to Hoek’s summary based on extensive experimental and practical engineering experience [
18,
19], a method for calculating
and
that can more realistically reflect the strength characteristics of the rock mass was chosen here. In the subsequent process, only the shape of the failure mechanism is changed without adjusting the values of
and
. This operation ensures that the selected
and
are always representative of the bearing capacity characteristics of the rock mass.
One advantage of this method is that it can find the only optimal equivalent MC linear envelope that represents the trend of the HB nonlinear damage envelope. The results were compared with those of Yang and Yin [
12], which verified the present work.
The present work was mostly focused on the evaluation of ultimate bearing capacity [
12], and this paper extends this work to the calculation of the ultimate bearing capacity of a foundation under the influence of an earthquake. Additionally, this research further gives the five types of typical rock foundations on the strip foundation bearing capacity upper bounds and predictive failure mechanics. These failure mechanics can reflect the characteristics of each different rock mass. Lastly, the influences of surface overloading, the unit weight of a rock mass, the strength properties of a rock mass, and seismic action on foundation bearing capacity were studied. A set of seismic uniaxial compression bearing capacity coefficient tables summarizes the results of theoretical calculations. For practical application, the engineer can select the bearing capacity coefficients from the tables for different seismic intensities and multiply them by the uniaxial compressive strength,
, of rock to get the upper limit of bearing capacity under different seismic intensities.
3. Kinematic Analysis of Strip Foundation on Rock Foundation under Seismic Action
The rock mass considered in the analysis obeyed the modified HB failure criterion, which was simplified by using the generalized tangent technique. The inertia forces caused by seismic motion was also taken into consideration to investigate its effect on the ultimate bearing capacity of foundation. It has been shown [
24] that the kinematic analysis of structures is an effective method for solving the upper limit of the bearing capacity. The upper limit theorem shows that in any kinematically allowed virtual velocity field (the field is compatible with the velocity at the boundary of the rock mass), the rate of work done by the actual external forces is less than or equal to the rate of internal energy dissipation within the rock and soil mass itself due to friction. The power of external force considered in the following analysis includes the vertical load on the foundation, the ground overload, the inertia force of the weight of the rock mass, and the earthquake acceleration. After loading, the work done by the frictional force on the velocity discontinuity surface inside the rock causes the internal energy to be consumed to balance the work done by the external force. Meanwhile, in order to obtain the smallest upper limit solution, it is necessary to simulate as many kinematically allowed velocity fields as possible. The analysis was performed in this study using a generalized Prandtl failure mechanism to find the smallest possible upper bound solution using multiple optimizations.
The seismic effect of a strip foundation placed on the surface of a rock foundation (D = 0) is analyzed in the following section considering a case corresponding to the classical problem of a shallow foundation on a semi-infinite horizontal rock medium.
3.1. Generalized Prandtl Failure Mechanism
The Prandtl failure mechanism is extended to rock foundations obeying the modified HB criterion, as shown in
Figure 3. As the seismic action is considered in the analysis, there is a tendency toward movement to the right in the horizontal direction.
The right side of failure mechanism consists of three parts, which include:
- -
The rigid wedge ABC (defined by the angular parameters , ) under the base of the foundation with velocity, along the direction at an angle of with BC for rigid body motion.
- -
The sector ACD (defined by the angle ) delineated by the logarithmic helix CD with A as the focal point, the side length of logarithmic helix shear zone is , where the length of AC is . The velocity increases exponentially from on the AC side so that on the AD side.
- -
The rigid wedge ADE in the rock masses on the side of the foundation, which carries out rigid body motion along the direction of angle with the velocity .
- -
The rest of the rock body remains stationary.
3.2. Calculation of Work Done by External Forces
As the rock mass below the edge of the body BCDE shown in
Figure 3 remains stationary, BCDE is a velocity interruption line. According to the flow law, the velocity of each point along this line must be at an angle of
with the line. Since AC is a velocity interruption line, the velocity
(velocity of the rock masses at the right of AC) is perpendicular to AC, when the body moves. Therefore, the value of
is
, and
, the change of velocity vector across AC, is at an angle of
with AC. The compatible velocity diagram at AC is shown in
Figure 4. AD is not a velocity interruption line, so the motion of rigid wedge ADE and the sector area ACD can be kept continuous, and the wedge ADE will be rigidly translated with velocity
. The velocity vector triangle consisting of
,
, and
mentioned in this paragraph and the direction of
are shown in
Figure 4.
The work done by external forces in assumed failure mechanism includes the self-weight action of the rock mass, ; the vertical load, , borne by the foundation; the surface overload, ; and the related pseudo-static inertia force simplified by the seismic action. Since the effect of vertical seismic acceleration is ignored in this consideration, therefore, the work done by the external forces include:
- (1)
work done by the vertical load, :
- (2)
work done by rock mass gravity, :
- (3)
work done by surface overload, :
Then the total work done by the external forces applied to the mechanism can be written as:
where
,
,
,
,
,
,
, and
are the functions of the angle parameters
,
,
, and
, which are shown here:
The functions , , and reflect the influence of the unit weight of rock mass; function reflects the influence of surface overload; functions , , , and reflect the influence of the horizontal seismic acceleration.
3.3. Calculation of Internal Energy Consumption
As the internal energy dissipation in the mechanism comes from the velocity jump on the velocity interruption lines BC, AC, CD, DE, and the shear energy dissipation inside the sector area ACD, the internal frictional resistance of the mechanism consists of:
- (1)
shear energy dissipation inside the rock mass:
- (2)
energy dissipation for velocity discontinuity on the velocity discontinuity line BC:
- (3)
energy dissipation for velocity discontinuity on velocity interruption line AC:
- (4)
energy dissipation for velocity discontinuity on velocity discontinuity line CD:
- (5)
energy dissipation for velocity discontinuity on the velocity interrupted line DE:
Finally, the internal energy dissipation rate of the failure mechanism can thus be rewritten as:
where
,
,
, and
are the functions of the angle parameters
,
,
, and
, which can be calculated as:
3.4. Calculation of Upper Bound of Mechanism Bearing Capacity
The work done by the external force on the mechanism and the energy consumed internally by the drag force are expressed in this subsection as functions of four angles:
,
,
, and
, which are
and
. According to the geometry in
Figure 3, the range of values for the four angular parameters is limited to
.
.
.
.
An upper limit of the upper load,
, that can be carried by the strip foundation located on the surface of the rock mass under the influence of seismic acceleration in the Prandtl failure mechanism is then expressed as:
where
;
; and
.
The upper bound, , of the bearing capacity of strip foundation could obtained by adjusting the three control angle parameters of failure mechanism , , .
The calculation process in this section can be illustrated by
Figure 5:
4. Verification
Verification of the present work was carried out in two parts. In
Section 4.1, a comparison with the analytical results of Yang and Yin [
12] is first presented. In
Section 4.2, the ultimate bearing capacity of strip foundations placed on five typical rock masses is calculated using this method, and the calculated results are compared with those of the finite element model.
4.1. Verification against Existing Theoretical Results (Analytical Solutions)
The estimation of the upper limit of the ultimate bearing capacity of a strip foundation on a rock foundation, such as conducted in studies by Yang and Yin [
12] and Saada [
27], is usually based on the assumption of a multi-wedge body failure mechanism. The control parameters of such mechanisms include the angle parameter,
, of the rigid body under the foundation; the top angle,
, and bottom angle,
, of each wedge; and the equivalent friction angle,
, which represents the nature of the rock mass itself. The computational accuracy of such mechanisms is mainly determined by the total number of divided wedges,
. To demonstrate the validity of the assessment method based on the generalized Prandtl mechanism provided in this study, a hypothetical failure mechanism with
,
,
,
,
, and
was selected without considering the seismic effect (assuming
), and the trend of the upper limit of the bearing capacity with the variation of
was demonstrated and then compared with the experimental results of Yang and Yin [
12]. The comparison results are shown in
Table 1.
Table 1 shows that the ultimate bearing capacity calculated by the present method is larger relative to the upper bound estimate of the bearing capacity of Yang and Yin [
12]. The maximum error does not exceed 10%, which indicates that the upper bound estimate of the bearing capacity in this paper is reliable. The reason for this discrepancy is that the equivalent friction angle selection method recommended by Drescher and Christopoulos [
25] and Collins et al. [
26] was adopted in the study by Yang and Yin [
12], while the equivalent
and
calculation methods summarized by Hoek [
18,
19] based on practical engineering experience were chosen here.
This equivalence friction angle generation, conducted by fitting the nonlinear HB damage criterion envelope, was used in the analysis of the present work to solve Equation (2), as recommended by Evert Hoek et al. [
18,
19]. The purpose of adapting this process was mainly to balance the area of the nonlinear HB curve above and below the linear MC envelope, so that one could find the fitting line that can best represent the change trend of HB strength envelope:
in which
, and:
where
is the depth of layer, which is assumed to be 100 m in this study.
Since the equivalent MC envelope is always on the upper side of the nonlinear HB failure envelope, the equivalent friction angle, , and equivalent cohesive force, , determined by the equivalent MC envelope, can be used to solve the upper limit of bearing capacity. In the following calculation process, the values of and do not change, ensuring that the equivalent friction angle, , is always the angle that represents the direction of the HB criterion envelope. In the following calculation, by changing the shape of the failure mechanism itself (namely, adjusting the three control angle parameters, α’, α, and δ), the minimum of the upper bound of the ultimate bearing capacity can be found.
One advantage of this method is that the minimization process of ultimate bearing capacity is accomplished by changing the shape of the mechanism itself. The method can find the equivalent friction angle, , and the corresponding equivalent cohesive force, , which best represent the characteristics of the rock mass after determining the type of rock mass (namely, determining the rock mass parameters GSI, and ).
4.2. Verification against Numerical Analysis Results
Different rock masses have unique mechanical characteristics, and the HB criterion uses the parameters
,
, and
to describe the characteristics of different rocks. In the following, five typical rocks provided by Hoek [
18,
19] were selected as examples to seek the supremum (the minimum values of the upper limit) of the ultimate bearing capacity, and the corresponding failure mechanisms when the supremum is obtained are shown in
Table 2. More parameters for typical rocks are provided in the literature [
18] for reference. The results presented in this section are the supremum of the bearing capacity without considering overload and seismic action (
,
). Since only the load carrying capacity calculation case cited in the previous section was carried out in Yang and Yin’s study [
12], the predicted results in this section will be compared with the numerical calculation results of the ABAQUS finite element model.
As shown in
Figure 6, this model is a simulation of the ultimate load of a strip foundation on a rock mass. The width of the strip foundation was
B = 0.5 m, and to eliminate the influence of size effect, the foundation was therefore set as a 5 m × 5 m square rock mass. The model had 918 nodes, 845 elements, and the element type was CPE4. The size of the rock unit decreases as the distance from the base of the bar increases. In the X direction (horizontal), the unit size increases from left to right with a minimum of 0.05 m (left) and a maximum of 0.5 m (right). In the Y direction (vertical), the unit size increases from top to bottom with a minimum of 0.05 m (top) and a maximum of 0.5 m (bottom). The ultimate bearing capacity obtained from each numerical simulation is given in the following table. When comparing the prediction results of this study with the numerical simulation results, the error between them is less than 5%.
The failure mechanisms corresponding with the supremum of the ultimate bearing capacity on the same kind of rock (with the same
,
, and
) remain unchanged when the consideration of overload and seismic action is added in the subsequent sections. The critical failure surfaces for different rock mass are shown in
Figure 7,
Figure 8,
Figure 9,
Figure 10,
Figure 11 and
Figure 12.
- І
Intense shear zones
Figure 8.
Critical failure surface (; ; ; ).
Figure 8.
Critical failure surface (; ; ; ).
- II
Brecciated shear/faults
Figure 9.
Critical failure surface (; ; ; ).
Figure 9.
Critical failure surface (; ; ; ).
- Ш
Sericite with low quartz
Figure 10.
Critical failure surface (; ; ; ).
Figure 10.
Critical failure surface (; ; ; ).
- Ⅳ
Sericite with similar quartz
Figure 11.
Critical failure surface (; ; ; ).
Figure 11.
Critical failure surface (; ; ; ).
- V
Sericite with high quartz
Figure 12.
Critical failure surface (; ; ; ).
Figure 12.
Critical failure surface (; ; ; ).
A large GSI value (>25) indicates a high quality rock mass, while a larger denotes a stronger and more complete rock mass. Therefore, sample I is a poor quality rock mass, while II and III have relatively similar strength and rock mass, and samples IV and V have very good rock integrity and very high quality. As can be seen from the table, the bearing capacity results predicted in this study match the rock masses of the samples.
Comparing the failure mechanisms of the five types of rock masses, it can be determined that:
- 1.
The calculated value of the equivalent friction angle, , is greater for rock masses with good properties, and the ultimate bearing capacity increases with an increase in .
- 2.
The shapes of the slip surfaces of different masses are similar when reaching the failure, but with the increase in ultimate bearing capacity that can be provided, the depth of the mobilized rock mass increases and the overall volume increases.
- 3.
The angular parameters of the failure mechanism can basically be determined within a general range: the range of values for can be set at 55° to 70°, the better the rock mass, the larger the value taken; can be set at 87°; and can be set at 90°. The shape of the damage mechanism of the rock foundation can be roughly depicted using this set of parameters.
5. Parametric Analysis
This section focuses on the effects of surface overload,
, rock self-weight,
, and horizontal seismic coefficient,
, on the seismic bearing capacity of the foundation. The conclusion is consistent with the existing results [
6]; that is, the elevation of the surface overload and the self-weight of the rock mass is beneficial for improving the ultimate bearing capacity of the foundation, while an increase in the horizontal seismic coefficient will lead to a sharp decrease in the bearing capacity.
5.1. Effect of Surface Overload, q0, and Rock Mass Gravity, γ
The effects of surface overload, , and rock mass gravity, , on the upper limit of bearing capacity, , was investigated on a rock mass much like the intense shear zones above, where , , , and .
The variation trend of the upper limit of ultimate bearing capacity of foundations when
changes from 0–50 kPa (fixed
) was first studied.
Figure 13 shows the variation of the upper limit with surface overload,
, in the static case (
) and the seismic case (
). The results were as expected: the ultimate bearing capacity of the foundation decreases under the seismic action. Inspection of
Figure 13 suggests that the decrease in bearing capacity during the earthquake was significant and became more dramatic with an increase in the surface overload,
.
Table 3 shows the differences between the estimated upper limit of the bearing capacity for the static case (
) and the earthquake case (
).
As can be seen in
Figure 14, the effect of the unit weight,
, was lower compared to the effect of the surface overload,
, on the bearing capacity, which is also consistent with the results of a study by Saada [
27]. For the determined overload,
, the unit weight,
, rose from 20 kN/m
3 to 24 kN/m
3, while the ultimate bearing capacity rose by less than 0.1 MPa. Meanwhile for the determined unit weight,
, every 10 kPa rise in the value of the surface overload caused the ultimate bearing capacity to rise by 0.25 MPa.
In addition, it can also be seen in
Figure 13 and
Figure 14 that the variation in the upper bound of the bearing capacity,
, exhibits a linear dependence on the surface overload,
, and the unit weight of the rock mass,
.
5.2. Effect of Seismic Action and Rock Properties
The calculation of bearing capacity can be rewritten into the following form by analogy with the classical form of foundation bearing capacity of Terzaghi [
12]:
where the dimensionless parameters
,
, and
are the bearing capacity coefficients of uniaxial compressive strength of rock, surface overload, and self-weight of rock mass, respectively. To facilitate application in practical geotechnical engineering, it is further extended to allow for the surface overload,
, and the unit weight of the rock mass,
, to not be considered.
where
is defined as the seismic uniaxial compressive strength bearing capacity coefficient to further facilitate the use of the table in subsequent research. The upper limit of bearing capacity calculation formula can be reduced to
.
This leads to the expression for the seismic uniaxial compressive strength bearing capacity coefficient:
. The following
Table 4,
Table 5,
Table 6,
Table 7 and
Table 8 summarize several sets of
values for the five types of typical intact, unweathered (taking
D = 0) rocks mentioned above (
22, 20, 15.5, 14, and 25, respectively) at different seismic strengths.
In practical engineering, the seismic uniaxial compressive strength bearing capacity coefficients at different seismic intensities are selected and multiplied with the uniaxial compressive strength, , of this type of rock to obtain the upper bound of the bearing capacity. For each rock, the parameter , which characterizes the mass of the rock mass, is taken to vary in the interval from 5 to 80 in the calculation.
5.3. Effect of Horizontal Seismic Coefficient
Figure 15 shows the variation of the upper bound of the bearing capacity for the
intense shear zones with increases in the horizontal seismic coefficient. The parameters of the rocks in the figure are shown in
Table 2, and the surface overload
.
is taken into consideration. It can be clearly seen that the bearing capacity decreases very severely when
increases and is approximately linearly related to the horizontal seismic coefficient. This significant drop indicates that seismic action has a huge weakening effect on the bearing capacity of the foundation, when
rises from 0 to 0.2. The predicted value of the ultimate bearing capacity,
, drops from 10.8 MPa to 7 MPa.