Assessment of the Bearing Capacity of Foundations on Rock Masses Subjected to Seismic and Seepage Loads
sustainability
Article Assessment of the Bearing Capacity of Foundations on Rock Masses Subjected to Seismic and Seepage Loads
Rubén Galindo 1,* , Ana Alencar 1, Nihat Sinan Isik 2 and Claudio Olalla Marañón 1
1 Departamento de Ingeniería y Morfología del Terreno, Universidad Politécnica de Madrid, 28040 Madrid, Spain; [email protected] (A.A.); [email protected] (C.O.M.) 2 Department of Civil Engineering, Faculty of Technology, Gazi University, 06560 Ankara, Turkey; [email protected] * Correspondence: [email protected]
Received: 20 October 2020; Accepted: 30 November 2020; Published: 2 December 2020
Abstract: It is usual to adopt the seismic force acting as an additional body force, employing the pseudo-static hypothesis, when considering earthquakes in the estimation of the bearing capacity of foundations. A similar approach in seepage studies can be applied for the pore pressure’s consideration as an external force. In the present study, the bearing capacity of shallow foundations on rock masses considering the presence of the pseudo-static load was developed by applying an analytical solution for the Modified Hoek and Brown failure criterion. Calculations were performed adopting various inclinations of the load and the slope on the edge of the foundation, as well as different values of the vertical and horizontal components of the pseudo-static load. The results are presented in the form of charts to allow an affordable and immediate practical application for footing problems in the event of seismic loads or seepages. Finally, and to validate the analytical solution presented, a numerical study was developed applying the finite difference method to estimate the bearing capacity of a shallow foundation on a rock mass considering the presence of an additional horizontal force that could be caused by an earthquake or a seepage.
Keywords: Hoek–Brown criterion; seismic load; seepage; analytical method; pseudo-static load; finite difference method
1. Introduction Natural events such as earthquakes and floods generate impulsive loads and seepages in the ground, endangering civil constructions. Infrastructures must be built to resist natural risks in adequate safety conditions and be economically viable. Therefore, an optimized design that reliably considers the negative effects that these natural phenomena introduce to the foundations is highly advisable to guarantee the sustainability of the infrastructures. Today, research in the area of seismic bearing capacity is very much in demand because of the devastating effects of foundations under earthquake conditions. A high number of failures have occurred where field conditions have indicated that the bearing capacity was reduced during seismic events. In the estimation of the bearing capacity, when the effect of the earthquake is considered, it is usual to adopt the pseudo-static hypothesis where the seismic force acts as an additional body force within the soil mass. The vertical and horizontal acceleration are applied both on the ground and in the structure. Thus, the limit conditions can be evaluated by introducing pseudo-static equivalent forces, corresponding with the inertial forces in the soil during the seismic excitation. Such an approach is based on the hypothesis of a synchronous motion for the soil underneath the footing, a hypothesis that is acceptable only in the case of small footing widths and large values for the soil stiffness. According to
Sustainability 2020, 12, 10063; doi:10.3390/su122310063 www.mdpi.com/journal/sustainability Sustainability 2020, 12, 10063 2 of 21 numerous studies carried out about this issue in the field of geotechnical engineering, it is known that the bearing capacity is considerably reduced mostly due to the horizontal component. These studies are primarily based on (a) the limit equilibrium method [1–4], (b) the method of the characteristics [5], and (c) the limit analysis method using numerical analysis [6–8]. Limit equilibrium methods have evolved and incorporated different factors, generalizing the application hypotheses to solve the most general possible configurations. Thus, Sarma and Iossifelis [1] determined seismic bearing capacity factors for soils with seismic acceleration acting on the structure only and using the limit equilibrium technique of slope stability analysis with inclined slices. Later, Richarts et al. [2] considered the inertial mass of the ground using a simplification of the Prandtl failure surface to eliminate the fan-shaped transition zone and thereby average its effect concentration. This effect can be considered as a retaining wall with the active lateral thrust from the region below the foundation pushing against the passive resistance from the external region. Then, Choudhury and Rao [4], using the limit equilibrium method, studied the seismic forces for shallow strip footings embedded in sloping ground with a linear failure criterion. They considered pseudo-static forces acting both on the footing and on the soil below the footing and considered in the analysis a composite failure surface involving a planar and logspiral. For its part, Kumar and Rao [5] applied the method of the characteristics using pseudo-static forces for a Mohr–Coulomb criterion. The analytical methods were limited to simplified configurations where the inertia of the soil mass was not included and had to be completed with empirical or numerical terms. Adopting the conventional pseudo-static approach, Yang [7] used a non-linear failure criterion to evaluate the seismic bearing capacity of strip foundations on rock slopes by means of the generalized tangential technique, in which the non-linear strength is replaced by an “optimal” tangential Mohr–Coulomb domain in a limit analysis framework where the upper-bound solutions are obtained by optimization. A numerical study was performed by Raj et al. [8] using finite-element limit analysis for shallow strip foundations embedded in homogeneous soil slopes, applying a pseudo-static approach to consider the seismic action on both the soil and foundation. Nova and Montrasio [9] introduced a satisfactory description of the dynamic soil–structure interaction by means of the macro-element concept (the system as a unique, non-linear macro-element with a limited number of degrees of freedom), which has been applied by many researchers to different problems of shallow foundations [10,11]. Applying a similar approach, in recent years, studies have been published where the pore pressure has been included as an external force, analyzing the slope stability in soil [12] and rock [13], and affecting the bearing capacity of the shallow foundation on soil [14–16] and in jointed rock mass [17]. In foundations with large widths such as dams, the seepage tends to present an almost horizontal flow in the influence area of the foundation, as has been possible to verify in real cases [18]. In the case of seismic load, the horizontal acceleration is the same in the whole model, while in the case of seepage, the horizontal acceleration can vary once there is a loss of force along the water path. Depending on the direction of the groundwater flow, it can act as a passive (resistant) or active force [16], once it is in the same direction to the stress path or not. Thus, the wedge formed below the footing is asymmetric. Additionally, Veiskarami and Kumar [14] show that with an increase in the hydraulic gradient, the nature of the failure patterns becomes more non-symmetrical; for a non-symmetrical failure pattern, the footing would have a greater chance of failing in overturning rather than under simple vertical compression. It must be pointed out that the traditional formulations of the bearing capacity are based on the Mohr–Coulomb parameters (the cohesion and friction angle) that are very efficient in the field of soil mechanics, with a linear behavior. In rock mechanics, the current methods for estimating the bearing capacity adopt the Hoek and Brown failure criterion [19] (valid only for rock masses not excessively fractured) and Modified Hoek and Brown failure criterion [20,21] (valid under general conditions of degree of fracturing of the rock mass). Both failure criteria are non-linear and applicable to the rock Sustainability 2020, 12, 10063 3 of 21 mass with a homogeneous and isotropic behavior. The linearization of the failure criterion implies incorporating approximate methods and requires iterative procedures to ensure an optimized upper or lower bound for the solution; therefore, it is desirable to be able to address the problem using the non-linear criterion directly. In addition, most of the formulations are limited to flat ground, the need to analyze the bearing capacity of shallow foundations on sloping ground of moderate slopes being very common in dam and bridge foundations. Finally, although the numerical solutions allow solving complex problems with singular considerations when a seismic load or filtration acts, in the face of non-linear criteria, a complete analysis of numerical convergence is necessary, which complicates the practical applicability and the rapid design of foundations in rock masses. An analytical method for the calculation of shallow foundations that solves the internal equilibrium equations and boundary conditions combined with the failure criterion was proposed by Serrano and Olalla [22] and Serrano et al. [23], applying the Hoek and Brown [19] and the Modified Hoek and Brown failure criterion [20], respectively. It is based on the characteristic lines method [24], with the hypothesis of the weightless rock, strip foundation and associative flow law. The formulation of the bearing capacity proposed by Serrano et al. [23] introduces a bearing capacity factor, which makes the failure pressure proportional to the uniaxial compressive strength of the rock (UCS). In the present study, the analytical formulation of Serrano et al. [23] and design charts were developed to study the bearing capacity when there is an increase in load induced by forces of seismic origin or filtration in rock masses, where it is necessary to use a non-linear failure criterion and the own weight of the ground is generally negligible compared to the resistant components, considering, as usual, the possibility of the inclination of the ground to the edge of the foundation. Besides, a numerical model was created through a finite difference method, assuming a similar hypothesis for the analytical solution, and it was observed that the results obtained by both methods were quite similar.
2. Problem Statement
2.1. Mathematical Model As is generally known, in rock mechanics, the non-linear Modified Hoek and Brown failure criterion is the most used, and it is applicable for a rock mass with a homogeneous and isotropic behavior, meaning that by the inexistence or by the abundance of discontinuities, it has the same physical properties in all directions. In this research, the Modified Hoek and Brown failure criterion [21,25] was used, and it was formulated as a function of the major principal stress (σ3) and minor principal stress (σ1) according to the following equation: σ σ σ a 1 − 3 = m 3 + s (1) σc · σc
The uniaxial compressive strength (UCS) is σc, while the parameters m, s and a can be evaluated with (2)–(4) and depend on the intact rock parameter (mo), quality index of the rock mass (geological strength index (GSI)) and damage in the rock mass due to human actions (D), which in shallow foundations, is usually equal to zero. GSI 100 − m = mo e 28 14 D (2) · − · GSI 100 9 −3 D s = e − · (3)
1 1 GSI 20 a = + e −15 e −3 (4) 2 6 − Serrano et al. [23] proposed an analytical formulation for estimating the ultimate bearing capacity of the strip footing for a weightless rock mass, based on the characteristic method, which allows solving the internal equilibrium equations in a continuous medium together with the boundary equations and Sustainability 2020, 11, x FOR PEER REVIEW 4 of 22
Serrano et al. [23] proposed an analytical formulation for estimating the ultimate bearing capacity of the strip footing for a weightless rock mass, based on the characteristic method, which Sustainabilityallows solving2020, 12 , the 10063 internal equilibrium equations in a continuous medium together with4 of the 21 boundary equations and those that define the failure criterion. This solution is based on the Modified Hoek and Brown failure criterion [20], taking into account the associated plastic flow rule. those that define the failure criterion. This solution is based on the Modified Hoek and Brown failure According to this analytical formulation, the ground surface that supports the foundation is criterion [20], taking into account the associated plastic flow rule. composed of two sectors (Figure 1): Boundary 1 (free), with the inclination i1, where the load acting According to this analytical formulation, the ground surface that supports the foundation is on a surface is known (for example, the self-weight load on the foundation level or the load from composed of two sectors (Figure1): Boundary 1 (free), with the inclination i1, where the load acting installed anchors), and Boundary 2 (foundation), where the bearing capacity of the foundation on a surface is known (for example, the self-weight load on the foundation level or the load from should be determined (acting with the inclination of i2). installed anchors), and Boundary 2 (foundation), where the bearing capacity of the foundation should be determined (acting with the inclination of i2).
Boundary 2: Foundation
Ph i2 Boundary 1: Free surface i1
휋 휌2 1 ( + ) 4 2 Rankine-2 region 휋 휌1 ( − ) 4 2
Prandtl region Rankine-1 region
FigureFigure 1. 1.Mathematical Mathematical model model of of the the bearing bearing capacity capacity of of the the strip strip footing. footing.
TheThe solution solution based based on on the the characteristic characteristic lines lines method method requiresrequires thetheequation equation of ofthe the Riemann Riemann invariantsinvariants ( I(aI)[a) 26[26]] fulfilled fulfilled along along the the characteristic characteristic line: line: 퐼 (휌 ) + 휓 = 퐼 (휌 ) + 휓 (5) I푎a(ρ11) + ψ11= Ia(푎ρ2)2+ ψ2 2 (5) 1 휌 ρ 퐼푎(휌) = 1 ∙ [푐표푡𝑔(휌) + 푙푛 (푐표푡𝑔 ( ))] (6) Ia(ρ) = cotg(ρ) + ln cotg (6) 2 ∙2푘k· 2 2 · InIn this this equation, equation, the the instantaneous instantaneous friction friction angleangle atat the the boundary boundary 22 ( ρ(ρ2)2) isis thethe onlyonly unknown,unknown, becausebecause the the other other variables variables can becan defined be defined at Boundary at Boundary 1: the instantaneous 1: the instantaneous friction angle friction at Boundary angle at 1 (ρ1) and the angle (Ψ1) between the major principal stress and the vertical axis in this sector (Figure1). Sustainability 2020, 12, 10063 5 of 21
Thus, expressing Ψ2 (the angle between the major principal stress and the vertical axis in Boundary 2, as indicated in Figure1) as a function of ρ2, it is possible to estimate the ultimate bearing capacity. Through the analytical method [23], the bearing capacity was obtained by (7). P = βa N ζa (7) h · β −
The resistant parameters βa and ζa were applied to make dimensionless the calculation of the Modified Hoek and Brown failure criterion. βa represents the characteristic strength, which has the same units as the UCS and was used to make the pressures dimensionless, while ζa (the “tenacity coefficient”) is a dimensionless coefficient that, multiplied by βa, corresponds to the tensile strength.
! 1 s m (1 a) k (1 a) = = = = βa Aa UCS; ζa ; Aa · 1− ; k − (8) · (m Aa) a a · 2
Aa, k and the exponent a are constants for the rock mass and depend on the rock type (m), UCS and GSI. Nβ is the bearing capacity factor, and it can be calculated, according to the problem statement, as follows. The angle of internal friction (ρ1) can be obtained by iteration from the load at Boundary 1. From the value of ρ1 and by the iteration of (5), the value of the internal friction angle at Boundary 2 (ρ2) can be calculated. Finally, knowing ρ2, the bearing capacity factor (Nβ) can be calculated, and using, again, parameters βa and ζa, the ultimate bearing capacity (Ph) was estimated as an expression that depended on the instantaneous friction angle at Boundary 2 (ρ2), the inclination of the load on the foundation (i2) and the exponent of the Modified Hoek and Brown criterion (a; k = (1 a)/a): − 1 1 sin(ρ ) k a (1+sin(ρ )) N = ( ) − 2 · 2 ( ) β cos i2 k sin(ρ ) sin(ρ ) cos i2 r· 2 2 2 (9) a (1+k sin(ρ2)) + 1 · · sin(i2) − sin(ρ2)
2.2. Consideration of Pseudo-Static Load: Mathematical Transformation In the pseudo-static approach, static horizontal and vertical inertial forces, which are intended to represent the destabilizing effects of the earthquake or seepage, are calculated as the product of the seismic/seepage coefficients and the distributed load applied to the boundaries. In the case of the rock mass, the weight collaboration is usually negligible compared to the resistance of the ground, and therefore, the inertial forces are applied both to the foundation and to the free boundary. The vertical seismic/seepage coefficient kv is supposed to be a fraction of one horizontal kh, and in particular, the vertical acceleration is thus assumed to be in phase with the horizontal acceleration. The present study is divided into three parts: (a) The first one considered the horizontal (kh) and vertical (kv) components of the pseudo-static load on both boundaries, with a free boundary inclined by α at the edge of the foundation, which resembles the hypothesis of a seism. (b) In the second part, only the horizontal component (kh) on the foundation boundary was adopted (it being possible to consider both the horizontal and vertical components on the free boundary depending on the direction of the seepage), with the free boundary inclined by α at the edge of the foundation. This hypothesis is more similar to the presence of a seepage (both hypotheses are represented schematically in Figure2, and they are solved and shown in new charts including additional horizontal and vertical loads). (c) The final section is the comparison of the analytical result with that obtained numerically through the finite difference method. Sustainability 2020, 12, 10063 6 of 21 Sustainability 2020, 11, x FOR PEER REVIEW 6 of 22
FigureFigure 2. 2.Scheme Scheme ofof thethe pseudo-staticpseudo-static load load acting: acting: ((aa)) SeismicSeismic loadload (F(Fss);); ((bb)) SeepageSeepage (F(Fww).
TheThe applicationapplication ofof thethe analyticalanalytical methodmethod [[23]23] withwith anan increaseincrease inin thethe horizontalhorizontal andand//oror verticalvertical loadsloads producedproduced byby inertialinertial forcesforces cancan bebe carriedcarried outout byby meansmeans ofof aa parametricparametric transformationtransformation fromfrom thethe incidenceincidence anglesangles ofof thethe loadsloads inin thethe staticstatic configurationconfiguration toto thethe finalfinal configurationconfiguration includingincluding thethe seismicseismic or seepage loads. loads. Thus, Thus, for for a agener generalal case case of ofa pseudo a pseudo-static-static force force on onthe thetwo two boundaries, boundaries, the thestarting starting point point is the is theload load acting acting in the in thestatic static hypothesis hypothesis (subscript (subscript 0), 0),as is as represented is represented in Figure in Figure 3a3. a.In Inthis this case case,, the the inclination inclinationss of ofthe the load load on onthe the foundation foundation (p) (andp) and of the of the load load on onthe the free free boundary boundary (q)
(areq) are 𝑖02i 02andand 𝑖01i01, ,respectively. respectively. However, However, considering the pseudo pseudo-static-static load, the inclinationsinclinations ofof thethe loadsloads inin bothboth boundariesboundaries areare didifferentfferent ((i2i2 forfor thethe foundationfoundation boundaryboundary andand ii11 for the freefree boundary).boundary). FigureFigure3 3bb allows allows the the deduction deduction of of the the mathematical mathematical transformations transformationsof of these these angles angles from from the the static static configurationconfiguration to to the the final final pseudo-static pseudo-static configuration configuration as a function as a function of the angle of the (α angle) of the (α free) of boundary the free andboundary of the horizontal and of the (k h horizontal1 or kh2, depending (kh1 or k onh2, thedepending boundary) on and the vertical boundary) (kv1 or andkv2 , vertical depending (kv1 onor thekv2, boundary)depending components on the boundary) of the pseudo-static component load.s of the These pseudo transformations-static load. are These expressed transformations in (10) and (11).are expressed in (10) and (11). tan (i ) + k ( ) = 02 h2 tan i2 푡푎푛 (𝑖02) + 푘ℎ2 (10) 1 kv 푡푎푛 (𝑖2) = − 2 (10) 1 − 푘푣2 (1 kv1)tan(α + i01) tan (α + i1) = (1 −− 푘푣1)푡푎푛(훼 + 𝑖01) (11) 푡푎푛 ( + 𝑖1) = 1 kh1tan(α + i01) (11) 1 − 푘ℎ1푡푎푛(훼 + 𝑖01) Therefore, for the pseudo-static calculation, the analytical formulation of the characteristics method expressed by (5) can be used using the load inclinations on each boundary obtained through the transformationsBoundary indicated 2 in (Foundation) Equations (10) and (11)Boundary as a function 1 (Free of the) horizontal and vertical components of the added inertial load and of the angle of inclination of Boundary 1. pv aIn the case of the seismic load, it is consideredi02 that kh1 = kh2 and kv1 = kv2, while in the case of the i01 seepage load, kv1 = 0; thus, for clearerph notation, it is denoted that for the seismicqv load, the horizontal and vertical components of the pseudo-static load are kh and kv, respectively, and for the seepage load, the horizontal component of the pseudo-static load on the foundation boundaryqh is called ia.
Sustainability 2020, 11, x FOR PEER REVIEW 6 of 22
Figure 2. Scheme of the pseudo-static load acting: (a) Seismic load (Fs); (b) Seepage (Fw).
The application of the analytical method [23] with an increase in the horizontal and/or vertical loads produced by inertial forces can be carried out by means of a parametric transformation from the incidence angles of the loads in the static configuration to the final configuration including the seismic or seepage loads. Thus, for a general case of a pseudo-static force on the two boundaries, the starting point is the load acting in the static hypothesis (subscript 0), as is represented in Figure 3a. In this case, the inclinations of the load on the foundation (p) and of the load on the free boundary (q)
are 𝑖02 and 𝑖01, respectively. However, considering the pseudo-static load, the inclinations of the loads in both boundaries are different (i2 for the foundation boundary and i1 for the free boundary). Figure 3b allows the deduction of the mathematical transformations of these angles from the static configuration to the final pseudo-static configuration as a function of the angle (α) of the free boundary and of the horizontal (kh1 or kh2, depending on the boundary) and vertical (kv1 or kv2, depending on the boundary) components of the pseudo-static load. These transformations are expressed in (10) and (11).
푡푎푛 (𝑖02) + 푘ℎ2 푡푎푛 (𝑖2) = (10) 1 − 푘푣2
(1 − 푘푣1)푡푎푛(훼 + 𝑖01) Sustainability 2020, 12, 10063 푡푎푛 ( + 𝑖1) = (711 of) 21 1 − 푘ℎ1푡푎푛(훼 + 𝑖01)
Boundary 2 (Foundation) Boundary 1 (Free)
pv a i02 i01 ph qv
qh
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Boundary 2 (Foundation) Boundary 1 (Free) b pv-kv2pv
ph+kh2pv qv-kv1qv
qh-kh1qv
푝ℎ +푘ℎ2·푝푣 휋 푞ℎ −푘ℎ1·푞푣 푡푎푛 (𝑖2) = 푡푎푛 ( − − 𝑖1) = (1 − 푘푣2) 푝푣 2 (1 − 푘푣1) 푞푣
FigureFigure 3. 3. SchemeScheme of of the the seismic seismic load load estimation: estimation: (a) static (a) configurationstatic configuration and (b) pseudo-staticand (b) pseudo inclination-static inclinationon Boundary on1 Boundary (external boundary1 (external of boundary foundation). of foundation). Note: In this Note: figure, In the this subscripts figure, the “v” subscript and “h”s refer“v” andto the “h” vertical refer to and the horizontal vertical and projections horizontal of projection the load. s of the load.
3. DesignTherefore, Charts for for the Estimation pseudo-static of Bearing calculation, Capacity the analytical formulation of the characteristics method expressed by (5) can be used using the load inclinations on each boundary obtained through 3.1. Calculation Cases and Representation of Analytical Results the transformations indicated in Equations (10) and (11) as a function of the horizontal and vertical componentsOnce the of mathematicalthe added inertial transformation load and of ofthe the angle load of anglesinclination in the of boundaries,Boundary 1. for the problem presentedIn the in case Figure of the3a, accordingseismic load to, (10) it is and considered (11) forthe that seismic kh1 = kh and2 and seepage kv1 = kv load2, while has in been the carriedcase of out,the seepagethe analytical load, formulationkv1 = 0; thus, usingfor clearer the method notation of, theit is characteristicdenoted that linesfor the can seismic be applied. load, Thethe horizontal results are andpresented vertical as component graphs, whichs of allowthe pseudo the estimation-static load of the are bearing kh and capacitykv, respectively, considering and thefor presencethe seepage of a loadpseudo-static, the horizontal load. component of the pseudo-static load on the foundation boundary is called ia. The charts are clustered according to the exponent “a” of the Modified Hoek and Brown criterion, 3.the Design inclination Chartsα of for Boundary Estimation 1 and of kBearingv of the Capacity foundation boundary, and they were developed based on io2 and the horizontal component of the pseudo-static load of the foundation boundary (kh). It is noted 3.1.that Calculation for high confining Cases and pressures Representation in Boundary of Analytical 1, it is Results not always possible to obtain a bearing capacity value;Once this the limit mathematical is demarcated transformation by the non-equilibrium of the loa line.d angles in the boundaries, for the problem presentedFor the in developmentFigure 3a, according of the new to (10) charts, and three (11) for values the seismic of the GSI and (geological seepage load strength has been index) carried were out,adopted the analytical (8, 20 and 100),formulation which generated using the exponents method of “ a the” of characteristic the Modified lines Hoek can and be Brown applied. criterion The resultsequal toare 0.5, presented 0.55 and as 0.6 graphs from, (4).which Based allow on the (7), estimation the values of of the the bearing rock type capacity (mo) andconsidering the uniaxial the presence of a pseudo-static load. The charts are clustered according to the exponent “a” of the Modified Hoek and Brown criterion, the inclination α of Boundary 1 and kv of the foundation boundary, and they were developed based on io2 and the horizontal component of the pseudo-static load of the foundation boundary (kh). It is noted that for high confining pressures in Boundary 1, it is not always possible to obtain a bearing capacity value; this limit is demarcated by the non-equilibrium line. For the development of the new charts, three values of the GSI (geological strength index) were adopted (8, 20 and 100), which generated exponents “a” of the Modified Hoek and Brown criterion equal to 0.5, 0.55 and 0.6 from (4). Based on (7), the values of the rock type (mo) and the uniaxial compressive strength of the rock (UCS) do not influence the normalized graphs, and to perform the calculations, mo = 15 and UCS = 1 MPa were adopted. In the graphs developed for kv > 0 on the foundation boundary, representative of the seismic load, four kh values were adopted and correlated two values of kv, two slope angles for the free boundary (α) and three initial inclination angles for the load on the foundation boundary (io2), representing the inclination angle of the load without considering the pseudo-static load. In Table 1, the values used in the analysis are indicated. Sustainability 2020, 12, 10063 8 of 21 compressive strength of the rock (UCS) do not influence the normalized graphs, and to perform the calculations, mo = 15 and UCS = 1 MPa were adopted. In the graphs developed for kv > 0 on the foundation boundary, representative of the seismic load, four kh values were adopted and correlated two values of kv, two slope angles for the free boundary (α) and three initial inclination angles for the load on the foundation boundary (io2), representing the inclination angle of the load without considering the pseudo-static load. In Table1, the values used in the analysis are indicated.
Table 1. Geometric parameters adopted in the model (kv > 0).
kh kv = kh, kh/2 α (◦) io2 (◦) a 0.1 0.1, 0.05 0.5, 0.2 0.2, 0.1 0, 20 0, 10, 20 0.55, 0.3 0.3, 0.15 0.6 0.4 0.4, 0.2
On the other hand, in the charts developed with kv = 0 on the foundation boundary, representative of the seepage loads, three values of horizontal load were used, in those cases called ia (additional inclination); three values of the slope (α) and another three values of io2 were also used, which are shown in Table2.
Table 2. Geometric parameters adopted in the model (kv = 0).
ia α (◦) io2 (◦) a 0.1, 0.2, 0.3 0, 5, 10 0, 10, 20 0.5, 0.55, 0.6
It is emphasized that kh (the horizontal component of the seismic load) and ia (the horizontal component of the seepage load) were in proportion to the vertical load applied; the values expressed in Tables1 and2 are the relation between the horizontal and the vertical load. The final inclination of the load in the foundation boundary, in the cases of seepage loads, can be obtained directly as a simplification of (10): tan (i2) = tan(io2) + ia (12)
3.2. Charts for Estimation of Bearing Capacity Considering kh and kv > 0 (Seismic Case)
These design charts allow obtaining the bearing capacity factor Nβ (9), and they are presented in Figures4–6. Each graph represented corresponds to determined values of the exponent “ a” of the Modified Hoek and Brown criterion of the rock mass (a function of the GSI of the quality of the rock mass), the inclination α of the free boundary and the ratio of the vertical and horizontal components of the pseudo-static load (kv/kh = 1 or 0.5). In each graph, different curves corresponding to the angles of the inclination of the static load on Boundary 2 (io2) and horizontal component of the pseudo-static load (kh) are presented, so that in the abscissa, the known value of the normalized main major stress is presented, normalized on Boundary 1 (σ*01), estimated through (13). It is dimensionless and corresponds to the load acting on Boundary 1, and its value depends on the inclination angle of the load i1 obtained by (11) (Figure1). σ1 σ01∗ = + ζa (13) βa Among the graphs, it is easy to appreciate that as there is a greater slope for the free boundary (α), a lower kv/kh ratio and a lower exponent “a” from the Modified Hoek and Brown failure criterion, the value of Nβ follows a declining trend. Sustainability 2020, 12, 10063 9 of 21 Sustainability 2020, 11, x FOR PEER REVIEW 10 of 22
a = 0.5 / = 0 / kv = kh a = 0.5 / = 20 / kv = kh 15 1 i2=0; kh=0.1 15 i2=0; kh=0.1 Static Case 1 2 i2=0; kh=0.2 1 2 i2=0; kh=0.2 3 i2=10; kh=0.1 4 i =0; k =0.3 3 i2=10; kh=0.1 10 2 2 h Static Case 5 i =10; k =0.2 10 2 h 1 4 i2=0; kh=0.3 3 6 N i2=20; kh=0.1 5 i2=10; kh=0.2
N 2 i2=0; kh=0.4 7 6 i2=20; kh=0.1 5 4 i2=10; kh=0.3 3 i2=20; kh=0.2 5 4 i2=0; kh=0.4 5 7 6 i =10; k =0.4 5 i2=10; kh=0.3 8 2 h 6 7 Non-equilibrium i2=20; kh=0.3 i2=20; kh=0.2 8 boundary 7 9 Non-equilibrium 0 9 i2=20; kh=0.4 8 9 boundary i2=10; kh=0.4 0,01 0,10 1,00 0 8 i2=20; kh=0.3 Normalized main major stress, *01 0,01 0,10 1,00 9 i2=20; kh=0.4 Normalized main major stress, *01
a = 0.5 / = 0 / kv = kh/2 a = 0.5 / =20 / kv = kh/2 15 15 1 i2=0; kh=0.2 1 i2=0; kh=0.2 2 i =0; k =0.3 Static Case 2 i2=0; kh=0.3 2 h 3 3 i2=10; kh=0.2 i2=10; kh=0.2 10 1 10 Static Case 4 4 i2=0; kh=0.4
i2=0; kh=0.4 N 5 i2=10; kh=0.3 5 i2=10; kh=0.3 1
2 6 i2=20; kh=0.2 N 6 i =20; k =0.2 2 h 2 5 3 5 7 i2=10; kh=0.4 7 i2=10; kh=0.4 3 4 4 i2=20; kh=0.3 i2=20; kh=0.3 8 5 8 5 i =20; k =0.4 i =20; k =0.4 2 h 6 Non-equilibrium 2 h 6 7 8 boundary 7 Non-equilibrium boundary 0 0 8 0,01 0.10 1.00 0.01 0.10 1.00
Normalized main major stress, *01 Normalized main major stress, *01
Figure 4. Seismic charts for estimation of the bearing capacity factor Nβ in the case of an exponent from Modified Hoek and Brown failure criterion: a = 0.5. Sustainability 2020, 11, x FOR PEER REVIEW 11 of 22 Sustainability 2020, 12, 10063 10 of 21
Figure 4. Seismic charts for estimation of the bearing capacity factor Nß in the case of an exponent from Modified Hoek and Brown failure criterion: a = 0.5.
Figure 5. Seismic charts for estimation of the bearing capacity factor Nß in the case of an exponent from Modified Hoek and Brown failure criterion: a = 0.55. Figure 5. Seismic charts for estimation of the bearing capacity factor Nβ in the case of an exponent from Modified Hoek and Brown failure criterion: a = 0.55. Sustainability 2020, 12, 10063 11 of 21 Sustainability 2020, 11, x FOR PEER REVIEW 12 of 22
ß FigureFigure 6. Seismic 6. Seismic charts charts for for estimation estimation of of the the bearing bearing capacity capacity factor factorN Nβ inin thethe casecase of an an exponent exponent from from Modified Modified Hoek Hoek and and Brown Brown failure failure criterion: criterion: a = a0.6=. 0.6. Sustainability 2020, 12, 10063 12 of 21
Table3 shows the results of the bearing capacity ( Ph) of some studied cases. It should be noted that the value of Ph is not directly proportional to Nβ, meaning that a higher value of Nβ does not necessarily mean that the value of Ph will be higher. Besides, the same value of Nβ is associated with different values of Ph depending on the other geotechnical parameters, as shown in (7). On the other hand, it is observed that under equal conditions (only varying the GSI and, consequently, the parameter “a”), the greater the GSI, the lower the value of Nβ; however, as expected, the bearing capacity of the rock mass is higher.
Table 3. Bearing capacity estimation by analytical method [23](mo = 15 and uniaxial compressive strength (UCS) = 1 MPa).
β P GSI a α ( ) σ* i ( ) k k a ζ N h ◦ 01 o2 ◦ h v (MPa) a β (MPa) 8 0.6 0 0.1 0 0.2 0.2 0.1051 0.00062 6.31 0.664 20 0.55 0 0.1 0 0.2 0.2 0.1309 0.00122 5.42 0.709 100 0.5 0 0.1 0 0.2 0.2 1.8750 0.03556 4.93 9.174 8 0.6 0 0.1 0 0.2 0.1 0.1051 0.00062 6.70 0.704 20 0.55 0 0.1 0 0.2 0.1 0.1309 0.00122 5.73 0.750 100 0.5 0 0.1 0 0.2 0.1 1.8750 0.03556 5.21 9.698 8 0.6 0 0.1 10 0.2 0.2 0.1051 0.00062 3.77 0.397 20 0.55 0 0.1 10 0.2 0.2 0.1309 0.00122 3.28 0.429 100 0.5 0 0.1 10 0.2 0.2 1.8750 0.03556 3.01 5.576 8 0.6 0 0.1 10 0.2 0.1 0.1051 0.00062 4.29 0.451 20 0.55 0 0.1 10 0.2 0.1 0.1309 0.00122 3.72 0.487 100 0.5 0 0.1 10 0.2 0.1 1.8750 0.03556 3.41 6.326 8 0.6 20 0.1 0 0.2 0.2 0.1051 0.00062 4.21 0.442 20 0.55 20 0.1 0 0.2 0.2 0.1309 0.00122 3.77 0.493 100 0.5 20 0.1 0 0.2 0.2 1.8750 0.03556 3.53 6.560 8 0.6 20 0.1 0 0.2 0.1 0.1051 0.00062 4.47 0.470 20 0.55 20 0.1 0 0.2 0.1 0.1309 0.00122 3.99 0.522 100 0.5 20 0.1 0 0.2 0.1 1.8750 0.03556 3.74 6.940 8 0.6 20 0.1 10 0.2 0.2 0.1051 0.00062 2.50 0.262 20 0.55 20 0.1 10 0.2 0.2 0.1309 0.00122 2.28 0.299 100 0.5 20 0.1 10 0.2 0.2 1.8750 0.03556 3.51 6.513 8 0.6 20 0.1 10 0.2 0.1 0.1051 0.00062 2.84 0.299 20 0.55 20 0.1 10 0.2 0.1 0.1309 0.00122 2.58 0.338 100 0.5 20 0.1 10 0.2 0.1 1.8750 0.03556 2.45 4.529
It should be noted that in each graph, an area is indicated, in the lower right part, in which it was not possible to obtain the mechanical balance due to excessively high load conditions on the free boundary.
3.3. Charts for Estimation of Bearing Capacity Considering Only a Horizontal Pseudo-Static Load on Foundation (Seepage Case) The same representations were realized as in the previous section, where the load capacity factor Nβ of the analytical Equation (9) can be obtained in Figures7–9. In this case, each graph represented corresponds to a determined value of the exponent “a”, the inclination α of Boundary 1 and considering a value of kv = 0 on the foundation boundary. In the same way, in each graph, the different curves correspond to different inclination angles for the load on Boundary 2 (io2) and the horizontal component of the seepage pseudo-static load (ia). In this case, of the seepage load, the abscissa of the normalized main major stress normalized on Boundary 1 (σ*01) estimated through (13) corresponds to the transformation of the original inclination of the load from the static configuration to the pseudo-static situation, where the horizontal and vertical components will appear according to the seepage trajectories considered. SustainabilitySustainability2020, 12, 10063 2020, 11, x FOR PEER REVIEW 13 of14 21 of 22
a = 0.5 / = 0 15 1 i2=0; ia=0.1 Static Case 1 2 i2=0; ia=0.2
2 10 3 i2=10; ia=0.1 3 4
4 i2=0; ia=0.3 N 5 5 i2=10; ia=0.2 5 6 i2=20; ia=0.1 6
7 Non-equilibrium i2=10; ia=0.3 8 boundary 0 7 i2=20; ia=0.2 0,01 0,10 1,00
8 i2=20; ia=0.3 Normalized main major stress, *01
a = 0.5 / = 10 15 1 i2=0; ia=0.1
2 i2=0; ia=0.2 Static Case 3 i2=10; ia=0.1 10 1
2 4 i2=0; ia=0.3 5 i2=10; ia=0.2 N 3 4
5 i2=20; ia=0.1 5 6 i2=10; ia=0.3 6
7 7 i2=20; ia=0.2 8 Non-equilibrium 8 0 boundary i2=20; ia=0.3 0,01 0,10 1,00 Normalized main major stress, * 01
Figure 7. SeepageFigure 7. charts Seepage for estimationcharts for estimation of the bearing of the capacitybearing capacity factor N βfactorin the Nß case in the of case an exponent of an exponent from Modified from Modified Hoek Hoek and Brown and Brown failure failure criterion: criterion:a = 0.5.a = 0.5.
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FigureFigure 8. 8.Seepage Seepage charts charts forfor estimationestimation ofof thethe bearingbearing capacitycapacity factor Nβß inin the the case case of of an an exponent exponent from from Modified Modified Hoek Hoek and and Brown Brown failure failure criterion: criterion: a =a =0.550.55.. SustainabilitySustainability2020, 12 2020, 10063, 11, x FOR PEER REVIEW 15 of16 21 of 22
a = 0.6 / = 0 a = 0.6 / = 5 15 15 1 1 i =0; i =0.1 1 2 2 a Static Case Static Case 2 1 i2=0; ia=0.1 2 i =0; i =0.2 3 3 2 a 4 3 i2=10; ia=0.1 4 2 i2=0; ia=0.2 10 4 i2=0; ia=0.3 10 5 5 3 i2=10; ia=0.1 5 i2=10; ia=0.2
6 N
N i =20; i =0.1 4 i2=0; ia=0.3 6 6 2 a i2=10; ia=0.3
5 i2=10; ia=0.2 5 7 7 5 7 i2=20; ia=0.2
i2=20; ia=0.1 8 i2=20; ia=0.3 6 8 8 Non-equilibrium i2=10; ia=0.3 Non-equilibrium boundary boundary 0 0 7 i2=20; ia=0.2 0.01 0,10 1.00 0,01 0,10 1,00 8 i =20; i =0.3 Normalized main major stress, *01 2 a Normalized main major stress, *01
a = 0.6 / = 10 15 1 1 i2=0; ia=0.1
Static Case 2 2 i2=0; ia=0.2
3 3 i2=10; ia=0.1 4 10 4 i2=0; ia=0.3
5 5 i2=10; ia=0.2
N 6 6 i2=20; ia=0.1 i2=10; ia=0.3 7 5 7 i2=20; ia=0.2
8 i2=20; ia=0.3
8 Non-equilibrium boundary 0 0,01 0,10 1,00
Normalized main major stress, *01
FigureFigure 9. Seepage 9. Seepage charts charts for estimation for estimation of the of bearing the bearing capacity capacity factor factorNβ in N theßin case the case of an of exponent an exponent from from Modified Modified Hoek Hoek and and Brown Brown failure failure criterion: criterion:a = a0.6. = 0.6. Sustainability 2020, 12, 10063 16 of 21 Sustainability 2020, 12, x FOR PEER REVIEW 17 of 22
In this case, since the seepage is typical of dam foundations, it is more realistic to consider cases of 4. Numerical Validation moderate ground slopes, showing the charts for slopes of the free boundary (α) equal to 0◦, 5◦ and 10◦. In order to compare the results obtained by applying the chart proposed in the present study with4. Numerical those numerically Validation estimated (through FDM: finite difference method), the same hypotheses (weightlessIn order rock, to compare strip foundation the results and obtained associative by applying flow law) the chart were proposed used in the in the rock present-foundation study model.with those numerically estimated (through FDM: finite difference method), the same hypotheses (weightlessThe 2D rock,models strip were foundation used to calculate and associative the cases flow by law) the finite were useddifference in the method, rock-foundation employing model. the commercialThe 2D models FLAC weresoftware used to[27], calculate applying the cases the by planethe finite strain difference condition method, (strip employing footing). the Twocommercial-dimensional FLAC nume softwarerical [27 models], applying have the been plane used strain by condition many researchers (strip footing). to solve Two-dimensional problems of foundationsnumerical models under have dynamic been usedloads by [28] many and researchers in rock masses to solve [29]. problems Figure 10 of foundationsshows the model under used dynamic and whereloads [28 the] and boundaries in rock masseswere located [29]. Figure, at a 10 distance shows that the modeldid not used interfere and where with the boundaries result; in all were the simulationslocated, at a, distance the associative that did flow not interfere rule, weightless with the result;rock mass in all and the simulations,smooth interface the associative at the base flow of rule,the foundationweightless rockwere mass adopted. and smoothThe modified interface Hoek at the–Brown base ofconstitutive the foundation model were available adopted. in FLAC The modified V.7 was used,Hoek–Brown which corresponds constitutive to model (1). available in FLAC V.7 was used, which corresponds to (1).
FFigureigure 10. 10. 2D2D mod modelel used used in in the the calculation calculation through through FDM FDM (FLAC (FLAC 2D). 2D).
ItIt is is assumed assumed that that the the bearing bearing capacity capacity is reached is reached when when the continuous the continuous medium medium does not does support not supportmore load more because load because an internal an internal failure failure mechanism mechanism is formed. is formed. In the In case the case under under study, study, due due to the to thepresence presence of a verticalof a vertical and a and horizontal a horizontal force, the force, vertical the forcevertical was force considered was considered unknown; unknown therefore,; therefore,in the calculation, in the calculation, a constant a constant horizontal horizontal load was load applied, was applied while, thewhile vertical the vertical load increasedload increase untild untilreaching reaching failure. failure. Thus, the Thus, inclination the inclination of the load of applied the load was applied also unknown, was also because unknown, it depended because on it dependthe ratioed between on the ratio the vertical between and the the vertica horizontall and the components. horizontal components. Therefore,Therefore, to to estimate the bearing capacity for for a determinate load load inclination inclination,, it it is necessary to carry out a series of calculations to find the corresponding combination of the horizontal (σ ) and carry out a series of calculations to find the corresponding combination of the horizontal (σh) and vertical (σv) components. Figure 11 shows the results for vertical loads obtained for the case studied vertical (σv) components. Figure 11 shows the results for vertical loads obtained for the case studied (kv = 0, io = 0 , mo = 15, UCS = 100 MPa, GSI = 65 and foundation width B = 2.25 m) as a function of (kv = 0, io2 2= 0°, ◦mo = 15, UCS = 100 MPa, GSI = 65 and foundation width B = 2.25 m) as a function of the equivalentthe equivalent load load inclination. inclination. Sustainability 2020, 12, x FOR PEER REVIEW 18 of 22
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Figure 11. Vertical load (σv) as function of the load inclination on foundation boundary (i2).
The vertical load was applied through velocity increments, and the bearing capacity was determined from the relation between the stresses and displacements of one of the nodes; in this case, the central node of the foundation was considered. In Figure 12a, the displacement of the Figure 11. Vertical load (σv) as function of the load inclination on foundation boundary (i2). centralFigure node 11. Verticalof the foundation load (σv) (abscissa)as function with of therespect load to inclination the load applied on foundation to the ground boundary from (thei2). foundationThe vertical is represented. load was Inapplied this figure through is observedvelocity increments, that the maximum and the loadbearing that capacity the ground was Thedeterminedsupports vertical is limited from load the to was relation the appliedasymptotic between through value the stresses of the velocity curve and displacements represented. increments, of one and of the the nodes; bearing in this capacity case, was determinedthe centralA fromdditionally, node the ofrelation thea convergence foundation between wasstudy considered.the was stresses carried In Figureout,and consisting displacements 12a, the displacementof the analysis of one of theof ofthe central the values nodes; node of in this case, theofthe thecentral bearing foundation ca nodepacity (abscissa) of obtained the foundation with under respect the different towa thes considered. load increments applied of to Inthe the Figure velocity ground 12aused. from, theWith the foundationdisplacement a decrease in is of the represented.the value of the In this velocity figure increments, is observed the that result the maximum converged load towards that the the ground final value supports that is is the limited upper to central thelimitnode asymptotic in of the the theoretical foundation value of metho the curve d(abscissa) (27b). represented. with respect to the load applied to the ground from the foundation is represented. In this figure is observed that the maximum load that the ground supports is limited to the asymptotic value of the curve represented. Additionally, a convergence study was carried out, consisting of the analysis of the values of the bearing capacity obtained under the different increments of the velocity used. With a decrease in the value of the velocity increments, the result converged towards the final value that is the upper limit in the theoretical method (27b).
Figure 12. Cont.
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b 200
150
100
50 Bearing capacity(MPa)Bearing 0 1,00E 10-5- 05 1,00E 10--606 1,00E 10-7- 07 1,00E 10-8 -08 1,00E 10-9- 09 Velocity increments (m/step)
Figure 12. Estimation of bearing capacity by FDM (i = 0.47): (a) Stress–strain curve at a point of the Figure 12. Estimation of bearing capacity by aFDM (ia = 0.47): (a) Stress–strain curve at a point of the ground; (b) Convergence analysis. ground; (b) Convergence analysis. Additionally, a convergence study was carried out, consisting of the analysis of the values of the In the example studied, a seepage case was studied, where kv = 0, io2 = 0°, mo = 15, UCS = 100 MPa, bearing capacity obtained under the different increments of the velocity used. With a decrease in the GSI = 65, α = 0° and B = 2.25 m were adopted (which did not influence the result because of the value of the velocity increments, the result converged towards the final value that is the upper limit in assumption of a weightless rock mass). In addition, with io2 = 0°, i2 = arctan(ia) (see (12)). Table 4 the theoretical method (27b). shows the results obtained numerically (PhFDM) and analytically (PhS&O) (first chart proposed Figure 7) In the example studied, a seepage case was studied, where kv = 0, io2 = 0◦, mo = 15, UCS = 100 MPa, considering different values of ia. According to the margin error ratio observed in Table 4, less than GSI = 65, α = 0 and B = 2.25 m were adopted (which did not influence the result because of the 5%, it can ◦be concluded that the two calculation methods have very similar results. assumption of a weightless rock mass). In addition, with io2 = 0◦, i2 = arctan(ia) (see (12)). Table4 shows the results obtained numerically (P ) and analytically (P ) (first chart proposed Figure7) Table 4. Numerical and analyticalhFDM results for bearing capacityhS&O (kv = 0, io2 = 0°, mo = 15, UCS = 100 MPa, consideringGSI diff =erent 65 and values B = 2.25 of m).ia. According to the margin error ratio observed in Table4, less than 5%, it can be concluded that the two calculation methods have very similar results. PhFDM PhS&O |푷풉푺&푂 − 푷풉푭푫푴|
Table 4. Numericali2 = and ia analytical results for bearing capacity푬풓풓풐풓 (k = 0, (i%)==0 , m = 15, UCS = 100 MPa, (MPa) (MPa) v o2 ◦ o 푷풉푺&푂 GSI = 65 and B = 2.25 m). 0 259 270 4.07 |PhS&O PhFDM| i2 = ia PhFDM (MPa) PhS&O (MPa) Error (%)= − 0.1 233 230 P1.30hS&O 0.20 187 259 191.5 270 4.07 2.35 0.1 233 230 1.30 0.30.2 155.2 187 155.7 191.5 2.35 0.32 0.3 155.2 155.7 0.32 In the numerical calculation, to estimate the bearing capacity, a stress path was formed until Inwas the reached, numerical taking calculation, into account to estimate the whole the wedge bearing of capacity, the ground a stress below path the was foundation. formed untilTherefore, was reached,the graphic taking output into account of the the vertical whole component wedge of the of groundthe total below stress the tensor foundation. at failure Therefore, was used to the graphicunderstand output how of the the vertical failure component mechanism of change the totald stressdepending tensor on at i failurea. was used to understand Figure 13 shows that the stress turned horizontally with an increase in ia, the rotation being how the failure mechanism changed depending on ia. larger when the horizontal force was wider. Additionally, it is noted that the vertical stress was well Figure 13 shows that the stress turned horizontally with an increase in ia, the rotation being largerdistributed when the horizontal in the case force that wasia = 0. wider. Additionally, it is noted that the vertical stress was well distributed in the case that ia = 0. Sustainability 2020, 12, 10063 19 of 21 Sustainability 2020, 12, x FOR PEER REVIEW 20 of 22
FigureFigure 13.13. The variation of the vertical component of the total stress tensor obtained by FDM under the foundation.
5. Conclusion Conclusionss AnAn optimized optimized design design that that adequately adequately considers considers the the negative negative effects effects that that natural natural phenomena phenomena such assuch earthquakes as earthquakes and floods and introduce floods introduce to foundations to foundations is highly advisable is highly to advisable guarantee to the guarantee sustainability the ofsustainability infrastructures. of infrastructures. Applying aa pseudo-static pseudo-static approach approach that that considers considers the seismic the seismic force and force the andseepage the as seepage an additional as an bodyadditional force, body a series force, of parameterized a series of parameterized charts for estimating charts for the estimat bearinging capacitythe bearing of shallow capacity foundations of shallow onfoundation rock massess on were rock proposed. masses were The charts proposed. were The calculated charts through were calculated the analytical through solution the analyticalproposed bysolution Serrano proposed et al. [23 by] by Serrano means ofet al. a previous [23] by means mathematical of a previous transformation mathematical of the transformati angles of incidenceon of the in theangles boundaries of incidence of the in pseudo-staticthe boundaries loads of the produced pseudo- bystatic the loads inertial produced action. by the inertial action. Each chart waswas made according to the exponent “ a” of the Modified Modified Hoek and Brown criterion, the inclination α of the free boundary and kv of the foundation boundary (kv = 0 in the seepage case), the inclination α of the free boundary and kv of the foundation boundary (kv = 0 in the seepage case), and they were developed based on io and the horizontal component of the pseudo-static load of the and they were developed based on io22 and the horizontal component of the pseudo-static load of the foundation boundary (called k in the seismic case and ia in the seepage case). It is noted that for foundation boundary (called kh hin the seismic case and ia in the seepage case). It is noted that for high highconfining confining pressures pressures in Boundary in Boundary 1, it is 1, not it is always not always possible possible to obtain to obtain a bearing a bearing capacity capacity value value;; this thislimit limit is demarcat is demarcateded by the by non the- non-equilibriumequilibrium line. line. As expected, it was observed that the bearing capacity decreaseddecreased with an increase in the pseudo-staticpseudo-static load and the original inclination of the load. It waswas also observed that the value of the bearing capacity was not directly proportional to the bearing capacity factor N ; therefore, a higher bearing capacity was not directly proportional to the bearing capacity factor Nβß; therefore, a higher value of N is not associated with a greater value of P . In addition, the same value of N generates value of Nβß is not associated with a greater value of Phh. In addition, the same value of Nβß generates different values of P , depending on the other geotechnical parameters. different values of Phh, depending on the other geotechnical parameters. A validation validation using using the the finite finite diff erence difference method method was carriedwas carried out in a out particular in a particular case. The numerical case. The andnumerical analytical and results,analytical according results, according to the example to the studied,example showstudied, a variationshow a variation of less than of less 5%. than In 5%. the numericalIn the numerical graphic graphic output output of the verticalof the vertical component component of the total of the stress total tensor, stress ittensor is observed, it is observed that the that the stress turned horizontally with an increase in ia. In addition, it is noted that the vertical stress was well distributed in the case that ia = 0. Sustainability 2020, 12, 10063 20 of 21
stress turned horizontally with an increase in ia. In addition, it is noted that the vertical stress was well distributed in the case that ia = 0.
Author Contributions: Conceptualization, R.G.; methodology, R.G.; validation, R.G.; writing—original draft preparation, A.A.; writing—review and editing, R.G. and A.A.; supervision, N.S.I. and C.O.M. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by Universidad Politécnica de Madrid. Acknowledgments: The research described in this paper was financially supported by the Universidad Politécnica de Madrid by the grant with reference VMENTORUPM20RAGA of the university’s own program for carrying out research and innovation projects. Conflicts of Interest: The authors declare no conflict of interest.
References
1. Sarma, S.K.; Iossifelis, I.S. Seismic bearing capacity factors of shallow strip footings. Geotechnique 1990, 40, 265–273. [CrossRef] 2. Richards, R.; Elms, D.G.; Budhu, M. Seismic bearing capacity and settlement of foundations. J. Geotech. Eng. 1993, 119, 662–674. [CrossRef] 3. Kumar, J.; Kumar, N. Seismic bearing capacity of rough footings on slopes using limit equilibrium. Geotechnique 2003, 53, 363–369. [CrossRef] 4. Choudhury, D.; Rao, K.S.S. Seismic Bearing Capacity of Shallow Strip Footings Embedded in Slope. Int. J. Geomech. 2006, 6, 176–184. [CrossRef] 5. Kumar, J.; Rao, V.B.K. Seismic bearing capacity of foundations on slopes. Géotechnique 2003, 53, 347–361. [CrossRef] 6. Kumar, J.; Ghosh, P. Seismic bearing capacity for embedded footings on sloping ground. Géotechnique 2006, 56, 133–140. [CrossRef] 7. Yang, X.L. Seismic bearing capacity of a strip footing on rock slopes. Can. Geotech. J. 2009, 46, 943–954. [CrossRef] 8. Raj, D.; Singh, Y.; Shukla, S. Seismic Bearing Capacity of Strip Foundation Embedded in c-φ Soil Slope. Int. J. Geomech. 2018, 18.[CrossRef] 9. Nova, R.; Montrasio, L. Settlement of shallow foundations on sand. Géotechnique 1991, 41, 243–256. [CrossRef] 10. Le Pape, Y.; Sieffert, J.P. Application of thermodynamics to the global modelling of shallow foundations on frictional material. Int. J. Numer. Anal. Methods Geomech. 2001, 25, 1377–1408. [CrossRef] 11. Di Prisco, C.; Pisanò, F. Seismic response of rigid shallow footings. Eur. J. Environ. Civ. Eng. 2011, 15, 185–221. [CrossRef] 12. Veiskarami, M.; Fadaie, S. Stability of Supported Vertical Cuts in Granular Matters in Presence of the Seepage Flow by a Semi-Analytical Approach. Sci. Iran. 2017, 24, 537–550. [CrossRef] 13. Saada, Z.; Maghous, S.; Garnier, D. Stability analysis of rock slopes subjected to seepage forces using the modified Hoek–Brown criterion. Int. J. Rock Mech. Min. Sci. 2012, 55, 45–54. [CrossRef] 14. Veiskarami, M.; Kumar, J. Bearing capacity of foundations subjected to groundwater flow. Geomech. Geoengin. 2012, 7, 293–301. [CrossRef] 15. Kumar, J.; Chakraborty, D. Bearing Capacity of Foundations with Inclined Groundwater Seepage. Int. J. Geomech. 2013, 13, 611–624. [CrossRef] 16. Veiskarami, M.; Habibagahi, G. Foundations bearing capacity subjected to seepage by the kinematic approach of the limit analysis. Front. Struct. Civ. Eng. 2013, 7, 446–455. [CrossRef] 17. Imani, M.; Fahimifar, A.; Sharifzadeh, M. Upper Bound Solution for the Bearing Capacity of Submerged Jointed Rock Foundations. Rock Mech. Rock Eng. 2012, 45.[CrossRef] 18. Wang, M.; Chen, Y.F.; Hu, R.; Liu, W.; Zhou, C.B. Coupled hydro-mechanical analysis of a dam foundation with thick fluvial deposits: A case study of the Danba Hydropower Project, Southwestern China. Eur. J. Environ. Civ. Eng. 2016, 20, 19–44. [CrossRef] 19. Hoek, E.; Brown, E.T. Empirical strength criterion for rock masses. J. Geotech. Eng. Div. ASCE 1980, 106, 1013–1035. 20. Hoek, E.; Brown, E.T. Practical estimates of rock mass strength. Int. J. Rock Mech. Min. Sci. 1997, 34, 1165–1186. [CrossRef] Sustainability 2020, 12, 10063 21 of 21
21. Hoek, E.; Carranza-Torres, C.; Corkum, B. Hoek-Brown failure criterion—2002 Edition. In Proceedings of the NARMS-TAC, Mining Innovation and Technology, Toronto, ON, Canada, 10 July 2002; Hammah, R., Bawden, W., Curran, J., Telesnicki, M., Eds.; pp. 267–273. Available online: https://www.rocscience.com/help/ rocdata/pdf_files/theory/Hoek-Brown_Failure_Criterion-2002_Edition.pdf (accessed on 1 September 2020). 22. Serrano, A.; Olalla, C. Ultimate bearing capacity of rock masses. Int. J. Rock Mech. Min. Sci. Geomech. 1994, 31, 93–106. [CrossRef] 23. Serrano, A.; Olalla, C.; González, J. Ultimate bearing capacity of rock masses based on the modified Hoek-Brown criterion. Int. J. Rock Mech. Min. Sci. 2000, 37, 1013–1018. [CrossRef] 24. Sokolovskii, V.V. Statics of Soil Media; Jones, R.; Schofield, A., Translators; Butterworths Science: London, UK, 1965. 25. Hoek, E.; Kaiser, P.K.; Bawden, W.F. Support of Underground Excavations in Hard Rock; AA Balkema: Rotterdam, The Netherlands, 1995. 26. Serrano, A.; Olalla, C.; Galindo, R.A. Ultimate bearing capacity at the tip of a pile in rock based on the modified Hoek–Brown criterion. Int. J. Rock Mech. Min. Sci. 2014, 71, 83–90. [CrossRef] 27. FLAC. Manuals, Complete Set; Itasca: Minneapolis, MN, USA, 2011. 28. Galindo, R.; Illueca, M.; Jiménez, R. Permanent deformation estimates of dynamic equipment foundations: Application to a gas turbine in granular soils. Soil Dyn. 2014, 63, 8–18. [CrossRef] 29. Benito, J.L.; Moreno, J.; Sanz, E.; Olalla, C. Influence of Natural Cavities on the Design of Shallow Foundations. Appl. Sci. 2020, 10, 1119. [CrossRef]
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