Seismic bearing capacity of strip footings on rock masses using the Hoek-Brown failure criterion

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1 Accepted Manuscript Seismic bearing capacity of strip footings on rock masses using the Hoek-Brown failure criterion Amin Keshavarz, Abdoreza Fazeli, Siavosh Sadeghi PII: S (15)1-X DOI: 1.116/j.jrmge Reference: JRMGE 1 To appear in: Journal of Rock Mechanics and Geotechnical Engineering Received Date: 1 July 15 Revised Date: 1 September 15 Accepted Date: 6 October 15 Please cite this article as: Keshavarz A, Fazeli A, Sadeghi S, Seismic bearing capacity of strip footings on rock masses using the Hoek-Brown failure criterion, Journal of Rock Mechanics and Geotechnical Engineering (15), doi: 1.116/j.jrmge This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

2 Seismic bearing capacity of strip footings on rock masses using the Hoek-Brown failure criterion Amin Keshavarz*, Abdoreza Fazeli, Siavosh Sadeghi School of Engineering, Persian Gulf University, Bushehr, Iran Received 1 July 15; received in revised form 1 September 15; accepted 6 October 15 Abstract: In this paper, the bearing capacity of strip footings on rock masses has been studied in the seismic case. The stress characteristics or slip line method was used for analysis. The problem was analyzed in the plane strain condition using the Hoek-Brown failure criterion. First, the equilibrium equations along the stress characteristics were obtained and the rock failure criterion was applied. Then, the equations were solved using the finite difference method. A computer code has been provided for analysis. Given the footing and rock parameters, the code can calculate the stress characteristics network and obtain the stress distribution under the footing. The seismic effects have been applied as the horizontal and vertical pseudo-static coefficients. The results of this paper are very close to those of the other studies. The seismic bearing capacity of weightless rock masses can be obtained using the proposed equations and graphs without calculating the whole stress characteristics network. Keywords: rock mass; bearing capacity; strip footings; stress characteristics; Hoek-Brown failure criterion footings. Saada et al. (11) assessed the seismic bearing capacity of strip 1. Introduction footings near rock slopes. They used the limit analysis method and proposed graphs to obtain the bearing capacity in static and pseudo-static cases. Bearing capacity of strip footings is an old geotechnical problem. Several Stress characteristics or slip line methods introduced by Sokolovski et al. equations and graphs have been proposed to calculate the bearing capacity of (196) has been successfully used to solve many geotechnical problems, strip footings on soils. Because the behaviors of soil and rock are different, these including the bearing capacity of foundations (Veiskarami et al., 1; Serrano et equations or graphs have some limitations in calculating the bearing capacity of al., 15), lateral earth pressure (Peng and Chen, 1), bearing capacity of rock masses. pipelines (Gao et al., 15), unsaturated soils (Vo and Russell, 1) and Serrano and Olalla (199) studied the bearing capacity of strip footings on reinforced soil structures (Jahanandish and Keshavarz, 5; Keshavarz et al., weightless rock masses using the Hoek-Brown failure criterion (Hoek and Brown, 11). In this method, equilibrium equations along the slip lines are solved, and 198). Using the slip line method, they presented some equations and graphs to thus the stress state is known at any point in the failure zone. compute the static bearing capacity of strip footings. To consider the Hoek- In this paper, the seismic bearing capacity of strip footings on rock masses is Brown failure criterion into analysis, they also proposed a parameter named evaluated using the stress characteristics method. The Hoek-Brown failure instantaneous friction angle. Later, they developed a method to calculate the criterion (Hoek and Brown, 198) is used. As explained below, using this bearing capacity of strip footings on rock slopes (Serrano and Olalla, 1996). In criterion instead of modified Hoek-Brown simplifies the equations and the closed their studies, rock mass was weightless and the surcharge was considered. form solutions can be obtained for weightless rock masses. However, the Serrano et al. () used the modified Hoek-Brown failure criterion (Hoek et al., parameters of the Hoek-Brown failure criterion can be obtained from the 199) to evaluate the static bearing capacity of strip footings on weightless rock modified Hoek-Brown criterion using the empirical relationships. Seismic effects masses. are considered in the analysis as the pseudo-static earthquake coefficients. Yang et al. () employed the lower bound limit analysis method to Surcharge effects are considered in the calculation as well. Also, three examples calculate the bearing capacity of strip footings on rock masses. They used the are presented to illustrate the usage of proposed graphs and equations. However, Hoek-Brown failure criterion and neglected the weight of the rock mass. Using in this study, it is assumed that the rock mass is not highly fractured, the footing the limit analysis method and the generalized Hoek-Brown failure criterion, base and ground surface are horizontal and the surcharge is applied vertically on Merifield et al. (6) assessed the bearing capacity of strip footings on rock the ground surface. masses. They employed the finite element method with the lower and upper bound theorems of limit analysis and proposed some graphs to determine the bearing capacity. They also considered the rock mass weight to calculate the bearing capacity. Yang (9) used the upper bound theorem of limit analysis and the modified Hoek-Brown criterion to calculate the seismic bearing capacity of strip footing on rock slopes. He employed the optimization method to compute the upper bound solution. Using the slip line method and the Hoek-Brown, modified Hoek- Brown and nonlinear twin-shear criteria, Zhou et al. (9) evaluated the effect of the intermediate principal stress on the ultimate bearing capacity of strip *Corresponding author. Tel: ; addresses: keshavarz@pgu.ac.ir, amin_keshavarz@yahoo.com. Theory.1. The Hoek-Brown failure criterion Hoek and Brown (198) introduced a rock mass failure criterion as σ σ σ = m + s (1) 1 b σ c σ c σ 1 and σ are the major and minor principal stresses at failure, respectively; and σ c is the uniaxial compressive strength of the rock; and s are constant parameters that depend on the characteristics and degree of fracturing of the rock mass. In Mohr circle of stress, σ1 = p + R, σ = p R, p is the average stress (p=(σ 1+σ )/) and R is the radius of the Mohr circle. Eq. (1) can be rewritten as

3 p R = β + ζ + 1 β β β = σ c / 8 and ζ = 8 s / m (Serrano and Olalla, 199). Therefore, the b rock mass strength can be defined with β and ζ. These parameters can be determined as (Hoek and Brown, 1997): RMR 1 mb = m exp a RMR 1 s = exp b a=8 and b=9 for undisturbed rock mass, and a=1 and b=6 for disturbed rock mass; m and RMR are Hoek-Brown parameter for sound rock and Bieniawski s rock mass rating index, respectively... Stress equilibrium equations Under plane strain conditions in the x-z plane, the unknown stress components at any point in the rock mass are σ x, σ z and τ xz (Fig. 1). The equilibrium equations are σ x τ xz + = X x z σ z τ xz + = Z z x X and Z are the body and/or inertia forces in x and z directions, respectively. Assuming that and K v are the pseudo-static earthquake coefficients in the horizontal and vertical directions, respectively, then, X = γ and Z = γ (1 Kv ), γ is the unit weight of the rock mass. σ Fig. 1. The stress components. σ + The yield condition for a homogeneous medium can be written as (Booker and Davis, 197): ( σ σ τ ) ( ψ ) f,, = R F p, = (5) x z xz ψ is the angle between the positive x axis and the direction of σ 1 (Fig. 1). The stress components can be derived from the Mohr circle as σ x = p + R cos( ψ ) σ z = p R cos( ψ ) τ xz = R sin( ψ ) () () () (6) Using Eqs. () and (6), two stress characteristics directions can be found. The equilibrium equations of stress along these two directions can be written as (Jahanandish and Keshavarz, 5; Keshavarz et al., 11): (1) Along the σ + direction, we have dz = tan( ψ m + µ ) dx sin ( m + µ ) F dp + d ψ = [dx sin( µ ) cos( m) cos( m) dz cos( µ )] X + [dx cos( µ ) + dz sin( µ )] Z () Along the σ direction, we have ο z z τ xz ψ σ x µ µ σ 1 x σ (7) dz = tan( ψ m µ ) dx sin[( m µ )] F dp + d ψ = [dx sin( µ ) + (8) cos( m) cos( m) dz cos( µ )] X + [dx cos( µ ) dz sin( µ )] Z 1 F tan( m) = F ψ F cos( µ ) = cos( m) p Serrano and Olalla (199) proposed the concept of instantaneous friction angle, ρ, which can be expressed as R sin ρ = (1) p Using Eqs. (9) and (1) the following equations can be obtained: m =, cos( µ ) = sin ρ (11) Also from Eqs. () and (1) we have 1 sin ρ R = β sin ρ p = β (.5cot ρ ζ ) (9) (1) In the stress characteristics method, each point in the medium is described with four parameters: x, z, p and ψ, x and z are the coordinates of the point. Writing Eqs. (7) and (8) in finite difference form, the unknown information at any point C can be found from points A and B, BC is the positive and AC is the negative stress characteristics (Fig. ): za zb xatmm + xbtmp xc = tmp tmm zc = zb + ( xc xb ) tmp A ψ C = A A1 Bmp ( ψ C ψ B ) pc = pb + S mp (1) tmp =.5[tan( ψ C + µ C ) + tan( ψ B + µ B )] tmm =.5[tan( ψ C µ C ) + tan( ψ A µ )] A A1 = X[ Smp ( xc xb ) Cmp ( zc zb )] + Z[ Cmp ( xc xb ) + Smp ( zc zb )] A = X[ Smm ( xc xa) + Cmm ( zc za)] + Z[ Cmm( xc xa) Smm ( zc za)] A = Amm [ pb pa + ( A1 + ψ BBmp ) / Amp ] Bmmψ A A A = Amm Bmp Amp B mm Amp =.5(cos ρc + cos ρb ), Amm =.5(cos ρc + cos ρ A) Bmp = FB + FC, Bmm = FA + F C Cmp =.5(sin ρb + sin ρc ), Cmm =.5(sin ρa + sin ρc ) Smp =.5(cos ρb + cos ρc ), Smm =.5(cos ρ A + cos ρc ) (1) The trial and error procedure is used to compute the properties of point C using Eq. (1). For the first try, the properties of point C are assumed to be equal to those of points B and A in the positive and negative directions, respectively. Then the new properties are obtained for point C. This procedure is continued until the differences between the calculated properties of point C in the last two steps are small enough... Boundary conditions To solve any problem with the stress characteristics method, the boundary conditions must be firstly obtained. Fig. shows a typical stress characteristics network. The zone OABCD is the failure zone which will be obtained after solving the problem. OD is the ground surface and vertical surcharge q is applied on this boundary. OA is the footing boundary with length equal to the footing width, B. The ultimate bearing capacity is the applied vertical pressure on this boundary. A B σ σ +

4 Fig.. The unknowns at point C can be found from points A and B BC is the positive and AC is the negative stress characteristics...1. Boundary conditions along the ground surface On the boundary OD (Fig. ), x and z are known, and p and ψ are unknown. Shear stress (τ ) and normal stress (σ ) on this boundary are equal to: σ = q(1 K ), τ = qk (15) v h The angle ψ on this boundary can be obtained as (Keshavarz et al., 11): 1 p sinδ ψ =.5 sin δ R (16) p and R are the average stress and radius of the Mohr circle on this boundary, respectively, and 1 Kh δ = tan 1 Kv (17) For the static case, a more general equation for ψ is proposed by Serrano and Olalla (199). Considering the Mohr circle of stress and using Eq. (1), R can be written as 1 sin ρ R = ( p σ ) + τ = β sin ρ (18) ρ is the instantaneous friction angle on the ground. Employing Eq. (1) one can have p = β (.5 cot ρ ζ ) (19) Substituting Eq. (19) into Eq. (18) and after some algebraic simplification, the following equation can be obtained: τ 1 sin ρ (.5cot ρ σ ) + = β sin ρ () σ = ζ + σ / β. To calculate ρ, Eq. () must be solved. If it is assumed that w = sin ρ, then Eq. () can be written in the following form: Aw + 8w + Cw + 1 = (1) τ A = + + σ ( σ + 1) β C = 6 σ The solution to Eq. (1) is AC 6 AC w = sin ρ = + S.5 S A A A S AC 1 S =.5 Q + + A A Q Q = = C + 1A 1 = C 7AC If τ =, then the solution to Eq. (1) is ρ = sin σ +... Boundary condition along the footing-rock interface () () () (5) Along the footing-rock interface, i.e. the boundary OA (Fig. ), z is known but x, p and ψ are unknown. The normal and shear stresses along this boundary are τ f = qukh (6) σ f = qu (1 Kv ) Similar to boundary OD, the angle ψ of this boundary is ψ f =.5 π sin p sin δ δ Rf Fig.. A sample of stress characteristics x (m) network. p f is the average stress along the footing-rock interface. For the static case (δ=), Eq. (7) would be a special case of the equation proposed by Serrano and Olalla (199). Using the Mohr circle of stress, the ultimate bearing capacity can be calculated as pf + Rf + ( Rf pf ) tan u = 1 Kv + tan δ q z (m) D.5 1 σ c / (γb) = 1 K =Κ h v =, q= = 1.7, s=.6.. Bearing capacity of weightless rock mass δ (8) The bearing capacity of strip footings on rock masses can be written as (Kulhawy and Carter, 199; Merifield et al., 6): q q (1 K v ) C q u cn σ 1 f f (7) = σ (9) N σ is the bearing capacity factor for rock mass. For weightless rock mass (γ=), the stress distribution beneath the footing is uniform, and the bearing capacity can be obtained by solving the singularity point without calculating the whole stress characteristics network. At the singular point (Point O, Fig. ), dx=dz=. Therefore, for this point, Eq. (8) is written in the following form: sin( µ )dp + Rdψ = () Using Eq. (1), the solution to Eq. () is (Serrano and Olalla, 199): O B q u Κ h q u (1 K v ) B A

5 ψ =.5[cot ρ ln tan(.5 ρ)] + C (1) C is the integration constant. The following nonlinear equation can be obtained by integrating from the ground surface to the footing-rock interface: ln[tan(.5 ρ )] cot ρ + ψ = ln[tan(.5 ρ )] cot ρ + ψ () f f f Substituting ψ f into Eq. () we have 1.5cot ρf ζ ln[tan(.5 ρf )] ln[tan(.5 ρ)] cot ρf sin sin ρf sin δ + 1 sin ρ f 1.5cot ρ ζ π + cot ρ sin sin ρ sin δ = 1 sin ρ ( ) It can be seen that Eq. () is a nonlinear equation for ρ f. This equation can be solved using the FZERO function in Matlab or Excel SOLVER. The bearing capacity factor for weightless rock mass can be obtained using Eqs. (1), (8) and (9) as N 1 sin ρf.5cot f + (.5cot f ) tan m sin b ρf cosδ σ = 8 1 Kv + tan δ ρ ζ ρ ζ δ () The bearing capacity coefficient of weightless rock mass can be computed in the following way. Firstly, according to the geometry properties, surcharge and other parameters, ρ is computed from Eq. (). Then Eq. () is used to obtain ρ f. Finally, N σ is calculated using Eq. (). These calculations can be done by hand or using Excel..5. Solution procedure for the stress characteristics network To consider the weight of the rock mass in analysis, the whole stress characteristics network must be solved. The solution procedure is similar to the conventional stress characteristics or slip line method. As shown in Fig., the network includes three zones: OCD, OCB and OAB, namely, passive, Goursat and active zones, respectively. First the passive zone and then the Goursat and active zones are computed. Solution starts from the ground surface boundary (OD). This boundary is divided into a finite number of points. Having the boundary condition, the network properties (x, z, p and ψ) are known at these points. The network at zone OCD can be solved using the information on line OD. Point C is the last point of this zone to be solved. Because the values of p and ψ are different at the left and right sides of point O, this point is a singular point. To solve the network in the zone OCB, this point must be considered first. Trial and error procedure is needed to solve this singularity problem. First a value of ψ is assumed at the footing-rock interface (ψ f). Then the Goursat zone is divided into a number of finite sections, and using Eq. (), the values of ψ at these points are calculated and a new value of ψ f is obtained. This process is repeated until the differences between the values of ψ f in the last two steps are small enough. Getting the information at point O and line OC, the zone OCB can be computed. The zone OAB is calculated using the information along the line OB. At boundary OA, ψ depends on p and the stress distribution beneath the footing can be calculated based on the information on negative characteristics. The ultimate bearing capacity is considered as the average of the normal stress along the footing-rock interface. In this manner, assuming the length OD, the characteristics network is solved and the length OA, which is the footing width, is obtained. Therefore, for a finite width B, the required length of OD must be selected using trial and error method.. Results and discussion Serrano and Olalla (1996) used the stress characteristics method to compute the static ultimate bearing capacity of weightless rock masses. The results of this paper for the static case and weightless rock are the same as the results of these researchers. Yang et al. () applied the lower bound limit analysis method to evaluate the bearing capacity of the weightless rock masses. Table 1 shows the results of this paper together with the results of Yang et al. () for σ c=5 MPa. As can be seen, the results of two methods agree well with each other. Table 1. Comparison of the results of this study with those of Yang et al. () for weightless rock masses (σc=5 MPa). s qu (MPa) Yang et al. () This study Kulhawy and Carter (199) proposed the following equation to compute the bearing capacity coefficient for weightless rock masses (N σ) using a simple lower bound analysis: Nσ = s + s + m s (5) b Table presents a comparison between the results of N σ in this paper and other studies. It can be seen from Table that the results of this study are very close to those of Serrano et al. () and Merifield et al. (6). Kulhawy and Carter (199) assumed a very simple stress field for the lower bound method. Therefore, their results are very conservative. Table. Comparison of the bearing capacity coefficient for weightless rock mass N σ (q=kh=kv=, s=1) mb Merifield et al. Kulhawy and Carter Serrano et al. This study (6) (199) (Eq. (5)) () The variation of static bearing capacity factor N σ for different values of s and is shown in Fig.. As can be expected, the bearing capacity increases with increasing s and. This figure can be used as a design chart to easily determine the bearing capacity factor N σ. N σ q=, = s N σ =5 Fig.. Variation of the bearing capacity factor N σ with s and mb Using the procedure mentioned above, a computer code is written to solve the problem and compute the network. If the weight of the rock mass is ignored, as discussed in the previous section, the ultimate bearing capacity can be obtained without solving the whole characteristics network. Fig. 5 shows the effect of horizontal and vertical seismic coefficients on N σ. The horizontal and vertical seismic coefficients can be positive or negative, and the worst case that produces smaller N σ is selected here. It is obvious that the

6 bearing capacity decreases with increasing seismic coefficients. As can be seen, K v has a lower effect on N σ than. Fig. 6 demonstrates the effects of the surcharge and unit weight on N σ. The bearing capacity increases with increasing γ. The higher the surcharge, the greater the value of N σ. For the selected parameters in Fig. 6, when γ changes from zero to kn/m, the bearing capacity coefficient increases by about 19% and the average value of N σ/n σ is about 1.. N σ K v = = =.1, K v = =.1 =., K v = =. =., K v = =. =., K v = K v =. =1, q= kpa σ c =5 MPa 1 1E- 1E s Fig. 5. Effects of horizontal and vertical pseudo-static coefficients on the bearing capacity factor N σ. N σ γ= γ=1 kn/m γ= kn/m B=1m, =1, s=e- σ c =1MPa, =.1, K v =. 1 5 q (kpa) Fig. 6. Effects of unit weight and surcharge on the bearing capacity factor N σ. The results of the bearing capacity factor N σ for different values of, s and are shown in Table. This table can provide a reference for design. Table. The bearing capacity factor N σ for different parameters (σc=1 MPa, B=1 m, γ= kn/m, q=, Kv=) Kh s mb=.1 mb=1 mb=1 mb= mb= mb= N σ As mentioned above, for weightless rock mass, the load distribution beneath the footing is uniform and footing width (B) does not have any effect on the bearing capacity. However, in the case of γ, N σ depends on B. Fig. 7 shows the effects of B on the bearing capacity factor, N σ. It can be seen that increasing B leads to increase in N σ, but this effect is minor and can be ignored. For the parameters indicated in Fig. 7, if B increases from.5 m to m, the average of N σ for all values of will increase by about 7%. N σ s=e-, σ c =1 MPa, =.1, K v =, q=5 kpa, γ= kn/m 1 = = =15 =1 =5 = B (m) Fig. 7. Effects of B on N σ. To better understand how to use the proposed figures and tables, some examples are provided as follows. When the surcharge is zero, the static bearing capacity can be found from the proposed figures (Example 1). But if the surcharge and/or pseudo-static coefficients are not zero, the bearing capacity of the weightless rock mass can be obtained from the presented equations in the previous sections (Examples and ). (1) Example 1 A strip footing is located on a disturbed rock mass with σ c= MPa, RMR=65, m =1 and γ=5 kn/m. The static bearing capacity of this footing without consideration of the weight of rock mass (q u) and with consideration of the weight (q u) can be calculated as follows. Using Eq. () and assuming a=1 and b=6, the values of and s would be.81 and.9, respectively. If is assumed approximately to be 1, according to Fig., N σ would be about.71. Therefore, q u=n σσ c=1. MPa. When the weight of the rock mass is considered, assuming B=1 m, from Table, N σ can be approximated to be.7 (although the unit weight is different) and thus q u=1.6 MPa. In this example, after solving the whole stress characteristics network using the developed program, q u = 1. MPa, and q u = 1.6 MPa. Therefore, in this example, the bearing capacity of the weightless rock mass is % smaller than the rock mass with γ=5 kn/m. () Example Similar to Example 1, but assuming that the footing is located at depth m below the ground surface, the static bearing capacity of the rock mass can be computed as follows. The effect of the depth can be considered as a surcharge equal to q= 5=5 kpa. Therefore, β=σ c/8=.6 MPa and ζ=8s/ m =.1. Using Eq. (15), σ =5 kpa and τ =. Thus, σ ζ σ / β = + =.. From Eq. (5), ρ =.879. Substituting these parameters into Eq. (), the following nonlinear equation is obtained: ln[tan(.5 ρ )] ln[tan(.5 ρ )] cot ρ + π + cot ρ = (6) f f After solving this nonlinear equation, ρ f would be equal to.91. Therefore, from Eq. (), N σ=.787 and q u=15.65 MPa. In this example, it is impossible to calculate the bearing capacity of the rock mass with weight without solving the whole network. Using the computer program, q u and q u would be equal to MPa and 16.7 MPa, respectively. () Example b

7 Same as Example, this example assumes that =. and K v=, and the seismic bearing capacity is obtained as follows. From Eq. (15), σ =5 kpa and τ =1 kpa. Using Eq. (), ρ would be equal to.879. After solving Eq. (), ρ f=.87. According to Eq. (), N σ=.56, and therefore q u=11. MPa. Using the developed computer program and solving the whole network, the values of q u and q u are MPa and 11.5 MPa, respectively.. Conclusions In this paper, the stress characteristics method has been employed to evaluate the bearing capacity of the strip footing on the rock mass in the seismic case. The Hoek-Brown criterion has been used for the rock mass. Earthquake effects have been considered in the analysis as the pseudo-static horizontal and vertical coefficients. A computer program has been developed, and it can solve the problem and compute the stress characteristics network. The stress distribution in the footing-rock interface is obtained after solving the problem. The effects of several parameters on the bearing capacity have been evaluated. Furthermore, some equations, tables and graphs are provided, and can be used to compute the bearing capacity of the weightless rock masses. The comparisons between the results of this study with those in the literature show the accuracy of the proposed method. The effect of the horizontal earthquake coefficient on the bearing capacity is significant, but the vertical pseudo-static coefficient has a minor effect. If the rock mass unit weight is Conflict of interest The authors wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome. References Booker J, Davis E. A general treatment of plastic anisotropy under conditions of plane strain. Journal of the Mechanics and Physics of Solids 197; (): 9-5. Gao FP, Wang N, Zhao B. A general slip-line field solution for the ultimate bearing capacity of a pipeline on drained soils. Ocean Engineering 15; 1: 5-1. Hoek E, Brown ET. Practical estimates of rock mass strength. International Journal of Rock Mechanics and Mining Sciences 1997; (8): Hoek E, Brown ET. Underground excavations in rock. London: The Institution of Mining & Metallurgy, 198. Hoek E, Wood D, Shah S. A modified Hoek-Brown failure criterion for jointed rock masses. In: Proceedings of the International ISRM Symposium on Rock Characterization. American Society of Civil Engineers, 199. p Jahanandish M, Keshavarz A. Seismic bearing capacity of foundations on reinforced soil slopes. Geotextiles and Geomembranes 5; (1): 1-5. Keshavarz A, Jahanandish M, Ghahramani A. Seismic bearing capacity analysis of reinforced soils by the method of stress characteristics. Iranian Journal of Science and Technology, Transactions of Civil Engineering 11; 5: Kulhawy F, Carter JP. Settlement and bearing capacity of foundations on rock masses and socketed foundations in rock masses. In: Engineering in Rock Masses. Oxford, UK: Butterworth-Heinemann Ltd., 199. p Merifield RS, Lyamin AV, Sloan SW. Limit analysis solutions for the bearing capacity of rock masses using the generalised Hoek Brown criterion. International Journal of Rock Mechanics and Mining Sciences 6; (6): 9-7. Peng MX, Chen J. Slip-line solution to active earth pressure on retaining walls. Géotechnique 1; 6(1): Saada Z, Maghous S, Garnier D. Seismic bearing capacity of shallow foundations near rock slopes using the generalized Hoek Brown criterion. International Journal for Numerical and Analytical Methods in Geomechanics 11; 5(6): 7-8. Serrano A, Olalla C. Ultimate bearing capacity of rock masses. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts 199; 1(): Serrano A, Olalla C. Allowable bearing capacity of rock foundations using a non-linear failure criterium. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts 1996; (): 7-5. considered in the analysis, the bearing capacity increases. This increase depends Serrano A, Olalla C, Gonzalez J. Ultimate bearing capacity of rock masses based on the on the values of the parameters and the errors induced by neglecting the rock unit modified Hoek-Brown criterion. International Journal of Rock Mechanics and Mining weight can change from 1% to %. It is obvious that neglecting the unit weight Sciences ; 7(6): of the rock mass is conservative. The ultimate seismic bearing capacity of the Serrano A, Olalla C, Jimenez R. Analytical bearing capacity of strip footings in weightless materials with power-law failure criteria. International Journal of Geomechanics 15. weightless rock mass can be computed using the introduced equations in the doi: 1.161/(ASCE)GM paper. Sokolovski VV, Jones D, Schofield AN. Statics of soil media. Butterwooths Scientific, 196. Veiskarami M, Kumar J, Valikhah F. Effect of the flow rule on the bearing capacity of strip foundations on sand by the upper-bound limit analysis and slip lines. International Journal of Geomechanics 1; 1(): 18. Vo T, Russell AR. Slip line theory applied to a retaining wall unsaturated soil interaction problem. Computers and Geotechnics 1; 55: Yang X, Yin JH, Li L. Influence of a nonlinear failure criterion on the bearing capacity of a strip footing resting on rock mass using a lower bound approach. Canadian Geotechnical Journal ; (): 7-7. Yang X. Seismic bearing capacity of a strip footing on rock slopes. Canadian Geotechnical Journal 9; 6(8): 9-5. Zhou XP, Yang HQ, Zhang YX, Yu MH. The effect of the intermediate principal stress on the ultimate bearing capacity of a foundation on rock masses. Computers and Geotechnics 9; 6(5): Dr. Amin Keshavarz is currently an assistant professor of Civil Engineering at School of Engineering of Persian Gulf University, Iran. He received his BSc degree in Civil Engineering from Persian Gulf University in 1997 and his MSc and PhD degrees in Civil Engineering (Soil Mechanics and Foundations) from Shiraz University, Iran in and 7, respectively. His research interests cover stress characteristics and zero extension line (ZEL) methods, soil dynamics and geotechnical earthquake engineering and stability analysis of reinforced and unreinforced soil slopes and retaining walls.