FIG. 2.(a)Velocity Field of M1Mechanism;(b)Velocity Hodographs
FIG. 1.Failure Mechanism M1for Static Bearing Capacity Analysis
tion can be interpreted as an expression of the virtual rate of work principle.This obrvation has often been made (Davis 1968;Mroz and Drescher 1969;Michalowski 1989;Salenc ¸on 1990;De Buhan and Salenc ¸on 1993;Drescher and Detournay 1993).The equivalence of the two approaches plays a key role in the derivations of the limit loads for nonassociative mate-rials.Recent theoretical c
onsiderations made on translational failure mechanisms (Drescher and Detournay 1993;Michal-owski and Shi 1995,1996)allow one to conclude that for a nonassociative material,the limit load can be obtained by the u of the flow rule associated with a new yield condition in which c and are replaced by c *and *as follows:
cos ⌿sin tan *=(1)1Ϫsin ⌿sin
cos ⌿cos
c *=
c
(2)
1Ϫsin ⌿sin
消防知识安全Hence,the results prented in the prent paper can be ud for nonassociative material provided the internal friction angle and the cohesion c are replaced with *and c *calculated from (1)and (2),respectively.Failure Mechanisms M1Mechanism
The M1mechanism is shown in Fig.1.This mechanism is symmetrical,and it permits the calculation of the bearing ca-
pacity in the ca of no-ismic loading.The wedge ABC is translating vertically as a rigid body with the same initial downward velocity as the footing.The downward movement of the footing and wedge is accommodated by the lateral movement of the adjacent soil as indicated by the two radial shear zones.The angles ,␣i ,and i (i =1,...,n )are as yet unspecified.Since the movement is symmetrical about the footing,it is only necessary to consider the movement on the right-hand side of Fig.1.
The radial shear zone BCD is compod of n triangular rigid blocks.As shown in Fig.2(a),all the triangles move as rigid bodies in directions that make an angle with the disconti-nuity lines d i (i =1,...,n ).The velocity of each triangle is determined by the condition that the relative velocity between the triangles in contact must have the direction that makes an angle to the contact surface.The velocity hodographs are shown in Fig.2(b).The velocities so determined constitute a kinematically admissible velocity field.
As shown in Fig.3,the external forces contributing to the incremental external work consist of the foundation load,the weight of the soil mass,and the surcharge q on the foundation level.The increme
ntal external work for the different external forces can be easily obtained;the calculations are prented in Appendix I.
Energy is dissipated at the discontinuity surfaces d i (i =1,...,n )between the material at rest and the material in motion and at the discontinuity surfaces l i (i =1,...,n )within the radial shear zone.The incremental energy dissipation per unit房间的歌词
FIG. 3.Free-Body Diagram for M1Failure Mechanism
FIG. 4.Failure Mechanism M2for Seismic Bearing Capacity Analysis
FIG. 5.(a)Velocity Field of M2Mechanism;(b)Velocity Hodo-
graph
length along a velocity discontinuity or a narrow transition zone can be expresd as
⌬D =c ⌬V cos
(3)
L where ⌬V =incremental displacement or velocity that makes an angle with the velocity discontinuity according to the associated flow rule of perfect plasticity;and c =cohesion parameter.Calculations of the incremental internal energy dis-sipation along the different velocity discontinuities are given in Appendix I.
Equating the total rate at which work is done by the force on the foundation,the soil weight in motion,and the surcharge loading [(26)in Appendix I]to the total rate of energy dissi-pation along the lines of velocity discontinuities [(33)in Ap-pendix I],it is found,after some simplifications,that an upper bound on the bearing capacity of the soil is
P B S 0
q =
=␥N (,␣,)ϩqN (,␣,)ϩcN (,␣,)cS ␥S i i qS i i cS i i B 2
0(4)
in which the static bearing capacity factors ␣i ,i ),N (,␥S N qS (,␣i ,i ),and N cS (,␣i ,i )can be expresd in terms of the (2n ϩ1)as yet unspecified angles (,␣i ,i ).They are given as follows:
N =Ϫ(f ϩf )
(5)␥S 12N =Ϫf (6)qS 3
N =2(f ϩf ϩf )
(7)
cS 456The ultimate static bearing capacity of the foundation is ob-tained by minimization of q cS [(4)]with regard to the mech-anism’s parameters.However,in practice,the minimum values of the three factors ␣i ,i ),N qS (,␣i ,i ),and N cS (,␣i ,N (,␥S i )are determined independently of
查看内存条型号each other,and therefore their u errs on the safe side (e Static Bearing Capacity Factors in the fourth ction).M2Mechanism
The M2mechanism is shown in Fig.4.This mechanism is nonsymmetrical,and it permits the calculation of the bearing capacity in the prence of ismic loading.As is well known,an earthquake has two possible effects on a soil-foundation system.One is to increa the driving forces,and the other is to decrea the shearing resistance of the soil.In this paper,only the reduction of the bearing capacity due to the increa in driving forces is investigated under ismic loading condi-tions.The shear strength of the soil is assumed to remain un-affected by the ismic loading.On the other hand,the earth-quake acceleration for both the soil and the structure is assumed to be the same:Only the horizontal ismic coeffi-cient K h is considered,the vertical ismic coefficient often being disregarded.Finally,the earthquake load on the structure is reprented by the ba shear load acting at the foundation
level and an eccentricity for the vertical foundation load.The moment due to the ismic load on the structure is not con-sidered.Only the ba shear load will be taken into account.Except for the triangular area directly below the ba of the footing,the M2nonsymmetrical mechanism is similar to the right-hand side of the M1mechanism.Wedge ABC is trans-lating as a rigid body with a downward
velocity V 1inclined at an angle to the discontinuity line AC (Fig.5).The foun-dation is assumed to move with the same velocity as wedge ABC (i.e.,V 1).The rest of the mechanism is similar in form to the right-hand side of the M1mechanism.
As shown in Fig.6,the external forces contributing to the incremental external work consist of the force acting on the footing,the weight of soil in motion,the surcharge loading,and the different inertia forces.The inertia forces concern
FIG. 6.
Free-Body Diagram for M2Failure Mechanism
FIG.7.Critical Slip Surface for =45؇and n =12
TABLE 1.N ␥S Value versus Number of Rigid Blocks n for =45؇from M1Symmetrical Mechanism
n (1)N ␥S (2)Reduction
(%)(3)2741.93—3447.9439.64384.2814.25359.50 6.46347.19 3.47340.16 2.08335.76 1.39332.820.910330.770.611329.270.512328.140.313327.270.314
326.59
0.2
the ba shear load,the inertia forces of the soil in motion,and the surcharge loading.Energy is dissipated along the lines l i (i =1,...,n Ϫ1)and d i (i =1,...,n ).Calculations of the incremental external work and the internal energy dissi-pation along the different velocity discontinuities are given in Appendix II.
Equating the total external rate of work [(44)in Appendix II]to the total internal rate of energy dissipation [(49)in Ap-pendix II],it is found that the value of the upper bound on the bearing capacity is
P B E 0
q =
=␥N (␣,)ϩqN (␣,)ϩcN (␣,)(8)
cE ␥E i i qE i i cE i i B 2
儿童谜语大全
0in which the ismic bearing capacity factors i ),
N (␣,␥E i N qE (␣i ,i ),and N cE (␣i ,i )can be expresd in terms of the (2n )as yet unspecified angles (␣i ,i ).They are given as fol-lows:
1
虎皮鹦鹉怎么分辨雌雄N =Ϫ(g ϩK g )
(9)␥E 1h 2sin(Ϫ)ϩK cos(Ϫ)
1h 11
N =Ϫ
(g ϩK g )
(10)qE 3h 4sin(Ϫ)ϩK cos(Ϫ)
1h 11
N =
(g ϩg )
(11)
cE 56sin(Ϫ)ϩK cos(Ϫ)银耳汤功效
1h 1From the equations,it is clear that only the factor in-N ␥E cludes the soil inertia.The N qE factor includes the inertia of the foundation load and the surcharge loading;however,the N cE factor only includes the inertia of the foundation load and thus corresponds to the ca of a footing subject to an inclined load.
As in the static ca,the minimum value of q cE gives the ultimate ismic bearing capacity of the foundation.However,in practice,the minimum values of the three factors N (␣,␥E i i ),N qE (␣i ,i ),and N cE (␣i ,i )are determined independently of each other,and therefore their u errs on the safe side.NUMERICAL RESULTS
The most critical bearing capacity factors can be obtained by minimization of the factors [(5)–(7)and (9)–(11)]with regard to the mechanism’s parameters.The minimization pro-cedure can be performed using the optimization tool available in most spreadsheet software packages.In this paper,one us the Solver optimization tool of Microsoft Excel.Two computer programs using the Visual Basic programming language that resides in Microsoft Excel have been written to define the static and ismic bearing capacity factors as function of the mechanism’s parameters [(5)–(7)and (9)
–(11)].Initial values need to be assigned to the different angular parameters.The solver tool is then invoked to ‘‘minimize’’the bearing capacity factor ‘‘by changing’’the angular parameters,‘‘subject to’’the constraints {ϩ␣i =(cf.Fig.1)and ␣i ϩi Նn ͚i =1i ϩ1[cf.Fig.2(b)]}for the M1mechanism and ␣i =(cf.
n {͚i =1Fig.4)and ␣i ϩi Ն[cf.Fig.5(b)]}for the M2mech-i ϩ1anism.The method of minimization ud is the general re-duced gradient method.Additional information on Solver op-tions and algorithms can be found in the Microsoft Excel Solver’s help file and at the website
In the following ctions,we prent and discuss in succes-sion (1)the static bearing capacity factors N qS ,and N cS N ,␥S given by both the M1and M2mechanisms;and (2)the ismic bearing capacity factors N qE ,and N cE given by the M2N ,␥E nonsymmetrical mechanism.
Static Bearing Capacity Factors
First,the results given by the M1symmetrical mechanism will be prented and compared to tho given by other ex-isting solutions.Second,the results of the M2nonsymmetrical mechanism for K h =0will be prented and compared to tho given by the M1symmetrical mechanism.This permits us to estimate the difference between results when considering a nonsymmetrical mechanism for a cent
rally loaded footing.Table 1prents the factor obtained from the M1mech-N ␥S anism for =45Њand for various values of n (the number of the triangular rigid blocks).It can be obrved that the upper-bound solution can be improved by increasing the number of rigid blocks.The reduction in the value decreas with the N ␥S n -increa and attains 0.2%for n =14.It should be mentioned that the same trend is also obrved for the N qS and N cS factors.Fig.7shows the critical slip surface obtained from the nu-merical minimization of the factor for =45Њand for n N ␥S =12.It can be obrved that the critical failure mechanism obtained by the computer program is compod of two radial shear zones sandwiched between an active triangular wedge under the footing and a Rankine passive wedge.It should be noted that the radial shear zones are not bounded by log-spiral slip surfaces as is the ca of the Prandtl mechanism.Finally,note that all subquent calculations are made for n =14.Table 2prents the N qS ,and N cS factors obtained from N ,␥S the computer program for ranging from 0to 50Њ.
To check the effect of the superposition method,one cal-culates the ultimate load P direct obtained by direct numerical minimization of P S [(4)]and compares it to the one obtained by the superposition method P superposition using the N qS ,and N ,␥S N cS factors.For =30Њ,c =10kPa,q =10kPa,B 0=1m,and ␥=18kN/m 3,one obtains P direct =726.13kN/m and
TABLE 2.N qS ,and N cS Values from M1Symmetrical Mech-N ,␥S anism
(1)N ␥S (2)N qS (3)N cS (4)0— 1.00 5.151— 1.09 5.382— 1.20 5.643— 1.31 5.914— 1.43 6.195— 1.57 6.506— 1.72 6.827— 1.887.178— 2.067.549— 2.267.9310— 2.478.3611— 2.718.8112— 2.989.3013— 3.279.8214 1.62 3.5910.3915 1.95 3.9510.9916 2.32 4.3411.6517 2.75 4.7812.3618 3.25 5.2713.1319 3.82 5.8113.9620 4.49 6.4114.8621 5.267.0815.8522 6.157.8416.92237.198.6818.09248.409.6219.37259.8110.6920.772611.4611.8822.322713.3913.2324.012815.6714.7625.882918.3516.4927.953021.5118.4630.243125.2620.7032.793229.7123.2635.623335.0226.1938.793441.3729.5642.343549.0033.4446.333658.2137.9350.823769.3543.1355.913882.9149.1961.683999.4856.2768.2540119.8464.5875.7741144.9974.3684.4042176.2385.9594.3543215.2799.73105.8744264.39116.20119.2945326.59135.99134.9946405.97159.91153.4647508.04189.00175.3148640.42224.59201.3249813.64268.44232.4950
1,042.48
322.88
270.09
TABLE 4.Comparison of Prent Factor with Other Upper-N ␥S Bound Solutions
(1)Prent solution (M1)
(2)
Chen (1975)
Prandtl1(3)Prandtl2(4)Prandtl3(5)15 1.9 2.7 2.3 2.120 4.5 5.9 5.2 4.6259.812.411.410.93021.526.725.031.53549.060.257.0138.040
119.8
147.0
141.0
1,803.0
FIG.8.Comparison of Prent Factor with Results of
读书后感N ␥S Other Authors
TABLE 3.Comparison of Prent Factor with that of Other
N ␥S Authors (1)Prent solution (M1)
(2)
Caquot and Ke ´ril (1953)
(3)
Meyerhof (1963)(4)Vesic (1973)(5)20 4.49 4.97 2.87 5.39259.8110.4 6.7710.883027.5121.815.6722.43549.048.037.1548.0340119.84113.093.69109.4145
326.59
297.0
262.74
271.76
P superposition =680.58kN/m,which indicates that the superpo-sition effect errs on the safe side.
Comparison of Results with Existing Solutions
N ␥S Factor.As is well known,there are a great many solutions for in the literature bad on different methods N ␥S and the differences among them are sometimes substantial.Becau of the great nsibility of the factor to the friction N ␥S angle,particularly for >30Њ,the tendency today,in practi
ce,is to u the values given by Caquot and Ke ´ril (1953),Mey-erhof (1963)[cf.(12)],and Vesic (1973)[cf.(13)]
N (Meyerhof)=(N Ϫ1)tan 1.4(12)␥S qS N (Vesic)=2(N ϩ1)tan
(13)
␥S qS where N qS is given as follows:
2
N =exp(tan )tan ϩ(14)
三亚风景图片qS ͩͪ
42
The values given by Caquot and Ke ´ril and the expression suggested by Vesic are being increasingly ud.Table 3and Fig.8show the comparison with the aforementioned au-thors.The maximal difference between the prent solution and that of Caquot and Ke ´ril is smaller than 10%for Յ45Њ.
On the other hand,rigorous upper-bound solutions for an associated flow rule Coulomb material are propod in the literature.Chen (1975)considered three symmetrical failure mechanisms referred to as Prandtl1,Prandtl2,and Prandtl3and gave rigorous upper-bound solutions in the framework of the limit analysis theory.Prandtl1is compod of a triangular ac-tive wedge under the footing,two radial log-spiral shear zones and two triangular passive wedges.Prandtl2differs from Prandtl1only in that an additional rigid body zone has been introduced.Finally,Prandtl3rembles cloly the Prandtl1mechanism;however,each shear zone is now bounded by a circular arc.The upper-bound solutions given by the prent M1mechanism and tho given by the three aforementioned mechanisms propod by Chen are prented in Table 4.It is clear that the prent upper-bound solutions are better than
