Limit Analysis in Soil Mechanics.pdf

{3 = 0° {3 = 9° {3 = 18° {3 = 27°

70°

a =

90°

a =

a=

110°

KA

Error (070)

KA

Error (%)

KA

Error (%)

0.372 0.421 0.484 0.571

0.81 0.71 0.83 0.88

0.171 0.186 0.205 0.233

0.50 0.43 0.44 0.39

0.065 0.069 0.073 0.081

0.31 0.15 0.14 0.25

{3 =

0°

{3 = 9° {3 = 13.5° {3 = 18° {3 = 22.5°

a =

70°

a =

90°

a=

For an actual design work, at least four practical aspects must be considered. They are the loading and strain conditions, the soil - structure interface friction, the progressive failure and scale effect, and the cohesion and surcharge effects. 4.7.1 Loading and strain conditions

The soil parameters for analysis and design are generally obtained by testing in laboratories soil samples taken from the ground. For the results from a test to be useful, not only the initial stress condition of the sample representing an element in the ground of interest should be recognized but also the loading and strain conditions (or the stress paths) should be carefully simulated to those to be expected in the field. Lateral earth pressures acting on long retaining walls are generally considered as plane strain problems. For lateral earth pressure analyses, the strength parameters should therefore be obtained from plane strain compression (active case) or plane strain extension (passive case) test. However, in many cases, only triaxial compresion test or direct shear test results rather than plane strain test results are available. Modification of the strength parameters is therefore frequently required before they can be entered into calculation. A relationship between the triaxial cP-value, cP tx ' and the plane strain cP-value, cP ps ' can be derived from the corresponding stress-dilatancy relations proposed by Rowe (l969a). From Rowe (1962), the stress-dilatancy relation for the triaxial and plane strain loading cases can be expressed as: (4.28)

b. Error on Kp-values introduced by pure-friction idealization of interface material

acceptable ranges. It is therefore suggested that the pure friction idealization which gives a 'safe' estimation of the upper bound, can be adopted for practical purposes.

110°

Kp

Error (%)

Kp

Error (%)

Kp

Error (%)

14.38 24.41 31.45 40.24 51.17

-2.23 -2.41 -2.48 -2.48 -2.10

35.41 60.28 77.78 99.69 126.93

-1.81 -1.68 -1.57 -1.53 -1.63

111.81 191.31 247.25 317.14 404.01

-0.56 -0.56 -0.49 -0.41 -0.52

where IT; and lT~ are the principal stresses, D represents the dilatancy and is equal to (l - dvld€a)' in which dv is the volume decrease per unit volume and d€a is the axial strain due to particle slips, and cPf is the frictional component of cP with its value varies from 1>p.' the angle of mineral-to-mineral friction, to cP cv ' the angle of internal friction at the critical state. In Eq. (4.28), Rowe (l969a) suggested that D can be taken as 2 and 1> f can be taken as cPp. in a triaxial compression test if the sand tested is at its densest state. The upper limit of the stress ratio corresponding to the densest state can then be expressed as:

134

135 (4.29)

Also, by taking D ~ 1 and cf>f = cf>Cy for the case that the sand is at its loosest state; the lower limit of the stress ratio corresponding to this state can be taken as: ,

,

al/a3

=•. tan2(45 + 0

I,;.) 2'1'CY

Noticing that a;/a~ = tan2 (45 0 + ~cf» and by combining Eqs. (4.28) to (4.33), the general expressions for the peak triaxial compression and the peak plane strain compression l/>-values can .be expressed, respectively, as:

(4.36)

(4.30)

For the plane strain compression case, Rowe (l969a) suggested that D can be taken as 2 for the densest state and 1 for the loosest state as in the triaxial cCimpression case. He also suggested that l/>f can be taken as l/>CY for sand at any relative density. Hence, the upper and the lower limits of the stress ratio for the plane strain case can be expressed respectively as: . (4.31)

and

(4.37)

where D is given by Eq. (4.34) and cf>f is given by Eq. (4.35). Hence, cf>tx and cf>ps can be related as:

and (4.32) where The stress ratios corresponding to the intermediate densities can be obtained by interpolation. In the general stress-dilatancy equation, Eq. (4.28), D can be expressed in terms of the angle of dilation, P, as:

+ :cf>CY) - 1) + icf>cY) + 1 ------------

~

1.0

(4.38)

(4.33) Here, P, or consequently, D, is dependent on the relative density (RD), the stress path, and the strain condition. Analysis of the data obtained by Cornforth (1964) for medium-to-fine, well-graded blasted sand reflects that D for both the plane strain compression and the triaxial compression situations can be approximated by:

D

= 0.64(RD)2 + 0.36(RD) + 1.0

(4.34)

The value of l/>f presented in Eq. (4.28) is also dependent on the relative density, the stress path, and the strain condition, in general, although l/>f was found independent of the relative density and equal to cf>CY in the plane strain compression and extension cases. According to the results obtained by Rowe (1962) for a medium-to-fine sand, the value of l/>f for the triaxial compression case can be approximated by: l/>f

=

cf>1'

+

[l - 0.463(RD) - 0.537(RD?J(cf>CY - cf»

(4.35)

They can also be related by tan l/>ps = 'l7t tan cf>tx' Some results plotted as '17 and 'l7t vs. (RD) for cf>I' = 15 0 to 35 0 are shown in Fig. 4.13 for reference. A linear approximation between cf>CY and l/>I' will be given later in Eq. (4.41). Since it has been reported (Rowe, 1969a) that in both plane strain compression and extension, cf>f = cf>CY' Eq. (4.38) originally derived for the compression case, is also valid for the extension case, although the values of l/>f and D may be different from those given by Eqs. (4.34) and (4.35). Analysis of Cornforth's results (Cornforth, 1964) shows that for the triaxial extension test, D can be approximated by:

D = 1.23(RD)3 - 0.79(RD?

+ 0.49(RD) + 1.0

:5

1.93

(4.39)

This gives D-values somewhat lower than those estimated by Eq. (4.34) for given relative densities. This tends to indicate that there is more dilatation in the compres-

'. 137

136 1,28 Plane Strain and

1.24

Triaxial Compressions

1.20

II>

~

0-

1.16

-e- -ec: ~

c: ~

II

1.12 1.08

~ 1.04 1.00 0

0.4

02.

RELATIVE

DENSITY,

0.6

0.8

1.0

RD

1.24 Plone

1.20

Strain and

Triaxial

Compressions

sion test than in the extension test. The angle of dilatation, P, of a given soil is therefore expected to be larger if sheared under triaxial compression condition than if sheared under triaxial exte~sion condition. Although, there is no information available on the plane strain extension D-va1ue, it is practically accepted that Eq. (4.39) is also valid for this case in lieu of the fact that Eq. (4.34) is reported to be equally valid for both triaxial and plane strain compressions (Rowe, 1969a). As far as ~f is concerned, it has also been reported that for both the triaxial compression and triaxial extension cases, ~f = ~p.' when the soil is in the densest state, and, ~f = ~cv' when the soil is in the loosest state. However, there is no adequate information so that a relation similar to Eq. (4.35) can be developed for soils at intermediate density for the extension case. Since D or P for the triaxial extension case is different from that for the triaxial compression case, it is expected that cPf is also different for the two cases. Equation (4.38) is therefore not strictly applicable to the extension case, since ~f is an indeterminate value. Another fact worth noting is that in the active earth pressure case, failure is induced by lateral unloading (or plane strain compression), whereas in the passive pressure case, failure is induced by lateral loading (or plane strain extension). As has just discussed, Eq. (4.38) seems not strictly applicable to the passive case. This is because under unloading shear, the sample generally has more 'brittle' behavior. Consequently, both cPf and D may be different from those given by Eqs. (4.34) and (4.35) as developed for the loading case. Fortunately, the variations of both ~f and D are expected to follow quite similar trends for given loading and strain conditions. Unless complete information can be obtained, Eq. (4.38) with D and cPr given by Eq~. (4.34) and (4.35) is suggested for estimating the corresponding factor fOr the cP-value to be used in the analysis of active and passive earth pressures. In case that only the direct shear cP-value, cPds' is available, the correction can be made based on the equation developed by Rowe (l969a):

1.16

-ei-i

tan~ds =

cos~cv

(4.40)

For ~p. = 15° to 40°, which covers all non-metal materials, the experimental data reported by Horne (1969) reflect that the relation between cPcv and cPp. can be fairly approximated by:

II

t::"

tancPps

1.12 1.08 1.04

(4.41)

1.00 0

0.2

0.4 RELATIVE

0.8

0.6 DENSITY

RD

Fig. 4.13. Variation of cf>p/cf>'x and tan cf>p/tan cf>tx with cf>. and relative density of sand.

1.0

For most earth materials, ~p. "" 25° to 30°, the value of cP cv ranges approximately from 31.5° to 36° . Hence, once the type of minerals forming the soil and the relative density of the material is known, both cP tx and ~ds can be properly corrected to give cP ps that can be adopted for the theoretical analyses.

138

139

4.7.2 Soil- structure interface friction As pointed out by Davis (1968), the strength parameter cf> on the velocity characteristics in a soil mass is different from the Mohr-Coulomb cf>-parameter. If the Mohr-Coulomb value for the plane strain case is cf>ps and the angle of dilatation is P for a soil mass, the corresponding cf>-value on the velocity characteristics in the soil mass is given as Eq. (3.39b), Le.: sincf>ps

cOSP

= tan-I ( cf>k 1 - sincf>ps sin P

)

= sin-I

[(D - 1)/(D

+ 1)]

(4.45) Furthermore, similar to Eq. (4.42), 0, ops and Pw can be related as: (4.46)

(4.43)

In general, the value of a can be evaluated by a direct shear testing with the soil to be used as the backfill sliding over the wall material. The angle of dilatation for the interface material, Pw' can also be measured in this manner. For obtaining strict upper bounds by the limit analysis method, both a and Pw are required, since a nonassociated flow rule should be applied to the interface material. However, the value of P is quite often not given, the following rules are suggested for estimating Pw ' (a) For 'sand - smooth steel' interface, a :5 cf>JL is probably the case. Purely frictional soil-wall sliding predominates the interface movement. In this case, Pw can be taken as zero. (b) For 'sand-rough steel' interface, cf>JL :5 a :5 cf>eY is the possible situation. A linear interpolation between II w = 0, corresponding to a = cf>JL' and the pw-value corresponding to a = cf>eY is suggested. Similar to Eq. (4.40) as proposed by Rowe (l969a), the relation between 0, the angle of wall friction in a direct shear test, and the corresponding plane strain value, 0ps' can be approximated by: tano = tanops coso eY

°

(4.42)

It is noted from the expression that cf>k :5 cf>ps' On the soil-structure interface, which is a velocity characteristic if the wall friction is fully mobilized, the maximum a-value corresponding to the perfectly rough, soil-to-soil sliding situation is 0max = cf>k' Hence, the angle of wall friction, 0, is always smaller than the Mohr-Coulomb A--value , 'l'ps' A- unless when P = 'l'ps A- ,in which case the velocity characteristics is the 'I' same as the stress characteristics. The value of P, if not measured, can be calculated from Eqs. (4.28) and (4.33), since cf>r can be estimated from Eq. (4.35) if the relative density of the material is known, and according to Eq. (4.33), the P can be written as: P

°

case. Note that for the constant where 0ey is the a-value corresponding to Pw = volume test, Pw = and ops = 0eY' Equation (4.44) reduces to tan ~ = sinops = sinoey.But since a = cf>JL is assumed as Pw = 0, we have tancf>JL = smo ey . Consequently, Eq. (4.44) becomes:

(4.44)

By assuming a = cf>eY and solving Eqs. (4.45) and (4.46) simultaneously, the Pw value corresponding to a = cf>eY' denoted by Pwo ' can be obtained. The pw-value for a given a can then be estimated by: (4.47)

(c) For 'sand - smooth concrete' interface with cf>eY :5 a :5 cf>k' the movement is approaching from soil-wall sliding to soil-soil sliding. In this case, Pw :5 P. Also, the aey-value is controlled practically by the sand grain-to-sand grain sliding and 0ey = cf>eY can be assumed. By solving Eqs. (4.44) and (4.46), considering 0ey = cf>eY' the pw-value can be reasonably estimated. (d) For 'sand - rough concrete' interface, a can be assumed as equal to cf>k' although Brumund and Leonards (1973) reported that a was as high as the triaxial cf>-value for the same kind of interface. The movement is practically a soil-to-soil sliding and the pw-value can be taken as P.

4.7.3 Progressive failure and scale effect In a direct application of the Mohr-Coulomb cf>-parameter, cf>ps' to plane strain stability problems, we implicitly assume that the strength of the soil along the failure surface is fully mobilized everywhere along the surface. This is probably the case in most laboratory tests in which the tested specimen is assumed representative of a soil element in the soil mass. This is because the specimen is generally so small that the strain is practically considered uniform along the failure surface, although boundary restrains do exist in almost all tests. In a soil mass, the strains along the failure surface are seldom uniform and failure of the soil mass is generally of progressive nature. At the instant of failure, the maximum shearing resistance available

141

140 on the failure surface must be, on average, somewhere between the peak state and the ultimate state as explained before (see Fig. 3.24). In most stability analyses, such as the limit analysis and the limit equilibrium method, the overall equilibrium of the soil mass involving deformation is considered. An average mobilized rP-value, rPm' rather than the peak rP-value, rP ps ' should be adopted in the analysis. The selection of cP m should be based on the progressive failure consideration so that the cP-value so chosen is corresponding to the average strain or the average stress level in the soil along the failure surface. This can only be obtained from the comparison of the theoretical analysis with the model test results. However, it involves another uncertainty, the scale effect, when applying the model test results to the field. This is because different soil masses involved in models of different size will result in a different extent of the progressive effect. Consequently, the cP -value that fits the theories will be different if the models are of different scale. F~rther­ more, both the failure mechanism involved and the interface roughness have a great influence on the extent of the progressive failure in a soil mass. Therefore, in the selection of proper cPm-value for design purpose, problem characteristics (e.g., active or passive case in lateral earth pressure problems) and interface roughness as well as the size of the structure (e.g., wall height in lateral earth pressure problems) should be considered. Rowe (l969b) recommended a method of considering the progressive failure effect in the selection of cPm-value for the stability analysis of granular soils. With the A. = 'l'mo' A. where assumption that. when the wall is perfectly rough 0 = rP ps and 'I'm . cP mo IS the maXImum cP-value to fit the current failure theory. He introduced a socalled 'progressive index' Itp' It is defined as:

0max = rPk situation, which is a perfectly rough situation for the case of concrete walls and sand backfills. For the case of steel walls and sand backfills, the maximum possible value of 0 may be taJ
1.0 ,-----,-------,--:===......__----, (4)

0,9 ton"( sinr"COSV ) 1- slnof1,. Sin v (3)

(4)

&

__ ----€l

..... 08

l (4.48)

Itp

'

-e-

(2) I~I~

(3W

He suggested, based on model studies on dense sands, that It at field scales can be taken as 0.4 for active pressure and 0.8 for passive pressure ftr practical design based on classical failure theories. He also claimed that these values are applicable to sands at intermediate densities. In order to look more closely at the effect of wall height " H on the ""'p-value , which has a great influence on the result of stability analysis, especially for the passive pressure case in which the progressive failure effect is great, some test results on models of different size are analyzed. Very limited data from Rowe and Peaker (1965), Rowe (1969b), and Kerisel (1972) are available for this purpose for the passive pressure case. As discussed previously, the maximum possible o-value for a perfectly rough wall corresponding to a soil-to-soil sliding condition should be equal to cPk' It is therefore proper to redefine cP mo as the cPm-value corresponding to II

IIl

--~

Steel Wall Soil-to- Wall Sliding

I

II

I~ 0.7 "-

8m,,'

0I>~

tari '( sin 01>,,) V'O

x

Q)

(3)/

-0

.E Q)

>

0.6

0\

~2)

(I) Tschebolarioff (1953)

'in Ul

(2) Rowe

~

'"e CL

(3) (4)

0.5 '-----_ _-----"-

o

1.0

a Johnson

a Peaker (1965)

Rowe (1969 b) Tcheng (Kerisel,1972)

--'-2.0

- J -_ _--.J

3,0

4.0

Wall Height, H , meters

Fig. 4.14. Progressive index as function of wall height for passive translational wall movement case.

142

143

Hence, this discrepancy should be recognized in interpreting/the information on the progressive index as function of wall height from model test results. It was found by Rowe (1969b) that the tPm-value of a soil mass subjected to active· pressure or passive pressure is approximately equal to its corresponding triaxial tPvalue, tPtx' if the wall is perfectly smooth, Le., 0 = O. Furthermore, it was found that the tP-value decreases linearly as the friction component, tan 0, increases, with its ultimate value equal to tP mo ' Based on these two findings, the tPm-value corresponding to an arbitrary wall friction, 0, can be approximated by:

,.,

(4.49)

where tP mo can be estimated from Fig. 4.14 or similar relations and from Eq. (4.48). In practice, the o-value, which is a function of the wall movement, may not

c- ¢

Soil

Y1.

sin

=

a

Vo cos P V3 = V, e'I/"o,

Y2L Vo

= ~

cos P

o. Active Case

be always full mobilized: By Eq. (4.49), a proper tPm-value corresponding to the given o-value can be selected for analysis and the lateral pressures can then be properly estimated.

4.7.4 Cohesion and surcharge effects In many cases, the backfill may possess cohesion although free-draining material is generally preferred. Presence of surcharge on the slope of the backfill is not uncommon either. Both cohesion and surcharge have certain influence on the lateral earth pressures. Their effects can be included in the lateral earth pressure evaluations. The versatility of the upper-bound limit analysis enables the effect of cohesion and surcharge being included in the calculations with rather little difficulty. For practical purposes, the soil-wall systems and their associated mechanisms of failure with pure frictional interface idealization as shown in Fig. 4.15 are considered. By including the internal energy dissipations along AB, BC, CD and in the radial shear zone OBC as contributed by the cohesion component, represented by the cparameter, the dissipation along the interface OA as contributed by the adhesion ca' and the potential energy change or external work induced by the surcharge, q, acting on OD in the analysis for cohesionless soils as stated in Section 4.4, the lateral earth pressures for the generalized case can be evaluated. Detailed derivation for the case of ca = 0 can be found in the book by Chen (1975). If the adhesion is expressed as a ratio of c, e.g. ca = Ac, the lateral earth pressures for a mixed soil backfill with uniform surcharge can be expressed as:

h

PA = Pp

=h

H2 (NA-y) + qH(NAq ) + cH(NAc )

(4.50)

H2 (Np-y) + qH(Npq ) + cH(Npc )

(4.51)

where c- Soil

f.la.PlnO'

NA-y

=

cos(e + tP)

[ .

sin2a cos(e - 0) costP

SIne

cos(a - e) +

cos(e + tP)

-------'-=-----'--'---

costP(1 + 9 tan2tP)

[cos(a - e)[3 tantP + e-31/- tanq,( - 3 tancf> cos1/; + sin1/;»)

Vo

y..!.- = sin Va V3 b.

a cosp

Vo, = cos(a·p) Vo cos P

+ sin(a -

e) [1

+ e- 3lPtanq,(-3 tantP sin1/; - coslf)]]

= V, e I/"o,

Possive Cose

Fig. 4.15. Log-sandwich failure mechanisms for lateral earth pressure analyses in c-q, soils subjected to uniform surcharge.

+

cos(e + cf» cos(a - e - If) sin(a + {3 - e - If)e- 3 Y, tanq,J cos(a + (3 - cf> - e - If)

(4.52)

144

145

N Aq =

N Ac

=

cos(Q + ¢) cos(a - Q - 1/;)e- 2 Y, tane/> sina cos(Q - 0) cos(a + (3 - ¢ - Q - 1/;) - 1 sina cos(Q - 0)

[A cos(a -

or Q)

sina

+

. SIllQ

Ca

A= -

·c

cos(Q + ¢) sin(a + (3 - Q - 1/;)e- 2 Y, tane/> cos(a + (3 - ¢ - Q - 1/;)

+-=-=-~_-:...:~~_---.:...-_--=:-_-:...:_--

I)J

cos(Q + ¢) (e- 2 Y, tan

N p ,¥ =

cos(Q - ¢) sin2a

[ . SIllQ cos(a - Q)

cos(Q + 0) COS¢

(4.54)

+ cos(Q - ¢) cos(a - Q - 1/;) sin(a + (3 - Q - 1/;)e3 Y, tan
Pq

N pc =

sina cos(Q + 0) cos(a + (3 + ¢ - Q - 1/;)

1

sina cos(Q + 0) cos(e

[A cos(a -

Q)

sina

cjJ) sin(a + (3 -

cos¢cv

(4.59)

The actual value of ca or A should be carefully evaluated by a direct shear test with the backfill material placed over the wall material. If the progressive failure effect is taken into consideration, the c-parameter should be modified. However, little is known on this aspect. Assuming that thecparameter varies in the same manner as the coefficient of internal friction tan¢ does, the average mobilized c-value, cm' can be estimated as: tan¢m

cm = - cps tan-l'f'ps

+ sin(a - Q) [1 + e3 Y, tan
= __co_s--:.(Q=---_.---'¢--'.)_c_o_s-'--(a_ _------"Q_-_..-'-.1/;)=---e_2 Y,_.._ta_n_e/>_

~

cos(Q - ¢)

+ -----'-----'----cos¢ (1 + 9 tan2¢)

[cos(a - Q) [- 3tan¢ + e3Y, tan
N

(4.58)

(4.53)

(4.55)

(4.56)

+ SIllQ .

(4.60)

where cps is the c-parameter corresponding to the plane strain condition. For practical purposes, the Cps-value can also be taken as the triaxial c-value, ctx ' since the cohesion characteristics of a soil, like pure friction, is almost independent of loading and strain conditions. If only the direct shear c-value, Cds' is available, cps can be estimated by Eq. (3.39a) with ¢ replaced by ¢ps' c replaced by cps' and ck replaced by Cds' if 11 is known. To obtain the most critical values of P A and P p , maximization and minimization, respectively, are required. The optimization should be performed with respect to the entire equation rather than to the individual terms in Eqs. (4.50) and (4.51). That is: (4.61) (4.62)

e - 1/;)e2 ,p tan

+_.-:::--'----------'-'--------'--------'-=-----'-----cos(a

+

+

(3

+

¢ - Q - 1/;)

_c_os....::(.:::-e_----:¢--'.)-,:(:-e2_y,_t_an_
sin¢

(4.57)

It should be noted that the value of the soil- wall adhesion, ca' is a function of both soil properties and characteristics of wall face in contact with the backfill. The maximum possible adhesion is ck as given in Eq. (3.39a), when there is a soil-to-soil interface sliding. For an idealized purely frictional soil- wall interface, 11 = II w = 0, the adhesion can be defined, according to Eq. (3.39a), as:

In this chapter, we have showed why the upper-bound limit analysis method can be applied to cohesionless soils for obtaining reasonably accurate estimates of the lateral earth pressures despite the fact that the normality condition required in the limit analysis is not actually observed in cohesionless soils during plastic flow. Both theoretical justifications and actual comparisons of the results of analysis confirm this applicability. By properly taking into account the four practical aspects discussed in this section, the upper-bound limit analysis method can be adopted for the analysis and actual design of rigid retaining structures (Chen and Chang, 1981). By the same principle, the analysis can also be extended to include the earthquake

147

146 forces\bY introducing a seismic coefficient and by using concept. This will be presented in Chapter 5.

the pseudostatic analysis Chapter 5

References Bruinund, W.F. and Leonards, G.A., 1973. Experimental study of static and dynamic friction between sand and typical construction materials. J. Test. Eval., 1(2): 162-165. Caquot, A. and Kerisel, J., 1948. Tables for the Calculation of Passive Pressure, Active Pressure, and Bearing Capacity of Foundations. Gauthier-Villars, Paris. Chen, W.F., 1975. Limit Analysis and Soil Plasticity. Elsevier, Amsterdam, 638 pp. Chen, W.F. and Chang, M.F., 1981. Limit Analysis in Soil Mechanics and Its Applications to Lateral Earth Pressure Problems, Solid Mechanics Archives, Vol. 6, No.3, Sijthoff & Noordhoff International Publishers, Alphen aan den Rijn, The Netherlands, pp. 331 - 399. Chen, W.F. and Rosenfarb, J.L., 1973. Limit analysis solutions of earth pressure problems. Soils Found., 13(4): 45 - 60. Conforth, D.H., 1964. Some experiments on the influence of strain conditions on the strength of sand. Geotechnique, 14(2): 143-167. Davis, E.H., 1968. Theories of plasticity and the failure of soil masses. In: l.K. Lee (Editor) Soil Mechanics: Selected Topics. Butterworth & Co., U.K., pp. 341-380. Finn, W.D., 1967. Applications of limit analysis in soil mechanics. Proc. J. Soil Mech. Found. Div., ASCE, 93(SM5): 101- 120. Habibagahi, K. and Ghahramani, A., 1977. Zero extension theory of earth pressure. J. Geotech. Div., ASCE, 105(GT7): 881- 896. Hettiaratchi, R.P. and Reece, A.R., 1975. Boundary wedges in two-dimensional passive soil failure. Geotechnique, 25(2): 197 - 220. Horne, M.R., 1969. The behavior of an assembly of rotund, rigid, cohesionless particles, Ill. Proc. R. Soc. London, Ser. A, 310: 21- 34, James, R.G. and Bransby, P.L., 1970. Experimental and theoretical investigations of a passive earth pressure problem, Geotechnique, 20(1): 17 - 37. Kerisel, J., 1972. The language of models in soil mechanics (translated). Proc. 5th European ConL on Soil Mech. and Found. Eng., Madrid, Vol. 2, pp. 3-30. Lee, l.K. and Herington, J .R., 1972. A theoretical study of the pressures acting on a rigid wall by a sloping earth or rockfill. Geotechnique 22(1): 1- 26. Rowe, P.W., 1962. The stress-dilatancy relation for static equilibrium of an assembly of particles in contact. Proc. R. Soc. London, Ser. A, 269: 500- 527. Rowe, P.W., 1969a. The relation between the shear strength of sands in triaxial compression, plane strain, and direct shear. Geotechnique, 19(1): 75 - 86. Rowe, P.W., 1969b. Progressive failure and strength of a sand mass. Proc. 7th Int. ConL on Soil Mech. and Found. Eng., Mexico, Vol. 1, pp. 34-349. Rowe, P .W. and Peaker, K., 1965. Passive earth pressure measurements. Geotechnique, 15(1): 57 -78. Scott, R.F., 1963. Principles of Soil Mechanics. Addison-Wesley, Reading, MA, 523 pp. Sokolovskii, V.V., 1965. Static of Granular Media. Pergamon Press, New York, NY, 232 pp. Terzaghi, K., 1943. Theoretical Soil Mechanics. John Wiley & Sons, New York, NY, 510 pp. Tschebotarioff, G.P. and Johnson, S.C., 1953. The Effects of Retaining Boundaries on the Passive Resistance of Sand. Princeton University, Princeton, NJ.

RIGID RETAINING WALLS SUBJECTED TO EARTHQUAKE FORCES* 5.1 Introduction Although seldom reported and documented, numerous failures of rigid retaining walls in areas of intensive seismic activity have been attributed to earthquake effects. Earthquake can endanger the stability of a soil - wall system by either increasing the driving forces acting on the wall or by reducing the resistance of the backfill and/or the foundation soils. For most moderate earthquakes in which the seismic acceleration is no more than 0.3 g, the mechanical properties of most soils will probably not change considerably (Okamoto, 1956). It can be practically assumed that there is no strength reduction in the foundation soils that controls the movement of the wall and in the backfill that influences the magnitude of lateral earth pressures, unless that the soil is cohesionless, not very permeable, and submerged under water. Hence, the assessment of seismic lateral earth pressures or changes in lateral earth pressures as the result of an earthquake is of more practical significance in most aseismic designs of retaining walls. Since the earthquake motion is of an oscillatory nature, dynamic analysis of lateral earth pressures is certainly more realistic. However, dynamic analysis involves many uncertainties, e.g. the extent of soil mass effectively participating in vibrations, that are not yet wholly understood. Furthermore, providing the necessary information for a dynamic analysis and performing such an analysis are relatively expensive. Quasi-static analysis using the seismic coefficient concept is therefore of greater practical value in many cases, although the assessment of the seismic coefficients still relies highly on past experience. The well-known Mononobe-Okabe analysis of seismic lateral earth pressures proposed by Mononobe and Matsuo (1929) and Okabe (1926) is generally adopted in current practice for aseismic design of retaining walls. The analysis is a direct modification of the Coulomb wedge analysis. In the analysis, the earthquake effects are replaced by a quasi"static inertia force whose magnitude is computed on the basis of the seismic coefficient concept. • This chapter is based on the Ph.D. thesis by M.F. Chang (1981) and the paper by Chang and Chen (1982).

148 As in the Coulomb analysis, the failure surface is assumed planar in the Mononobe-Okabe method, regardless of the fact that the most critical sliding surface may be curved. Similar to Coulomb's, the Mononobe-Okabe analysis may underestimate the active earth pressure and overestimate the passive earth pressure. In the passive earth pressure case in which the most critical sliding surface is usually curved, the overestimation may be very serious, especially when the soil- wall interface is rough and the backfill surface is steeply sloped. The Mononobe-Okabe analysis has been further modified by Prakash and Saran (1966) for assessing seismic active earth pressure for walls retaining horizontal c-cjJ soils. Uniform surcharge and tension cracks at top of the back fill are included in their formulation. However, the seismic acceleration is assumed to act in the horizontal direction and the failure surface is assumed planar as in the Coulomb or Mononobe-Okabe analysis. In this chapter, the upper-bound method of limit analysis applied previously to lateral earth pressure calculations is extended to include the earthquake effects. Translational wall movement and log-sandwich mechanism of failure as suggested by Chen and Rosenfarb (1973) are assumed in the formulation. Quasi-static representation of earthquake effects using the seismic coefficient concept is adopted. The direction of seismic acceleration is taken as arbitrary and the most critical direction is found by iteration. Earthquakes have unfavorable effects of increasing active and decreasing passive lateral earth pressures. To investigate how the lateral earth pressures are affected, extensive numerical results based on the limit analysis method reported by Chang and Chen (l982) are presented in dimensionless forms. For the active earth pressure case, the coefficient of seismic active pressure, K AE , is adopted. For the passive earth pressure case, the coefficient of seismic passive earth pressure, K pE , is used. In order to see the validity of the limit analysis method, some KAE-values and KpE-values for various soil- wall conditions and seismic accelerations are calculated and compared with solutions by the well-known Mononobe-Okabe analysis. The difference between them is discussed. Some parametric studies have been conducted to investigate the effects of the parameters involved in the analysis of the calculated K AE- and KpE-values. The variations of K AE- and KpE-values with the cjJ-parameter for different levels of earthquake, or for different seismic coefficients, are presented. The increase in the KAE-value and the decrease in the KpE-value as the result of an earthquake is noted. The effect of the angle of wall friction on the K AE- and KpFvalues is also given. The influence of the geometrical factors on the K AE- and KpE-values is mentioned. The effect of the direction of seismic acceleration on the results is carefully discussed. Afterwards, the effects of possible presence of uniform surcharge and cohesion in the backfill on the KAE-values and the KpE-values, as well as the effect of seismic

149 , forces on the most critical sliding surface are c~refully' studied and discussed. Their importance in geotechnical engineering is also described. Finally, the earth pressure tables that provide both static and seismic active and passive earth pressures are presented in terms of dimensionless coefficients (Chang, 1981). 5.2 General considerations In the theoretical formulation for seismic lateral earth pressure analyses, the upper bound limit analysis method is adopted. A general soil- wall system with translational wall movement and a cjJ-spirallog-sandwich mechanism of failure proposed by Chen and Rosenfarb (1973), as shown in Figs. 5.1 and 5.2 are assumed.

Wz=J. dWz A

sina

V, = cosp Vo

v.01-- cos(a-p)v. cosp 0 Fig. 5.1. Log-sandwich mechanism for seismic passive earth pressure analysis.

.w~ o

V,

sin a

= cosp

Vo

_ cos(a-Pl Vo,- cos p Vo

W

VB' =V,e-B't.,,,, V3 = V,e-""·''''

Fig. 5.2. Log-sandwich mechanism for seismic active earth pressure analysis.

151

150

The backfill material is assumed to possess some cohesion (c > 0). A uniform surcharge q is assumed to act on the surface ofthe backfill sloped at an angle (3. The soil- wall interface is assumed inclined with its repose angle equal to a. An earthquake has two possible effects on a soil- wall system. One is to increase the driving force. The other is to decrease the shearing resistance of the soil. The reduction in the shearing resistance of a soil during an earthquake is in effect only when the magnitude of the earthquake exceeds a certain limit and the ground conditions are favorable for such a reduction. The evaluation of such a reduction requires considerable knowledge in earthquake engineering and soil dynamics. Research conducted by Okamoto (1956) indicated that when the average ground acceleration is larger than 0.3 g, there is a considerable reduction in strength for most soils. However, he claimed that in many cases, the ground acceleration is less than 0.3 g and the mechanical properties of most soils do not change significantly in these cases. In this chapter, only the increase in driving force is to be considered. The shear strength of the soil is assumed unaffected as the result of the seismic loading. In the quasi-static analysis of seismic lateral earth pressures, a constant seismic coefficient, k, is assumed for the entire soil mass involved. A seismic force, which is equal to k times the weight of a soil mass, is assumed to act at the center of gravity of the sliding soil mass. The seismic force is assumed to act in a direction at an angle () from the horizontal as shown in Figs. 5.1 and 5.2. Quite often, the direction of the seismic acceleration may be essentially horizontal. In some situations, however, the vertical component can be very large, for example, at locations near the epicenter ofan earthquake. In general, the relative magnitude of the horizontal and vertical acceleration components varies from one case to another. A general direction of the seismic force is therefore assumed in the present formulation. In case that the magnitudes of the horizontal and vertical seismic coefficients, k h and k y, respectively, are known for a given earthquake, the direction () = tan - I (ky/kh) is then fixed. The seismic lateral earth pressure due to that particular earthquake can then be evaluated. In most analysis for design purposes, however, () is generally unknown. The ()-value should be optimized to give the most critical condition.

5.3.1 Calculations of incremental external work The incremental external WQrk. due to an external force is the external force multiplied by the corresponding incremental displacement or velocity. The incremental external work due to self-weight in a region is the vertical component of the velocity in that region multiplied by the weight of the region. The seismic force which is assumed constant in each region, can be divided into a vertical component and a horizontal component. The incremental external work contributed by this force in a region can be obtained by the muliplication of the force components and the corresponding velocity components in that region. They should be added to those due to self-weight. In the following derivatives, the variables involved are defined as the following: "( = unit weight of the backfill material c{J = angle of internal friction of the backfill material o = angle of soil- wall interface friction k = seismic coefficient () = inclination of the seismic coefficient with the horizontal H = vertical height of wall. All the others such as the geometrical factors, a, (3, e, 1/;, ()f, and the incremental displacements or velocities, vO' VI' v3 ' Vo' etc. are as defined in Figs. 5.1 and 5.2. In zone OAB, we have: AWOAB = - (WI - kWI sin() vly _ "(HZ cos(e - cf» tane 2 sin a cos c{J

k sin () cos(a - e)

(5. I)

Note that Eq. (5.1) is similar to Eq. (4.15) except that k

= 0,

Pw

= 0, and t

=

c{J. In Zone OBC, we have:

=

J[- (dWz -

k dWz sin()vO'y

+

k dWz cos() vo'x l cIA

A

"(HZ

cosz(e - c{J)

In the passive case, the decrease in the passive earth pressure as the result of an earthquake is of major concern. In this case, the possible critical direction of the seismic force, k W (k = resultant seismic coefficient) is pointing away from the wall, as shown in Fig. 5.1. Herein, as in the lateral earth pressure analyses for the static cases, Chen and Rosenfarb (1973), the seismic passive earth pressure can be easily derived by considering the equilibrium of external work and internal energy dissipation.

[(I _

- k cos() sin(a - e)] Vo

AWOBC

5.3 Seismic passive earth pressure analysis

+ kWI cos() vlx

2

(1

+

aZ )

[

•

(I - k sm() [cos(a - e)

sina coszc{J cose

[ea'" (a cos1/; + sin1/;) - aJ + sin(a - e) [ea"'(a sin1/; - cos1/;) + 1]] - k cos() [sin(a - e) [ea"'(a cos1/; + sin1/;) - aJ - cos(a - e) [ea'" (a sin1/; - cos1/;) +

1]]]

Vo

(5.2)

153

152

where a = 3 tan1>. In Zone OCD, we have:

AWOCD =

(W3

-

kW3 sinO)

v 3y

+ kW3 cosO

q (aD) (1 - k sinO) V3y -

_ ['YH2

v 3x

q (OD) k cosO v3x

cos(e - 1» e3,ptan¢sin(cx + (3 - e - 1/;) + qH e2 ,ptan¢]

2

sincx cos1>

cos(e - 1» [(1 - k sinO) cos(cx - e - 1/;) - k cosO sin(cx -

e-

1/;)]

_ _ _ _ _ _ _ _'-----_-=----_.:....:.;;c--

cose cos(cx + (3 + 1> - e - 1/;)

V

o

(5.3)

With this background, the incremental energy dissipation along a velocity discontinuity is simply ADL multiplied by the length of the discontinuity or the total frictional force multiplied by the.incremental displacement for the pure friction energy dissipation along the soil- wall interface. It should also be noted that for the case that there is a radial shear zone, such as zone aBC in Fig. 5.1, the energy dissipation in the radial shear zone is the same as that along the spiral arc BC (Chen, 1975). It is noted that the internal dissipation is totally independent of the seismic coefficient. Along OA, there is: ADOA

Due to the seismic passive earth pressure PpE' using Eq. (4.18), we have: (5.4)

=

PpE sino VOl

=

(PPE sino + ca s:cx)

+ ca (OA)

VOl

cos~:s~

(5.8)

e) Vo

Along AB, there is:

The total incremental external work is the summation of these four parts of contributions, Eqs. (5.1) to (5.4). That is:

C

H tane

(5.9)

Vo

Along BC, there is: (5.5)

ADBC =

5.3.2 Calculations of incremental internal energy dissipation According to Finn' (1967) or Cilen (1975), the incremental energy dissipation per unit length along a velocity discontinuity, or a narrow transition zone, can be expressed as:

J

L

[c VO' cos1>l dL

c H cos(e - 1» (e2,ptan¢ - I)

= j.

=

c Av cos1>

AD CD =

C

(CD) v3 cos1>

+

C H cos(e sin(cx (3 - Q - 1/;) e2,ptan¢ ---=---'--':---'--::--'-----.,---=--'-----;:--- Vo

(5.6)

Where A v is the incremental displacement or velocity which makes an angle c/J with the velocity discontinuity according to the associated flow rule of perfect plasticity, and c is the cohesion parameter. On an idealized soil- wall interface, where adhesion and pure friction present, the following equation, which is modified from Eq. (5.6) can be adopted: (5.7)

where Sf is cohesion parameter, c, adhesion parameter, ca' or pure friction component of a unit interface force, and AV L is the tangential component of the incremental displacement along the velocity discontinuity.

(5.10)

Along CD, there is:

1»

ADL

Vo

sm1> cosQ

cosQ cos(cx

+

(3

+

1> -Q - 1/;)

(5.11)

From Zone OBC, we have: AD

OBC

=

AD

BC

=

c H cos(Q - ¢) (e2 ,ptan¢ - I) 2 sinc/J cose

V

0

(5.12)

The total incremental energy dissipation is the summation of the five parts given above, Eqs. (5.8) to (5.12). That is: (5.13)

154

155 K pE = N p

By equating E[W]ext in Eq. (5.5) to E[LlD] in Eq. (5.13), we have:

(5.14) where Np'Y' N pq and N pc are passive earth pressure factors. They are given as: N P'Y

cos(e - cjJ)

=

sinza cos(e sin(a - e)J

+ 0) coscjJ

+

[(1 - k sinO) sin(a - 12)

+

+

+ si~1/;)

- aJ [(1 - k sinO)

+ [ealb(a sinif; - cosif;) + IJ

k cosO cos(a - e)JJ

cos(Q - cjJ) sin(a + (3 - Q - if;)ea,p [ . (1 - k smO) cos(a + (3 + cjJ - 12 - 1/;)

cos(a - Q - 1/;) - k cosO sin(a - 12 - if;)J) where a

1 . sma cos(Q

'YH

q

+

,2c

(5.19)

- Npc

'Y H

The most critical KpE-vafue can be obtained by minimization with respect to 12 and if; angles in Fig. 5.1. The most critical failure surface is then defined by the 12 and 1/; which give minimum K pE .

In the active case, an increase in the active earth pressure as the result of earthquake is of major concern., The possible critical direction of a seismic acceleration is therefore pointing toward the wall, as assumed in Fig. 5.2. The O-value is considered positive when the seismic force has a downward vertical component. This is different from that assumed in the passive case, in which the O-angle is positive when the seismic force has an upward component. Similar to the formulation for the seismic passive earth pressure, the seismic active earth pressure can be derived as: (5.20)

(5.15) where NA'Y' N Aq , and N Ac are active earth pressure factors. They are given as:

= 3 tancjJ.

[cos(e - cjJ) eZ,ptan¢ [(1 - k sinO) cos(a - 12 - 1/;) ] -: kcosO sin(a ' 1 2 --' if;)] N pq = cos(a + (3 + cjJ - 12 - if;) sina cos(Q +- 0) ( N pc =

2q

- Np

k' smO) cos(a - 12) - k cosO

cos(e - cjJ) [[eQlP(a cos1/; (1 + a Z) coscjJ

cos(a - 12) - k cosO sin(a - e)J

+

5.4 Seismic active earth pressure analysis

[(1 -

( . sme

'Y

+

[A cos(a 0)

.

12)

sma

+

cos(e + e/» _ (sin Q[(1 sinza cos(e - 0) cose/>

(5.16)

+ (3 - 12 - if;)eZ,ptan¢ cos(a + (3 + cjJ - Q - if;)

[(1

cos(e - cjJ) sin(a

sine/>

1)]

-

+ k sinO) cos (a

-

12)

+ k cosO sin(a - e)J

+ [e-a,p( - a sinif; - cosif;) + IJ [(1 + k sinO) sin (a - 12) (5.17)

- k cos0 cos (a -

12 )JJ

+

cos(e

cjJ) sin(a

+

(3 - 12 - 1/;) e-a,p

+ -----'-----,--------:--------'-----:-:--cos(a

+

(3 - e/> - Q -

[(1 + k sinO) cos(a - 12 - if;) + k cosO sin(a The seismic passive earth pressure can also be expressed in terms of an 'equivalent' coefficient of seismic passive earth pressure, K pE , as:

where a

=

Q -

if;)

if;)])

(5.21)

3 tane/> is the same as in Eq. (5.15).

cos(e

+

e/» e-Z,ptan¢ [(1

+ k sinO) cos(a - 12 - 1/;)

(5.18) cos(a in which

Q)

+ k cosO sin(a - 12)] + cos(e + e/» [[e-a,p( - a cos1/; + sin1/;) + aJ (1 + a2 )cose/>

.

SIne

+--=--.:....;--~---.-.:-:---..::_~-­

+ cos(e - cjJ) (eZlbtanq, -

+ k sinO) cos (a

+ (3 - e/> -

Q -

+ k cosO sin(a - 12 - 1/;)] (5.22) if;) sina cos(e - 0)

156

157

-1

--;-----:------:::sina cos(e 0)

[A cos(a -

e)

sina

+ sine

0.6 ~--~--~--~-----,

a

cos(e + rf» sin(a + (3 - e - 1/;) e- 2,ptanq, +----=-=-------'--:---:---:------'----;-:--cos(a + (3 - rf> - e ~ 1/;) cos(e + rf» (e- 2 ,ptanq, -

sinrf>

1)]

:9QC>

.f3." O·

1>" 40·

0.5 -

Limit Analysis }

____ M-O Analysis

Practically Identical

(5.23)

where A = calc is the same as in Eq. (5.17). If expressed in terms of 'equivalent' coefficient of seismic active earth pressure, K AE , Eq. (5.18) becomes: 02

(5.24) in which:

0.1

LO---O:-':.I-----;;0';;-.2--~0;;';.3;----;Q~.4 Horizontal Seismic Coefficient, kh

(5.25) The most critical K AE-value can be obtained by maximization with respect to e and 1/;-angles in Fig. 5.2. The e and 1/; at which the KAE-value is maximum determine the most critical sliding surface. Two computer programs for assessing seismic lateral earth pressures have been developed based on Eqs. (5.14) to (5.25). Details of the program documentation can be found in the thesis by Chang (1981). 5.5 Numerical results and discussions

In the presentation of the results of seismic active and passive earth pressure analyses, dimensionless coefficients K AE and K pE are adopted. As mentioned previously, the present limit analysis solutions are valid when there is no reduction in soil strength due to an earthquake. The results presented are meaningful generally only when the seismic acceleration a :5 0.3 g, since most soils will either liquify or seriously weaken if a > 0.3 g (Okamoto, 1956). It is only for purely theoretical interest that, in some cases, results are presented for seismic coefficients up to 0.4. However, only when k :5 0.3 are the results recommended for practical use.

Fig. 5.3. Some (KAE)n-values by limit analysis and Mononobe-Okabe analysis (vertical wall and horizontal backfill).

5.5.1 Comparison with Mononobe-Okabe solution .

,

The Mononobe-Okabe analysis, which is an extension of the Coulomb's analysis, has been experimentally proved by Mononobe and Matsuo (1929) and Ishii 'et al. (1960) to be effective in assessing the seismic active earth pressure. It is generally adopted in the current aseismic design of rigid retaining walls. The MononobeOkabe solution is therefore practically acceptable at least for the active pressure case although its applicability to the passive pressure case is somewhat in doubt. S~me results on seismic active and passive earth pressures as obtained by the present limit analysis method are compared with the Mononobe-Okabe (M-O) solutions. They are shown in Figs. 5.3 to 5.6. For the active case, the KAE-values obtained by the two methods are ~r~cti:allY identical for most cases (Figs. 5.3 and 5.5). This is true even when the WallIS mclmed and the slope angle of the backfill is larger than zero, as shown in Fig. 5.5. The fact that the most critical, or potential sliding surface for the active case is practically planar, as shown in Fig. 5.7, is responsible for this consequence. For the passive case, the most critical sliding surface is much different from a planar surface as is assumed in the M-O analysis (Fig. 5.8). The KpE-values are seriously overestimated by the M-O method. They are, in most cases, higher than

-I 159

158 IOO.-----~--~---.,_--___...

BO

100.-----~---."..--~.--------,

_ _ Limit Analysis

_ _ limit Analysis

___ M-O Analysis

___ M-6 Analysis

60 N

:I:

>-

~ o..ll. 40

_3Q!,..

30'

20

o~~·~ ~ o

0.1

0.3

0.2

Horizontal Seismic Coefficient

-10"

0.4

,kil

Slope Angle of Backfill

Fig. 5.4. Some KpE-values by limit analysis and Mononobe-Okabe analysis (vertical wall and horizontal backfill).

,f3

.----~---,_---,---___;

Ii

'" = 40' 8 = 20' 'h= 0.20 (a) a

Seismic

__

~

i.O

20'

Fig. 5.6. Comparison of KpE-values by limit analysis and Mononobe-Okabe analysis (general soil-wall system).

_ _- 7

12

10'

0'

• _7° ,

Static .11

lfd 11'00

tY. =- 0.° t

7°,.

I

KAE= 0.65

KA II 0.42

'70"

O.B

Seismic

P.t

_ _ Limit Analysis

= 10°,

Static

N

J\

'l:. 0.6

-,N ....

_..t......cJQ..lL.n._ 1

=200,IjI.

"'d =- 0°,

KAE = 0.37

= 2°,

KA = 0.22

.60°

.ad.= 50°

dtl

(b) a = 90' ,

f3

= 10'

{3 =5' 0.2

L_~=====~a~:--JII!li_:;'" =_~-=-

% = 5', '/J.

oL-_ _-'-_ _-'-_ _--'-_ _---' -20'

0

_10

0°

Slope Angle of Backfill •

10°

KAE =0.20

=12', KA = 0.10

20'

13

Fig. 5.5. Comparison of KAE-values by limit analysis and Mononobe-Okabe analysis (general soil-wall system).

(c)

a=1I0',

f3

=5'

Fig. 5.7. Effect of seismic forces on failure mechanism in active pressure analysis (q, = 40·,0 k h = 0.20).

20· ,

160

161 0.7r---~--~---~----,

p. = 4S' • '/t.=IS' Kp = 8.36

Pd= 4S'. '/td 10' ,Qd,=,Q.=2S'

KpE = 7.82

.// /.

StatI.

P. =48° , 'It.= 3QO Kp = 16.26

Pd =48"

(b) a=90',

{3 =10'

I

Y,d" 25°

KpE = 15.06

0.2

~ =Pd =~

Kp (e)

a =110', {3 =5'

'

'1'. .. t d

=40°

32.92, KPE=29.54

Fig. 5.8. Effect of seismic forces on failure mechanism in passive earth pressure analysis ( = 40°, 0 = 20°, k h = 0.20).

those obtained by the limit analysis. This is especially the case when the wall is rough (Fig. 5.4) and the angle of wall repose is large (Fig. 5.6). For smooth walls, the potential sliding surface is practically planar and the two methods give almost identical results. 5.5.2 Some parametric studies

In the analysis of seismic lateral earth pressures on rigid walls retaining cohesionless soil, the parameters involved include the unit weight of soil, 'Y, the angle of internal friction, cjJ, the angle of soil- wall interface friction, 0, the slope angle of the backfill, {3, the angle of wall repose, a, the height of the wall, H, the seismic coefficient, k, and the direction of the seismic acceleration, e, if there is no uniform surcharge presented. Since the dimensionless lateral earth pressure coefficients KAE and K pE are generally selected to represent the lateral earth pressure, 'Y and Hare irrelevant to the problem for the case of cohesionless backfill (c = 0) and zero surcharge (q = 0).

--8=4>/3 --8=4>/2 -----8= 2#3 2S'

30·

35°

Angle of Internal Friction

40· I

45'

"

Fig. 5.9. Variation of KAE-values with -angle for earthquakes of different level.

Parameters cjJ and k - internal friction angle and seismic coefficient Figures 5.9 and 5.10 show the variation of K AE- and KpE-values with parameter cjJ fo!: diff~rent horizontal seismic Cgeffi,ci~nts, kh~' The KAKVlj.!Ut'( decreases as cjJ increases for a given kh-value. On the contrary,the.KpE-value.increases as cjJ increases for a given kh-value. When there is an earthquake, the KAE-value increases and the KpE-value decreases. As the magnitude of an earthquake becomes larger, the K AEvalue increases and the KpE-value decreases furthermore. When Figs. 5.9 and 5.10 are replotted as shown in Figs. 5.11 and 5.12, where the increases of pressures due to an earthquake are normalized by the corresponding static pressures, the percentage of increase in the K AE-value and decrease in the KpE-value as the result of increase in the magnitude of an earthquake, or kh-value, is much more clear. It is found that for the active case, the increase in K AE is more obvious for denser soils with higher cjJ-values than for looser soils with lower cjJvalues. While, the decrease in K pE is more obvious for looser soils than for denser soils in the passive case. Parameter 0 - interface friction Figures 5.9 to 5.14 also show the parameter 0 affects the K AE and K pE values. For the active case, the KAE-value may become smaller or larger when the o-value increases, depending on the ..p-angle and the kh-value as shown in Figs. 5.9 and

163

162

5.13. However, as shown in Fig. 5.11, when the a-value is increased the percentage of change in KAE-value as the result of an earthquake is seen also to increase. It is

25r----,----,------,---'

I

flo.

H 8 20

25r---~---~---r;----,

- - 8- .p/3 --8-#2 ~ 15

------ 8 - 2.p/3

~

--

o..'fI.

.

"

':11:0,.

10 c

~

j

10

c

e

-----8.2.p/3 - - 8 • .p/2 - - 8 . .p/3

if.

5 Angle of Internal Friction ,

rp

fig. 5.10. Variation of KpE-values with qI-angle for earthquakes of different level.

0.1

0.3

0.2

Horizontal Seismic Coefficient

t

0.4

kh

fig. 5.12. Decrease in KpE-values as the result of seismic forces. 140

HlsT H .

1.2 r-----,----.-....,......,..-.,.-,r------,

120

~E

~

>2

K••- fJlEd

~

J

1.0

a-90°, fJ=oo

100

r

2

0.8

KA • (KAE )llhao

80

.K,.E" K,.E- K.

II 60

--- 8 -2/3.p

-IN

- 8 - V2.p

--o'!I

"f

.l!

J

.p - 40"

N

~0.6

.5

a '90",

} 0.4

40

----- 8 • 2.p/3 - - 8 • .p/2 - - 8 -.p/3

0.2

20

oOk---~0;,..I---~0"'.2~--~0"'.3~--O....,l.4

0 0

0.1

0.2

0.3

0.4

Horizontal Seismic Coefficient ,Kh

fig. 5. I I. Increase in KAE-va!ues as the result of seismic forces.

Horizontal Seismic Coefficient

,K h

Fig. 5.13. Effect of slope angle on KAE-values for 0

i



and 0

<1>/2 cases.

164

165

therefore expected that, in most cases when k h

> 0, the KAE -value is larger when

ais high than when ais low. However, it should be noted that if the normal component (KAE)n is considered, the value decreases as a increases, unless the kh-value is very high (Fig. 5.3). For the passive case, the KpE-values increase as the a-value increases, whether there is an earthquake or not. This is true even when the magnitude of earthquake is high. This is because that the percentages of decreases in the KpE-value as the result of an earthquake, although larger for larger a-values, are not much different for the cases of lower a-values and for the cases of high a-values as shown in Fig. 5.12. The general trend that K pE increases with increasing avalues can also be seen clearly from Fig. 5.4 and Fig. 5.14. More results on the effects of parameter aon the K AE and K pE value are given in Figs. 5.15 and 5.18.

Parameters a and (3 - wall geometry and backfill shape The geometry of the wall and backfill as reflected by the angles a and (3 in Fig. 5.1 or 5.2 have considerable effects on the magnitude of the lateral earth pressures. Figure 5.5 shows that for a given kh-value and soil condition, the KAE-value increases as the slope angle, (3, increases and the angle of wall response, a, decreases. The (3-effect is larger as the kh-value becomes higher as shown in Fig. 5.13.

For the passive case, the KpE-value increases as the (3-value and the a-angle increases as shown in Fig. 5.6. The (3-effect is practically unaffected by the variation in k h (Fig. 5.14).

Parameter 0 - direction of earthquake force As mentioned previously, the direction of the resultant seismic acceleration varies from one earthquake to another. Although Housner (1974) claimed that k v :::: Gj) k h for most earthquakes, current practice tends to assume that the seismic acceleration is essentially horizontal (0 = 0°). The effect of this assumption on the results of analyses depends on how much the most critical direction differs from the horizontal one, how the actual seismic acceleration differs from the horizontal and what is the magnitude of the earthquake. Figure 5.15 shows that for the active case, the KAE-value obtained based on the assumption 0 = 0° is not much different from the optimized KAE-value obtained when the seismic acceleration assumes the most critical direction, i.e. 0 = Ocr' This is probably because the Ocr-values are found to be essentially equal to zero, especially if the a-angle is high, as shown in the figures. It may therefore be concluded that

1.2

50,--------:-:=--------, 1.0

~

30

0.8

-7---------

f3

f3/io.

10

k

1.0

=0.20 5'

---- e • eo, _e. O'

0.8

____ e· ec,

.

= 20'

~N

----- --

ep = 40' 8 = 2/3 ep

k = 0.20

~E

_8=/2

----"

ep = 40' 8 = ep/2

{

--- 8 =2/3 40

1.2 (b)

(0)

~':J.O.4 ------

--~-;;O'='"----

0.2~e~":":=~8:"·_ _~a!!.:=:!I~10~·';,5.;------~2. 0.2~8:::. ...~7~'--.-':a;.::=:.!111!!2.0:.·5~.:-----22".~ _ 00~':-----:5:-:.---ILO.,----"------' 0 ::----=------:-::-----,'::,---___=_:' 15°

0'---------·-- --_.0.2 0.3 o 0.1 Horizontal Seismic Coefficient

I

Slope Angle of Backfill

0.4

Kh

Fig. 5.14. Effect of slope angle on KpE-values for {,

= cp/2 and {, = t cp cases.

•

{3

20°

0°

5°

10°

Slope Angle of Backfill •

15°

20°

f3

Fig. 5.15. Effect of direction of seismic acceleration on KAFvalues for general soil-wall system: (a)

{, = cp/2;

(b) {,

= t cp.

166 in a given earthquake, even though its vertical seismic acceleration, ay = kyg, may be significant compared to its horizontal acceleration, ah = khg, its effect on the KAE-value may be neglected for practical applications. The effect of ky on the KAE-value is also shown in Fig. 5.16. It can be seen that the KAE-value is increased only in the order of 7070 even when the ky is as high as 2/3 kh • For this reason, Seed and Whitman (1970) recommended that the influence of ky can be neglected in practical designs of retaining walls. Figure 5.16 shows also the following points. First, the effect of ky is a maximum when kh is around 0.2 and decreases as kh further increases. Second, the ky-effect is smaller when the backfill is sloped than when it is horizontal. Third, the

167 downward inertia force ay (ky > 0) tends to increase the K AE-value while the upward action ay (ky < 0) tends to decrease the K AE-value. It is of interest to note that if the resultant seismic acceleration, a = (a~ + ai'ii, is considered and is assumed to act horizontally, the difference between the K AEvalue so obtained and that obtained by optimization with respect to (j = tan - I (ky/kh ) will even have a less value than that shown in Fig. 5.16. This is due partly to the fact that the most critical surfaces are practically the same for both cases. Also, there is no change in the magnitude of the resultant acceleration. Figure 5.17 shows how the normalized KAE-value, and ~O vary with (j for an earthquake of different magnitude. Here ~O is defined as: (5.26)

The ~o has a unique maximum in the whole range of (j. It was found that as k increases, the ~o-value becomes smaller. The difference between (KAE)O *0 and (KAE)O = 0 becomes larger. However, the maximum ~o-values, (~O)cr' which are of primary interest, are not much different from each other. They are all very close to one. Also they all occur at nearly the same (j-angle for a given soil- wall system. In most cases, (KAE)o = 0 can be taken as (KAE)O=O' For practical purposes, the vertical acceleration, ay, can be neglected if the actual acceleration is nearly horizontal. Otherwise, the resultant acceleration can be used

1.2 1.0

-90·

1./

O. 180· 8~-180 Kg 90.

,p:

0.4f::::;:===::::::::::;'---

a =90·, {3 =0· =40",8=20·

('i9)cr = 1.012

0.2

1.015 1.006

for K= 0.10

0.20 030

OL~_~_-,-~~~-~-~~=-~-~:--~c:-

-180·

-120·

-60·

O·

Direction of Seismic Acceleration

Fig. 5.16. Effect of yertical seismic acceleration component on normalized KAE-yalues.

60· t

120·

180·

B

Fig. 5.17. Variation of normalized K AE-yalues with direction of seismic acceleration for earthquakes of different magnitude.

169

168 and assumed to act horizontally «(J = 0) without much sacrifice in the accuracy of the determination of KAE-values. For the passive case, the effect of the direction of seismic acceleration on the KpE-values are partly shown in Figs. 5.18. Unlike the active case, the KpE-value corresponding to the optimized (J-angle, (J cr' is considerably smaller than that corresponding to (J = O. This is due to the fact that the (Jcr-values are much larger than zero in all the cases investigated. In fact, (Jcr is close to 90° when the jJ-value is high. In this case, the vertical acceleration, a y , may play even a more important role than the horizontal component, ah' does. Figure 5.19 shows the effect of ay on the KpE-values. It is clear that the effect becomes larger as both the k h and k y values become higher. For the case of kh = 0.3 and k y =0 t k h , the effect is in the order of 25070. The ky-effect therefore cannot be simply neglected for the passive case. Figure 5.19 also shows that the 1I~ -value becomes smaller as jJ becomes larger, where the 1I~.y is defined as: y

This is probably because, as jJ increases, the (KpE)k = 0 increases in a faster rate than the (KpE)k * 0 does, even though (Jcr becomes larger as those shown in Fig. 5.18.

v

Similar to the active case, if the resultant seismic acceleration is adopted and assumed to act horizontally, the difference between (KpE)e = 0 and (Kp0e = e becomes smaller. However, the difference is still too significant to be neglected i~ practice. Figure 5.20 shows the variation of the normalized KpE-value, and ~~ with (J for different levels of earthquake. Here ~~ is defined as: (KpE)e

*0

(KpE)e

= 0

(5.28)

1.300r----.----.---~-.------,

K,

1.250

(KpE)ky

*0

(KpE)k y

= 0

(5.27)

1.200

K,=iKhl ep = 40°.8 =20°

100 (a)

(b)

f.?1

• 40· 8 • 12 " • 0.20

-8/

..

:r

60

PeE 8

W //

a

""

-I'" ,

r

H

2

1.100

80

/

__ 8=0·

60

___ 8= 8..

>-

K pE = PPE I~

100r------.----,..----~--___,

lH

(8=-26.565°

a =90°

1.150

80

= %Kh I

(8=-33.691°)

__ 8=0· ___ 8 = 8"

..

et "

;;

40

40

---

a. ~gOO

_----7{)0-0

a. =90

_----SOD

(1=70°

----

------

o 8cr= 63°

_-----83° 72·

15· Slope Angle of Backfill

, /3

20·

~~

Kg

_---

k,gt~ __ --820

.----a=70o _----68°

0.850 K,

_---e2°

',K, = ~ Kh

~ = K sin e

K,=iKht (8 =26.565°)

0.800

B~~_---72°

K.

0.75 0 '-10· Slope Angle of Backfill •

15·

20·

/3

o

--J.

0.1

t

(8 = 18.435°)

0.2

Horizontal Seismic Coefficient.

=~

kh t

-"-(_8_=3_3_.6_9_1°--1)

-'--

0.3

0.4

kh

Fig. 5.18. Effect of direction of seismic acceleration on KpE-yalues for general soil-wall systems: (a) 0

=

12, (b) 0

=

t .

Fig. 5.19. Effect of vertical seismic acceleration component on normalized KpE-yalues.

171

170 It is found that, similar to the active case, there are unique minimum ~~ values, (~~)cr for different k-values for a given soil- wall system. They all occur at nearly the same O-angle. However, the (~~)cr values are quite different from one another and are all less than one. That is, in all cases, (KpE)e = e is less than (KpE)e = o' In actual practice, unless the seismic acceleration is nee~rly horizontal, the K pE value should be assessed by optimization with respect to O. This is of great importance especially when the retaining structures of concern are located close to potential epicenters where the vertical· component of the seismic acceleration may be larger than the horizontal component. If this fact is not taken into consideration, use of (KpE)e = 0 will give unsafe designs. It is noted from Figs. 5.15 and 5.18 that the Ocr-value are almost unaffected by the change in a-values for both the active and the passive cases.

Active case - surcharge effect The KAE-value for the case when there is a surcharge q, (KAE)q *- 0' is given by Eq. (5.25) as:

(5.29)

and (KAE)q = 0

K =0.30

.

-i7 K =0.20

1> x

~

1.2 K=O.IO

~

" -'"

'R' 1.0 0.8

(~eO)er = 0.940 0.873 0.798

for K=O.IO 0.20 0.30

Direction of Seismic Acceleration ,

n

~et>

.1. -

'I' -

(5.30)

.1.

'I'er

eer =56°-58"

[NA-ylQ

we have:

= Qet>

f

=

fer (5.31)

Note that if NA-y and N Aq as given in Eqs. (5.21) and (5.22) are substituted into Eq. (5.31), the resulting equation for l1 q is independent of a, if we use the samecritical_ sliding surface for both N A-y and N Aq calculations. In general, l1 q is a function of ¢, a, {3, k, and q/'YH. The effect of surcharge on the normalized (KA~q *- 0 value is shown in Fig. 5.21. In general, 7/ q increases linearly with q/'YH. The rate of increase is larger as a and {3 get higher. Figure 5.22 shows that the l1 q is totally independent of afor the case (ex = 90 0 , (3 = ¢12) investigated. The unique critical surface as found is responsible for this finding. The l1q-value, although it increases slightly as ¢ increases, can be considered as independent of the ¢-angle for practical purposes. Figure 5.23 indicates that the magnitude of an earthquake has no effect on the 7/ q for the particular case (a = 90 0 , (3 = 20°, cP = 40 0 , 0 = 20 0 ) investigated. This is because that when the critical sliding surface is planar, NA/NA-y = sina/sin(a + (3) and Eq. (5.31) becomes:

8

l1 q Fig. 5.20. Variation of normalized KpE-values with direction of seismic acceleration for earthquakes of different magnitude.

= 0'

N + -2qNAq ] _ _ [ A -y 'Y H e - eet> f - fer

Quite often, a soil- wall system is subjected to a surcharge and the backfill may be cohesive. The presence of surcharge and/or cohesion may influence the lateral earth pressures considerably. It is therefore worthwhile to investigate how the lateral earth pressures are affected by the surcharge and the cohesion in the backfill. We shall first check the surcharge effect in the active earth pressure case.

:::'"

[NA -yl,.,~ --

If (KAE)q *- 0 is normalized with respect to (KAE)q

5.5.3 Surcharge and cohesion effects

o

=

2q sina 1 + 'YH sin(a + (3)

'* f(k

h)

(5.32)

173

172 3.5r--~~-~--~--~--,

30

HI

wtr

y • 120 Iblfl' ·40·, 8 ·20· H • lOft , K .0.20 c .0

'.lE

KAE•

~E I~Y H2

o :,. 2.5 _ _Q

1

=60 0

Active case - cohesion effect From Eq. (5.25), the KAE,value for the case when there is a cohesive c in the backfill, (KAE)c,* 0' can be expressed as:

90·

"-

In general, it may vary slightly with the kh-value when the critical surface is not exactly planar. Summarizing Fig. 5.21 through Fig. 5.23, it is interesting to note that 1J q is essentially a function of a and (3 only. This is consistent with Eq. (5.32), since for the active earth pressure case, the most critical sliding surface is often almost planar. We shall now check the cohesion effect in the active earth pressure case.

~

120·

1i 2.0

{3.

'"

1.5

(5.33) for

a

=60° a 90°

1.0""0==--::"0.:-1--o:::'-.2=---:oc'::.3:-----:Q~4--0---.J5

and

q/yH

(5.34) Fig. 5.21. Effect of surcharge on normalized KAE-values for soil-wall systems of different geometry.

The normalized (KAE)c

'* o,value, 1Jc' can be expressed as:

2.2

2.2

{3 2. 0 ~

'"S

~ 1.8 "-

~

~

~/.

- '30·

~

~

k g,/

H I U ; . 1 2 0 Ib/lt' '/ 8 / a'90·, {3'12 'i.E a H = lOft, Kh =0.20, //. C' 0 KAE = PAE'rr H2 . 1(/

1.6

r;.

g

50·

.~

KAE '" P"'E / tyH

H '" 10ft t c=o

2

::::'""

?t

~

for 011

for all Ith-values

8 - values

!'" 1.4

0

14

a

0

-.: 1.8 ~ -;: ),6

./

40·

y' 120 Ibllt' '40·,8. 20· a =90 o ,f3= 20°

2.0

ff

~

1.2

12

0.1

0.2

0.3

04

0.5

q/yH

Fig. 5.22. Effect of surcharge on normalized KAE-va!ues for soils of different compaction and interfaces of different roughness.

0.1

0.2

0.3

0.4

0.5

q/y H

Fig. 5.23. Effect of surcharge on normalized KAE-va!ues for soil-wall systems subjected to earthquakes of different magnitude.

174

'l/ c

175

(KAE)c

*0

(KAE)c = 0

05

8= 0' 12

(5.35)

i

ip

'i' 0

l!>

Note that N Ac is independent of the seismic coefficient, k, as shown in Eq. (5.23). The A-value in N Ac can be taken as A = cos cP cv ' if cP ;;::; ¢cv or as A = cos ¢, if ¢ < ¢cv' A typical value of Ais 0.836. This corresponds to a ¢-angle at critical state, cP cv = 33.3°, which is typical for most siliceous sands. Figure 5.24 shows that 'l/ c decreases as c/'YH increases, since the cohesion has a negative effect on the active earth pressure. The rate of decrease in 'l/ c becomes larger as ex increases. However, as (3 = ¢, the cohesion effect is practically constant when c/'YH ;;::; 0.10. This is probably because as (3 = ¢, the cP-angle rather than the c-parameter predominantly controls the active earth pressure. Figure 5.25 shows that 'l/ c is essentially independent of 0 and is slightly affected by the ¢-angle. However, for practical purposes, the 'l/c-value can be treated as independent of both 0 and cPo

)

..... ~

40'

0.836

-0.5

~

Y =120 Iblft' a '90',f3 = 4>12

-1.0

H = lOft. ",,'0.20

_ _~_ _~_ _~_---.l 0.10 0.15 0.20 0.25 ely H

-1.5L_~

o

0.05

Fig. 5.25. Effect of cohesion on normalized KAE-va!ues for soils of different compaction and interfaces of different roughness.

1.0

0.5

'i'

-"

~

;

.;"'

1

l

"-

~

~ -0.50

~

" #-

-1.00

0

-05

-1.50

__ __ __ 0.10 0.15 0.20

1.5L_~

-2.00Lo--o::':f)-=-5----,-o.'::Io--o~.I5=----=-0.~20,------0---.l.25 clyH

Fig. 5.24. Effect of cohesion on normalized KAE-values for soil-wall systems of different geometry.

o

~

0.05

~

~_---.l

0.25

clyH

Fig. 5.26. Effect of cohesion on normalized KAE-va!ues for soil-wall systems subjected to earthquakes of different magnitude.

177

176 The 7J c versus c/"(H curves for different kh-values as shown in Fig. 5.26 reflect that the 7Jc-value is seriously affected by the earthquake magnitude. This is because in Eq. (5.35), N Ac is totally independent of k, while N A-y is dependent on k. It seems that the cohesion has a larger effect on the K AE-value when there is no earthquake (k = 0) than when there is an earthquake (k > 0). The larger the earthquake is, the less the cohesion affects the active earth pressure for a given c/"(H value. In summary, the 7Jc-value can be considered as practically independent of the r/Jand a-angles only. We shall now examine the surcharge effect in the passive earth pressure case. Passive case - surcharge effect Similar to the active case, the normalized (KpE)q expressed as: , 7J q

(KpE)q

'* 0

(KpE)q

=

'* 0 values, or 7J~-value, can be

(5.36)

0

In general 1J~ is dependent on r/J, a, {3, k and q!"(H as well as critical sliding surface is seldom the same for the cases (KPE)q unique in the passive case.

a,

Passive case - cohesion effect Finally, for the passive case, the cohesion effect can be investigated by calculating the normalized (KpE)c 0 or 7J~, for different soil- wall conditions and earthquake conditions. Note that, similar to the active case, 7J~ can be expressed as:

'*

since the most

'* 0 and (KPE)q = 0

2.2r---~--~--------"

)'=

Figure 5.27 shows that 7J~ increases linearly with q/"(H. The rate of increase is larger as a and {3 becomes smaller. This is contrary to the active case. The effect of surcharge OJ;! ~he 1J~-values for soils of different compaction and interfaces of different roughness is shown in Fig. 5.28. It seems that the 1J~-value is dependent on both r/J and a. The rate of increase in 1J~ with q/"(H, or the effect of surcharge on the 1J~-value, is larger as r/J and a get smaller. The surcharge therefore has larger effect on the KpE-value for the case of looser soil and smaller a-value than for the case of denser soil and larger a-value. Figure 5.29 shows that the kh-value, or the magnitude of an earthquake, has little effect on the 1J~-value. This is because the most critical sliding surface, in contrast to the active case, is generally far from being planar. However, the difference in 1J~ for different k h-values is practically negligible.

120 Ibltt'

¢= 40'.8. 20' H = lOft.". =0.20

c=o

, 7J c

(KpE)c

'* 0

(KpE)c =

(5.37)

0

The effect of cohesion on the 7J~-values for different soil - wall conditions is shown in Figs. 5.30 and 5.31. It is obvious that the 1J~-value depends on the geometry of the wall and backfill, or the angles a and {3. It also depends on the strength factors r/J and a. In general, the 7J~-value increases linearly as c/"(H increases for a given soil - wall system. As a and {3 decrease, the 7J~-value increases. For given a and {3 values, the 7J~-value increases as r/J and adecrease. The cohesion effect is more obvious in the looser soil than in the denser soil. Figures 5.32 shows the effect of cohesion on the 1J~-value for earthquake of different magnitude. It is clear that the 1J~-values, although affected by the kh-value, can be considered as practically independent of the kh-value, which is different from that for the active case. 5.5.4 Seismic effects on potential sliding surface

0.2 q

0.3

0.4

05

I)'H

Fig. 5.27. Effect of surcharge on normalized KpE-va]ues for soil-wall systems of different geometry.

The seismic acceleration generated by earthquakes not only imposes extra loading to a soil mass but also shifts the sliding surface to less favorable positions. Consequently, in addition to the change in the lateral earth pressures, the most critical or potential sliding surface is also altered.

179

178 2.2

'1' ·120 Ib/ll' a' 90' ,{3 '4>12 H -10ft, " '0.20 c •0

2.2

KpE

2.0

= PPE

/~y

H

'"

2

~

'" :::: "

,

0

:::: '"

1.8

'"

--""30'

~

'"

40'

~ ...!!

l,

H·IOll , " '0.20 1.8

__ a '60'

0

a

~

r '120. IbllI' cP' 35' ,8 '17.5'

2.0 0

50'

90'

1.6

120'

~

1.6

'a

t:'"

1.4

1.2 c/yH I.O~:::::"_~

o

0.1

_ _~_ _~_ _~_----' 0.5 0.4 0.2 0.3 q/'1'H

Fig. 5.30. Effect of cohesion on normalized KpE-va!ues for soil-wall systems of different geometry.

Fig. 5.28. Effect of surcharge on normalized K PE-values for soils of different compaction and interfaces of different roughness. 2!1r--------~---,---,

r ·120 Ib/ll' a' 90',.8' cP/2 H'IOll,'h' O.20

2.2 '1"120 Ib/ll' "', 40' ,8 • 20· a' 90",.8.20' H = 10ft t C .. 0

2.0

'ii'

i

1.8

KpE' PpE li'1'H

~ 2.0

-"

'" ~

i'"

2

1 =.

'"

l '1

'h '0.30 0.20 0.10 0

1.6

!!

"

~

1.8

_ _ '" • 30' , X • 0.866 35'

0.836

40'

0.836

1.6

1.4

1.4

1.2

0.1

0.2

03

0.4

0.5

q I'1'H

Fig. 5.29. Effect of surcharge on normalized KpE-values for soil-wall systems subjected to earthquakes of different magnitude.

005

0.10

0.15

0.20

025

clr H

Fig. 5.31. Effect of cohesion on normalized K PE-values for soils of different compaction and interfaces of different roughness.

181

180

5.5.5 General remarks

2.2

The upper-bound technique of limit analysis of perfect plasticity is applied to determine the seismic laterai earth pressures in a quasi-static manner. Here, as with most limit equilibrium methods, the present analysis gives no information on the point of action of the resultant seismic lateral pressures. This point will be further discussed in the following chapter. The magnitude of the lateral pressures as calculated by the present method is, however, fairly reasonable for the case of translation wall movement.

y = 120lb/ft = 35·. 8= 17.5· a = 90·. {3= 17.5· H = lOft

2.0

_ . _ Kh =0.30

0.20 0.10

o '~ 1.4

5.6 Earth pressure tables for practical use 1.2

1.0~_-..L-_ _~_~-=--_."..,..,_---l.

o

0.05

0.10 0.15 c/yH

0.20

0.25

Fig. 5.32. Effect of cohesion on normalized KpE-values for soil-wall systems subjected to earthquakes of different magnitude.

Figures 5.7 and 5.8 show some typical changes in the potential sliding surface as the result of an earthquake with k = 0.20. It is interesting to note that the potential sliding surface becomes more extended when earthquake presents, especially in the active case. This conforms with the experimental results of Murphy (1960). For the passive case, it seems that the change in the potential sliding surface is not as much as that in the active case, although it is also found more extended in the earthquake case. The change in the potential sliding surface as the result of earthquake has also been noted by Sabzevari and Ghahramani (1974). The prediction of the potential sliding surface is of iIp.portance when it is necessary to back calculate strength parameters from field test or actual failures. It is also influential in some geotechnical designs, such as in the design of bulkhead anchorages and earth anchors. Although, as was pointed out by Chang and Chen (1982), the potential sliding surface as reflected by the limit analysis is not representative of the actual failure surface, the results as shown in Fig. 5.7, especially deserve special attention. In aseismatic design of most anchorage systems, not only the change in the lateral earth pressures has to be aware, but also the change in the potential sliding surface has to be carefully considered. The retaining structures have to be anchored well outside the potential sliding surface, which is more extended in the case of earthquake, for the safe design of these structures.

While earth pressure tables and charts suitable for the static design of retaining walls are generally available, tables or charts for seismic lateral earth pressures are scarce. Seismic active and passive earth pressures are required in a seismic design of retaining walls subjected to earthquake forces. Development of seismic earth pressure tables is of practical value for retaining wall design in earthquake environments. The upper-bound limit analysis of active and passive earth pressures as developed in this chapter is used to generate earth pressure tables that provide both static and seismic active and passive earth pressures. The effect of earthquake forces is taken into account in a quasi-static manner using the seismic coefficient concept. Dimensionless coefficients of active and passive earth pressures are presented in the forthcoming. To reduce the number of tables and charts, we generate this information under the following conditions: 1. The seismic acceleration acts in the horizontal direction (fJ = 0°). 2. There is no surcharge acting on the surface of the backfill (q = 0). 3. There is no cohesion in the backfill material (c = 0). The coefficients of the generated active and passive earth pressures are obtained for the following cases: 1. -parameter, = 20°,25°,30°,35°,40°,45°,50°. 2. Seismic coefficient (acts horizontally). k = 0, 0.05, 0.10, 0.15, 0.20, 0.25, 0.30. 3. Angle of wall repose, ex = 60°, 75°, 90°, 105°, 120°. 4. Slope of backfill (normalized with respect to <1». {3/ = 0, t, t, !, t, i, 1. 5. Angle of wall friction (normalized with respect to <1». O/cP = 0, t, h f, 1. The tables generated are listed in Appendix A developed originally by Chang (1981). For the tables to be of practical value, procedures were developed for extending the present listed earth pressure coefficients to cases other than those specified before. They are given in the following subsections, following the work of Chang (1981).

183

182 5.6.1 Correction for direction of seismic acceleration

thermore, there is a general tendency that the 71o-value increases slightly as a, {3 and

odecrease. For practical purposes it is recommended that no correction is required In general, the direction of the resultant seismic acceleration during a given earthquake is not necessarily horizontal. The seismic acceleration may assume any direction depending on the characteristic of an earthquake and the distance of the structure from the epicenter. For this reason, it should be assumed that the seismic acceleration in an actual lateral pressure analysis for design purposes can act in any direction. The lateral earth pressures corresponding to the most critical direction of seismic acceleration are then used for actual design. To develop a way of correlating the seismic active and passive earth pressure coefficients for the special case of horizontal acceleration (IJ = 0), (KAE)O = 0 and (KpE)o = 0' to those corresponding to the optimized acceleration direction (0 = Ocr)' (KAE)O = 0 and (KpE)o = 0 ' some sensitivity analyses were performed. The ratio of (KAE)Oc~ 0 to (KAE)O : 0 is denoted as 710' and that of (KpE)o = 0 to (KPE)O = 0 as 71~' cr cr Some typical results of sensitivity analyses for the active earth pressure case are shown in Figs. 5.33 and 5.34. It is clear that the 710-value is sensitive not only to the geometrical factors a and {3, but also to the strength factors ¢ and o. Further, as pointed out previously, the 71o-value is also a function of the seismic coefficient k. Development of simple correction factors for this case is therefore of great difficulty. Fortunately, the 71o-value is generally in the order of no more than 1.015. In many cases, the 710-values are very close to unity. This is because the critical acceleration angle 8cr does not deviate much from zero, as shown in Fig. 5.15. Fur-

0

(710

=

1) unless a

< 90°,

{3 =:;; 4>12, and 0 =:;; ¢12, then 710

5.33).

. . .

.

1.01 can be used (Fig.

In reality, the angle of wall friction, 0, is seldom less than ¢12, the active earth pressure coefficients as listed in the tables can be used for practical purposes without corrections in most cases. The results of sensitivity analyses for the passive earth pressure case are presented 1.010 0 I

k = 0.20

_Cb

/31¢ = 0.50. 81¢ = 0.50

w

«

:.::

-~

_Cb•

¢ = 45° 40° 35° 30° 25°

1.005

w

«

:.::

•Cb

s::--

0 60

75

90

Angle of Wall Repose,

Fig. 5.34. Sensitivity of '16 to changes in

1.02

C/

and

105

a,

120

degree

q,.

0.90

I

0

k = 0.20

_Cb

_Cb•

q, = 40 ° • /31 q, = 0.50

w

«

w n.

:.::

:.:: :;

Cb I _Cb

=

'" N & /31t/> -, = 1/4

1.01

PIt/> = 1/2 /31q,=3/4

w

« :.::

•Cb

r::-

0 0

0.25

0.5

1

---- ---0.75

Normalized Angle of Wall Friction, 81t/>

Fig. 5.33. Sensitivity of '16 to changes in

C/,

fI and

o.

~ I

a= 90° 81'" = 0.5

0.80

&

a = 120°

_Cb

l!::.

105° 90° 75° 60°

w n. :.::

•

~

-.;;:

t/>= 25° 30° 35° 40° ..... 45°

a

= 90°

{31 t/> = 0.5

50

k = 0.20

¢ =40°. 81¢ = 0.50

0.70 1.0

0

0.5

0.25

Normalized Slope of Backfill. /31

Fig. 5.35. Sensitivity of '1; to changes in

C/

and fl.

0.75

if>

1.0

184

185

in Figs. 5.35 and 5,36. It is found that the l1~-value decreases as (3 and ¢ increase. In general, 1]~ is not very sensitive to the variation in ex and O/¢ as long as ol > j or and ex 2= 90° as shown in the figures. Hence, for developing the correlation factors, ex and O/¢ can be kept constant. The fact that the l1~-value varies with ex and O/¢ when 0 :s; j and ex < 90° can then be taken care by a modification factor. Based on the facts that the 1]~-value decreases as the seismic coefficient k increases, and that 11~ varies with (31 and ex as shown in Fig. 5.35, the correction factors for estimating (KpE)o = 0 r from (KpE)o = 0 have been developed. They are summarized in Figs. B5.1 to BS .6 in Appendix B for practical use. To take care of the variation when ex < 90° and O/ :s; j, a modification factor

or the soil - wall interface is fairly smooth (0 < ¢12). For practical purpose, fl.~ can be taken as 0.97 if ex < 90° and 0 :s; j ¢, and 1.0 for other cases. The fact that the' critical acc.eleration angle ecr for passive earth pressure cases is much different from zero on the horizontal direction, and in many cases close to 90°, is probably the reason for the large difference between (KpE)o = Ocr and (KpE)o = o' The 1]~-value can be as low as 0.60 when the -angle and the seismic coefficient are high. The correction factors as shown in Figs. B5.1 to B5.7 should, therefore, be applied when the seismic acceleration of an earthquake force may assume any direction other than the horizontal.

5.6.2 Correction jor the presence oj surcharge as shown in Fig. B5.7, Appendix B, is provided. With this information, the KpE-value corresponding to e = ecr' (KpE)o = ocr can be estimated from the KpE-values listed in the earth pressure tables, (KpE)o = 0' by the following relation: (5.38) where 1]~ is obtained from Figs. B5.1 to B5.6 and fl.~ is given in Fig. B5.7. In fact, as it can be seen from Fig. B5.7, fl.~ is generally greater than 0.97, unless ex < 75°

A uniform surcharge q acting on the surface of the backfill of a soil - wall system will ind~ce additional loading on the wall. The consequence of its presence is to cause both the active and the passive earth pressures to increase. The coefficients of lateral earth pressures should therefore be corrected if q =1= O. An attempt has been made to find a correlation between the K AE and K pE corresponding to q = 0 case, (KAE)q = 0 and (KpE)q = 0' and those corresponding to q =1= 0 case, (KAE)q 0 and (KpE)q 0' based on an optimization with respect to the whole expressions for the lateral earth pressures as shown below:

*

*

(5.39) 0.85

(5.40)

k = 20 0 I

9'> = 40°

0.84

~Cb

w

ll.

0.83

l.::

a=900

<; Cb

--

I

PI9'>

= 0.50

:as=-----------'Q!"!:!:~

---:=.---- - - - - - - - -

0.82

*

,,1f":75°

I

~Cb

w

0.81

ll.

l.::

,

1::-Cb

0.80

0.79 0

0.25

0.5

Normalized Angle of Wall Friction, Fig. 5.36. Sensitivity of '1~ to changes in ex and D.

0.75

SI

rP

where NAy' N Aq , N Ac ' N py ' N pq , and N pc are earth pressure factors, 'Y, is the unit weight of the backfill, and H is the height of the retaining wall. It was found that there is no simple general correlation between (KAE)q = 0 and (KAE)q * 0 and between (KpE)q = 0 and ~KpE)q o' This is probably because. the ratio l1 q = (KAE)q * o/(KAfi)q = 0 and l1 q = (KpE)q * o/(KpE)q = 0 are functlOns of all variables involved, although it was found practically independent of the seismic coefficient k. A study has been conducted to see if the K AE and K pE can be practically approximated by an optimization with respect to each individual term in the expressions for K AE and K pE . The corresponding approximate values can be expressed as:

1.0

(5.41)

187

186

KpE =

Min [Np

-y

1+

(5.42)

2q Min [N pq ] + 2C Min [NpcJ 'YH

",H

The results of this study are shown in Figs. 5.37 and 5.38 for the case of zero cohesion (c = 0). It is interesting to see that for both the active and the passive cases, the 71 -value and 71' -value as obtained by Eqs. (5.39) to (5.42) differ <:nly slightly. For t&e active case; the approximation gives higher KAE-values, i.e. K AE > K AE • This is because individual optimization will allow each component to take its most critical failure surface while overall optimization will require every component to assume a compromised failure surface. For the same reasoning as the active case, individual optimization gives lower

KpE-value than overall optimization does. ~his is clearly shown in Fig. _5.3~. Fortunately, the difference between KAE and K AE and between K pE and K PE IS very small. The zero or slight change in the most critical sliding surface between the two cases of optimization, as noted in Figs. 5.37 and 5.38 is mainly responsible for this consequence. Furthermore, use of KAE and KpE instead of K AE and K pE for design purpose is on the safe side. Therefore, the active and passive earth pressures obtained based on optimization with respect to the individual components are practically acceptable at least for the case of c = O. Based on these findings, correlations between earth pressure factors N Aq , N pq , and N A-y' Np-y are of interest. In fact, the earth pressure tables give directly the N A-y and Np-y-values, since from Eqs. (5.39) and (5.40), K AE = NA-y and K pE = Np-y 2.0_---....----.,...----,----,------,

3.5~-----r----,----.------r----'

y=120 Ib/ft

o

4>=40 c=o

~3.0

1&1

......

o

K

AE

2 =P.AE 1-21- vH I

"

-.'" 1.8

,

1&1 IL

1&1


o

_ _ Max (N

Max

Aq

~

o iI-

-.'"

<}

>

... o

2q

(N.4y+ yH

0

{3=20 ,k h =0.20

:ll: ......

11-

-."'. 2.5

y=120 Ib/ft 3 0 t/> =400 , 8 =20 C =0 , H=IO'

o 0

0

8=20 ,H=IO' 0 {3=20 , Kh=0.20

-.

ct ~

3

1.6

o II

NAq>

-.'" ~

II

:=......

-."'2.0 1&1

1.4

o

ct

11-

:=......

"nI'"

o

IL

:ll:

1t-

~

~

II

-i!

~ 1.5

1.2

~

0.1

0.2

0.3

0.4

0.5

q/yH Fig. 5.37. Difference in 'lq by optimization with respect to N Aq and NAy + (2q/-yH)NAq·

0.1

0.3

0.2

0.4

0.5

q lyH Fig. 5.38. Difference in 'l~ by optimization with respect to N pq and N py + (2ql-yH)Npq.

189

188

for the case of q = c = O. Hence if the correlations between NAy and N Aq and between N p and N pq can be established, K AE and K pE can be estimated based on Eqs. (5.41) and (5.42) with NA'Y and Np'Y obtainable from the earth pressure tabl~s. The correlations between N A and N Aq , and Np'Y and N pq are completely llldependent of the q/ 'YH-value. In'Y the special case when the most critical sliding surface is planar, it can be proved that the ratios a q = N Aq/NAy and a~ = NplNp'Y can be expressed in the simple form: sina sin(a + (3)

(5.43)

where a is the angle of wall repose and (3 is the slope angle of the backfill. The O!qand a~-values are dependent on the geometry of the soil - wall system only. Hence, if the most critical sliding surface is practically planar, Fig. 5.39 as developed based on Eq. (5.43) can then be used for obtaining the correlation factors a q and a~ for the active and passive cases, respectively. In general, the sliding surface is not planar, especially for the passive case, the a q - and a~-values will no longer depend only on a and (3. Some sensitivity studies have been conducted to investigate the effect of several variables involved. The results are shown in Figs. 5.40 to 5.45. It is found the a q is practically independent of the seismic coefficient k and so does the a~-value (Figs. 5.40 and 5.41). For a 2.0..-------.-----..-----.,-------.,

5-----....---.......- - - , - - - - - - - , - - - - ,

, I

4

t

J

• •.I ./ '.

. ., .. , /. / .

:": ~~. '" ~Possive .. //-:·~Active '/.

.

{3=1/J

~

« ~ 1.01t=========~i=======t========~

-

-:t

II.,.

a=90· 1/J=40·,8=1/J12

a

a q =a

/2

o

.,.

sina = ----sin (a + {3)

q

OL-

--.1

o

0.1

-'-

....J'l.--

.....J

02

0.3

0.4

Horizontal Seismic Coefficient, k h Fig. 5.40. Sensitivity of

"'q

to changes in k h •

"CI C

Cl

1.0.------y-----.------y--------,

2

1

z~ .......,. .po II .,.

{3=0 ~

;-

0.8

!---------------

1/J/2

.......,. 0.6

a

Do Z

.11.,.

0.41-

a

0.21-

0L._ _..L_ _---JL-_ _....l-_ _--1_ _- - - ' o 10 20 30 40 50 Slope Angle of Backfill 1 {3 ,degree Fig. 5.39. Correlation factor

"'q =

"'~ for active and passive planar failures.

OL-

o

a=90 • 1/J=40·.8=1/J/2 L-

0.1

----JL....-

0.2

L-

0.3

Horizontal Seismic Coefficient, k h Fig. 5.41. Sensitivity of "'~ to changes in k h .

---J

0.4

190

191

essentially independent of the strength parameter, ¢, and the soil- wall interface friction angle, o. For the passive case, the Q!~-value is, in general, dependent on Q!, {3, 4J and o. Based on these sensitivity studies, charts for the correlation factor Q!q similar to that shown in Fig. 5.42 have been developed. They are shown in Figs. B5.8 - B5.l3 in Appendix B for practical use. The Q!~-value is found varying erratically with {3. They were plotted in the form shown in Figs. B5.14 - B5.19 for design purposes. With the availability of these charts for the correlation factors Q!q and Q!~, the active and passive earth pressure coefficients for the case of q =1= 0 and c = 0 can be

easily calculated using the following equations modified from Eqs. (5.41) and (5.42): (5.44)

(5.45)

2.0r-------r-------r-------r-----~

2.o',....----..,..-----,-----,------..------.

~

======J !... 1.2f=:::::::~;~;=====:9~0~0 t z

0

o.e

60

II

o

90

~

1.6

~ :

z

1.0

0

60

0

I--------b-------=~------~

...

"

for all 8-values

o

¢=40°. 8o=¢/2

0,4

O......-----'------'-------L.-------'

Kh=0.20 0.20

30

0040

0.60

o.eo

Normalized Slope Angle, f3/¢ Fig. 5.42. Sensitivity of

O
to changes in

0<

35

40

Angle of Internal Friction. ¢

1.00

Fig. 5.44. Sensitivity of

O
to changes in <1>,

0<

45

50

• degree

and 0.

and (3.

8::: 0 I.°r-====C:::======::C======~;:-l 0

0.8 )...

:- 0.6 ..... ~

z

o"

.C'

0.2

¢ := 40 0 I(h

:=

,

8= ¢ /2

0.20

0'------'------''------..L.------1------l

o

0.20

0040

0.60

o.eo

1.00

0.4

0.2

IX=

90 .13= 0

o3-':"'0-:-------'3'-5-----4O..l------4LS-----5..J0 Angle of Internal Friction.

Normalized Slope Angle, f3/¢ Fig. 5.43. Sensitivity of O<~ to changes in

0<

and (3.

¢/2

I(h=0.20

Fig. 5.45. Sensitivity of

O
to changes in



and 0.

¢ • degree

193

192 where N Ar = (KAE)q = 0 and N pr = (KpE)q earth pressure tables in Appendix A.

= 0

can be directly obtained from the

5.6.3 Correction for presence of cohesion As noted previously, the contribution of cohesion to the lateral earth pressure is not gravity-related and is therefore independent of the earthquake forces. Thus, any attempt to correct the K AE- and KpE-values as the result of the presence of cohesion by considering the relations between (KAE)c '" 0 and (KAE)c = 0 and between (KpE)c '" 0 and (KpE)c = 0 will not be practical. Development of N Ac and N pc for various soil- wall conditions may be more useful, if, as in the case of q *- 0, Eqs. (5.39) and (5.40) are approximated by Eqs. (5.41) and (5.42) for the case of c *O. For this approximation to be valid, superposition must hold.

In developing earth pressure theories, most investigators tend to take for granted that superposition holds, regardless of the fact that the potential sliding surface in an overaIl optimization may. be guite different from those in an individual optimization (Prakash and Saran, 1966). Figures 5.46 and 5.47 show the difference in both the magnitude of K AE and K pE and the potential sliding surfaces, represented by ecr and ,pcr' between the two optimizations. It is found that for the passive case, both the magnitude in K pE and the mechanism of failure, denoted by r; and:t. 'f . "cr Y'cr 1 an average IS taken, do not differ considerably between the two cases. This is probably because the rather curved potential sliding surface as detected in the case of cohesionless backfill is not much different from that would be detected if the backfill were purely cohesive. The compromised sliding surface is therefore not much different from either one of the above two cases. 2.0r----,.----,-----.----......--------.

1.0,...;;::-----,,-----,-----,-------,,-------.

o

y=120 Ib/tt 3 "'=30°, 8=17.5° q=O ,H=IO'

1.8

/3= 17. 5°

, kh=O

0

II

_u

...

III

:.::

1.6

.....

Min. ( NPy+ :~ N pc ) Min. (N pc)

0 ij. _u

...

III

:.::

a=60°

1.4

Pcr=40", tcr=15°

II _ u

Pcr =2~ 0. "'cr =5°

~

1.2

_5.0'--

o

..L-

n05

-L-

-L.

--I._ _---..::.....J

0.10

0.15

0.20

0.25

C/yH Fig. 5.46. Difference in '1c by optimization with respect to N Ac and N A > + (2c/-yH) N Ac '

I """'--_-...J'--_ _---'L.-

o

0.05

L.-

0.10

0.15

L.-_ _..........I

0.20

0.25

C/yH Fig. 5.47. Difference in '1~ by optimization with respect to N pc and N p > + (2c/-yH) N pc '

194

195

For the- active case, it is clear from Fig. 5.46 that the compromised mechanism of failure for the case of an overall optimization differs considerably from that for the case of optimization with respect to the cohesion component, N Ac ' alone. The compromised potential sliding surface, as in the case of c = 0, is practically planar. Only when both the a-angle and the cI'YH-value are so high that the cohesion has a predominant influence on the resultant KAE-value, then the sliding surface changes to essentially the logarithmic spiral shape. The most critical sliding surface is found to be the logarithmic spiral shape in the case of optimization with respect to N Ac ' The difference in TIc = (KAE)c *- O/(KAE)c = 0 for the two cases of optimization can be considered in some situations. The fhidings as shown in Figs. 5.46 and 5.47 reflect that superposition is not always valid. That is, K AE and K pE as given by Eqs. (5.39) and (5.40), are not always equal to KAE and KpE as given by Eqs. (5.41) and (5.42) for the case of c o. For the approximation of Eqs. (5.41) and (5.42) to be practically valid, the NAc-value as obtained by individual optimization should be modified at least for a ~ 90° in the active earth pressure case. Since the N Ac- and Npc-values are totally independent of the seismic coefficient and the cI'YH-value, development of N Ac and N pc tables or charts is much simplified. Furthermore, the ratios a c = NAc/NAy and a~ = Npc/Np-y are found dependent. not only on the seismic coefficient, but also on the geometry of the soil- wall system and the strength parameters cP and o. The direct development of tables or charts for N Ac and N pc factors, instead of a c- and a~-values is therefore easier and of more practical value. Based on these considerations, tables showing. the N Ac- and Npc-values for various soil- wall conditions have been developed. In retaining wall design, materials as close to cohesionless as possible are generally selected for the backfill. Although the backfill can be slightly cohesive, the cP-angle is generally high. For this reason, the N Ac and N pc values for cP = 25° to 45° have been calculated. However, to include the case when c-value is appreciably high, the N Ac- and Npc-values for cP = 20° have also been generated. They are given in Appendix B for practical use. In preparing these tables, the ratio of soil- wall adhesion ca to c-parameter, A = c/c, is take as cos cP for cP < 33.3° and as 0.836 for cP ~ 33.3°. To account for the discrepancy in the NAc-value resulting from an individual optimization of N Ac alone, a modification factor!L c = [NAc]Max [K l[NAc]Max [N I is recommended. It is presented in a graphical form as shown in F1g. B5.20 in pendix B. In practice, however, a is seldom larger than 90° when the wall is built for retaining purpose. This modification is, therefore, not necessary in most cases. Against the background of this information, the seismic lateral earth pressures for the case of c*"O and q = 0 are approximated by the following equations modified from Eqs. (5.41) and (5.42):

*"

Ap-

(KAE)c

*0

(KpE)c

*0

:::::

:::::

2c

NA'y

+

Np'y

+ 'Y2c

'Y

(!L~Ac)

H

(5.46)

N

(5.47)

Pc

H

where N A'y = (KAE)c = 0 and Np'y pressure tables in Appendix A.

(KpE)c = 0 are obtained from the earth

5.6.4 Correlation for mixed effects from acceleration direction, surcharge and cohesion

In a general situation in which a given earthquake force of unknown direction of acceleration (8 0) is presented and, in addition, there are uniform surcharge and cohesion on a soil- wall system, a correlation for the mixed effect from these three components has to be considered. As far as the cohesion component is concerned, since it is independent of the magnitude of an earthquake, the superposition as shown in Eqs. (5.46) and (5.47) is unaffected by the direction of the seismic acceleration. Also, the cohesion component, which is not gravity-related, is not to be significantly affected by the presence of surcharge either. Hence, for the case of q 0 and c 0, the following equations, combined from Eqs. (5.44) and (5.45), and Eqs. (5.46) and (5.47) are valid:

*"

*"

*"

(5.48)

(KpE)q

* 0,

C

*0 ::::: NP'y (

1

+

2c 2 q ,) a q + 'Y N pc H H

'Y

*"

(5.49)

To account for the effect of 8 0 on the seismic lateral pressures when a uniform surcharge is presented, an investigation on how the presence of surcharge affects the correction factors TIe and Tlo has been conducted. It is found that for the sample case investigated, cP = 40°, 0 = 20°, k = 0.20, q/-yH = 0.10, the average TIe-value is 1.012, 1.007, and 1.001 respectively for a = 75°, 90° and 120° when {3 :::; cPl2. As {3 increases, the lJe-value becomes close to one. The lJe-values show practically no difference from the recommended values as given earlier. Hence, the correction factor TIe is practically unaffected by the presence of a surcharge. For the passive case, the Tlo-values obtained based on q/-yH = 0.10 are compared with the recommended !LoTIo-values, which are based on q/'YH = 0, as shown in Fig. 5.48. It is interesting to note that, here, as in the case of q = 0, the Tlo-value is practically the same for walls with different values of a-angle. The calculated Tlo-values for q/'YH = 0 are practically identical with the recommended !L~lJo-values shown

196

197

earlier. It can therefore be concluded that the 'Y/~- and JL~'Y/~-values as recommended for q = 0 case are equally valid for the case of q =1= O. Based on these findings, the following general equations are suggested for practical use for the case of 0 = Ocr' q =1= 0, and c =1= 0: (5.50)

0.90

...------,-------,--------r-------,

0.88

¢=40

•

8=20°

0.86

w

a. ll::

0.84

a a

w

a. 1\

}

= 90°

..:!..... = 0 10 YH .

a = 120°

ll::

-<:1)

0

= 75

0.82

f::'

0--0

Recommended

fl'e

7J ~

Based on ql YH = 0

0.80

1..-

.1-

o

5

....L.

....1.

15

10

Slope Angle of Backfill.

Note that: (a) 'Y/e can be taken as 1.0 for most cases, or as 1.01 if Ci :s; 90°, (3 :s; (j :s; ¢12; (b) 'Y/~ can be obtained from Figs. B5.1 to B5.6 and JL~ is given in Fig. B5.7; (c) Ci q and Ci~ can be obtained from Figs. B5.8 to B5.19; (d) JLc can be obtained from Fig. B5.20 and N Ac and N pc are given in Appendix B and (e) NA-y and Np-y are given in Appendix A. For special cases in which the c-value is appreciably high, the .p-value may be low. N Ac and Npc-values for.p = 20° are given in Appendix B. Since backfills with cvalues are seldom acceptable in most retaining wall design, correction factors needed in Eqs. (5.50) and (5.51) were not developed for the .p = 20° case. It is recommended that they can be estimated by extrapolation, or can be taken as those corresponding to .p = 25° for a slightly conservative design. ¢12 and

References

k = 0.20 0

(5.51)

---'

20

f3

Fig. 5.48. Recommended correlation factor p,~ ~~ as compared with calculated ~~-values based on q/-yH = 0.10.

Chang, M.F., 1981. Static and seismic lateral earth pressures on rigid retaining structures. Ph.D. Thesis, School of Civil Engineering, Purdue University, West Lafayette, IN, 465 pp. Chang, M.F. and Chen, W.F., 1982. Lateral earth pressures on rigid retaining walls subjected to earthquake forces. Solid Mechanics Archives, Vol. 7, Martinus Nijhoff Publishers, The Hague, The Netherlands, pp. 315-362. Chen, W.F., 1975. Limit Analysis and Soil Plasticity. Elsevier, Amsterdam, 630 pp. Chen, W.F. and Rosenfarb, J.L., 1973. Limit analysis solutions of earth pressure problems. Soils Found., 13(4): 45 - 60. Finn, W.D., 1967. Application of limit analysis in soil mechanics. Proc. J. Soil Mech. Found. Div., ASCE, 93(SM5): 101-120. Housner, G.W., 1974. Strong ground motion. In: R.L. Wiegel (Editor), Earthquake Engineering. Prentice-Hall, New York, NY, pp. 75 -91. Ishii, Y., Arai, H. and Tsuchida, H., 1960. Lateral earth pressure in an earthquake. Proc. 2nd World Conference on Earthquake Engineering, Tokyo, Vol. I, p. 211. Mononobe, N. and Matsuo, H., 1929. On the determination of earth pressures during earthquakes. Proc. World Engineering Conference, Vol. 9, p. 176. Murphy, V.A., 1960. The effect of ground characteristics on the aseismic design of structures. Proc. 2nd World Conf. on Earthquake Engineering, Tokyo, pp. 231- 247. Okabe, S., 1926. General theory of earth pressure. J. Jpn. Soc. Civ. Eng., Vol. 12, No. I. Okamoto, S., 1956. Bearing capacity of sandy soil and lateral earth pressure during earthquakes. Proc. 1st World Conf. on Earthquake Engineering, California, pp. 27 - I through 27 - 26. Prakash, S. and Saran, S., 1966. Static and dynamic earth pressures behind retaining walls. Proc. 3rd. Symp. on Earthquake Engineering, Roorkee, India, pp. 277 - 288. Sabzevari, A. and Ghahramani, A., 1974. Dynamic passive earth-pressure problem. J. Geotech. Eng. Div., ASCE, IOO(GTI): 15-30. Seed, H.B. and Whitman, R. V., 1970. Design of earth retaining structures for dynamic loads. Proc. ASCE Specialty Conf. on Lateral Stresses in the Ground and Design of Earth Retaining Structures, Ithaca, NY, pp. 103 -147.

I 198

199 Coefficients for active earth pressure K A for'" = 20° and k = 0.20,0.25,0.30, and
APPENDIX A: SEISMIC EARTH PRESSURE TABLES FOR K A AND K p (Chang, 1981)

.........................

Coefficients for active earth pressure K A for'" = 20° and k = 0,0.05,0.1 and 0.15

."

:.. 1.:.~~;~~ .. ~.:

,I.

1/6

61.

1"'3

......................... :.t•• .. ..

.

~.:

1"'2

.77

90.00

.60 .49

.B6

.S3

.63

.71

.51

.5'

ai $

:.~~;~~ ~.: :~;.:

2...3

'I' 'I.

5/1;

- - K A~lIALUES ---

75.00

:..!.:..;~:~~ ..~ . :...;;;.:

1.01 1.09

.S?

.56

.Ba

.74

1..3

1"6

0/$

1"2

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5"

.96

1.06

90.00

1.00

.67.73

105.00

90.ll0

120.00

.56

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60.00

.98

1.10

lll5.00

.55

.45

.59

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.63

.65

1.61

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3.39 2.86

.3333

2.45

1"3

1.4B

1.98

1.07

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18.47

60.00

13.63

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10.93 6.76

19.36

9.21

1.95

6.23

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1.74

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19.89

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1.01 .68

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.60

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InS. 00

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2.n5

6.69

B.97

105.00

3.41

120.00

.52

.57

.65

1.S8

5.97

B.19

120.00

2.78

60.00

1.04

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.75

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105.00

.37

.38

120.00

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.74

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SO. DO

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1.50

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1.20

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2.45

1.56

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2.77

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1.30

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.65

4.95

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3.59

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.'18

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1.09

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60.00

1.30

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1.62

7.36

1.55

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1.01

1.24

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1.03

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1.17

1.41

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3.01

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2.75

4.06

1.59

1.96

2.64

4.08

6.18

3a.29

1.36

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2.72

4.08

19.63

3.11

4.67

10.44

27.24

11.48

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25.41

10.68

1.36

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1.18

2.32

8.42

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12.79

7.09

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120.00

2.94

11.88

5.22

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2.00

3.22

15.31

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1.02

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1.12

1.34

75.00

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5.75

4.35

120.00

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1.02

1.16

1.44

2.42

11.23

60.00

7.B2

75.00

75.00

.65

.70

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.57

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120.00

.38

.41

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.6667

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1.04

75.00

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.53

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1.20

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1.51

75.00

1.76 1.59

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1.13 .69

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1.02

2.62

11.91

6.27

120.0n

2.38

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1.19

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1.47

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1.S9

1.(1000

60.00

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1.12

2.78

1l.!l9

2.20

3.53

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1.21

105. on

1.0000

3.!l7

75.00 90.0n

12.29 .63

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1.26

11.19 9.01

18.38

B.36

1.61

2.15

3.29

5.07

24.63

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15.95

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13.61 10.55

1.27

1.68

2.54

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1.04

1.37

1.75

13.76

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24.65

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8.33

105.00

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120.00

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2.76

2.47

..:..._

.

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1.75

2.35

3.6'1

.•:,,:

.

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5'. --- K -V~:"U£S - A

5.50

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1.51

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1.02

1.24

1.53

10.64

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13.88

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15.96

105.00

32.48

.40

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1.17

60.00

29.n3 1.42 2.38 13.75 10.09

.Go;

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75.00 1.61

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32.57

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9n.00

13.05

105.00

12.27

60.00

120.00 6.18

105.00

.75

120.00

.70

32.76 1.01

10.01

1.68

2.06

SO.OO

1.37

75.00

1.n3

1.24

1.50

90.00

.85

1.01

1.25

1.90

2.32

105. no

2.49

3.1B 19.73

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90.00

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.<19

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13.17 11.95

30.91

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1.70

2.51

7.23

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.
75.00

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16.33

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---1(;.. -VALUES - -

a

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.:••

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k.:•• :~; . : 1.08

1.59

1.02 2.95

2.~

- - K;.. -lJALUES -

K!\VAl..UES ---

10.04

11.07

1.12

105.00

11.32

24.18

1.0B

1,43 1.04

3.B7 25.92

9.51

2.06

61. 9-

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1.97

22.72

2.87 .~

;~ ~.:

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1.33

4.29

4.57

2.13

20.79

8.26

60.00

120.00

105.00

105.00

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1.09

........................ :,d••: ..•f.:.. .~~; ~~

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60.00

90.00

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'1.30

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75.00 .93

...

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n.oo

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1.20

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1.75

10.98

75.nO

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105.00

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105.00

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120.00

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2-'3

3.0'1

4.53 2.86

1.21

1.01 .84

1"'2

- - "'A'VALUES ---

3.83

1.B5

75.0ll .53

1.23

75.0n

--- K;..-lJALUE5 ---

14>

1.19

- - K;..-UALUE5 ---

0.

60.00

23.29

34.13

13.63

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1.0000

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.CC

1.2('

201 200 25° and k

Coefficients of active earth pressure K A for q,

_

~

»

:.t..:.. ~~~~~u~.=

Coefficients for active earth pressure K A for q, 0.05

0.05, 0.10, 0.15 and 0.20

.:..

; .: .

~;

.. :

tu:.~;~~~ ~.

:~~

.

...

90.00

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.2S

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.53

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.85

.95

.59

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IcO.D')

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1.1i

5.';:::

105.00

.56

6.94

4.34

120.00

.33

5.90

3.39

1.93

.311

."

.52

90.00

2.42

105.00

."

120.00

.30

'I,o .3333

...

60.00 7:5.0(1

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90.00

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1.51

7.::::;

3.27

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5.15

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1.79 18.96 16.27 3.56

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30° and k

Coefficients of active earth pressure K A for ¢

.19

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105.00 120.00

k '"

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.66

Coefficients of active earth pressure K A for

0.15, 0.20, 0.25 and 0.30

35° and k

1.09

1.ao

3.27

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.72

.33 .2S

1.08

1.32

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2.76

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.23

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24.2"

1.68

1.0000

Ell.ClO

l.tS

207

206 Coefficients of active earth pressure K A for'" k = 0 ........ ~

~

..t.:. ~~:~~ "k.: 1"3

= 40° and k = 0.20,0.25 and 0.30, and for'" = 45° and

B/41

Vi!

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1.73

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1.08

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105.00

.23

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120.00

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1.70

5.21

1.21

1.08

3.17

60.011 75.011 90.011

1/8

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1.39

2.69

10.5('

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105.00

2.09 .25

1.2B

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1.31

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3.78

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2.39

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60.00

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1.70

31.71

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105.00

4.49

1.40

3.79

120.011

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3:i.73

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1.53

1.86

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3.19 1.13

1.45

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105.00

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3.59

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2.51

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13.36

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75.00

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.

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1.62

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13.58

75.00

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90.00

.47

105.00

120.00

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60.00

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75.00

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.09

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.12

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1.43

120.00

1.13

1.97

16.51

60.00

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3.36 .12

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1.78 1.40

75. liD

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105.00

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120.00

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60.00

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2.53 1.3::

11.33

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1.16

2.54

105.00

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1.37

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90.00

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1.10

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1.17

1.5.2

75.00

3.31

3.31

3.51

29.95

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60.00

1.20

1.18

23.06

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1.46

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1.54

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90.00

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105.00

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120.00

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1.06

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1.35

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1.21 .78

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1.S7 2.73

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11.56

2.64

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1.18

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1.21

3.82

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60.00

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3.46

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1.53

1.11

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120.00 1.20

1.30 3.81

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2.42

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1.54

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90.00

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5.28

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1.47

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5.47 3.11

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....

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1.59

. 75.00

120.00

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2.65

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75.00

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105.00

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0.05,0.10,0.15 and 0.20

...........................

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:..!.:.~~;~~ ..~. :.. :~;.:

--- K" -l..'A:.UES - -

60.00

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r •••

;:~ . :

45° and k

Coefficients of active earth pressure K A for '"

.09

.26

1.55

1.22 1.71

.35 120.110

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8.51

209

208 45° and k

Coefficients of active earth pressure K A for
........................ :••~.: . ~;;~e ...~.: ..;;~.: ,,. 1"3

"3 1'.

1.00

1.41

2.55

."

2l.ES

.SI

1.16

2"3

e/.

25.74

1.65

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'I.o

1/3

.3333

60.00

.25

2.98

.S7

1.70

3.ll6

E;;.37

.97

1.70

13.94

.76

.93

75.00

.3333 75.00

.61

.19

.21

.23

.a8

.42

."

.61

.76

.SSG?

1.66

GD.Do

so.oo

60.00

."19

.35

... 1.47

1.62

.16

.73

.87

1.117

1.41

.48

.55

.65

.83

.56

2.33

).fJ

." ...

."

3.16

."

.47

."

1"-2

."

120.00

1.17

75.00

."

'1.25

.27

.01

.30

.01

7.~B

6S.62

18.05

.35 .26

.31

80.00

.5'

75.00

.31

.34

.39

.47

.11

120.00

..,

75.00

.35

.40

.20

.22

105.00

.11

.12

2.29

1.23

.6S

60.00 2.18

.79

16.00

."

13.39

.sooo

120.00 16.=5

.56

.45

75.00

.03

2.77

.16

3.77

4.e9

9.30

.56

.SSS7

.03

1/'

2'3

1.43

.58

.72

.41

75.00

90.00

.42

.21

120.00

1.67

.,. ."

.3333

75.00

12a.00 .5000

.48

.25

11'3

."

60.00

.3S .>B

90.00

."

.0;'

105.00

.51

.58

2/3

.S>

.67

."

60.00

.51

•5'

.0'

.DB

105.00

.0<

G.13

1.0000

.02

.02

.76

.89

.8:'

1.04

1.31

.43

,50

.65

1.62

.17

.19

.2i?

.27

.62

90.00

.18

.07

.oa

.22

105.00

.08

.10

.04

.75

.91

1.77

1.20

018

.75

2.18

19.09

.75

.30

.42

1.00

6.47

.52

3.19

.3333

60.00

.56

.65

75.00

.35

.39

120.00 1.14

7.69 3.5:3

.5000

.93

.44

.52

.,.

1.90

75.00

.47

.54

.24

.27

.06

.07

.06

.r>

60.00

12(1.00 .7B

.S7

2.38

20.85

.24

.05

.57

.69

1.23

10.29

.19 .OS

1.85 .75

." 1.74 .51 .23

90.00

2.78 .87

.27 .14

.03

.03

.14

."

6.26

75.00

105.00

2.38

'"

SO.OO

.55

..,

75.00

.37

.41

eo.oo

.25

60.00

.61

1"3

.74

.30

.64

.96 .'13

.72

.86 .50

1.05

.69

75.00

.41

90.00

.25

.38

2.32

105.00

.15

.21

120.00

.07

60.00

.82

13.06

75.00

.45

.79

." 1.0000

8.93 1.~7

2.43

20.19 7.GII

.15 .10

2.13

1.51

90.00

.27

105.00

.15

120.00

.07

6tl.OO

1.94

..DB

20.49

.75

11.22 6.20

2.69

1.22

1.58

2.S4

25.65

1.52

12.65

.22

5.77

a.sa

.61 .36

.18

.21

.09

.10

.54

.67

.08

.61

2.7B

2.35

.91

75.00

.66.77

go.OO

.j3 .18

.44

6.76 2.65

.10

.14

2.92

1.34

1.61

2.41

4.45

38.6(1

.77

16.06

.22

2.Stl

1.27

14.00

.34 .• 17

105.00 120.00

1.26

.11

60.00

1.07

...,

.35

11.:39

31.36

2/3

."

.17

24.78 1.36

4.53

.DB

.90

.07

3.0S

.32

1..2

120.00

.76

2.49

---K ·'JALUES --A

.09

1.55

.28

.71

1.12

.43

.07

.55

.05

.3333

15.54 5.!!&!

.16

.73

.17

1.03

.15

.45 .15

.77

.38

."

.26

1.58 2.3a

.11

.23

10!:i.oo

.S'

58.S1

75.00

1.36

.12

.e~

6.62

105.00

4.Cl5

.OS

.06

.Bl

1.07 .47

75.00

1.27

4.59

.05

.05

.08

.59

3.71

.1G

.12

120.00

10.05

1.91

1.86

.27

.15

90.00

.18

.21

1.03 .61

.76

.r>

.09

'I.

."

1.19

2.41

.05

e/$

2/3

.DB

.06

1.62

.83

.59

...

.79

.45

. . :..t.~.;~~~~ ..~.~ .. ~~;.:

.24

105.00

."

75.00

.08

.67

.J<

120.110

2.75

1.49

,15

.<9

."

. •51

911.00

.02 .6667

41.<:'5 10.90

1..3

.17 .07

<.71

--~ ·UALUES - -

.48

.63

.48

.16

.OB

1"2

.30

.28

2.!j0:

.,.

.....•.................. ,,.

.32

.DB

.84

1.22

.:•.!.:.;~;~~.~.: .. ;~~.:

.18

.02

."

.OS

SO.OO

120.00

.42

.71

.02

.21

1.76

1.01100

105.00

.OS

.34

.02

.4S

75.011 1.00

--- KJ\-UALUES - -

.17 .02

1.37

SO.OO

..........................

.62

.05

1.79

120.00

.03

81.G!]

:... ~.:.;;e;ee ..~.:.• ;~;.:

.65

.05

4.03

8.77

1.17

.15

7.28

.02

.13

.50

.14

.53

23.67

,02

7.9~

.12

4.25

.28

.39

.01

1.22

105.00

.5'

.49

.BI

.07

1.04

.3333

.S8

.so

.27

.34

.os

.81

11.20:

SO.OO

.76

.51

.15

.sa

.47

1.2~

.04

1.73

1.0000

.31

... 75.00

.5B

.06

.87

.14

.DB

120.00

.65

.51

.05

.71

lEO.OO

.'13

.34

.60

90.00

1.0000

.72

60.00

.0'1

.60

.17

.>B

." ."

7~.DO

.17

.Il

.26

." .5000

.07

."

.84

.Ga

~.OD

.01

.06

.36

.68

.30

.56

12a.00

.05

.10

.91

1.84

.115

.31

1.63

3.BB

.13

.05

.19

.70

7.02

.26

.Il

75.00

1.23

12.B3

.B3

105.00

90.00

.95

2.34

.52

." .25

16.54

1.14

.E3

5/6

1.45

120.00

.04

.04

.IS

.B8

.Il

.04

120.00

31.30

.37

105.00

--- K -liALUES - A

60.00

.23

2.77

.53

.31

.71

.J<

6.09

."

120.00 .3333

.48

.76

.67

................................ :..t.=.;~;~~.....~.:.~.:

.38

.64

.<'

.14

75.00

4.1t

2..3

.61

.33

.51

1.05

1.61

60.00

",

5/6

--- XJ\-UALUES ---

105.00 .95

.15

.BI

.S<

.Il

2/3

120.00

1.95

.52

1<'2

--- KJ\ -L!:lLUES ---

.52

." .13

. . :•• t.:.;e;e~ ••:.:.. ~~;.:

.........................

.37

120.00

0.10, 0.15, 0.20 and 0.25

50° and k

Coefficients of active earth pressure K A for
:•.t•• :tl;~~~e .. ~.:... ;~~.:

--- KJ\-UALUE5 ---

--- K ~UALUES - -

.ae

50° and k

0.25 and 0.30, and for
.99

7.3<

&.54

3.0B

.14

3.14

1.45

3.72

6.09

11.04

93.93

1.l6

1.70

3.00

24.63

.7:l

l.29

9.45

..,

.54

.10

.11

.30 .16

4.00 4.01

1.e9

211

210

.........._

:..~.:.;~;~~ .. ~.:

..-.

SO.OO

.so

75.00

....s,

60.00

.11

.19

.s,

.:

.

;:~.:

1....2

.82 .52

1.03

...

.22

.26

.12

.14

.80

.S8

.31

.35

1.50

.64

1.2B

.29 1.9S

7S.011 90.00

105.00

.28

.17

120.00 SO.OD

75.00 90.00

.Sl>G1

...

105.00

.17

120.00

."

12G.OG

3.50

30.44

1.91

15.02

.3333

60.GO

3.79

2.10

2.30

2.22

2.42

2.62

2.B6

3.1S

3.47

3.79

4.75

5.2<1

5.75

2.22

2.4<1

2.39

2.61

36.19

1.51

."

2.00

.79

.11

.5000 75.00

16.62

." .13

8.B4

105.00

3.35

120.00

1.S1

.29

45.82

.6S67

.72

.90

1.30

2.30

19.06

75.ao

.<10

."l9

.S7

1.19

8.71

90.0a

.58

.11

.13

.30

4.61

1.75

3.42

4.37

7.e8

13.15

111.49

1.10

2.16

1.37

.52

1.0000

1"2

1.8S

90.00

1.97

2.1'1

105.00

2.50

2.73

2.00

2.15

3.95

4.31

2.05

2.25

1.95

2.16

2.36

3.01

3.33

3.65

120.00 .5000

.SS67

'1.50

60.00

1.93

75.00

2.07

5.51

2.B6 3.54

3.91

4.9G

5.43

6.96

7.80

8.67

2_30

2.67

3.99

2.S6

4.<12

.1211.011

4.99

5.57

Gil. 1111

2.02

2.27

75.011

2.20

2.<15

911.00

2.65

2.96

3.57

1211.011 GO.Oa

3.99

2.99

3.06 3.26

2.'16

2.62

I.Ba

2.65

1.83

4.'12

las.OO

75.00

2.<15

90.00

3.02

3.39

4.17

4.S9

2.61

3.07

2.85

2.26

2.54

2.B3

3.12

2.70

3.03

3.37

3.72

120.00

75.00

.... 51 5.55

90.00

7.14

7.73

105.00

11.<16

12.44

120.00

4.21

4.78

5.37

8.12 3.33

2.62 3.35

2.<12

3.23

2.30

2.52

5.05

J05.00

2.85

2.<13

2.68

3.18

3.51

120.00

4.22

<1.74

3.53

75.(1)

1.S8

2.22

4.211

90.00

.5000

2.61

'I.o

3.5<1

75.00

3.83

1.54

5.27

4.90

5.31)

5.72

2.96

3.19

3.0B

1.49

I.GB

1.86

6.29

3.711

4.21 5.82

S.39

.3333

4.59

75.00

1.48

1.66

1.B3

1.61

1.Bl

2.02

2.23

2.50

120.00

2.70

3.12

60.00

1.55

1.79

75.00

'1.95

1"2

2.07 2.09

105.00

2.30

2.67

120.00

3.30

3.8B

2.0<1

3.<10

3.74

<1.18

4.59

5.79 10.18

5.a2

5.'16

6.98

7.60

1.0000

2.19

5.90 2.46

2.73

2.36

2.63

2.91

2.83

3.15

3.49

3.80

4.24

2.15

2.8S

J211.00

3.63

4.18

4.75

5.32

5.92

60.00

1.70

1.94

2.19

2.45

2.72

3.70

J.U

2.22

6.77

7.39 3.30

.5000

3.44

1.78

2.04

4.1I6

90.00

5.42

105.00

2.71

3.11

120.00

4.01

4.65

60.1I0

1.75

2.02

3.54

.6667

3.73 4.0B

75.00

1.87

4.45

90.00

2.22

105.00 12a.00 1.0000

3.60 3.95

2.16

5.30

5.98 2.89

2.44

2.74

2.90

3.25

4.41

5.14

1.87

2.20 2.<10

2.74

2.91

3.32

3.76

3.39

3.95

4.53

5.14

90.00

7.28

105.00

3.J8 3.SB

3.04

7.'11

BolS

3.19

3.50

3..35

3.67

3.98

4.36

3.39

2.06

5.26

7.1S

5.30

75.00

4.BO

<1.73

3.27

3.6<1

60.00

6.62

4.3<1

2.59

2.83

2.38

6.03

4.32

120.00

S.32

9.J~

3.22

3.59

3.97

3.47

3.85

4.2<1 5.15

6.16

.

:.!

6.<12

7.10

10.20

11.32

2.44 2.77

3.05

3.95

4.37

2.65

4.Bl

5.26 3.00

2.26

2.50

2.75

2.31

2.56

2.S1

2.37

4.46

GO.OO

5.06

3.01

3.78

7.30

9.68

3.0'1

3.37

3.72

4.07

3.31

3.66

4.02

4.39

3.24

3.64

4.05

4.47

4.91

5.37

<1.46

5.01

2.33

2.64

90.00

2.56

7.02

7.44 8.B7

11.92

1.0000

2.0B

2.37

2.59

2.9'1

3.23

3.74

60.00

1.43

1.70

7S.00

1.46

1.73

1.95

2.20

2.81 3.52 5.10

2.45 2.48

2.97 2.7'1

90.00

3.01 3.<15

2.05

2.47

2.86

3.25

3.65


<1.49

2.90

3.SS

4.16

4.78

5.41

6.07

6.75

2.10

2.38

2.11

2.43

2.22

2.54

.5000

60.00

1.75

75.00

2.59

3.18 3.27

105.00

1.63

75.00

1.73

1.92

120.00 2.21

2.51

2.B2

3.13

2.9S

3.27

3.45

.6667

60.00

3.49

90.00

4.SS

105.00

120.00

7015

60.00

1.73

75.00

1.9a

2.29

4.7J

5.69

90.0G 105.00

2.98

3.15

3.60

3.36

3.76

'1.06

<1.5<1

5.S4 8.73

1.81

2.23

5.04

2.74

<1.32

60.00

1.56

1.94

1.71

2.10

3.07

3.40

3019

3.SS

5.00

5.57

3.36 3.S3

75.00 90.00

2.60

2.37

120.00 loOOGO

2.S1

7.67 1.49

75.00

105.00

120.00

2.15 2.37

105.00

4.31

60.00

90.00

4.54

1.71

1.77

120.00

2.66

120.00

6.16

1.48

4.61

4.13

4.17

90.00 105.00

6.97

2.75

3.19

2.56

4.20

3.21

5.17

2.37

6.31

75.00

3.58

2"3

3.Bl

9a.00

3.29

1"2

2.00

5.67

3.<19

.6667

lr.J

--- Kp-VALUES ---

1.61

120.00

2.94

3.22

8.60

.

.. ..

:.;e;~e ~.: ;~~.:

."

2,IJ

2.22

7.58

2.97

75.0a

120.00

1.92

1.70

4.02

5.99

7.S0

6.54

2.10

60.00

IDS. 00

3.0'1

3.16

2.19

SO.OO

2.
5.8S 4.12

5.42

2.32

P

105.00

3.87

2.74

1"3

°

90.00

3.:37

50.00 105.00


2.09

2.50

105.00 5.2B 60.00

3.74

1.87

--- K -VALVES - -

60.00

90.00

3.01 3.60

2.33

8.81 .661i7

2.63

3.3S

1.65

75.00

9.49

'I. 2.45

2.44

2.20

3.61

2.37

2.95

611.00

105.00

3.30

5.9B

2.11

120.1I11

<1.84

3.56 3.08

3.S7

105.1I0

3.15

.3333

9.<14

2.30 2.20

2.92

2.92

5.51

60.00

2.60

2.50

3.06

•.•.....................

:3.10

3.76

9.12

3.78

1.99 2.39

2.10

2.25

5.57

2.95

75.00

2.:31

2.1)9

3.35

2.66

1.89

1.91

:••!.:. ;~; ~~ ..~.: •. ;:~.:

2.13

2.t!!

3.11

2.39

90.00

2r.J

7S.00

7.76

2.90

3."11

105.00

-VALVES ---

60.00

7.18

2.64

3.10

1.72 1.71

2.08

5.15

2.77

4." 1.0GOO

3.79

1.91

2.83

4.97

.

120.00

5.86

2.89

4.87

3.12

2.49

~

2.07

3.05


6.17 2.20

105.00 120.00

2.72

2.79

<1.16

S.16 2.52

1.74

7S.00

5.66 7.45

2.53

3.S2

5.02

2.S8

6.<11

3.88

<1.15

3.50

10.50

p

2.30 3.311

3.<19

--- K

4.69

S.05

4.03

3.17

1"3

2r.J

2.7S

3.02

3.60

4.02

.GSS7

2.45

.1.56

0.05 and 0.10

---

5.'15

2.7'1

2.23

!..~.;~;~e •• ~·~ .. ;~~.:

2.86

3.65

I.B9

90.00

10.12 3.19

...:.•,

3.59

2.47

3.33 3.69

4.39

2.58

3.29

2.38

75.00 '1.26

2.80

5.35

3.07

2.56

3_44

90.00

2.26

4.47

2.54

2.29

3.17

2.50

105.011

105.011

1.00aa

5.00

1.79

2.11

90.00 105.00

3.94

60.00

I.B5

75.00

3.20

2.'16

2.68

120.00

60.00

90.00

2.35

2.96

105.00

4.08

3.63

............................ :•.t•.;:.;~;~~.'. ~.: ... ;~~.: --xP -VALUES

3.12

2.84

11.22

20° and k

.5000

2.59

2.59

29.47

q,

7.93

2.23

3.37

2.34

1.55

120.00

5.14

3.37

1.79

5.27

3.58

2.28

75.00

2.96

21"3

- - K -UALUES --P

o

2.GI

2.76

3.25 4.73

1..-&

91. '10 2.44

2.62 2.57

90.00

2.09

.sr

5.89

3.11

2.15

2.S0

4.36

2.03

.36

3.9<1

1.73

120.00

7.~6

.15

3.18

SO.OO

105.00

9.00

2.8S

.13

3.00

120.00

5.76

2.58

.11

2.75

5.9B

7.05

6.48

4.7S

1.85

7.37

2.37

4.18

2.29

12.16

1.7'1

4.S0

6.BI

2.11

2019

6.44

5.B2

.76

1.70

4.45

2.67

1.9'1

1.99

3.BI

60.00

.33

SO.OO

<1.11

2r.J

p -VALUES - 2.11 2.27

1.78

5.BS

120.00

.27

75.00

2.91

1.80

3.45

3.76

.23

1r.J

2.70

75.00

1.94

90.00

105.00

105.00

105.00

1"6

2.50

1.78

3.08

3.10

8.05

.37

Coefficients of passive earth pressure K p for

75.00

3.77

5.22

120.00

.45

3.59

3.S3

4.49

105.00

2.78

60.00

3.0S

4.13

1.53

75.00

120.00

"'3

3.35

120.00

.3333

2.53 2.91

1.91

60.00

2.47

2.37 2.73

4.07

.53

2.32

2.22 2.55

2.02 90.00

7.64

.59

.20

120.00

105.00

1.46

2.17

2.07 2.38

8.41

.so

.21

90.00

3.49

3.94

2.02

0.15,0.20,0.25 and 0.30

.:.. ~.:.;~;e~ ..~.:..;;~.:.

~-- K

1.93

.29

.5'

105.00

7.36

8.14

1.88

1"2

2r.J -VALVES ---

2.21

.51

.10

.39

1.7B

.13

75.00

SO.DD

1.74

75.00

.26

1.17

2.31

60.00

.21

.95

Gil. DO

D

.11

50.00

1.0noo

13.31

.19

60.00

105.00

24.32

1.GO

.19

.n

.29

2.79

1"3

20° and k

:..t =.;~;~~ .. ~.:.. ;~:.:

~.:

--- xl'

q,

.........................

.

!.:.;~;~~ ~.:

'I. 61.

2/3

--- K -UALUf;5 --A

.30

50.00 105.00

120.00

1'-3

Coefficients of passive earth pressure K p for

Coefficients of passive earth pressure K p

Coefficients of active earth pressure KA

4.16

5.1<1

B.63 ·2.67

3.05

2.<17 2.97

3.83 3.G5

3.43

10S.00

3.90 5.31

7.2<1

4.<10

<1.07 4.91 G.72 10.64

213

212 Coefficients of passive earth pressure K p for q,

..... "._ :..

!.:.~;;~; ..~.:

,,'

al.

'I.

.3333

1/3

SO.OO

,..,

2.14

2.37

2.0E:

2.28

2.52

: ~.:.~;~~~ ..~.:

1"'2

2.59

1"2

2;'3

2.34

3.0S

50.00

3.38

2.46

2.75

3.34

3.76

4.19

S.U;

5.6B

120.00

5.23

5."

G.S7

8.33

5.24

10.20

60.00

2.20

2.50

2.83

3.17

3.53

3.91

<.29

75.00

2.45

2.79

3.14

3.52

3.92

4.34

4.77

90.00

3.96

4.37

4.99

5.S?

B.25

120.00 .5000 75.00

4.9S

5.5C1

75.00

2.7B

3.13

3.50

3.8S

4.30

3.0B

3.46

3.B7

4.30

4.75

7.98

8.84

105.00

15.02

120.00

3.<19

3.91

6.33

7.tH

90.00

7.39

9.32

10.36

105.00

2.93

3.37

2.93

3.38

3.B7

4.40

5.03

4.32

"l.39

4.94

7.22

120.00

9.53

11.0B

12.78

2.BS

3.40

3.95

75.00

3.45 4.62

105.00

4.65 5.39

6.96

8.15

9.42

12.17

14.26

16.52

.. REMARK: THE: KP-UALUE: IS OUE:? 1000

~HEN

,,'

1.0'3

IS.5B

IS.70

5.17

S.BS

6.55

5.33

S.OS

6.82

7.64

B.08 12.28

13.67

15.54

21.S4

24.48

3.03

3.44

3.BB

3.41

27.47

3.30

4.27

3.28

3.77

4.87

2.55 2.50

4.75 60.00

2.09

2.92

3.51

3.95

5.45

5.23

2.4ll

2.73

5.19

2.62

7.84

9.12

13.6t

15.98

1.-s

5.-s

105.00

3.99

120.00

6.47

5.5:

2.21

.5000

2.57

2.97

3.37

60.00

1.82

3.SS

75.00

1.90

&.09

B.90 3.47

3.B7

4.29

.3333

10.ll6 3.39

7.73 11.50 3.B3

75.00

105.00

2.94

120.00

4.50

60.00

13.02

14.65

<:.31

4.Bl

4.Bl

5.3i'

.5000

2.36 75.00 90.00

10.26 2.77

3.22

2.71 3.50

105.00

4.09

4.75

5.99

6.97

B.58 2.67

3.19

3.76

75.00

3.17

3.76

4.40

13.57

2.72

1115.00

3.79

120.00

6.10

3.111

3.53

5.45

6.20

B.03

9.19

7',02

5.B4

7.51

B.Bi!

13.07

15.39

• REMARK: THE KP-UALUE IS OtJ£R 1000

~HEN

6.77 10.23

5.03

5.74

5.B5

6.1>7

7.76 11.76 20.70

999.99 APPEFlRS

6.85

1.0'2

3.14

2.59

3.06

3.57

7.B9

90.00

3.33

3.93

11.77

105.00

4.82

5.73

6.72

120.00

9.68

11.43

5.89

6.75

2.0'3

4.2i'"

4.16

4.66 5.84

7.59

.3333

5.50

60.00

1.87

2.20

2.56

2.37

2.77

75.00

2.59

3.06

75.00

2.46

2.94

3.46

3.78

90.00

3.15

120.00

7.55

60.00

8.42

105.00

14.21

120.00

4.75

.5000

60.00

2.43

";.21

14.82

105.00

5.S1

120.00

9.83

4.01

17.35

60.011

5.83

S.48

3.82

4.27

111.112

4.79

5.65

G.59

4.28

105.00

7.18

. 8.49

9.93

120.00

12.45

14.811

17.37

4.95 7.Gl

• RE:t1AR": THE I:P-VALUE IS OVER 1000 tmi:N S9:).99 APPEARS

14.48

16.64

4.03

4.62

5.26

3.91

4.5a

5.19

5.ao

6.67

7.62

a.64

10.44

12.43

14.59

16.95

19.55

2.84

3.42


4.77

3.58

1.0000

7.00

60.00

2.30

7.44 12.96

14.85

105.011

6.45

19.65

22.78

26.16

120.00

l1.li!

7.411

2.46

2.15

4.50

5.10 8.13

9.2i'"

3.75

4.22

2.46

75.00

.3333

10.81 19.10

12.70 25.75

999.59 AP?EAR5

.

~.:

1.0'2

2.0'3

5"6

2.79

3.14

3.54

3.96

4.42

3.11

90.00

3.30

9.29 16.17

- - K -Uf-ll.UES - P

60.00 3.52 3.76

~HEN

1.0'3

'I.

2.75

2.52

2.88

6.19 7.80

!.:.:~;~~ ..~. :

al.

5"" -

6.35

4.71

• RE:MARK: THE KP-UALUE IS OVER 1000

.:..

6.12 2.49

7.3ll

4.33

~.: :;~.:

2.95

5.00

3.30

75.00

.

~

3.48

5.96

105.00

4.35

5.10

5.54

120.00

7.38

B.78

10.34

12.14

14.14

sO.on

2.57

3.03

3.55

4.1
<1.78

7.93 16.37

18.62 6.22

3.10 5.02

6.2:0

7.14

6.05

7.33

10.24

11.90

2.25

2.67

3.62

2.46

2.93

3.97

5.33

4.99

5.71 105.011 13.71 .5000 SolS

5.75

7.49

6.sa

10.53

120.00

U.24

13.52

16.38

15.50

60.00

2.62

3.3!!

4.02

75.00

3.42

'911.00

2.5ll 3.48

6.15

7.19

8.~

9.~

7.42

5.59

10.25

12.10

H.13

16.34

13.67

90.00

.6667 2.17

6.S)

3.39

4.0e

10.26

105.00

3.69

4.85

5.90

3.79 S.G5

6.54

7.03

e.27

9.G3

7.51

90.00

U.U

105.00

19.19

7.32

7.76

S.!:;

7.96

16.70

20.19

3.78

4.54

5.40

6.S1

7.94

9.45 15.44

9.37

10.91

15.15

17.71

28.65

39017 7.40

8.54

11.16

13.05

15.14

12.92 24.50

18.30

21.47

24.96

3.30

6.S7

8.18

3.82

4.55

5.37

6.2G

7.23

75.00

4.B7

5.99

7.28

B.74

10.39

12.23

14.2E

'M19

5.98

7.05

8.24

9.53

90.00

7.10

8.74

10.60

12.74

15.15

17.as

20.EO

105.00

6.05

7.43

6.94

21.a6

25.33

29.8S

120.00

10.38

15.53

3.77

4.&7

5.70

35.01

4.09

999.59 APPEARS

60.00

10.69 20.21

2.70

~HEN

1.0000

B.77 16.45

3.13

• REMARr;:: THE: r;:P-UALUE IS OVER 1000

6.30

5.45

2.50

2.91

4.67

6.36

15.00

5.10

4.B3

a6.S9

5.52

4.67 5.67

60.00

14.3a 23.06

3.26

75.00

13.21

4.74

3.34

5.17 lao. 00

lao.DO

60.00

13.96 ";.18 4.53

10.67

2.B2

11.22

4.53

1.0000

7.30

3.48

9.62

2.74

B.72

4.25 4.62

2.97

8.40

75.00

5.69

3.79

5.85

6.37 7.50

5.44

90.00

5.74

3.6B

3.53

4.57 11.713

a.s,,;

2.02

3.20

2.05 75.00

19.8S

5.61

3.28

75.00

2.75 3.04

4.17

5.28

5.33

S.37 10.58

4.84

2.13

120.00

5.52 9.117

3.54

3.65 .3333

3.35

4.74

4.12

U.92

2.5S

3.SS

7.70

2.98

105.110 9.78

4.11

3.02

3.18

4.02

2.34

7.72 15.02

60.00

so.OO

1.95

120.0ll .6667

5.96

4.43

2.19

1~.02

75.00

8.34

6.46

3.47

3.36

13.23

6.08

5.21

7.31

3.92

105.00

16.92 5.29

10.95

1.9ll

4.76

3.35

2.16

9.88 12.87

75.00

.50ll0 3.71

4.sa 6.37

3.38

5.11

7.45

6.59

6.513 2.16

2.23

7.59

2.77

6.7<:

60.00

3.76 5.71

2.49

2.68 4.70

12.48

3.27

-- 1),-UA'_UE5

3.es

-VALUES -

2.63

6.55

5"6

8.73

2.23

3.95

10.89

• REMARK: THE KP-UALUl:: IS OVEIl 1000 I-lHEN 999.99 APPEARS

3.08

1.97

120.00

11.46

H:.?7

90.00

2S.BB

7.97

5.41

7.53

13.45

2.83

1211.00

....

7.7";

4.59

3.65

5.71

3.82

5.20

11.90 20.69

4.09

2.69

2.42

2.24

9.86

20.44 1.00110

7.12

5.89

2.26

4.94

:... ~.:,,:;:~~

8.43 2.90

2.05

KpUIlLUES - -

3.33

3.24

60.00

.

3.04

2.80

4.23

.:

2.9J

2.50

3.70

4.81

6.11

993.99 APPEARS

a.SB

2.39

75.00

5.47

15.40

2.81

2.13

3.39

3.26

5.133 9.62

3.38

.6GS7

7.93

2.95

5.13

2.99

2.65

10.26

12.04

4.70

i.22

2.62

a.46

6.95

10.11

2.5~

75.00

.6661

13.79

4.95

6.95

5.95

3.B3

2.34

.5000

1.0000 5.22

2.56

6.00

2.27

2.11

5.71

15.72

Ill.61

2.39

60.00

15.44

4.22

5.10

6.33

2.25

5.22

7.05

4.21

120.00

4.22

11.86

1"3

2.a4

1115.00

9.03 7.48

c.02

90.00

3.S7 105.00 120.00

..

2.52

75.00

3.S2 5.06

B.77

~HEN

3.09

13.91

16.23

:.tu:.~;;~~" ~ ;.~~~~

7.0S

3.01

6.03

G.67

3.25

3.09

4.89

11.58

......._ 2.0'3

6.25

3.85

- - Kp-UAwrsI.SS

12.22

7.21

4.53

1.95

75.00

14.84 4.B2

8.15

10.66

• REMARK: THE: KP-UI'1LUE: IS OUi::P, 1000

999.99 APPEARS

5.57

2.86

3.9ll

105.00

13.27

1.72

SO.OO

5.32 8.53

5.29

9.21

2.79

9.11

2.99

.3333

2.S4 2.99

12.35

90.00

7.11

lB.99

1.0'2

11.78

5.35

.........................

BI¢;

10.39

120.00

14.62

3.11

60.00

105.00

9.111

4.25

8.03

:.. t.:.~;;~~ ...~.: .. ;~~.:

'I.

5.49

6.32

2.83

6.14

2.7S

90.00

4.35

5.38

5.53

le.lIe 1.00110

4.91

7.B9

17.85

2.52

4.18

75.00

5.05

.3333

3.B6

2.29

.5000

!l.SI .Se;G7

2.45 2.72

13.50

4.41

105.00

2.43 2.79

7.54

2.37

7.U;

60.00

...,

1.o'a

-- \ 75.00

105.00

2.15

12.05

2.GS

2.52

4.40

SO.OO

1.0'3

51.

30°

.

_

.. :.. ~~;.:

t:.~~;~~_ ~

at.

5.0'6

P

3.36 4.02

75.no SO.Oll

6.39

2.0'3

--- K -UALUES ---

3.6ll :U;6

4.08

10.G8

1.0'2

'I.

2:.B3 3.31

.:..

.20.

6.45

5.70 .3333

=

a/¢

2.0'3

2.48 2.99

'1>= 25.00 k

~

p

2.84

0.20, 0.25 and 0.30, and for q,

25° and k

.,

;.

- - K -UrlLUr5 - -

90.00

105.00

...................... .....

........................ .. ..;~~.:

.

~.:

-Kp-l,I.o)':"UES -

75.00

Coefficients of passive earth pressure K p for q, and k = a

0, 0.05, 0.10 and 0.15

25° and k

11.79 120.00

22:.70

28.01

34.18

49.17

• REMRKI THE I:P-UALUE IS OVER 1000 I!HEM 993.!J9 APPEARS

9.E5

214 Coefficients for passive earth pressure K p for q,

30° and k

60.00

2.44

2.12

.5000

SD.DO

2.91

3.39

105.00

4.20

01.95

120.Dll

7.0a

8.48

11'3

5"

p

2.77

3.101

3.51 3.SS

10.06

3.54

3.9B

3.9S

ol.51

4.53

5.19

5.92

6.75

7.82

9.01

11.69

13.SS

16.21

2.9<1

3.51

75.00

2.93

3.<19

4.13

6.50

90.00

3.ge

4.70

5.57

B.95

4.10

4.76

5.47

4.<15

2.75

3.13

75.0n

2.26

3.03

3.4B

12n.on

IB.r!: 6.25

.3333

6n.on

90.00

5.99

7.25

B.GS

10.33

le.IS

14.28

16.57

10.76

13. lEi

15.92

19.10

22.71

26.75

31.22

GO.Oll

3.31

J.SS

4.68

5.48

6.37

7.35

75.00

4.00

4.77

5.64

6.63

7.72

B.93

14.94

21l.50 39.10

33.39

19.65

23.66

2B.26

3.70

4.46

5.33

6.30

7.38

B.57

4.56

5.49

6.56

7.75

9.10

10.58

.120.110

13.10

1&.14

60.00

3.02

75.00

3.74

5.24

6.41

11'.02

12.97

15.13

105.00

8.41

10.34

12.57

15.14

16.05

21.32

24.S3

120.00

15.75

19.51

23.85

20.84

34.52

olO.91

4B.00

GO.OO

3.6S

".55

9.28

7.76

£.77

5.59

B.U

9.61

B.46 15.46

3.2<1

3.89

7S.0n

ol.67

3.20

11.2B

...67

15.55

12.51

105.00

4.13

5.n2

6.04

7.21

'.54

10.06

120.00

G.BS

8.5<1

10.<16

12.67

15.23

18.13

2.65

3.21

3.SS

4.5S

5.40

6.31

75.00

3.01

3.68


5.31

6.E9

7.41

3.95

4.B7

5.93

7.16

8.55

In.14

105.00

5.9&

7.45

9.20

11.20

13.51

16.16 3C.38

18.Ell

2.73

IS.SZ

120.00

7.S8

sO.on

3.S0

4.38

5.26

7.36

8.5B

ol.43

5.37

6.<15

7.G7

9.0<1

10.57

7.56

9.11

10.87

12.67

15.11

~.19

28.21

3<1.n2

40.55

6.20

5.03

60.00

10.00 18.79 4.42

3.52

6.25

5.47

'.67

8.03

9.57

11.29

B.4ol

10.16

12.09

1<1.25

5.82

7.U

8.GO

10.27

12.17

1<1.26

15.00

<1.52

5.S
6.94

90.00

G.83

8.46

10.35

12.51

14.98

17.74

2n.BO

90.00

&.55

8.18

10.0B

7.19

8.57

7.26

8.76

10.<16

12.38

14.E!S

11.13

13.82

16.93

20.<18

24.55

120.01l

12.72

16.52

21.nl

26.20

32.2B

39.2<1

47.20

3.09

4.02

7.15

9.40

11.26

3.92

5.08

9.75

11.81

14.16

34;77 67.56

120.00

90.00

2.73

3.22

3.77

4.39

5.0B

3.8S

4.64

5.50

6.48

7.59

3.69

1ll5.00

5.75

7.1<'

S.15

18.<13

120.00

10.72

60.00

3.n3

75.00

3.79

7.55

3.34

3.97

5.44

3.67

4.se

6.39

5.3<1

7.49

9.90

11.82

120.00

9.78

12.19

15.00

18.2ol

60.00

2.58

3.16

3.82

4.57

5.ol0

75.00

3.08

3.78

4.57

5.46

6.48

sO.on

4.19

5.16

12.06 14.54

IB.49

aa.61

7.91l

9.81

12.02

13.06

16.27

19.29

19.74

25.09

31.38

26.83

38.52 .6667

4.60

5.7B

7.15

8.73

10.57

12.64

15.0:

40.14

• REJ1ARKt THE KP-VALIJE IS OVER 1000 J.lHtN 999.99 APPEARS

14.25

22.05

90.no

7.94

14.56

6.26

9.63 17.92

15.06

10.72

10.31

7.76 1<1.31

24.79

S.93

.GGG7

6.20 11.2<1

12.76

7.36

SO. Oil

120.00

12n.00

4;.70 11.'<:9

1.0000

13.52

17.32

21.73

60.00

3.24

4.15

5.21

75.00

4.13

5.26

6.59

7.28

1.85

9.46

ll.2S

9.B9

11.91

14.20

14.23

17.47

2U.7i!

SO.OO

5.95

7.60

9.52

11.76

lol.36

17.30

29.16

3<1.65

105.00

9.16

12.53

15.77

19.53

23.89

28.87

120.00

18.65

2<1.06

30.38

<16.25

• REHARKI THE KP-UALut IS O\)ER 1000 l.lHEN 999.99 APPEARS

20.65

lol.10

13.B6

16.95

20.47

<1<1.58

54.69

28.22 66.36

11'3

1"2

2/3

s.-6

4.53

5.29

5.47

6.46

7.59

9.28

11.06

- xp -VqLUE"S -

SO.OO

7.53

75.00

2.73

3.25

3.87

90.00

3.59

4.3B

5.32

2.82

3.32

3.68

6.43

&.14

8.7<1

10.71

13.01

15.73

18.B5

105.00

5.57

6.93

8.5B

10.55

12.92

15.71

18.95

16.92

21.03

25.97

31.77

3S.4'

120.00

10.32

13.14

11;.5<1

20.73

25.74

31.10

38.69

3.7<1

<1.59

5.60

2.97

3.S8

<1.54

5.56

6.76

8.14

9.7<1

4.72

5.B4

7.16

5.75

7.10

8.S9

10.56

12.75

6.76

8.11

9.S5

60.00

8.73

10.55

12.65

?S.on

15.45

~.oo

11.8<1

14.96

18.77

23.33

2E1.S!!

3<1.e7

31l.7n

39.87

48.68

60.15

73.47

4.35

6.72

5.8B 8.58

13.62

79.66

97.<16

12.25

14.85

7S.no

5.30

S.81

B.S5

13.53

16.65

20.23

5.16

6.80

75.nn

7.V

9.53

sn.oo

11.68

15.33

Ins. 00

21.47

28.21

30.ol3

13.51

17.05

21.31

26.3<1

24.39

:?0.95

38.BS

4B.17

65.74.

82.78

12.31

15.67 25.2<1

13.35

16.2<1

16.en

2n.78

25.ol2

75.0n

5.1<1

79.63 6.3<1

7.9B

9.95

12.2S

8.50

10.75

13.45

16.57

sO.on

7.9n

10.29

13.25

16.94

21.19

1<1.0<1

18.47

23.89

30.57

3B.5~

21.53 29.99

l.noOO

60.00 75.00

48.36 105.00

157.06

4.97

120.00

17.57

12S.21

3.83

105.00

2<1.<19 39."7

30.37

80.00

32.12

19.73

• REI1ARKI THE KP-UALUE IS OVER 1000 WfEN 999.99 APPEFIRS

11.27

120.00 .6667

S8.S0

14.15

31.77

10.65

2!i.E6

12S.00

58.75 100.14

10.87

ol5.B7

51.56

19.79

12.1<1

:.76

90.nO

9.99

10.5'6

73.9<1

10.04

5.55

25.25

64.23

18.97

60.13

8.22

2n.76

6.<13

8.17

4S.3n

S.67

16.91

40.27

1
19.52 38.35

5.36

75.00

S.n7

30.<14

14.66 30.05

4.27

16.1E

15.08

105. no

11.52 23.32

12.03

31.26

10.SS

8.94 17.85

16.11

sn.oo

lG.Ol

3.94

8.50

120.00

13.3<1

11.67

4.63

105.00

8.2<1

2<1.00

120.00

2<1.35

.5000

JO.92

sn.oo

120.0n

I1.G9

_.

6.0S

105.00

1.0000

18.56 35.79

11.00

2<1.0~

6.12

14.71 28.28

9.28

.120.00

90.0n

32.Sb

B.12

12. III

58.59

90.00

12.C

20.93

6.00

105.00

4.~

1.2.53

In.53

4.S2

10.1l3

7.51

B.91

90.00

8.29

5.37

11.24

7.77

5.25

9.28

3.44

46.8S

9.32

6.20

7.79

18.53

75.00

7.47

8.99

6.77

<1.46

120.00 .5000

6.31

6.21

7.57

5.46

2.96

4.50

5.12

6.33

6.56

3.74

4.15

5.2<1

14,50

2.49

3.32

4.~

27.5S

60.0n

15.GO

3.47

5.35

.5n1l0

10.56

75.00

22.46

30.79

5.10

7.:?2

4.29

16.37

25.91

4.2n

G.19

18.07

13.8<1

21.57

3.41

5.1S

3.39

11.83

17.19

2.71

4.29

14.29

9.67

14.50

GO.OO

3.51

60.00

1.S7

11.67

32.74

2.S2

120.00

6.51

30.5'

9.27

17.E6 27.35

60.00

9.62

5.23

120.00

20.23 38.7<1

7.64

:,,:~;~~ ..,,~.: •.;~;.:

a

13.21

105.00

75.00

10.66

.3333 7.4';

JS.45

6.2~.

90.00

3.<17

10.;;<:: 1';.90 2;.;;0

7.55

6.51

75.00

6.63 10.19

2.31

B.;;2

I1.B8

5.BO

21.85

II'S

51'6

5.olB 6.50

16.61 31.6'1

6.19

__

5.41

3.91

15.41

:.t

2.84

8.75

2.64

5.31

2.41 2.78

13.45 25.48

4.9<1

.. REMI'lRK: THE KP-UALUE IS DUEll 1000 ms.E;N 999.99 APPEARS

.

11'3 11'2 21'3 ___ K -VQt.UE5 p 3.32 3.88 4.52

15.&5

5.02

Go.no

4.42

°

60.00

4.8&

5.SB

11.21

•• ••••e.:

7.47

15.00

6.75

120.00

13.21

6.43

3.63

6&.72

6.3<1

5.45

5.50

55.37

11.06

4.70

SO.OIl

45.<13

9.17

4.69

105.00

36.78

3.68

4.01

2.74

as.33

5.35

3.32

2.38

75.00

105.00

23.01

2.63

2.BE:

60.00

an.57 3<1.36

3.71

120.0n

lli.55

17.13 28.56

90.00

.I5.a5

7.87

14.11 23.46

105.00

13.45

6.47

105.00

10.25

11.<19 19.07

4.48

11.35

120.00

9.48

8.S4

5.10

3.71

9.23

1"6

<1.5<1

75.00

15.24

:.~ :.;~:~~ ~.:

51'6

2.9<1

7.30

. ••

.

In.75 15.71

60.00

12.00

.. REMARK: THE KP-UALUE IS OVER 1000 1JH0l 999.99 APPEARS

:••~.:.;~:~~ •••~:••:;2.:

105.00

7.95

&.<15

1.0000

<1.92

31.56

10.21

6.4n

3.99

16.sa 25.83

5.96

29.<13

4.01

11.18 20.90

8.3<1

57.12

4.46

8.99 16.&1

4.83

47.79

2.93

13.00

S.71

75.00

.66S7

5.50

3.85

24.64

3.39

105.00 120.00

S.58

5.31

20.44

2.35

20.16 38.37

1.25

<1.11

39.5<1

2.01

16.V 31.99

6.06

2.4B

16.75

60.nn

1<1.00 26.36

8.Bol

3.00

32.3<1

'l.S!

12.36

75.nn

13.57

3.11

U.34

7.05

90.00

26.11

P

5.16

sO.on

10.81

2.73

<1.22

1<1.9<1

20.74

2.59

3.<10

3.59

10.50

120.00

2.3a

7.38

2.69

75.00

12.42

8.84

105.00

21'3

6.25

12.52

6i'.72

Kp -UAl...UES - 2.71 3.53

25.21

5.24

7.38

1.0000

-

20.74

<1.36

10.39

34.B3

• REMARKI THE KP-UAl...UE IS O\)ER 1000 l-lHEN 999.99 APFEAR5

16.83

3.57

6.09

57.61

2.05

5.01

13.<17

2.B9

8.5<1

29.65

60.00

11.
In.S2

2.3n

4.97

48.49

1/2

9.~

4.10

10.46

8.16

60.00

6.9<1

25.00

1/3

8.74

120.nn

4.no

40.40

,'.

7.23

.5noo

5.55

20.85

al.

5.92

7.38 8.B7

4.36

17.22

01.

30.59

6.28

&.32

3.J6

33.27

5"6

2S.56

5.30

75.00

14.05

~~-

21.16

5.26

.3333

90.00

27.09

1"3

17.33

3.30

11.32

- - K -U~!..UES

13.99

13.67

21.12

..

11.16

2.6n

105.00

.

16.27

10.60

.120.00

999.99 APPEARS

10.19 13.69

&n.no

105. no

~N

7.28

11.41

<1.78

20.78

1<1.77

7.<13

9.<1<1

7.~

.66S?

5.38

7.72

3.80

17.<12

2.92

8.0<1

4.53

6.23

2.83

8.93

3.60

15.03

8.72

6.:n

3.91 3.7B

Ins. no

26.<19

6n.on

:••t..:..;~;~~u~ ..:••;t~.:

1.001l0

1<1.72 27.81

5.70

10.13 IB.29

15.nn

120.00 1.0n1l0

12.32 23.16

<1.15

B.G4 IS.
10.75

10.2<1 19.n7

3.5n

7.35

.50GO

8.<13

6.82 120.00

3.51

12.9<1

10.31

6.56

3.0& 3.30

6.2n

12.01

5.55

-VALUES-

10.76

2.54

22.35

2.92

2.65

5.19

6.<18

18.69

2.41

8.87

2.45

7.nl

so.on

2.27

<1.30

5.<17

12.69

6.65

1.91 2.02

7.23

16.0<1

2.67

4.95

75.00

3.53

<1.73

5.15

<1.56 5.10

S.80

13.69

10.27

-~

4.nl

3.B7

IDS. on

4.06

120.no

3.52

leo. no

11.62

5.<14

2/3

6.93

3.<15

4.70

.......... _

.66i7

7.71

9.76

3.76

3.08

90.00

B.18

75.00

• REtlARKI THE KP-VALUE 15 OVER 10110

.5000

2.68

4.olS

5.13 6.52

1/2

'UALUE5 - -

2.68

2.92

4.37 12.59

3.99

'.77

105.00

.500n

5.50

8.71

3.65 5.65

3.54

11'3 --- K

P

4.76

IDS. liD

7.12

3.31

2.82

120.00

90.00

2.n9

105.00

10.34

01.

-VAl.UES - -

51l.0n

".

al.

1"2

--- K

2.51

90.00

1.0000

2/3

GO.on

105.00

...67

1/2

2.GB

75.00

.3333

1,1'::;

35° and k

.......................... :..t.:.;~;~~ .•~.:•. ;;~.:

:..t:.;~;~e ..~.:..:;;.:

:..!.:.~: ~e ..~ .:..:~~.:

--- Kp -\JALUES - -

0.25 and 0.30, and for ¢

30° and k

•.......................

..........................

.........................

:..~.: .;~;e~.f..:.• ~~~.:

v.

Coefficients of passive earth pressure Kp for q, = 0 and 0.05

0.05, 0.10, 0.15 and 0.20

192.65

14.28

25.35

32.3E:

4a.15

59.3<1

102.81 6.6<1

8.68

1<1.09

17.Gl

21.73

7.04

9.29

12.08

IS.50

19.62

24.51

3Q.lN

11.28

5.00

1<1.92

19.41

24.93

31.58

39.<19

<18.73

2n.72

27.45

35.81

157.09

!S
<14.22 • REMARKt THE kP-UALUE IS OVER

11.12

5a.36 125.3:'

10nO~HEtI

999.99 APPEARS

216

217

Coefficients of passive earth pressure Kp for ¢

0.10,0.15,0.20 and 0.25

35° and k

......._..............••

......................... 1"6

BI,

B/~

5/5

2.80

2.G;

3.20

3.30

3.88

4.55

12.80

1~.00

15.66

20,39

25.49

J.se

5.52

6.74

8.17

9.82

7.02

8.65

10.S?

12.83

18.27

'22.9&

28.S1

37.75

47.B9

SO.04

74.34

10.0S

12.23

3.58

4.52

105.00

B.61

11.19

120.0D

17.17

22.60

29.<10

3.27

4.18

5.29

6.5B

3.87

4.SS

3.39

5.15

El.28

7.65

5.20

B.~

10.6

12.~

1...3

.3333

10.80

5.4~

13.27

90.00

3.72

4.85

7.62

10.01

28.24

37.62

38.54

79.50

6.24

12.31

7.91

9.91

12.~lB

Hi.75

20.04

25.24

2.84

4.43

5.48

6.73

8.19

9.90

75.00

3.47

5.56

6.94

8.66

10.56

12.9C

17.97

105.00

10.83

14.04

120.00

21.810

28.74

.5000 75.00

25.57

3.18

4.oS

4.08

6.48

8.51

6.80

9.05

11.85

15.31

10.8a

14.50

19.03

24.59

105.00

26.66

120.00

57.03

5.'"

~

.3333

10.99

.."

.5000

37.66

48.92

62.SB

79.36

9El.17

4.74

6.13

7.62

9.B6

12.31

15.2£0

13.2S

16.66

'20.65

9.74

12.70

17.35

22.87

6.33

8.35

14.07

10.64

20.53

75.M

4.79

32.55

90.00

7.33

120.00

'27.02

17.63

21.89

19.49

24.52

30.45

:31.37

39.49

14.02

1.0000

1-'3

3.74

4.09

5.05

6.eo

G.37

B.03

10.10

3.49 4.31 6.22

12.57

6.70

6.8S

.5000 75.00

3.08

4.00

3.34

5.15

36.50

6.63

8.45

20.32

27.58

e.49

3.04

3.18

3.99

15.57

105.00

4.S0

6.18

7.65

S.71

11.50

14.S9

3.20

60.00

2.69

3.42

90.00

4.60

.3333

4.21

.5000

10.66

60.00

S.13

75.00

3.79 5.57

7.S1

3.23

24.52

'30.63

75.00

el.2B

10.35

15.a9

.GSS7

60.00

SO.OO

7.03

9.43

20.69

26.24

32.85

12.4J

16.79

37.67

47.S0

SO.2l

IDS. on

35.26

BO.16

102.27

12E-00

J7.64

22.1E

16.12

6.14

SO.OO

4.50 S.28

8.54

90.00

10.01

13.64

18.21

23.88

105.00

25.03

33.47

44.05

120.00

53.43

71.65

14.83

30.78 3S.37 5S.9S

• REHFlRIi.l THE KP-UALUE IS OVER 11)00 IIHEN 899.33 APPEARS

19.23 39.1E

8.21

10.09

10.52

13.0e

5.74

9.22

11.24

12.45

15.49

19.17

'24.68

5.39

6.68

10.03

6.77

8.49

13.oa

10.01

H!..72

11.83

22.32

5.04

2El.35

35.65

10.09

12.47

1(1.59

.5000

13.30

J2.:n

15.37

28.71 12S.27

75.00

S.02

9n.00

'.5'3

8.02

105.00

17.44

17.76

23.51

17.64

23.32

39.79

52.76

69.17

89.3~

6.1S

7.93

10.18

12.96

1;.35

75.00

4.87

8.52

11.18

14.57

14.51

27.38

37.02

65.13

45.90

63.51

85.70

154.22

200.50

.B.04

10.08

12.5:

60.00

4.29

5.78

7.70

10.16

13.30

17.'H

21.9.9

14.86

19.65

75.00

14.07

8.es

'25.3~

78.51 Je.30

30.59

32.67

39.29

49.75 S2.04 197.91

• REMARK: THE l(f'-UALUE 15 OUEr. 1000 IIHEli 939.99 APPEARS

3.57

25.65

47.09

105.00

100.11

1'20.00

15.43

60.00

'15.47 5.17

14.07

19.57

31.69

44.82

60.00

5.70 5.92

IS.o4

120.00

44.03

60.00

5.04

75.00

7.41

10.34

90.00

12.70

17.94

105.• 00

25.38

60.00

7.'25

75.00

11.3'2

90.00

20.29

119.25 284.93

9.68

13.00

17.26

22.59

29.14

7.6'2

10.55

14.45

19.53

2S.oB

34.36

13.09

18.32

25.31

34.<12

46.20

61.06

79.32

70.a3

94.6a

167.54

226.18

300.20

391.16

101.60

129 57 1.0000

11'3

26.19

37.0'2

61.83

87.67

122.13

16.54

23.06

60.00

7.45

14.44

18.89

24.38

31.03

75.00

U.67

'23.10

30.31

39.17

49.90

90.00

55.94

72.37

92.30

105.00

91.13

laO.04

10.43

17.32

29.24

13.65

17.59

22.38

198.44

14.81

29.72 85.54 103.18

21'3

5a.65

7.31

10.20

7.63

10.12 14.77

86.OS

8.96

50.58 lea.30

22.73 40.84

105.00

59.91

84.29

120.00

143.55

202.19

50S.07

210.66 659.81

11'6

11'3

3.'28

4.01

2.66

11'2

12.25

75.00

3.16

3.9S

20.09

90.00

4.38

5.69

J3.07

4.95

6.'25

7.38

9.55

21'3

50'6

6.04

7.42

120.00

15.11

'20.90

28.74

39.03

52.55

6.04

7.89

10.22

60.00

3.50

4.60

75.00

4.83

6.23

4a.ea

90.00

.3333

9.91 15.89 39.51'

a3.20

13.17

24.35

16.61

4.91

69.9'2 13.18

16.85

19.05

24.68

86.80

65.5a

85.6"

105.00

13.62

19.10

26.54

36.37

43.26

65.87

203.53

120.00

30.60

43.74

61.62

85.1S

116.18

15t:i.12

10.07

13.33

17.47

22.66

19.63

25.85

14.S7

19.61

26.01

34.01

22.3~

92.34

121.07

caO.2S

289.32

17.26

22.76

23.60

34.60

45.20

46.20

69.74

94.63

60.00

4.10

75.00

5.76

126.32

48.19

42.81

56.97

55.41

76.71

102.60

116.61

15B.65

278.30

27S.35 381.04

670.14

7.55

7.SS

10.84

25.24

26.25

31>.75

92.34

122.73

61.38

8S.60

164.7n

221.71

233.35

60.00

8.86

9.45

12.86

17.26

22.94

30.03

10.12

14.06

19.25

26.07

17.5'2

24.52

33.88

46.11 3".43

90.00

1.0000

18.40

45.58

120.00

105.00

.6667

165.76 397.25

27.44

5.60

13.a9

80.54

166.36 226.'21

• REt1ARt:f TH( KP-UALUE IS OVER 1000 IIHEN 999.99 APPEARS

.5000

33.55 59.16

lS4.82

34.15

31.36

381.00

117.45

155.ao

19.39 24.91

73.67 132.69

146.99 '205.19

60.00

90.60

34.17

121.36

56.55 101.83

116.40

'26.S6

10.99

47.42

42.60 76.72

- - Kp-UALUES -

6.00

39.41

6;.02

36.40

31.58 56.83

.:..t.:.~~;~~ ..~;.:..;;~.:.

11'2

4.89

20.33

163.24

• REMARK: THE KP-UALut IS DUrP. 1000 I-IHEN 399.99 APPEARS

11.11

36.15

29.10

91.74 21£:.79

75.00

85.60

16.19

69.35 164.94

90.00

51.22 62.31

51.65 122.39

20.90

18.72

105.00

37.S8 8S.S2

33.06

6.10

120.00

32.35

12.97

6.35

105.00

19.73

24.00

32.78

.5000

7.73

38.57

90.00

39.0$

120.00 1.000n

11.12 13.13

60.00

75.00

12.13

4.32

l.nooo

13.58 '29.70

4.73

1'20.00

15.81

90.00

61.55

21.7;'

10.32 21.95

3.63

105.00

75.00

a5.S$

16.23

7.80 16.15

60.00

19.96

so.oo

75.00

12.07 15.72

75.'23

105.00

.3333

7.84

20.30

7.9S

21.42

5.gs

9.69

35.74

6o.Bl

31.B2

120.00

4.88 6.29

7.57

7.67

4.02

7.53

4.02 5.04

120.00

--- Kp-UALUES -

3.36

6.73

5.90

26.24

5.80

5.7B

1/6

5.-&

59.20

78.48

4.57

21'3

105.00

.:.t..:.~~;~~ .•~.: ;~~.:.

J9.34

8.33

.3333

11'2

45.97

• REMARK: THE KP-UALUE IS OVER 1000 IIHEN 999.99 APPEARS

".44

13.18

105.00

12.42

15.<12 31.07

11'3

11.83

120.00

35.BB

5.00

3.26

60.61

2-'3

3.sa

2.71

75.00 90.00

12.30

22.12

23.30

105.00

la.91

12.32

4.38 5.77

4.14

1.0000 19.36

57.52

as.57

36.74

8.02

34.78

H!.72

13.37 .2n.43

2.99

90.00

30.00

120.00 1.0000

6.03

7.59 14.94

59.67

12.88

4.63

.1lSS7

16.76

25.57

75.00

7.79

75.00 90.00

13.37 120.00

3.25

6.4S

10.55

4S.97

2.71

S.24

10.0B

7.67

90.00

2.24

105.00 27.39

12.44

18.ES

--- Kp -UALUES -

o 5.42

5.43

9.05

120.00

120.00

1...3

BI'

6.35

60.00

7.83

'24.37

105.00 10.87

34.27

6/;

4.96

105.00

:.t..:.;;;~~ ~.:.'.;~~.:

3.8S

4.37

3.80

........................ ••

10.51 21.16

120.00

5.34

16.1Q

• REMARK: TH( KP-UAi..U:: 15 OVER 1000 "''H~N 939.33 APPEARS

11.52

4.79

60.00

90.00

15.11 In.45

1"'2

3.10

2.77

lB.9B

4.24

20.72

li2.91

7s.on

.

75.00

3.36

14.56

= 0, 0.05

- - K -U..1LtJES p

11.27

lS.64

21.24

;~~.:

2.75

75.00

11.07

8.42

196.20

60.00

60.00

7.67 a.26

60.00

S.45 6.
1(1.73 16.55

3.61

- - Kp-UALUES - -

9.13

4.57

1'20.00

13.08

19.12

:••!.:.~~;~~ ..:.:

,,'

.3333

3.85

S.04

105.00

10.24

60.00

.6667

.. REMAn,,: THE KP-UALUi:: IS OVER 1000 IJHEN 999.!l9 APPEARS

.........

H!O.OO

8.04

17.91

GO.OO

1"'6

Cl

12.58

IS.35 20.76

6.48

105.00

2"'3

3.63 4.as

29.73

29.48

75.011

2.42

105.00

9.53 SO.OO

14.25

22.17

3.22 75.00

4.42

1/a p

11.17

40° and k

........................ :..!.:.~~;~~ ...~ :.•.. ~.:

--- K ·UAL!JE:5 ---

S.26

0.30, and for ¢

..

19.69

8.14

80.00

3.29

35° and k

:.~~;~~ ..~.:.. ;;~.:

- - Kp -UALUES - -

11.12

12.74

75.oa

.5000

2.78

2.32

6.21

9.28

6.74

60.0G

1/3

75.00


105.00

I,oj;

6/$

--- Kp -UALUES - -

2.35

.:..t

:..!.. :.~~:~e"*.k.: ... ;~;.:

:.!~.:.;;;?~.~ ..:... :~~.: 61,

Coefficients of passive earth pressure K p for ¢ and 0.10

1'2.29

50.80

45.83 61.84

105.~0

'24.57

35.29

49.8'2

6S.'20

120.00

57.83

83.53

118.48

165.0B

304.08

10.18

14.44

20.07

36.96

60.00

74.a~

75.00

134.78

90.00

10.97

15.84

42.61 40.21

55.99

76.64

1'27.01

48.9t·

57.34

19.66

28.40 58.49

83.03

115.70

158.44

213.47

06.72

140.12

199.17

277.75

380.45

5l2.78

• REMARKf THE ~P-UALUE IS OVEr. 1000 I-fHEN 99h.99 APPEAns

168.10

103.'20

136.72

219

................... _..... :.. ....k.:••: t:.:~;~~

1"'3

2/3

-- K

P

61l.GO

2.63

75.00

3.10 4.27

7.28

5.SB

al.

5"

4.se

G.07

8.22

1.69

9.48

15.97

12.32

3.84

4.8S

6.19

7.89

7.18

!t.
12.2S

23.13

30.59 70.18

92.84

13.2?

17.09

90.00

5.83

105.00

S.90

9.39

12.74

17.17

23.07

30.69

40.42

105.00

10.92

120.00

14.02

19.B6

27.S9

38.28

52.18

70.40

93.82

120.00

5.92

7.82

13.36

17.29

B.09

10.90

14.53

lS.2S

25.29

18.10

24.58

33.05

105.00

12.68

19.18

S6.54

88.92

105.00

24.36

83.52

115.67

157.51

211.26

120.00

&3.17

9.96

13.34

17.6S

SO.OO

5.'"

8.05

14.42

19.57

26.28

34.83

75.00

B.S5

12.89

24.79

33.99

45.99

61.41

90.00

16.19

S8.90

93.77

125.72

105.00

3S012

163.94

223.74

35.112

49.16

68.22

.84.35 115.95

156.8B

5.49 7.77

10.69

14.54

19.5B

12.9B

18.14

25.02

34.05

17.5a

22.S:O

45.80

60.72

25.58

31i.l'1

120.00

41.16

53.74

119.35

184.47

60.00

'.77

6.73

9.33

12.77

17.26

23.07

30.43

75.00

6.97

S.B9

13.85

19.11

as.os

35.02

"~..42

59.37

80.00

.5000

5.35

75.00

5.41

24.89

105.00

33.ea

46.02

sa.52

94.24

127.S9

1711.22

163.45 2i!5.30

305.73

62.17

408.01

~3.72

34.42

"9.04

120.00

55.74

81.45

11r..6S

8.83

9.S6

14.23

19.sa

27.45

37.18

10.81

15.47

22.05

30.91

42.56

57.68

.S6S?

6.59

9.20

12.67

17.26

23.19. 30.80

75.00

6.75

9.67

13.63

18.97

25.99

35.18

33.52

40.20

11.47

33.28

22.85 120.00

79.38

1.0000 75.00

76.92

10.25

19.00

103.19

13!:.42

3S.09

57.08

Bl.68

114.69

158.16

214.67

286.54

105.00

37.57

55.60

93.55

136.70

195.93

275.24

379.78

515.65 saS.47

120.00

89.78

133.19

~HEN

62.5Q

83.73

128.37

172.35

307.36

27.41

37.38

50018

30.67

42.49

57.98

77.89

192.68

272.55

55.01

90.00

IS OVEP. 1000

21.70

15.12

120.00

KP~VALUE

45.93

9.74

..............."

ai,

1"'6

1'3 ___ K

75.00

p

1'2 2'3 -UALUES -

3.97

4.94

4.n

6.13

60.00 75.00

104.36

140.24

SO.OO

215.97

290.18

105.00

518.52

120.00

20.64

65.74 154.27

236.85

420.43

S45.81

.5DOO

6.17

27.31

40.33

5B.23

82.68

115.26

15S.14

213.63

60.00

3.79

5.28

7.27

9.90

13.33

17.80

23.51

75.00

5.24

7.42

14.29

19.55

26.40

35.23

90.0D

12.36 16.49

SO.OO 75.00

17.51

24.56

24.20

'1.48 G.51

6.45 9.44

9.08 13.'10

12.57 IB.78

11.05

IG.20

23.34

32.95

105.00

2~.S4

32.qO

47.27

67.15

120.00

51.54

77.aB

112.39

90.00

19.62

9.51

25.92

SO.OO

17.S1

26.28

:l8.27

3S.14

54.09

78.89

8S.32

94.16

127.14

224.70

304.00

23.32

31.1s

35.34

189.12

7.85

9.07

12.77

17.16

1-20.00

26.17

3S.08

57.09

81.80

114.94

GO.OO

3.68

5.17

7.17

9.82

13.31

7.24

10.19

90.00

8.20

12.01

17.19

54.84

80.67

115.66

S.29

8.~4

1'1.48

34.29

.6667

47.51

60.00

4.33

75.00

6.2S

9.19

13.17

IS.n:

22.S1

90.00

93.86

128.99

174.24

105.00

21.01

31.sa

46.35

224.40

309.8S

417.sa

120.00

49.42

74.98

110.17

45.S4

S2.80

27.36

50.80 n.80

14.47

6.16

1.0000

20.98

17.21

.3333

llS.sa

22.26

321.84

128.65

20.02

31.62

IIB.B7

73.92

109.65

159.51

22S.77

99.89

IS1.09

224.23

95.7S

151.61

234.58

354.88

526.80

76S.S3

999.29

1113.23

638.79

967.78 999.S9

999.99

999.S9

40.8S

2':3

al.

5'6

32.a:; 27.41

1I~.34

17.87

23.77

60.00

5.30

7.83

11.47

16.71

Z4.11

35.62

75.00

8.24

12.44

18.69

'17.66

40.S2

62.79

90.00

12.19

39.21

SO.OO 7~.OO

55.2:"

1<1.89

4.22 6.01

90.00

10.53

16.03

24.25

105.00

22.37

35.13

54.15

485.92

S97.05

120.00

58.03

92.57

1401.51

34.46

4D.S3

60.00

5.16

7.72

11.38

12.21

18.47

56.85

83.67

U8.29

169.02

34.20

.5000

75.00

395.03

47.40

S9.SS 181.81

16.99

23.25

53.68

78.<11

221.<1S

333.24

491.81

16.68

24.22

34.63

27.57

40.7S

55.!:5

54.54

81.59

120.00 34.10

49.43

70.30

10.15

15.44

23.12

34.27

58.85

85.77

1'12.57

75.00

10.81

IS.93

25.0S

39.S0

59.17

79.86

119.08

170:1.17

24S.48

90.00

21.04

33.38

5a.D4

7S.67

48.03

77.05

120.90

195.86

129.46 208.64

328.29

505.52

47.83

73.<17

129.37

175.99

105.00 lZ0.00

78.60

122.20

185.29

278.50

409.01

565.03

133.56 21'1.84

49.54

331.79

506.68

758.27

S99.99

999.95

27.46

41.87

62.53

48.37

73.69

110.18

161.52

60.00

11.06

75.00

19.49

17.S2

131.55 231.87

.S6S?

105.00 120.00 1.0000

60.00

10.78

17.30

75.00

18.55

30.50

413.22

597.45

762.30

993.99

S:;3.S9

163.54

a3S.62

13<1.25

104.85

141.83

90.00

16.97

25.56

37.53

53.96

75.03

105.26

143.29

90.00

3S.75

63.42

98.79

150.62

225.29

330.35

474.31

150.14

2
334.45

293.42

105.00

34.51

52.58

17.63

111.38

157.13

217.S5

296.44

105.00

53.16

148.82

231.90

353.74

529.25

776.24

999.99

105.00

90.43

145.82

229.22

352.59

531.6S

7BS.85

959.99

l20.00

246.24

397.24

624.86

561.48

• REMARKI THE KP-UALUE 15 OUER 1000 l.lHElt 999.99 APPEARS

704.97

120.00

82.68

125.89

186.15

267.21

317.25 522.76

• P.£Ml'IRKI THE KP-VALUE IS OUER 1000 l.IH£N 999.99 APPEARS

712.20

120.00

253.71

405.58

S32.21

S99.99

.. REtlARKI THE KP-W1l.,UE IS OVER 1000 J.lHEN SS9.SS APPEARS

95S.99

S2.21

280.03

216.B8 520.97

38.59

71.79 125.16 25';.16

76.23

378.37

90.00

50.07 8S.20 176.:;:8

112.48 157.59

54.49

49.S0

172.!:5

341.35

23.18

93.67

711.15

105.00

39.69

34.04

261.7:;'

181.45

15.60

90.00

J4.7;: 16.73

124.10

26.36

47.99

5'6

27.51

8.86

110.42 256.65

77.50 122.83

54.74

.3333

10.09

10.32

85.45

79.S2

8.93

17.27

35.51

51.37

8.<13

6.75

62.99

37.78

3.70

6.42

11.10

25.85

42.29

2.-3

p -UALUE5 -

SO.OO

45.72

1.0000

1~

--~ K

4.S7

42.57

331.82

421.92

.

1'6

'I.

Kp -UH:'UES - -

2'12.31

.S6G7

326.25

., R£l1ARK: THE KP-UAl.UE 15 outp. 1000 J,lHEN 999.29 APPEARS

.

146.30

31.53

64.60

:.t••:.~;:~~ ..~.: :~;.:

94.S2

23.44

939.59

260.76

59.74

105.00

572.5S

90.00

120.00

_120.00

171.88 402.62

120.00

215.98

128.57

118.44

105.00

158.76

307.2<1

80.04

452.99

SO.OO

.5000

53.14

999.59

756.28 955.99

105.00

75.00

27.31 30.08

B4.65

73S.91

999.99 999.99

44.73

158.56 223.95

13.56 14.31

75.00

56.47

58.53

524.33

90.00

25.81

S8.
18.59

39.77

999.99

90.73

33.,24

34.95

10.S1

58.27

105.00 120.00

17.25

14.70

75.00

.5000

S2.10 S8.68

58.48

5.72

6.36 11.68

89.83

120.00

120.00 .EiS67

66.74

68.82

26.S3

75.00

120.00

75.00

48.94

221.43

.105.00

24.47

35.31

157.34

56. III

17.75

25016

48.78

17.56

90.50

256.S9

s.oa

17.69

33.S3

11.40

62.22

179.83

4.29

12.18

23.24

41.99

123.72

60.00

4.1.1

343.02 508.94

15.72

27.72

83.83

17.71

75.00

187.50

226.77

10.49

17.94

55.88

13.52

90.00

B4.05

147.10

6.91

11.35

22.90

10.25

25.55

54.89

93.60

60.00

16.54

7.73

44.29

35.1S

754.08

109.13

47.5S 81.22

lB6.73

1.0000

34.04

116.76

507.94

12.13

5.79

24.00

32.13

27.75 54.S8

1<:3.49

8.93

36.88

16.73 18.84

335.29

6.61

24.21

83.02

80.16

4.95

105.00

E51.11i

4BO.03

149.08

137.65 217.05

3.76

120.00

19.35

90.00

75.00

41.24

177.17

223.13 330.40

55.51

96.3<1

120.00

13.00 21.19

54.1!

105.00

10.1<1

95.78

38.76

IS.77

559.04

14.46

70.85

27.31

SS9.S9

10.99

30.87

19.0S

3:;.60

B.56

169.59 238.43

8.36

22.88

33.14

J'2

13.20

24.78

995.99

6.40

51.71

14.50

105.0D

1/.3

67.07 177.82

397.23

4.96

16.82

24.53

75.00

3.88

37.32

17.92

60.00

3.02

12.26

10.80

......

60.00

-

47.00 123.35

36.51

12.S6

30.18

32.58

7.94 B.45

.

"

15.43

14.14

84.11

24.24

9.65

2S.59

7.98

Sl.46

SO.OO

9'6

8.90

13.00

G.09 9.15

7.70

18.66

9.31

22.47

4.36

2/.3

H!.94

5.BS

11.02

10.64

6.32

6.15

120.00

4.24

12.07

117.79

223.18

...........

8.93

275.46

.. R£MARK: THE KP-VALUE 15 QUER 1000 I·IHEN 999.99 APFEARS

23.00

75.00

355.80

41.95

17.00

SO.OO

74.13

267.81

12.50

.3333

49.32

G.e5

74S.aS

125.60

32.16

3.91 5.71

11.14

377.58 599.59

18.23

9.15

17.50

105.00

35.37 81.71

8.28

75.00

75.00

.',

S.J6

IS1.61

161'.17

22.8S

3.73

13.44

268.5:; 506.16

141.55

2.91

10.26

';6.45 80.03 115.23

124.79

90.00

75.00

7.78

40.27

117.15

9.52

5.85

23.85

83.53

12.6S

4.3!i

lS.74 27.S4 55.02

.5000

-UA!.l!E5 _

4.96

90.00

120.00

473.69 666.05

58.21

7.64

3.27

328.95

p

3.04

105.00

245.3. 223.92

39.86

105.00

1.0000

149.S1

26.89

10.08

60.00

SO.OO

31.40 52.69

17.85

76.39

12.27

.3333

120.00

11.69

7.89

94.80

23.04

105.00

75.00

6.12

70.S3

16.66

40.62 92.02

S7.20

6.16

51.95

19.HI

75.00

245.52

4B.12

4.S3

37.80

12.00

29.ge 66.26

33.7S

4.82

27.14

8.S0 13.32

21.54

46.80

23.29

3.99

IS.2S

21.57

127.02

3.78

13.48

16.13

15.84

3.21

120.00

12.00

10.S6

2.98

30.78

8.93

84.39

37.14

2.58

105.00

13.81

56.09 98.0;7

-UALUES -

3.18

10.65

:..~.:.~;;~e.:..: .. :!~.:

1'2

_~_Kp

2.51

8.15

14S.S8

.

1/.3

!Y6

___ K

S.32

57.20

.

~

:N..~.:.:;:~~.,,~ . :•• ;~~.:

7.07

:.... !.:.~~;?~u~ . :.. ;~~.:

'I.

..R••••R

'

60.00

.120.00

.. REtlARK: THE KP-VALUE IS OllER 1000 miEN 999.9S Af>PEAP.5

999.99 APPEARS

••m

379.08

113.59

S.16

0,0.05,0.10 and 0.15

1'2 2'3 -l',l!.uE5 - -

32.69

75.00

46.99

224.85

S7.83 161.82

SO.OO

120.00

4.63

90.00

82.70

.3333

45° and k

90.00

60.00

105.00

., REt1ARK: THE

35.52

58.11

222.77 237.11

105.00

10.53

7.59

8.86

50.37

10.25

25.SS

120.00

eoa.50

4.00

60.00

.3333

4.95

15.7·1

13.22

2S.13

76.52

5.04

4.46

60.51

55.53

3.86

9.55

18.63

39.58

75.00

2D.80

5.138

42.5&

27.71

12.72

16.04

4.37

29.50

75.00

10.03

3.35

14.56

1.0000

3.90

6.82

10.97

24.12

3.06

SO.OO

10.23

8.21

17.0B

p

6D.00

75.00

7.85

S.lI

9.78

11.86

1'3 __.K

9.38

19.15

5.99

4.50

7.05

105.00

8/1

'I.

32.1i9

4.5<1

75.00 90.00

S.17

.•m

7.57

5.47

52.41

13.33

S.10

3.04

17.35

10.02

4.se

4.15

39.135

7.4;;

4.00

75.00

12.97

75.00

3.24

90.00

28.22

90.00

2.59

20.57

7.13

.5000

So'S

SD.DO

1'1.57

105.00

2.-3

- - Kp ~UALUES -

S.25

105.00

120.00

..........................

:..t.:..:·;:~~ ..~.: ....~.:

1~

1'3

'I.

~UALUES-

120.00

130.00

.3333

4.01

3.2S

.............................. ,...

:...~ ..:.~~:~~ ...~.: ...:;~.:

~~.:

u.

Ill.

'I. o

Coefficients of passive earth pressure Kp for ¢

0.15, 0.20, 0.25 and 0.30

40° and k

Coefficients of passive earth pressure Kp for¢

97.S6

., REMARK: THE: KP-UALUE IS OUER 1000 l.lHEN 5S9.99 APPEARS

993.99

.220

'221

Coefficients of passive earth pressure Kp for '" and k = 0

45° and k = 0.20, 0.25 and 0.30, and for '"

. . . . 0;

1/3 -

2.9S

o

1....2 K

p

~;~.:

1/3 --- K

J.SS

'l.sa

6.'15

7.53

10.70

15.27

14.57

21.72

32.24

47.48

93.04

124.47

183.79

8.40

11.95

17.06

24.23

12.85

18.90

27.62

8.51

11.30

3.B4

15.05

4.9B

105.00 120.00

22.70 4.13

.JJJJ

S.8B

5.86

7.41

120.00 34.11

.3333

50.00

4.05

5.80

75.00

5.71

8.55

120.00

.5000

56.21

60.00 7.82

34.35

53.49

90.52

142.73

220.67

7.60

11.e8

16.6"1

105.00

125.37

60.00

J20.00 to

2S6.98 725.24

105.00

.5000

35.30

60.00

190.02

12.73

21.04

335.93

502.89

738.52

7.48

11.18

11;;.60

24.43

35.72

51.55

4.95

27.39

120.00 73.17

135.56

125.06

190.54

285.10

418.57

517."l2

775.43

999.99

15.11

23.01

34.59

25.47 50.93

503.91

59.52

45.01 764.74

999.S9

959.93

93.97

136.97

120.00

121.27 200.22

75.00

41.47

29.28

6a.93 142.85

226.53

238.7&

389.19

617.50

....... " :.t*"':..

230.59

999.99

• REMARK: THE KP-VALUE IS OVER 1000

• oj)

2/.3

SIE

--- Kp-VALUES E.SI B.GS

11.59

4.98

10.47

'10

1/6

1"'3

6.E2

3.97

5.72

8.28

1'1'.94

17.20

5.55

8.38

12.59

18.77

27.82

.3333

24.77

139.67

60.00

59.48

75.00

75.00

32.81

52.14

81.41

52.24

86.41

139.16

219.02

27S.5B

105.00

751.59

120.00

11.54

17.82

27.30

336.81

508.04

82.23

123.49

191.05

288.03

425.9:;

33G.55

518.87

783.45

999.99

34.71

51.58

75.82

39.12

59.69

.5000

.5000

13.58 105.00

29.58

120.00

7;r.55

60.00

6.08

49.43

79.45

132.67 214.45

9.87

10S.00 120.00

19.05

31.35

50.31

7B.55

120.75

181.99

43.49

72.39

116.82

183.51

282.46

426.40

599.99

955.18

995.S9

599.99 999.99

999.99 APPEARS

142.08

2S.S5

46.20

72.79

90.00

34.99

58.29

94.25

148.70

222.G6

372.10

105.00

1.0000

12.84

21.76

1"'2

36.45

634.43 999.99

999.99

48.28

79.21

199.59

228.81

379.37

75.00

19.27

33.50

57.48

43.98

77.82

134.87

105.00

U8.05

210.52

120.00

371.90

SG5.22

999.99

72.13

125.71

5...-&

11.40 42.10

68.35

74.33

125.44

208.68

5.54

8.34

12.B2

19.82

298.92 999.99

120.00

829.18

995.99

999.99

S99.99

999.99

999.99

999.99

999.99

999.99

999.99

44.7..

75.00 90.00

1;.31 7d
105.00 120.00

40.55

70.79

47.39

71.77

60.00

5.30

8.12

.3333

20.0S

S9.45

242.57

75.00

573.53

90.00

75.20

131.89

226.70

999.99

105.00

69.67

125.03

222.57

120.00

234.22

414.05

714.80

999.99 SS9.S9

599.5:':

120.00

217.74

396.51

702.74

193.24

60.00

9.67

28.67

75.00

19.69

158.64

393.87

75.00

18.36

56.76

ao.oo

45.08

934.71

90.00

75.46

999.95

105.00

204.38

78.05

135.65 213.45

228.14

358.32

601.53

S9S.S5

.5000

999.99 275.11

20.18

98.06

119.89

2...3

5...-&

12.42

18.05

73.10

126.33

213.20

351.22

173.77

300.41

507.06

835.42 9SS.9!!

427.79 875.83

1.0000

31.72

50.15

999.99

90.00

999.99

105.00

599.99

120.00

IS OUER 1000 lJHO:N 999.99 APpeARS

47.23

645.31

91.38

16G.77 <155.40

SS9.99 205.72

134.02

229.26

383.51

999.99

999.99

999.59

599.99

999.9!;

61.05

104.72

175.35

286.41

455.45 933.47

999.99

599.59

75.00

39.42

71016

125.10

214.60

359.35

587.00

297.43

510.26

85<1.71

999.99

255.47

461.80

812.48

999.99

999.99

999.99

809.77

999.99 999.99

SS9.99

999.99

999.99 gs9.99

.. ..

.

3.40

1/6

1"'3

4.64

S.33

1/2

2/.3

75.00 90.00

sal.S4 ;-8.<12

.3333

8.75

12.59

5/6

209.37

353.92 32.03

IB.40

27.16

70.96

112.72

6.25 7.00

105.00

14.51

120.00

39.41

60.00

5.21

10.82

17.15

69.56

121.60

114.05

8.05

12.64

211.1'2

13.52

22.39

37.11

51.0!

233.46

31'6.52

40.77

73.47

129.94

225.18

381.99

633.5a

999.:9

228.23

407.35

709.45

999.99

999.99

9S9.S~

141.93

240.59

82.Se:

101.43

11.37

18.95

255.45

75.00

12.01

20.78

35.76

25.95

46.49

82.01

131.1f

.SIIOO

60.00

238.23 391.79

627.82

90.00

647.93

999.59

105.00

999.99 999.99

999.99

BO.B3

211.59

163.63 269.05

431.20

538.17

999.99

133016 229.71

999.59

390.28

69a.23

999.99

9.45

IS.55

28.49

4a.S9

17.88 411.68

999.99

75.00 90.00

999.99

599.99

99S.S9

999.99

999.99

• P.EtlARK; THE KP-UALUE IS DUER 1000 \.lliEI1999.99 APPEARS

999.59

132.31

230.16

109.36 201018 359.74

627017

344.99 636.25 9.9S.89

458.60 S60.36

1.0000

6~5.31

992.99

999.99

273.92 645.78

999.99

18.32

33.73

GII.34

105.23

37.52

69.11

123.66

215.66

164.24

294.00

512.87

448.50

803.13

89.09

272.63

217.as 155.26

74.28

16S.95 39S.SF

654.47

211.94

177.31

511.79 863.07

124.03 221.12

75.00

215.19 36'2.92 598.41

790.37

51.59

120.00 .6667

999.99

295.84

995.99 999.99

80.02

124.51

51.05

75.00

999.99

641.51 999.99

48.38

120.00

60.84

999.95

393.40

707.25

25B.I0 3B4.51

105.00

35.93

11.52

127.5B

100.44 141.42

!l99.9::

708.11

B48.58

50.58

!3"S.9!:

224.79 378.26

18.78

971.32

224.41 706.77

622.33

130.83 41~.05

354.35

374.54

131.71

959.99 909.99

74.67 E31.93

60.00

225.26 357.07

105.00

2S.45

le9.53

41.89

1'2.28

76.31 151.92

138.70

p

575.19

127.55

120.00

31.42 60.13

49.85

- - K .UALUE5 ---

350.73

120.00

267.69

~P-VALUE

34.74

........."

105.00

GO.oo

285.45

ega.59

3;7.13

230.94

S;!9.e:

180.95

347.55 564.91

:.t~:.;~~~~ ~.: ;;~.:

16.82

SO.52

999.99

363.13

156. IE:

140.92

11.95

999.99

19.98

112.08

106.88

• REMARK: THE KP-UALUE IS OUER 1000 tJHEN 999.99 APPEARS

27.016

268.41 842.17

25.74

68.64

57.12

93.S1

45.,;

543.61

12.27

207.57

1 ••

8.73

5"'6

-_

22.GB

28.86

;~;.: 1"'2

60.00

84.20

7.39 13.12

• REMARK: THE

845.07

959.99

12.73

S8.41

128.02 224.03 402.65

42.88 105.00

60.00

81G.5a

209.03

47.98

75.00

9S5.S6 999.99

905.89

171.44

49.47

60.00 75.00

71.53 223.34

280.76 "l41.96

173.35

123.52

19.11

60.00

104.36

468.21

24.25

100.£2

999.99

105.00

120.00

95.83

2a3.64

243.02

75.87 235.62

7.12

120.00

953.59

75.00

135.29

.G667

SSS.99

261.58

704.10

90.00

999.99

613.78

27.39

36.22

999.99 999.99

90.00

~

43.01 131.40

---Xp -UAU,-E5 -

110.02

30.78

SO.OO

27.37

406.85

105.00

..

17.03

.5000

8.65

17.33

25.03

250.51

362.58

:.!..:.;~~e~"," ~.: 2/3

105.00

942.06

9SS.99

..

p -VALUES - 8.50 11.9S

.3333

999.99 1~.n

133.48

EO.OO

599.84

705.43 999.99

29.04

999.99

• REMRKt THE KP-UALUE IS OVER 1000 llHEI1 999.99 APPEARS

415.44

711.11

999.99

511.S;;

239.32

60.00

599.99

345.99

17.22

120.00

538.56

229.30

90.00

224.02 370.81 134.85

,0"

7.55

75.00

112.17

B.20

345.39

418.13

117.1B

17.25

5.38

2<11.16

• REMARK: THE KP-VALUE IS Durn 1000 llHEN 999.99 APPEARS

380.67

9.B3

1.00PO.

72.02

1.0000

503.23

804.41

43.72

105.00

631.15

41.S3

105.00

.6667

90.00 20.29

120.00

105.00 120.00

592.40

90.00 105.00

277.12 e69.73

60.00

246.08

535.51

,.

4.65

105.00

EO.OO

55.08

4.6S

24.)E;

276.51

75.00

120.00

-- K

60.00

15.17

82.18

63.43

350.30

... 50.00 k..

&f. 17.28

75.00

33.79

~HEN

................ ................"

.

~;~~e ~.: ;~~.:

264.38 620.10

228."l2 223.84

999.99 APPEARS

117.63

105.00

129.88

4<1<1.01

73.03

35.26

37.44 87.70

88.70

180.12

11.13

147.42

.5000


1/2

9019 7.27

554.64

74.07

1/3

6.28 75.00 90.00

103.82

111.10

68.3B

73.25

~~~.:

- - Kp-VALUES

124.48 208.86 344.37

25.40

42.67

116

25.01

5.47

999.99

25.59

17.82

74.50

321.10

10.14

1.0000

120.00

S8.58

338.77

32.05

324.79

60.B3

217.13

16.24

185.37

12.11

179.02

G.26

.6667

105.00

272.04

105.00

282.17

a.56 13.56

.

" ,

&f.

58.52

219.84

90.00

2.90

40.53

52.81

75.00

IS.98

27.72

6.25 5.21

'f.

5/6

-

67.41

34.75

140.94

51.<18

29.92

24.S1

a8.4S

25.77

Ill.,m

18.84

17.13

33.58

IS.59

RtI1AR": THE KP-VALUE IS OUER 1000 "lHEN

.3333

101.53 271.58

32.72

18.39

21.20

21.45

22.34

219.54

204.43

44.94

53.91

7.59

87.03

75.50

105. Oil

•

24.33

183.59 'l9T.70

15.28 10.50

23.76

213

1/2

115.97

54.35

51.6S as.17

.£iSG?

1.0000

123.83

11.99 22.83

120.00

17.12 31.47

9.92

15.72 105.00

14.25

105.00

.

:..t:.;e~ee~.~.:

--- Kp-VIl!..UO:5

15.34

B.58

8.91

3.5B SO.110

p -VALUES ---

5.48

0.05, 0.10, 0.15 and 0.20

:.t.:.;e~~~ ..~.: .. ;~;.:

2....3

-VALUES -

50° and k

...........................

.

:...t :.~;~ ~e ..~~:

'f. 01.

Coefficients of passive earth pressure Kp for '"

SSS.S5

99~.9S

871.35

770.63 • REMARK: WE KP-VALUE IS OUEr. 1000 tJliEN 959.!l9 APreAR5

999.99

999.99

999.99

999.59

223

222 Coefficients of passive earth pressure K p for ¢

...........u

.. ell

II

50.00

k"..

:..t.:.:~;~~ ...~:..;;~.:

.25"

'"

oto GO.DO

4.63

75.00

6.20

9.12

B.8S

-

27.SS

60.00

3.35

4.&3

6.37

8.91

&.15

15.00

32.95

:

2"'3

t•.:•• ~~~~~

Blo

12.80

1...3

'10

21.31

-1.92

G.B5

10.67

17.04

27.48

44.55

72.11

115.51

90.00

6.69

10.S!

16.91

27.48

44.93

73.27

118.30

-1.65

-1.71

14.13

23.94

40.48

S8.S1

115.03

1911.]"

308.S4

105.00

13.15

23.52

40.1S

68.54

lls.01

IS3.4e

31S.s:i

90.00

-1.59

-1.64

120.00

38.21

S8.33

leG.S4

i!1l9.S4

357.10

595.74

971.03

120.00

37.13

G7.10

119.6B

209.71

350.28

606.02

999.99

105.00

-1.59

12.60

20.15

32.33

51.87

82.53

5.13

7.9S

8.0D

39.65

72.27

129.05

225.57

385.72 644.82

120.00

121.0S

224.54

404.65

710.79

899.99

11.22

Bol.JS

142.75 237.56,366.71

52.10

18.84 35.53

11.72 25.23


BI.
1<12.19

242.89

105.00

&5.95

122.04

219.S7

385.60

660018

120.00

206.14

384.05

S93.73

999.99

999. 9~

Si".73

131.45

230.62

394.81

105.00

106.38

197.99

357.43

6e8.33

75.00

5.04

7.89

90.00

15.57

27.S
105.00

38.52

71.07

120.00 117.61

220.84

B2.44

SO.OO

SSS.as

454.85

48.70

56.04

G26.1B

138.12 ;::79.72

222.S3

28.:U

31.52

17.85

as.S3 171.93

137.04

16.31

17.40

120.00

999.99

999.99

9.22

75.00

1.0000

999.99

166.90 278.68

60.00

3S.59

60.00 75.00

105.00

SO.OO

.3333

22.25

28.00

.6667

86.71

U>I.87

512.83

441.46 798.07

999.99

999.9S 99S.99

999.99

-1.55

-1.59

-1.G2

-1.6G

-1.4S

-J.52

-1.55

-1.89 -2.01

-2.09

~.OO

-1.60

-1.~

-1.~

-1.~

-l.n

-1.82

-l.SS

90.00

-1.43

-1.GO

-1.64

-1.68

-1.73

-1.80

-1.79

-1.S3

105.00

-2.01

-1.55

-2018

-2.31

120.no

-1.S4 -Z.30

120.00

-1.78

-1.85

-1.S2

Sl.1S

104.40

1~.7S

265.63

SO.OO

142.44

244.85

412.10

678.H:

75.00

-1.6S

-1.7'1

-1.BO

-1.85

-2.01

75.00

105.00

120.05

385.71

665.SS

SS9.9a

90.00

-1.S1

-1.SS

-1.71

-1.75

-1.88

90.00

120.00

200.35 377.83

120.00

-1.71

-1.78

60.00

-1.89

5.00 IS.92

SS9.99 28.13

31.02

'18.BO

55.6S 130.59


SSS.SS

75.00

139.40

228.54

283.13

'Iss. Sol

398.06

S71.2S

992.99

75.00

999.53

90.00

999.59

159.11

.S6S7

-1.91 -2.13

-1.73

-1.77

105.00

-1.64

182.10

307.29

50S.47

120.00

-1.72

121.85 215.95

373.0B

Ii2S.6e

999.59

SO. 00

-1.SO!

-1.7S

-2.13

513.'12

B87.25 999.99

999.99

~.n

~.n

230.04

'134.42

999.99

999.99

999.95

90.00

120.00

729.04

999.99

999.99 gea.59

999.99

999.99 999.93

105.00

-1.S9

-1.74

-1.79

120.00

-1.75

-1.81

-1.95

~oo

• REtulRK: TIlE KP-VALUE IS OVER 1000 l.lHEtl' 599.99 APPEARS

-1.81

-1.81

-2.20

-e.27

4l =

'" -1.55

-1.82

-2.20

120.00

.5000

-1.S7

30.00

-1.76

-1.23 -1.15

-1.26 -1.18

-1.S0

-1.70

-1.79

-1.39

-1.46

-1.51

90.00

-1.28

-1.83

-1.47

-1.50

-1.37

-1.40

-1.44

-1.47

-1.S1

-1.S7

-1.50

-1.54

-2.27

-1.90

-1.45

-2.19

-2.26

-1.62

-1.64

-1.48

-1.42

1.0000

-1.se

120.00

-1.50

75.00

-1.S2 -1.se

-1.55

-1.48

-1.51

-1.54

-1.92

-1.99 -1.64

-1.S9

-1.7'3

-I.SO

-1.64

-1.67

-1.57

-LSD

-1.S8

.'.

-1.53

-2.03

50.00

-1.47

90.00

-1.18

105.00

-1.05

-1.40

-1.64

75.00

-1.33 -1.22

-1.24

-1.26

120.00

-.92

-.94

-2.0S

SO.OO

-1.48

-1.59

-1.S8

-1.S6

-1.70

75.00

-1.27

-1.34

-1.39

-1.31

-.1.33

-1.35

-1.38

-1.24

-1.26

-1.28

-1.90

75.00

-1.42

-1.48

-1.S0

-1.35

90.00

-1.34

-1.39

-1.27

105.00 120.00 .5000

-1.38

-1.27

-1.29

-1.30

-1.94

-2.02

-2.09

-2.15

-1.64

-1.69

-1.7'1

-1.78

-1.'10

-1.42

-1.44

-1.32

.6SS7

-1.25

-1.2S

-1.30

-1.94

-1.91

-1.98

-2.04

2r.I

-1.51

-1.57

-1.63

-1.S9

-1.74

-1.79

-1.B3

90.00

-1.42

-1.47

-1.52

-1.55

-1.S0

-1.63

-1.S6

105.00

-1.38

-1.42

120.00

-1.30

-1.33

1.0000

-1.78

-1.10

-1.12

-.98

-.98

~1.00

-1.63

-1.08

-1.72 -1.43

-1.21

-1.25

-1.10

-1.J3

-.99

-1.8S -1.51

-1.25

-1.2B

-1.12

-1.14

-1.15 -1.02

-1.82

-1.SB

-1.93

-1.48

-1.51

-1.54

-1.(11

-1.46

-1.14 -1.02

-1.03

-1.B7

-1.93

-1.52

-1.04 -1.98 .-1.59

-1.17

60.00

-1.59

-1.88

-1.77

-1.85

-1.92

-1.S9

-2.04

~.OO

-1.~

-1.~

-1.~

-1.~

~.~

-1.81

-1.~

-1.33

-1.3S

-1.39

-1.0B

-1.10

-1.11

-1.70

-1.15

-1.34 -2.15

-1.78

-1.01

90.00

-1.23

-1.72

-.94

105.00

-1.47

-1.77

-1.74 -1.69

-1.22

75.00 -1.36

-1.37

11'2

-1.53

-1.'17

-1.34

-1.71 -1.66

.:

-1.08

-1.S2

-1.22

105.00

-1.95

-1.41

-1.20

-1.2B

-1.59

~;;~~ 1"'3

-1.20

-1.53

-1.7'1

__ N -VALVES Ac

-1.24

-1.4S

-1.66

-1.48 -1.45

.:••••••t••:••

-1.73

75.00

-1.61 -1.54

105.00

-1.17

-1.77

-1.B7 -1.70

-1.51

-1.96

-1.21

-1.67

-J.63

-2.12

-1.92

-1.S3

60.00

-1.59

-1.'19

-1.57

-1.22

-1.52

-2.04

-1.95

-1.14

-1.30

-1.B3

-1.46

120.00

SO.OO

-1.U'

-1.78

-1.95

105.00

IDS. 00

90.00

-1.59

-1.42

-1.14

120.00

-1.56

-1.45

~.07

-1.29

-l.sa

-1.53

-1.87

-1.93

-1.97

-1.S4

-1.39

~.oe

-1.91

-2.19

-1.S1

-1.78

•

-1.B'I

-2.12 -1.75

GO.OO

-1.90

20'3

-1.50

120.00

-1.85

-Z.3<

-1.7'1

-1.75

-1.89

120.00

75.00

-1.43

90.00

1"'2

75.00

GO.OO

-1.39

-1.8S

--~ N",c-UALVES - -

So.OO

120.00 1.0000

-1.70

-1.52

-2.03

75.00

-1.63 -1.19

-1.S5

-1.48

-1.$

75.00

.3333

.6667

-2.03

90.00 105.00

-1.59

105.00

-2.34

-1.79

•

120.00

-1.90

......................... ......................... 60.00

I .5000

-1.S7

-1.93

105.44

299.51

-1.84 -Z.05

105.00

90.00

84.32

59!:l.99

105.00

83.25 ISB.29

-1.9B

75.00

4S.no

So.oo

-1.45

-1.82

20.12

75.00

-1.'12 -1.90

.3333

11.42

35.30

-1.51

-1.71

24.51

17.53

• REMARK: THE KP-VALUE IS OU£R 1000 I-lHtN 999.99 APPEARS

-1.a3

-1.51

75.00

50.00

979.63

-1.78

-1.46 -1.42

90.00

999.99

999.99 999.99 S99.99

-1.6S

-1.'12 -1.38

75.00

-1.B5

105.00

999.99

619.65

-l.G3

90.00 105.00

-l.ss

141.60

616.11

369.90

-1.57

-t.ES

-1.61

75.00

9SS.92

194.79 355.12

215.63

-1.81

SO.OO

87.0n

103.'11

122.90

-1.77

-1.9S

.'.

-1.78

-1.9B

-1.73 -1.72

2"'3 NAc MI,IALUES - -

-1.8S

120.00

999.99

326.25

6a.04

-

60.00

-e.ea

-1.83

-1.G8 -1.67

1...3

'10 -2.21

84.'58

999.99

90.00

1.0000

-1.77

-2015

11.8.54

401.56 712.13

120.00 '!94.32

-2.00

I"'S

39S.51 999.99

S.56

Blo

2"'3

103.45

656•.07

31.40

V2

241.67 Ja9.44

105.00

IBO.58 302.41

52.74

75.00

225.S5

999.S9

999.99

236.59

12B.17

SSS.99

105.32

90.00

32.S4 62.44

660.~B

59.98

105.00

20.18 84.28

999.99 999.99

33.22

120.00 749.84

12.55 22.10

~:;~~

:

___ N -tJRt.tJES --AC

60.00

90.00

IS.00

.:••....t.;.. .....................•. ......:

...........................

V2

---l'p-VALUE5 -

'10

12.73

21.1"

1"'3

105.00

75.00

.6661

-U~LUES

II'S

Blo

213

10'2

___ K p 6.35

.:1333

.5000

1;3

Active earth pressure factor N Ac for.¢ .= 20°, 25°, 30° and 35°

.........................

••..•••

••••••••• u ••••••••••• ••

BIO

APPENDIX B: EARTH PRESSURE TABLES FOR N Ac AND N pc (Chang, 1981)

0.25 and 0.30

50° and k

120.00

-1.00

-1.03

SO.OO

-1.72

-1.79

-1.18

-1.20

-1.85

-1.91

-1.07

75.00

-1.43

-1.59

-1.64

-1.S7

90.00

-1.32

-1.37

-1.41

-1.44

-1.47

-1.50

105.00

-1~2S

-1.29

-1.32

-1.34

-1.37

-1.39

-1.52 -1.41 -1.24

225

224 40° and 45°

Active earth pressure factor N Ac for ¢

Passive earth pressure factor N pc for ¢

•.......................

........................... ,'.

'I. 01.

1/3

1,;2

213

01.

5'.

-1.62

-.99

.3333

120.00

-.72

60.00

-1.38

75.00

-1.17

-.8S

-.91

-1.50

-1.59

-1.23

60.0n

-1.28

-.77 -1.45

15.00

-1.56

-1.G5

-1.27

-1.32

105.00

-,94

-1.6S -1.32

-1.72 -1.35

-.95

-.97

-.79

-.BO

-1.72

-1.78

-l.BI

-1.38

.3333

-1.40

-.72

-l,B3·

-.99

-1.B7

.5000

-1.00

-~ao

-.8<1 -1.79

-1.57 -1.20

90.00

-.90

-.93

-.75

-.77

-.7B

-.79

-.80

-.56

-.58

-.59

-.59

-.60

-1.38 -1.18

-1.22

-.59

-1.20

-1.47

-1.50

75.00

SO.OO

-1.10

-1.15

-1.16

-1.21

-1.23

-1.25

90.00

-.B3

-.84

-.B6

-.87

-.B?

60.00

-1.72

-1.78

-1.83

-1.B8

-1.92

-1.95

-1.35

-1.44

-1.53

90.00

-1.24

-1.2B

-1.32

-1.35

105.00

-.1.13

-1.16

-1.19

-1.21

120.CO

-.65

-1.59

-1.60

-1.39

-1.40

90.00

-1.24

-1.25

IDS. DO

-1.02

-1.03

-1.21

-1.24 -1.07

20° and 25°

.',

"Ii

3.30

3.59

3.B9

75.00

3.40

3.71

4.06

4.65

90.00

4.24

11.63

7.80

8.2S

60.00

3.13

75.00

3.03

3.79

105.00

6.~

6.94 3.36

1/3 --- N

4.38

5.01

3.60

3.BS

4.11

4.36

4.66

.3333

4.2S

3.72

.4.8S

5.<16

6.17

6.92

7.76

6.41

7.22

6.c1

120.00

8.25

5a.oO

3.34

75.00

3.S5

4.38

17.59

4.96

5.59

3.85

6.29

7.S8

105.00

8.99

11 •.39

-1.29

12.7S 19.6J

4.73

5.37

6.07

S.BS

5.80

6.56

7.40

8.34

6.85

7.7S

2/3

<1.25

4.S2

5.00

4.42

4.80

5.22

5.66

5.95

7.33

7.94

4.03

4.41

4.82

4.63

5.06 B.SO

9.33

11.4B

12.46

6.71

13.52

7.73 60.00

3.24

75.00

3.63

B.21

9.22

9.77

10.33

10.92

4.113

4.73

5.03

5.35

0:.B5

3.73 3.B8

SO. 00

11.15 5.011

.5000

5.11

5.58

6.0B

5.40

5.90

6.43

7.00

5.71

6.a2

7.113

8.09

50.00

3.52

3.a8

4.27

75.00

11.10

<1.50

11.93

90.00

5.22

5.72

7.15

7.41 120.00

B.13

B.SII

9.17

3.$ 75.00

3.77

90.00 105.00

5.03 4.61 4.S4

5.27

5.61 7.32

6.0B

120.00 60.00

3.78

75.00

4.07

4.3a

4.60

9a.00

4.97

5.31

5.GG

105.ao

6.57

6.99

7.44

120.oa

6.03

7.76

8.24

10.81

11.44

.6657

13.32

5.25

5.71

6.01

S.5<1

B.50

S.2S

11.40

12.41

13.<18

14.62

4.09

4.50

<1.95

5.41

5.91

7.91

e.90 12.42

15.SB

18.37

21.<18

25.59

30.00

34.95

40;75

5.G9

6.75

9.42

11.07

5.77

120.00

15.20

60.00

4.25

75.00

75.00 105.00

.5000

1'2

5.53

2/3

al.

5/6

6.61

o

5.57

20.61

13.71

16.0S

19.29

22.<1<1

47.32

5<1.69

J6.98

19.62

24.03

27.S9

3S.4J

<12.31

51.02

.

t :••~;;~~ •.•••: 1/6

01.

1/3

1....2

2/3

5/6

--- Np,c -VALUES - -

SO;OO

3.52

4.25

S.19

6.46

a.09

10.17

7.90

9.<16

a.60

10.36

12.47

75.00

4.39

5.52

6.99

8.85

11.2<1

14.35

12.42

15.a3

18.20

90.00

6.52

8.32

10.63

13.6<1

17.51.

22.118

0:0.52


6.31

7.S6

9.29

11.27

13.6B

.6.B5

8.39

10.26

12.53

15.27

18.57

13.1S

37.71

27.61>

.:

5.1B

.3333

51.49

62.93

IB.3S

10.78

13.89

17.89

29.81

38.49

1~.00

19.~

~.40

~.~

42.~

$.~

71.n

~.~

60.00

4.72

S.OS

7.711

9.93

12.79

16.111

21.13

75.00

S.57

6.53

11.09

1<1.33

18.56

24.10

31.1B

67.<17

87.75

97.43

126.85

165.13 26.B2

27.64

90.00

17.60

22.SS

17.BB

23.<13

30.50

39.B2

76.87

120.00

33.58

43.96

57.33

7<1.80

29.72

49.79

50.36

28.11

34.48

42.21 9.00

11.01

13.43

IS.34

60.00

5.68

7.3S

9.55

12.35

15.9B

20.74

B.12

9.99

12.24

14.99

18.36

22.45

75.00

B.17

10.66

13.91

lB.18

23.62

30.76

40.01

12.11

14.92

18.33

22.4<1

27.42

33.58

90.00

13.07

17.21

22.45

29.34

38.29

<19.67

65.01

120.00

<13.32

56.56

7<1;18

126.27

164.22

214.
GO.OO

S.84

8.55

11.7<1

15.26

19.84

as.as

33.58

IG.38

21.54

28.28

36.S8

48.38

63.04

82.30

4.8a 75.00

7.55

.: 7.15

90.00

.5000

C4.~S

27.88 SO.OO

5.63 7.70

9.53

11.78

11.66

14.41

17.74

12.74

105.00

IB.SS

62.85

76.70

93.SS

10.53

12.91

15.81

19.3<1

21.67

2S.79

32.78

110.09

35.32

43.34

52.95

64.67

.S667

21.BO

75.00

105.00

33.33

13.85

13.36 21.96

60.00

35.92

11.73

8.59

105.00

32.19

-UALUES-

11.75

120.00

cB.82

.0

35.0:3

12.09

17.65

10.79

12.70

10.04

12.39

9.92

120.00

8.55

10.54

9.11

105.00

8.78

30.<18

14.06

7.77 7.27

15.06

8.35

13.13

9a.00

22.40

4.77

8.9<1

7.S5

<1.90

6.15

21.63

25.<15

10.56 12.07

16.,05

12.83

105.00

4.<14

19.14

75.00

B.09

1<1.48

19.69

6.74

6.Bl

120.00

6.93 8.B5

10.69

6.72

5.28

13.73

5.92 7.56 10.61

18.77

90.ao

<1.79

5.05 5.0:6

90.00

7.
6.21

60.00

105.00

9.41

75.00

:::15.43

lS.41

8.65

SO.OO

21.93

30.41:1

23.<13

7.94

So.oo

18.S5

26.21

1<1.6<1

7.28

7.25

16.19

22.52

20.96

6.68

5.96

13.90

19.32

13.05

S.10

4.98

11.B9

16.54

18.72

5.56

6.83

12.22

1<1.63

8.61

75.00

5.6a

10.52

10.80

16.70

90.00

4.66

7.76

9.26

11.60

5.57

6.42

9.6<1

S.65

7.91

7.62

120.00

3.71

8.30

.5.SB

10.30

4.65

10.12

5.28

4.83 6.76

14.88

75.00

.3333

IS.3S

9.12

t ••:•. ~~;~~"

IB.37

<1.52

13.23

1/3

12.29

7.15

9.10 12.43

15.90

10.16

8.97 12.34

10.46

5.25

11.0B

1.0000

6.16

7.90 10.76

3.65

8.C5

--- N

10.94

9.96

13.76

11.73

I'.

9.72

8.85

11.88

6.38

105.00

.5SS7

30.38 7.85

S.95

IDS. 00

24.53

5.55

120.00

75.00

17.94

2'3

-UALUES ---

4.B2 5.93

90.00

6.36 7.2<1

105.00

6.26

105.00

.5000 9.60

'0

9C.00

'I. 01.

5/6

-

9.73 3.SB

6.11

120.00

5.48

3.91

19.<11

5.37

60.00 75.00

.12.59 17.29

4.20

:

1....2

105.0a

.

1....2

cJ.SS

4.15

5.90

8.56

16.05

5.10

:

pcUALUE5

7.a7 7.S2

75.00

9.92

8.68

la.a8

60.00

60.0C

.3333

16.11

15.71

90.00

3.S7

12.12 7.2<1

11.39

10.26

::;~~

3.65

3.55

4.2B

................... " . :ll••••• ••:•• : ~

2/3

-1.26

105.00

75.00

-.S6

-.82

--- Npc: -U.::lLU::S - -

3.34

-1.34

1.0000

-1.05

12.69

G.
75.0C -1.17

11.40

S.77

6.77

75.00

-1.56

75.00

5.13

105.00

:u...1"•• :u;~;~~ .....: 1....2

-1.30

7.03

4.56

60.00

1/3

5.15

12a.oo

-1.26

6.3J

4.a4

14.01

'/6

--- N

SO.OO

5.B6

-1.32

-1.03

-1.77

. . . . . . . . . . . . . . . ~ •• " •• c •••

3.6G

-1.28

5.96

3.57

105.00

t...:..;;;~~

a

o

5.26

60.00

120.0a .5000

-.63

1.0000

-1.23

Passive earth pressure factor N pc for ¢

1....3

-.81 -.50

105.00

-1.97


17.19 .3333

-1.26

.6667

-1.44

<1.31

<1.04

-1.00

60.00

IcO.OO

-1.25

01.

pc

" ••••••••••••

:

,I.

V3

105.00 -1.65

-1.23

120.00

-.78

1/2

• • • 11

-.59 -1.50 -1.17

.105.00

1/3

3.47 5.23

-.77

.:

" ;~;~~

- - N -VALUES ---

60.aa

.G667

-1.00

.3333

-.77

-1.40

-1.39

.,.

-.76

-1.13

-1.34

,I.

-.74

-1.2B

-1.40

-1.71

-1.49

61.

-1.56

-.95

-1.07

-1.27

01.

-.92

75.00

75.00'

75.00

-.90

60.00

105.00 -.8t

.,.

,I.

5'.

-.54

-1.77

-1.04

1.0000

-1.51

-.97 -.~7

-.77

2"3

"

:•••..•t••:

75.aa

-1.09

120.00

1/2

75.00

-.78

120.00 .5000

-1.43

-1.0B

-1.02

1/3

--- NAc:-llALUES -

o

-1.S?

-1.24

105.00

'/6

01.

- - N -VALUES AC

-1.55

.."

:.....t...:..~;;~~ .....:

:......t ..:..:~~~~ .....:

30°. 35°. 40° and 45°

5.39 6.36

7.46

B.B9

10.24

12.14

6.47

7.07

7.60

8.26

9.S8

60.00

8.93

11.09

lS.90

20.77

12.57

16.60

28.59

37.37

10.1<1

12.61

15.61

19.25

23.69

29.08

35.63

75.00

1<1.49

19.16

as.24

33.13

43.39

56.7a

74.03

19.29

23.79

29.25

35.90

43.99

53.80

90.00

2<1.06

31.67

41.59

54.45

71.1B

92.89

121.111

56.08

73.$

95.83

125.08

67.93

83.11

101.60

124.09

151.36

12a.00

81.37

106.55

139.07

181.38

236.45

47.57

1.0000

9.50

144.10 271.85

15.61 25.65

25.48

84.82 160.23

75.00

120.00

13.71

64.66 122.62

90.00

13.18 IS.23

7.16

37.88

120.00

105.00

21.79

212.3!! 307.88

400.84

227

226

0.90

0.90f----~

~

j

pI'"

~

o.aof----+-------J',~"2'~<2I

:":::.. .!1'

II! 0.70f----t----t----+----j

..

"'"~

0

~~ lIZ 2.13 5/6

.P II

~

-

-~

PI'"

oao

o

-~

1/6

~

0.701----1--~_l---_+----"-1 --1

II

"

113 0.70 2.13

-~

l"

i:?

112.

-516

0.60

0.60

.L-_ _-:-L__~ 03 0.4

0.50L_ _-L_ _~l.:_--_:l::_--_:: o.50L_ _-L o 0.3 04 0 0.1

0.20

0.50L..- - - -.L..I----0-'-Z----0--'.3---0-.--'4 0 0 Seismic Coefficient I k

0500L----0.L.,----o.l..Z----0-J.L3---0-l.4 Seismic Coefficient

Seismic Coefficient • k

Se"fernie Coefficient , k

Figs. B5.5 and B5.6. Correction factor 50°.

1J;

, k

for estimating (KpE>o = Ocr from (KpE>o = 00' r/J

45° and

3.0

-

"'=205°

Z.5

...

;2 2.0 -

....

---

~

~

"

;

_ _ 8=0°

---

"'120

1.5

'"

1.0

1- l~ __._ ---; ------- ------=

12000

0.96f---+---+-~4_----+-----1

0·60

0.60

KpE = PPE/~ 0.50L...._ _---l-

o

0.10

Y HZ

-- ~

a= IOSo

a ....;anO

a

-=-'1::-::-_ _-='==--_--::-7. O.50L_ _---.J. 0.2.0

Seismic Coefficient • k

030

0.40

-!::__-----:~---::-!

0

0.3 Seismic Coefficient

for all 8

0.4 0.5

• k

205°

30°

35°

40°

Angle of Internal Friction

Figs. B5.1 to B5.4. Correction factor ~; for estimating (KpE>o = 0 from (KpE>o = 0°' r/J 35° and 40°. "

o

25°,30°, Fig. B5.7. Modification factor for

1J; when

Fig. B5.8. Correlation factor cxq (r/J = 25°).

cx

< 90° and

/j

~ues

0.20 0.4 0.6 Normalized Slope Angle ,

• ep

s1r/J.

"'i5'"

60°

0.8

PI'"

ID

229

228 3.0',---,---t---,---.....-,r--.....-, 4>=35·

3.0'r----r---.-------.----.-----,

2.51---+---+----1---+---1 2.Sif------I----I---l--....-1--....-1

2.Sf----'---+----I---.,,/-<"'-=,?-----1 - - 8=0·

3.0'~----r---~---,....---.-------,

1.2'r---r---,-----,--.,.-----, 4> =25·

p=SO·

13/4> -0

-1/3

pl2

-2/3

4>

~2.01-

2.0

--8=0·

-.

~ 2..0f---.___-~_+____:"L____i+.,L--+_____:h""

-2

"-

z'"

.!

"-

z:E

J" I. Sii===----+---::;...-\'-'-7''F---__=rt-:7'~:..--1--""7'''1 'c

O.S;I---+---f-----"k-..:.....,....::~;;;J

---_ ....

__ 8=0· ---I

#2 0.4

4>

for all 8- values

O.s,'--_ _.L-_ _...J...._ _-L-_ _-'-_ _....J o 0.2 0.4 as 0.8 1.0

O.s,'--_ _.L-_ _...J...._ _-L_ _....l...___ O.Si'--_ _.L-_ _...J...._ _-L_ _-'-_ _....J o 0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 O.S 0.8 1.0 Normalized Slope Angle

{3lcp

Normalized Slope Angle

t

Normalized Slope Angle

f3/ep

Fig. B5.13. Correlation factor

3.0

4>=40'

~

( z'"

" ,;

4> 1.5

1.5

...--------

for

all 8

0

0.2

0.4

0.6

Normalized Slope Angle

13/4>

1.0

(3/4>

-V3

rt 0.8

" as

O.S

~:

---2/3 ---I

4>

-8=0' O~

4>/2 4>

S values

0.2 0.4

O.S

Normalized Slope Angle

0.8

f3t.p

1.0

60'

eo·

75' Angle

02 lOS'

ot Wall Repose

t

120'

135'

t:t

Figs. B5.l4 to B5.16. Correlation factor "'~ : q, Figs. B5.9 to B5.l2. Correlation factor "'q: q,

--1/3

................ ,

=-_-LV3

--8=0· 4>/2

0.4

0.2

-1/3

---

z

.---0

.4>=35'

-0

.

for aR

0.8

1.2

:::

0.5

0.5

4> =30'

13/4>

4>

values

a

-0

1.0

1.0

t

(q, = 50°).

"'q

- - 8 =0' 4>/2

2.0

--8=0' 4>/2

2.0

Angle at Wall Repose

4> =45'

2.5

2.5

~/ep

I

1.2

3.0

0.2LS""O,,---;:'7s--:'----:--e.L0..,.'--.J10-S--:'--1.L2-0',---13.lS'

30°. 35°. 40° and 45°.

SO'

75 '

90'

25°. 30° and 35°.

120'

lOS'

Angl. of Wall Repose

."

135'

230

231

Chapter 6

SOME PRACTICAL CONSIDERATIONS IN DESIGN OF RIGID RETAINING STRUCTURES By M. F. Chang and W. F. Chen o!'

z

" 'a

6.1 Introduction

O!ll---t--">.....--+---'>..,~~.,->~

_8=0"

.p12 .p

0.2

O'----_....--l._ _-'-_ _.l-.._--'-_ _--'

60°

75"

90"

lOS"

120"

Angle of Wall Repose

0

L-_-L..,..-_--L.;-_ _L.:-_-..1---;-

135"

60"

75"

90"

lOS"

120"

...J.

135"

Angle of Wall Repose, a

,a

Figs. B5.17 to B5.19. Correlation factor C<~: ¢ = 40°,45° and 50°.

.p =50"

2.0r---,----,---,---,----,

1

1.8

q, a 8

-

a = 900)

----.

a = 105", for all

.,.....--

I

.a=12O°,

.8=ep

as specified

.p a 8 unless

value.

specifi~d

"" 1.61-----1---+---1-----+---1 :E

~

a;..

"..,

~ 1:41-----/---+---l--..-"'l·r:;:o;....----I ~

~

" ::l

-8=0"

--- .p/2 o

L60-:-"--7~5:--"-.p-9.J.~-:-"--I05L"----1.L20-:-"--'135" Angle of Wall Repose

I

0.8'----_--'-_ _--'-_ _'----_-..1-_---=-' 0.4 o 0.2 Normalized Slop. Angle

a

Fig. B5.20. Recommended. modification factor Ilc for

Ci ;;,:

90°.

•

PI.p

In the design of retaining structures, either ultimate-load-based or displacementbased, both the magnitude and the distribution of lateral earth pressure acting on the structure is of great importance. There are numerous methods available for assessing the resultant lateral earth pressures. Some of them have been extended to include the earthquake effects. However, most of the methods are developed based on translational wall movements and may not be suitable for assessing the lateral pressures corresponding to other modes of wall movement. At present, there is still a lack of well-developed theoretical methods for determining the distribution of lateral earth pressure, or the point of action of resultant lateral pressure. A method for assessing the distribution of lateral pressures, taking into consideration the mode of wall movement, is therefore urgently needed. Furthermore, the effect of wall movement on the magnitude of lateral earth pressures also deserves attention. In an attempt to assess the point of action of resultant lateral earth pressures and to investigate the effect of wall movement on lateral earth pressures, a so-called modified Dubrova method proposed by Chang (1981) is developed here based on the philosophy of Dubrova (1963) as described in the book by Harr (1966). In order that the method can be applied to general soil-wall systems subjected to earthquake forces, the well-known Mononobe-Okabe formula (Okabe, 1926; Mononobe and Matsuo, 1929) is used as the framework in the modified formulation. Dependence of strength mobilization on wall movement is carefully studied. Proper distributions for strength parameters cP and 0, which completely control the results of the modified Dubrova analysis, are suggested for three basic modes of wall movement based on the consideration of change in state of stress in the backfill during wall yielding. This chapter presents the modified Dubrova method and its numerical results. The effect of assumed distributions of mobilized cP and 0 is discussed. Changes in pressure distribution behind rigid walls, as the result of gradual wall yielding are included. Points of action for both static and earthquake cases as suggested and

232 adopted by previous investigators are reviewed. They are compared with those obtained by the modified Dubrova method. Effects of earthquake magnitude, strength parameters, and geometry of soil-wall systems on the point of action based on the modified Dubrova method are included. The validity of the developed method is carefully evaluated. It is well-recognized that not only the point of action but also the magnitude of resultant lateral earth pressure is highly dependent on the mode of wall movement. Some numerical results based on the modified Dubrova method are presented and compared with other theoretical or numerical solutions available for three different types of wall movement. Discrepancies are noted. Finally, problems are considered for some earth pressure theories for practical applications. Several analytical methods for assessing seismic lateral earth pressures and points of action of resultant forces are compared both qualitatively and numerically. Suggestions for selecting the analytical method for design of rigid retaining structures in seismic environments are also given. 6.2 Theoretical considerations of the modified Dubrova method In retaining structure design, not only the magnitude but also the point of action of the resultant lateral earth pressure has to be known. Most earth pressure theories are based on the limiting state concept. The backfill material which controls the lateral pressures acting on a retaining wall is generally assumed to behave as a rigid-plastic material. However, lateral pressure distribution is displacement related.' In spite of this, most earth pressure theories give no information about the point of action of the resultant pressure. The distribution of lateral earth pressure highly depends on the mode of wall movement. Its assessment generally requires a complicated step-by-step load deformation analysis (Potts and Fourie, 1986). Common practice still tends to assume that the distribution of the lateral earth pressure acting on a rigid wall is generally quasi-hydrostatic and consider the resultant force to act at the lower one-third point of the wall section, irrespective of the type of wall movement. It has been proved that this is not always true as described by Tschaebotarioff (1962). In many cases, the distribution is more parabolic-like. The distribution of lateral earth pressure is further complicated by the presence of earthquake. As in Coulomb's analysis, a quasi-hydrostatic distribution is assumed in the widely accepted Mononoke-Okabe method for the seismic lateral earth pressure determination. However, several researchers such as Matsuo (1941), Matsuo and Ohara (1960), and Ishii et al. (1960) showed experimentally that the distribution of seismic lateral earth pressure is not hydrostatic. Theoretical works of Prakash and Basavanna (1969) and Basavanna (1970) also showed that the point of application is higher than the lower one-third point for the active pressure case. These works, although provide ways of determining the point of action, do not take

233 into consideration the mode of wall movement and its effect on the degree of mobilization of soil strength. Based on the assumption, that the static-active failure Coulomb wedge holds for the dynamic condition, Nandakumaran and Joshi (1973) suggested a way of assessing the point of action of dynamic increment of active earth pressure. Unfortunately, the method is based on an unrealistic assumption. In general, the failure surface or the most critical sliding surface is flatter for the earthquake case than that for the static case as found by Murphy (1960) and as already shown in Chapter 5. It is clear that, at present, there exists no theoretically sound and yet simple procedure for assessing the point of action of resultant active pressure. Here, the upperbound limit analysis method, as in most earth pressure theories, gives no information on the lateral earth pressure distribution. The current practice of assuming the dynamic increment of earth pressure to act at the upper third point or the middle point of a rigid wall is not wholly justified. As far as the point of application of resultant passive pressure is concerned, Sabzevari and Ghahramani (1974) have developed a rigorous method of solving the problem using the method of associated fields of stress and displacement. They found that the point of action of dynamic forces differs from that for the corresponding static ones. They also claimed that the location of these points depends on the form of the wall displacement. The method has not been used extensively due to its dependence on large computers. The distribution of dynamic passive pressure has also been briefly treated by Ghahramani and Clemence (1980). It seems that the problem about the assessment of point of action still deserves more attention., The Dubrova method of redistribution of pressure (Dubrova, 1963) as described by Harr (1966) provides a way of assessing the earth pressure distribution for a specified wall movement. The method, which was originally developed for the case of vertical wall retaining cohesionless horizontal backfill, considers the degree of mobilization of strength as the result of the specified wall movement. It is an extension of the Coulomb's wedge theory with the effect of wall movement included. Detailed description of the method can also be found in Harr (1977). An attempt has been made by Saran and Prakash (1977) to extend the Dubrova method for assessing the distribution of seismic lateral earth pressures. However, no actual formulation of the pressure distribution was given. The work described here tends to generalize the Dubrova method using a similar idea as mentioned by them and assuming that the Mononobe-Okabe concept is valid for assessing seismic lateral earth pressures. The same idea is equally applicable if one makes use of the more rigorous formula as developed by the upper-bound limit analysis in which log-sandwich mechanism of failure rather than planar mechanism is assumed. However, since the most critical surface, which involves two variables, is also a function of the degree of mobilization of strength, the formulation becomes much more complicated. For

234

235

practical purposes, especially for the active case in which the most critical surface is almost planar, the Mononobe-Okabe formula is adequate. The Mononobe-Okabe formula is therefore adopted here as the basis for the formulation of the modified Dubrova equation. The theoretical consideration of the modified Dubrova method can be introduced as follows.

where =

= angle of internal friction of backfill =

In the derivation of Coulomb's formula and the Mononobe-Okabe formula, no wall movement was specified, regardless of the fact that the distribution of earth pressure is highly dependent on the mode of this movement. To take into account the effect of wall movement on the lateral earth pressures, Dubrova (1963) assumed that the backfill material mobilizes its strength to an extent in proportion to the corresponding wall movement. He hypothesized that the mobilized strength of the backfill, represented by 1/;, where II/;I :5 ¢, should vary along the height of the wall. However, he assumed that the angle of wall friction 0 is constant and related to ¢. The fact that I/; should vary with depth according to the mode of wall movement can be clearly seen from the works of James and Bransby (1970, 1971) for the static case. The velocity fields they predicted and observed clearly show the dependence of shear strain in the backfill on the wall movement. The theoretical work of Sabzevari and Ghahramani (1974) also showed that for the dynamic passive pressure case in which the wall rotates about the toe, the shear strain in the backfill is the largest at the top anf! decreases _as depth increases. It is well accepted that the angle of soil- wall interface friction 0, although generally assumed constant with depth, is also dependent on the wall movement. The extensive experimental work of James and Bransby (1970) remarkably showed that the angle of wall friction is maximum and close to the ¢-angle at the top and decreases to a very small value near the toe for the passive case and rotation about the toe. For this reason, it seems reasonable to assume that both I/; and 0 vary with depth in a same manner.

6.2.2 Formulation of the modified Dubrova method

angle of wall friction

= vertical seismic coefficient

+ k y )], k h is the horizontal seismic coefficient as shown in Fig. 6.1 i,(3,H = geometrical factors as defined in Fig. 6.1. Equation (6.1) is derived by the conventional limit equilibrium approach. By . replacing H with z, ¢ with I/;(z) and 0 with 0w(z), the accumulative resultant actIve pressure at depth Z is: ,;" tan -1 [kh/(l

Ok

6.2.1 Dependence of strength mobilization on wall movement

unit weight of soiL backfill

"I

¢ o ky

'YZ Z

(1 + ky ) cosz(1/; - i-Ok)

2 cos Ok cos 2i cos(ow + i + Ok) (1

(6.2)

+ ~A)2

where

I/;

= mobilized angle of internal friction which varies along the height of the wall

Ow

=

z

=

depending on the wall movement with its maximum value equal to ¢. mobilized angle of wall friction which also varies along the height of the wall and can be ta.ken as ml/;, with m = dow/dl/; :5 1. depth below the top of the wall.

H

H

Based on Fig. 6.1, the Mononobe-Okabe formula for the resultant active earth pressure, P AB' can be obtained by the equilibrium consideration (Okabe, 1926; and Mononobe and Matsuo, 1929), such as: (1 + k y ) cos z(¢ - i-Ok)

"1HZ

P AE

=-

2

cosOk cos

z. I

cos(o +

.

I

[(sin(¢ + 0) sin(¢ - (3 - Ok)}J]Z + Ok) 1 + cos «(3 - I') cos (0 + I. + 0) k

1_90° - a

1-90° - a

(6.1) Active Case

Passive Case

Fig. 6.1. Failure mechanisms assumed in modified Coulomb's analysis of seismic lateral earth pressures.

r

I 237

236 The unit active pressure at depth z can be obtained by taking derivative with respect to z using the following equation:

dPAE

aPAE

dz

az

aPAE d1f;

= -- = -- + -- -

PAE (z)

a1f;

yz(1

PPE(Z)

+ k v ) cos(1f; -

i-Ok)

dz

2(1

=

yz(l

cos(1f; - i-Ok)

+

~A)

~A cos(1f;

+ k) cos(1f; -

cos Ok cos 2i cos(ow

{

cosOk cos 2i cos(ow + i + Ok) (1 + ~A?

~A cos(1f;

(1f; + ow)]sin[~. (1f; .- (3 - Ok)]] t . cos«(3 - z) cos(ow + 1 + Ok)

= [Sin[ -

After differentiation, the unit passive pressure at depth

_ z (d1f;) (sin(1f; _ i-Ok) + cos(1f; - i-Ok) [(1 + m)

+

~P

(6.3)

dz

The result can be expressed as: PAE(Z) =

where

+i +

_ z(dlj;)(sin(lj; _ i-Ok) dz

~P cos(lj;

+ ow)

- (3 - Ok) - m tan(ow + i + Ok)])}

(6.4)

i-Ok)

z can

Ok) (1 - ~p?

cos(1f; - i-Ok)

+ cos(lj; - i-Ok) [_ 2(1 -

- (3 - Ok) - m tan(ow

be obtained as:

{

~p)

+ i +

(1 +

m)~p cos(lj;

+ ow)

Ok»)]}

(6.7)

where

where

~

=

+ ow) sin(1f; - (3 - Ok)]~ cos«(3 - I) COS (ow + i + Ok)

[Sin(1f;

A

Ok

=

tan -1 [kh /(1 + k v )]; withkv

It is noted that in the active case lj; ;::: 0, Ow ;::: 0 and in the passive case 1f; :5 0 and Ow :5 O. Consequently:

(6.8)

> 0 if acting downward and k h > 0 if poin-

~~~~~...'

-

.

Similarly, in the passive case, if we define k v > 0 when acting downward and k h > 0 if pointing toward a wall, as those assumed in the active case, the resultant passive earth pressure on the wall with height of H can be expressed as:

With this consideration, Eqs. (6.4) and (6.7) can be combined into one equation of ~A = ~k and ~P = - ~k are substituted into the two equations. The resulting equation is: p(z)

(1 + k v ) cos2(c/> + i + Ok) -------------------.,-(6.5) cos k cos 2'1 cos (~u - 1. - k - [sin(c/> + 0) sin(c/> + (3 + Ok)]t}2 cos«(3 - i) cos(o - i-Ok)

°)(I

°

By replacing H with z, c/> with - 1f;(z), and yz2 PpE = -

°

with - 0w(z), we have:

(1 + k v ) cos 2(1f; - i-Ok)

--------------

2 cosOk cos2i cos(ow

+i +

Ok) (1 _ ~p)2

=

cosOk cos 2i cos(ow + i + Ok) (1 + ~k?

- Z

+ where

{6.6)

-yz(1 + k v ) cos(lj; - i-Ok)

(:~)

~k cos(lj;

[sin(1f; - i-Ok) +

(

cos(1f; - i-Ok)

cos~l-+i ~~) Ok) ((1

- (3 - Ok) - m tan(ow + i + Ok»)]}

+ m)

~k cos(1f; +

ow) (6.9)

239

238

the upper sign is for the active case and the lower sign is 'for the passive case;

= do w/d1/' = oN , .. () k = tan- l [kh 1(1 + k v )]; k v > 0 if acting downward, k h > 0 If pomtmg toward • .. I m

the wall; for active case,

()k ~

0 is critical and for passive case,

()k ::5

0 IS cntlca ..

6.2.3 Distribution of mobilized strength parameters

As pointed out in Section 6.2.1, the degree of mobilization of the strength parameters cP and is highly dependent on wall movements. Both experimental investigations (James and Bransby, 1970) and theoretical works (James and Bransby, 1971; Sabzevari and Ghahramani, 1974) suggest that the mobilized c/J- and o-values, denoted as 1/' and Ow respectively, vary with depth in a manner depending on the mode of wall movement (Potts and Fourie, 1986). James and Bransby (1970), based on their observed failure patterns as shown in Fig. 6.2 and stress measurements along the interface, found that for the case of rotation about the toe into the backfill, both 1/' and Ow decrease as depth increases. They also found that for the passive case, rotation about the top, the degree of strength mobilization is the highest near the toe. Based on the rupture patterns postulated for the active case (Fig. 6.2), the variations of 1/' and Ow with depth for the active case may be similar to the passive case. It is therefore expected that both 1/' and Ow increase as depth increases in both active and passive cases for rotation about the top of the wall.

°

: .. [ 7 --~~

.....

",

I

-J

HI~V-

J

Rotational about Bottom

.rr 'jV'

Rotational about Tap

I

I

---J

_WT Y Horizontal Translation

Fig, 6.2. Typical passive rupture patterns reported by James and Bransby (1970) and postulated active rupture patterns.

Although the Dubrova method (Dubrova, 1963) does provide a way of including the important wall movement effect into earth pressure analyses, their results depend highly on the assumed. qistribution for 1/' and ow' Selection of proper distributions of 1/' and Ow is of prime importance. The fact that the 1/'- and 0w- values at a given depth are closely related to the wall displacement and the corresponding stress condition in the backfill at that depth should be carefully considered.

Dubrova's and Saran and Prakash's distributions of 1/'(z) and 0w(z) Recognizing the fact that the mobilized strength in the backfill should vary with depth, Dubrova (1963) assumed a linear distribution for 1/'(z), with the 1/'-value at the point of rotation, taken as 1/'a = 0, and the 1/'-value at the moving end, 1/'b = c/J for the case of active rotation of vertical wall away from horizontal backfill. He further assumed the 0w(z) is related to c/J and equals to the maximum ow-value, or the angle of wall friction 0. For the passive pressure case, the same 1/'(z)-distribution was adopted by Dubrova (1963) except that the 1/'-value is taken as negative in this case. The linear distribution of 1/'(z) as adopted by Dubrova (1963) was modified by Saran and Prakash (1977) for soil-wall system subjected to earthquake forces. They suggested that 1/' can be assumed to vary linearly from 1/'b = c/J at the moving end of an active rotating wall to 1/'a = {J + Ok at the point of rotation, where {J and ()k are the slope angle of the backfill and the direction of the resultant gravity force, respectively (Fig. 6.1). Instead of assuming Ow = all the way along the soil- wall interface, they recommended that 0w(z) be taken as m1/'(z) with m ::5 1. They believed that the mobilized o-value should be less than the mobilized cP-value. This is a reasonable assumption since the mobilization of is also dependent on the wall movement. The ow-value should be expected to vary in a similar way as the 1/'-value. Although the distribution of 1/'(z) as assumed by Saran and Prakash (1977) is much more generalized and the distribution of 0w(z) seems more reasonable than those assumed by Dubrova (1963), they gave no explanation to their assumed 1/'(z)distribution. The 1/'(z)-distribution was probably selected by them on the basis that the term inside the square root in ~A of Eq. (6.2) should be positive so that its solution would be determinate. In spite of its simplicity, taking 1/'a = {J + ()k simply for the sake of avoiding numerical difficulties without any physical reasoning may result in an improper 1/'(z)-distribution. Both Dubrova's and Saran and Prakash's distributions of 1/'(z) require further stronger physical grounds.

° °

Recommended distributions of 1/'(z) and 0w(z) - Static case, horizontal backfill For a 1/'(z)-distribution to be reasonable, it is better to select it based on the considerations of the changes of stress condition in the backfill as the result of a specified wall movement. Take the case of a vertical wall rotating about its toe away from a horizontal

240

241

backfill as an example. Before the wall starts tilting, it is reasonable to assume that the backfill is everywhere under the at-rest Ko-condition, if a naturally deposited soil mass is considered. As the wall starts yielding, the lateral displacement in the soil will be the largest near the top if the wall is rigid. It decreases as depth increases and reaches zero at the point of rotation. The consequence of this lateral yielding is to bring the backfill material from a Ko-condition to an active limiting equilibrium KA-condition or somewhat in between at locations where the shear strain is not high enough. In actuality, the average mobilized !/t-value depends on the shear strain within the soil mass, rather than the displacement of the wall, although the shear strain is related to the wall displacement. However, Dubrova (1963) assumed that the !/tvalue is directly related to the wall displacement. Therefore, it seems reasonable, for the case of rotation about the toe, to take the !/ta-value at the cf>-parameter corresponding to the Ko-condition, denoted by cf>o· That is, assuming Y;a = cf>o rather than !/ta = 0, as was done by Dubrova (1963), seems more reasonable. Taking !/tb = cf> at the top, however, is well accepted in this case. For the case of active rotation about the top, it can be assumed that the backfill behind the wall is under Ko-condition near the top and KA-condition near the toe. Similar to the case of rotation about the toe, the !/t-values can be taken as !/ta = cf>o at the top and Y;b = cf> at the toe. The Y;a- and !/tb-values similar to those selected for the active rotation cases are recommended for the passive rotation cases, except those values are taken as negatives in these cases. The reason for selecting cf>o ::5 Y; ::5 cf> for the case of (3 = 0 can al.so be readily seen from (a) in Fig. 6.3. The stress state everywhere in the backfill at any stage of wall movement should be in the dotted region for the case of outward active movement. (0)

{3=oo

T

O-f''''''--f----II-----.----

(b)

{3>oo

T

O-!"IC:::::~-+_----II---

Fig. 6.3. State of stress and mobilized q,-parameter behind a rigid wall rotating outward about the toe.

According to Jaky (1948), the Ko-value for cohesionless soil can be related to the cf>-parameter as: Ko

=

I - sincf>

(6.10)

Based on Fig. 6.3, the following equation can be obtained: .

(ul)6

(u3)6

smcf>o = - , - - - - - (ul

)6

+

(6.11)

(u3)6

where (ul)6 and (u3)6 are major and minor principal stresses corresponding to K _ condition. By definition, K o = (uh)6/(uy)6 = (u3)6/(ul)6. In general, K < 1 f~r o cohesionless soil. Consequently, Eq. (6.11) can then be reduced to: cf>o

1 - K o) = sin-I ( -I

(6.12)

+ Ko

By combining Eqs. (6.10) and (6.12), the cf>o-value can be estimated from the following expression: A. '/'0 -_

• - I( sm

sincf> ) 2 - sincf>

(6.13)

and

For ex.ample, the ~o-v~lue for sand with cf> = 40° is 0.36 the corresponding 0value IS 28.3°, WhICh IS about 0.7cf>. . . Si~ce the stress-strain behavior of soils is nonlinear, theoretically, the !/t(z)dIstnbution in the soil behind the rigid wall rotating either about the top or the toe should n?t .be linear. Nevertheless, the difference in the two extreme y;-values, Y;a and Y;b' IS m the order of 300/0. Furthermore, the strain field observed by James and Bransby (1970) and predicted by James and Bransby (1971) and Sabzevari and Ghahramani (1974) show, for the case of rotating about the toe into the backfill that the shear strain varies nonlinearly with depth in a manner very similar to th~ variation of shear stress with shear strain. This somewhat indicates that the variation in the stress level or the mobilized cf>-angle may be essentially linearly with depth. The measurement of James and Bransby (1970) also shows that 0 which is believed to vary with depth in a similar manner as Y; does, is linear wi~h depth along large portions of wall height. For practical purposes, it is recommended that linear variation between Y;a and Y;b can be assumed. Although, in general, the displacement required for a complete mobilization of soil-wall interface friction 0 is lower than that required for a complete mobilization of shear strength, or cf>-angle, experimental results of James and Bransby (1970) does

242

243

°

show that w decreases essentially linearly with depth for tli~ 6~se of passive, rotation about the toe. However, the mObilized o-angle, ow' in no cases can be larger than the 4>-angle based on the stress-dilatancy reported by Davis (1968) and Lee and Heringt~n (1972). It is therefore reasonable to assume that here 0w(z), as in 1/;(z), is linearly distributed and the magnitude is equal to m1/;, with m s; 1. The value of m = do w/d1/; can be taken as 0/4> since both 0w(z) and 1/;(z) are assumed to be linear with z. The o-value can be either measured in laboratories or estimated from the following equation proposed by Lee and Herington (1972), if v is known: tano

sin4> cosv - sino sinv

In order to assess the value of 1/;a' the other extreme case in which (3 = 4> is considered. It is clear that for this case the mobilized 4>-angle is equal to 4> under at rest condition, befo~e the wall starts,moving. That is !/ta = 4> in this particular case. Based on these considerations, it seems that as (3 changes from 0 to 4>, the 1/;-value corresponding to the at-rest condition, !/ta, changes from 4>0 to 4>. The !/ta-value for a given (3-value should therefore be somewhere in between 4>0 and 4>. It is obvious that the variation is dependent on the level of (3 with respect to 4>. For practical purposes, the following relation based on a linear interpolation is suggested for estimating the 1/;a-value for a backfill with slope angle of (3:

(6.14)

1/;a

=

4>0

+ (4) -

4> ) tan(3 o tan4>

(6.15)

where v is the angle of dilatation of the interface material. This equation has recently been confirmed by Potts and Fourie (1986).

"'tzl

Recommended distributions of 1/;(z) and Ow (z) - Earthquake case, inclined backfill

Earthquakes have two possible effects on a soil- wall system. They can either reduce the shearing resistance of soil or induce an additional driving force to the system. However, it has been generally recognized that the shear strength of a cohesionless soil is practically unaffected by the presence of most earthquakes unless the developed seismic acceleration is large, say larger than 0.3 g, and both the water table condition and soil characteristics favor the occurrence of liquefaction. The distributions of 1/;(z) and 0w(z) should, therefore, remain the same for both the static and the earthquake cases. This partly explains why the distribution of 1/;(z) proposed by Saran and Prakash (1977) is improper. When the ground surface of the backfill is inclined ((3 > 0), the state of stress in the backfill and its change upon wall movement is more complicated than when the backfill is horizontal. The Mohr circles for both cases of (3 = 0 and 0 < (3 < 4> are shown in Fig. 6.3. Point A in the figure represents the state of stress on a plane parallel to the ground surface at a given depth. Every Mohr circle representing the state of stress at different strain condition at the given depth should pass through point A. It is clear from the Mohr diagram that for the case of (3 = 0, the minimum mobilized 4>-angle, or 1/;a' is 4>0 as mentioned earlier. The possible state of stress in the backfill is located in the dotted region as shown in Fig. 6.3. That is,1/;o s; 1/; s; 4> in this case. As 4> > (3 > 0, the minimum !/t-parameter, !/ta, or the !/t-parameter before the wall starts yielding, is believed to be higher than the 1/;o-value, which representing the 1/;-value corresponding to the at rest Ko-condition for the case of (3 = O. This is because the deviatoric stress level is higher for the case of (3 > 0 than that for the case of (3 = O.

H

H

where

CPo = SIOI

._ [

sincp ]

2 _ sin

q,

Fig. 6.4. Assumed distributions of >fez) and 0w(z) for seismic active earth pressure analysis.

245

244 The maximum possible f-value, fb' is again equal to cP, which is independent of the slope angle. Hence the f-value at any stage of wall yielding should be in the range between fa' as given by Eq. (6.15), and rf>. That is, the state of stress should be in the dotted region as shown in Fig. 6.3b. For the passive case, negatives of those f a- and fb-values as recommended for the active pressure case are taken. As in the case of {3 = 0, a linear variation is again assumed between the two extremes for the case of (3 > 0. The distributions of fez) for active and passive rotations developed based on above-mentioned considerations are summarized in Figs. 6.4 and 6.5. It should be noted here that the f-value at any depth z at any stage of wall movement can be generally expressed as: (6.16)

"'(Z)

H

where H is the height of the wall, f b == cP, and fa is given in Eq. (6.15). The minimum possible f-value is fa and the maximum possible f-value is rf>. In order to satisfy the condition that the term inside the square root of ~A in Eq. (6.4) should be positive for the solution to be real, it requires for the active rotation case, that: (6.17)

Similarly, for the passive rotation case, the following condition is needed: (6.18) The condition required for the passive solution to be real, as given by Eq. (6.18), is generally easy to be satisfied. This is because Ifa Imin = rf>o and is positive in most cases. However, the condition as required for a real active solution as shown in Eq. (6.17), may be critical to some soil- wall systems with steep backfill slopes. Although (fa - (3) is positive, in general, the value may become very small or negative as {3 approaches rf>, since tan {3/tan rf> :5 {31c/>. The acceptable Ok-value (or the magnitude of earthquake a soil-wall system can be tolerated) may be very limited if the slope angle is high. The presence of earthquakes, therefore, can be very critical for a soil- wall system with a steep backfill slope.

6.2.4 Resultant lateral pressure and point of action

H

I/J,=-e/> b

where

~

J

. -I --.-".sine/> e/>o=sln

2 - SIO 't'

Fig.

6.5.

Assumed distributions of 1/t(z) and

Il w(z)

for seismic passive earth pressure analysis.

For purely rotational wall movements, the resultant lateral pressure is simply a direct integration of the pressure distribution calculated from the modified Dubrova equation, Eq. (6.9). The trapezoidal approximation of areas is adopted in the integration. To avoid serious error in the approximation, reasonably small segmental heights are used. The trapezoidal rule is adopted in calculating the resultant pressure and its point of action for each segment. The point of action of the total resultant lateral pressure is obtained on the basis that the summation of first moments of each individual trapezoid about the toe equals to the first moment of the corresponding resultant pressure about the toe. For a translational movement, f and Ow depend on the amount of translation. At the ultimate state, however, f = cP and Ow = 0, as commonly adopted, are reasonable. By differentiating PAE(Z) in Eq. (6.3) with respect to z, it can be found that the pressure distribution is theoretically linear as was assumed by Coulomb. However, Dubrova (1963) recommended that the translational wall movement can be considered as a combination of rotation about the top and the bottom. He suggested that the pressure distribution in a translational movement can be taken as the average of the distributions resulting from rotating about the top and rotating about the bottom.

I 246 Based on Dubrova's suggestion, the total resultant lateral earth pressure for the translational case can then be taken as the average of those for the two rotational cases, disregarding whether the backfill material is in an active state or in a passive state. Experimental results of Narain et al. (1969) on passive earth pressure for different wall movements coincidentally tend to support Dubrova's suggestion. On the other hand, the experimental and theoretical works of James and Bransby (1970, 1971) clearly show that the failure pattern for wall translation is a combination of that for rotation about the top and that for rotation about the toe (Fig. 6.2). They argued that the passive translational force should be greater than those sustained by walls which rotate either about the top or the toe. There are at least two reasons indicating that Dubrova's suggestion for combining rotational lateral pressures into translational lateral pressure is not justified. Firstly, the principle of superposition which may be good when strain is small so that the material still behaves elastically, is not necessarily applicable to the limiting equilibrium case. Secondly, unlike the passive earth pressure which increases as cP increases, the active earth pressure decreases as cP increases. pirect combination of the rotational active earth pressures for obtaining the translational active pressure is obviously not correct. Simply taking the average of the rotational pressure is not necessarily justified even for the passive case either. Based on these considerations, it is suggested that the unit translational active earth pressure can be obtained directly from Eq. (6.4) based on of = cP and Ow = 0. The unit translational passive earth pressure can be obtained from Eq. (6.7), by assuming of = - cP and Ow = - 0. Consequently, for the translational wall movement, the lateral pressure distribution is linear based on the modified Dubrova method for both static and earthquake cases. The resultant lateral pressure is equal to that of Coulomb's for both active and passive cases.

247 can be represented by the mobilized cP-parameter for cohesionless soils, is to backcalculate from stress measurements on model tests of different scale. However, this is expensive and generally not feasible. Another possible way is to assume the distribution based on the consideration of change of stress state upon wall yielding as described in Section 6.2. In order to see how the assumed distribution of cP-parameter, of(z) affects the calculated lateral pressure distributions, results obtained based on the recommended distributions, as shown in Figs. 6.4 and 6.5 and the distributions suggested by Dubrova (1963) and Saran and Prakash (1977) are compared. Figures 6.6 and 6.7 show the comparisons of static active and passive earth pressures based on 0.----,-----,-----,.-----,.-------. ~

\

\

\

\

\

\

,,

\

,

~ ~,l

2

\

\ N

3

\

0'-

H

''I.

o

, .

Ranklne'SI' Solution

III

..c:

Q. Q)

' [1 I "'o~:.'"

, ''I.

4

~

o

\

\

a;

Dubrova's concept of pressure redistribution on which the present modified Dubrova method is based is versatile in that the effect of wall movement on lateral earth pressure can be taken into account. This is done by assuming a distribution of mObilized strength according to the mode of wall movement. Unfortunately, the stress-strain behavior of soils is generally nonlinear. Even if the wall displacement is well known, the resulting distribution of mobilized strength, which controls the lateral pressure distribution, varies from one soil - wall system to another. Perhaps, an ideal way of obtaining the distribution of mobilized strength, which


\

r=.

' I......

tf

5

\

2

'"

,

-

Recommended Distributions

[l"'/ b

4

I

"'. ""

. . . . --.. . n

b

"""~

R.A.B.

6L---L.-L.L..J~~~.l..~-

o

• 8 - 0°

Dubrova's Distributions

_:>rp

\

a.

6.3.1 Effects of distribution of mobilized strength parameter on calculated lateral earth pressure

0

J - 0°

rp - 30°

'\~IO~'" \ H ',

'0

6.3 Some numerical results and discussions of the modified Dubrova method

P -

..... ..............

_ _...l.6

...l.-_~_

8

10

Normalized Pressure Intensity. p A (zl/y, ft

Fig. 6.6. Distributions of static active earth pressure based on different distributions of mobilized strength.

249

248 Dubrova's distribution and the recommended distribution of ~(z). Also shown in the figures are the Rankine solutions which also represent the solutions of modified Dubrova's method for translation wall movement. It is found that the calculated lateral pressure distribution is significantly influenced by the assumed ~(z)­ distribution. TheKK and Kp-values and the corresponding points of action, h a and hp ' for those distributions shown in Figs. 6.6 and 6.7 are summarized in Tables 6.1 and 6.2. For the case of rotation about the top, although the distributions of active and passive earth pressures are influenced by the assumed ~(z)-distribution, it seems that the resultant pressures are almost unaffected as shown in Tables 6.1 and 6.2. They

Or-----,------,-------r--------,

TABLE 6.1 K A -values and corresponding h.-values based on different distributions of mobilized strength ( = 30°, /j = 0°, (3 = i = 0°)

,8=1-0° ,!,
¢ _ 30° ,8 - 0°

2

3

c.

~

Recommended

~

Distributions

o q; .r::

Wall movement

Distribution of mobilized strength

KA

h/H

Rotation about the bottom

Dubrova's Recommended

1.001 0.500'

0.261 0.303

Rotation about the top

Dubrova's Recommended

0.333 0.333

0.456 0.371

Dubrova's Distributions

N

co

are practically the same as the Rankine solutions. However, for the case of rotation about the bottom or toe, the Dubrova distribution results in aKA -value two times the Rankine solution and a· Kp~value one-half that of Rankine. This tends to suggest that the distribution of ~(z) assumed is improper. Assuming ~ = 0 at the toe, which means no mobilization of strength at all at the point of rotation, seriously underestimates the soil resistance. This is mostly responsible for the too high K A and too low Kp-values obtained based on Dubrova's distributions. It is clearly shown in Figs. 6.6 and 6.7 that the lateral pressure distribution obtained based on the recommended distribution of ~(z) is much more reasonable. It should be noted that the difference between the two solutions may even be larger if a > 0 is conm ~(z), is sidered. This is because the 0w(z)-distribution recommended, 0w(z) much different from that assumed by Dubrova (1963), that is 0w(z) = a = constant. Even if two extreme values of ~ are fixed, the shape of the ~(z)-distribution in-

• KA = K o =

- sin = (I - sino)/(I

+ sino)

4

a. ell

o

""LJI ""b I

5

1/1. b

{j/

TABLE 6.2 Kp-values and corresponding hp-values based on different distributions of mobilized strength ( = 30°, /j = 0°, (3 = i = 0°) Wall movement

Distribution of mobilized strength

Kp

hp/H

Rotation about the bottom

Dubrova's Recommended

0.999 2.000'

0.444 0.369

Rotation about the top

Dubrova's Recommended

3.002 3.001

0.256 0.301

I

Normalized Pressure Intensity, Pp (zl/Y,tt

Fig. 6.7. Distributions of static passive earth pressure based on different distributions of mobilized strength.

• K p = Kf'o = (I

+ sino)/(I - sino)

250

251

fluences the calculated lateral pressure distribution too. In Fig. 6.6, active pressure distributions based on two different shapes of f-distribution as shown are plotted together with results based on the recommended linear f-distribution. In one case, the f-value at mid-height, fm' is assumed! (fb - fa) smaller than that corresponding to a linear distribution of which f m = t (fb + fa)' That is, f m = ! (f b + 3fa) is taken. In the other case, f m of! (fb - fa) larger than that corresponding to a linear distribution is assumed. That is f m = ! (3fb + fa) is taken. As expected, it is noted that the shape of calculated pressure distribution is different for each different shape of f-distribution. Consequently, the point of action of resultant lateral pressure is different for each distribution. However, it is interesting to note that the area enclosed by each pressure distribution curve, which represents the magnitude of the resultant force, is the same for these three cases.

It is apparent that the shape of the f-distribution seems to influence only the shape of the calculated pressure distribution, based on the modified Dubrova method. Effect of the shape of f-distribution on passive pressure distribution is also shown in Fig. 6.7. Results similar to the active case are found. The points of action and the K A - and Kp-values corresponding to the pressure distributions shown in Figs. 6.6 and 6.7 for three different shapes of f-distribution are summarized in Tables 6.3 and 6.4. It is apparent that the ha- and hp-values obtained based on a nonlinear f-distribution differ from those based on a linear fdistribution only by an order of 60/0. For the seismic case, Saran and Prakash's distribution, which is a direct extension 0

TABLE 6.3 Effect of shape of >f-distribution on h.- and KA -values

zt7 c9K /

/ /

Mode of movement Rotation about bottom

Rotation about top

>fm

ha/H

KA

~ (>fb

0.321

0.500

+ 3>fa) ~ ('h + >fa) ~ (3)fb + >fa)

0.303

0.500

0.286

0.500

-

~ (>fb

0.396

0.333

iii

0.371

0.333

'0

0.348

0.333

c.

+ 3 >fa)

t (>fb+

>fa)

} (3)fb + >fa)

3=

3

¢. Ifr (z)

¢ HICK¢ ¢o

0 I-

0,

CD al .J::

4

a..,

\ I I

C

d

>fm

hpiH

~ (>fb

Kp

5

+ 3>f.) ! (>fb + >fa)

0.347

2.000

0.369

2.000

~ (3)fb + >fa)

0.393

1.999

Rotation about top

} (>fb +

3>f.)

0.284

3.002

,

£' ,,

+ >fa) ~ (3)fb + >fa)

0.301

3.000

0.320

3.000

6

,

....

\

I

2

,,

:f." . . .

I

I

R.A.T.

Recommended Distributions

1',

I

0

! (>fb

,,

I

Rotation about bottom

1>

Saran and Prakash's Distributions

~

Mode of movement

¢-

N

0

TABLE 6.4 Effect of shape of >f-distribution on h p - and Kp-values

~ I - 0° , K h = 0.20 _ 0° 30° •

13 2

\

4

R.A.B.

................. .................

6

8

350 __ _..

10

Normalized Pressure Intensity. PAE(zl/y,ft

Fig. 6.8. Distributions of seismic active earth pressure based on different distributions of mobilized strength.

253

252

of Dubrova's distribution, is compared with the recommended distribution. Some results are shown in Figs. 6.8 and 6.9 for k h = 0.2. They show that the pressure distribution based on Saran and Prakash's 1/;(z)-distribution is different from that based on the recommended1/;(z)-distribution, especially for the case of rotation about the toe. However, it seems that the difference is less than that in static case, in general. This is because a non-zero 1/;-value is assumed at the point of rotation by Saran and Prakash (1977) for the seismic case. This allows consideration of some degree of mobilization of strength at the point of rotation. The KAE- and KpE-values and corresponding h a- and hp-values for those lateral pressure distributions shown in Figs. 6.8 and 6.9 are summarized in Tables 6.5 and 0

it

l~OK t ,/ I

::::

-

-·~l L: H

N

_¢

"iii ~

'0

3

OK

Q.

0

_¢

~ 0

CD

III

=1 = 0°, Kh= 0.20

Saran and Prakash's Distributions

"'_

rJ

IH

Recommended Distributions

-¢o

4

The Dubrova method enables lateral earth pressure distributions at different stages of wall yielding to be obtained. For developing such a kind of lateral pressure distribution, the change in state of stress as described in Section 6.2 is referred to. Before walls start yielding, the soil mass is everywhere under an at-rest condition. A uniform 1/;(z)-distribution, with 1/; = 1/;a might be appropriate. As the strength of the soil near the moving end is fully mobilized, 1/;b = ¢, is assumed for the moving end. At the point of rotation, however, the at-rest 1/;-value, 1/;a' as given in Eq. (6.15) is considered. For the special case of {3 = 0, the at-rest condition corresponds

TABLE 6.5 KAE,values and corresponding ha,values based on different distributions of mobilized strength (cf> 30°, IJ = 0°, (3 = i = 0°, k h = 0.20)

¢ = 30° ,8 = 0°

_¢' ¢o

f-

6.3.2 Pressure distributions at different stages of wall yielding

//'/(Y,¢)

/3 2

,/" ,/

6.6. Again, the KAE-values obtained by the two different distributions of 1/;(z) are the same for the case of rotation about the top, although the corresponding havalues are different. The differences in both the resultant pressure and the points of action, similar to the active case, are larger for the case of rotation about the toe.

Wall movement

Distribution of mobilized strength

K AE

ha/H

Rotation about the bottom

Saran and Prakash's Recommended

1.041 0.685

0.256 0.305

Rotation about the top

Saran and Prakash's Recommended

0.473 0.473

0,396 0.367

.s::

0. til 0

I I

5

TABLE 6.6 KpE,values and corresponding hp·values based on different distributions of mobilized strength (cf> 30°, IJ = 0°, (3 = i = 0°, k h = 0.20)

I I I

/

6 0

R.A.T.

-339 10

20

30

Normalized Pressure intensity, PpECz)/y,ft

Fig, 6.9. Distributions of seismic passive earth pressure based on different distributions of mobilized strength,

Wall movement

Distribution of mobilized strength

K pE

hp/H

Rotation about the bottom

Saran and Prakash's Recommended

1.039 1.673

0.461 0,375

Rotation about the top

Saran and Prakash's Recommended

2.630 2.630

0.276 0.299

40

!

254

255

to the Ko-condition, and 1/;a = cPo. On the basis of this consideration, the 1/;b-value at any stage of wall yielding may be expressed as: (6.19) The at-rest condition, which existed before any wall yielding, corresponds to am = O. The state at which the soil strength is fully mobilized at the moving end corresponds to am = I. Using Eq. (6.19) as the basis of assuming the distribution of 1/;(z) for different stages of wall yielding, typical lateral pressure distributions corresponding to am = 0, t, j, and I have been generated for both the static active and the static passive

if.

a

30

/)

a

00

cases. They are shown in Figs. 6.10 and 6.11. The gradual change in the pressure distribution as the result of gradual wall yielding is clearly illustrated in the figures. For reference, the translational pressure distribution for am = I, which corresponds to the ultimate state of all types of wall movement is also shown in the same figures. It is interesting to note from both Figs. 6.10 and 6.11 that the lateral pressure distribution for the case of rotation about the toe only changes locally from the original at-rest line as the wall continuously yields until the strength in the backfill is fully mobilized near the top, although it is believed that the pressure distribution may eventually approach the ultimate state if further movement is allowed. This can

°

{3al=OO

-N

{3= I = 0 0

= 2

N

-

'0 c.

{:.

2

o c. o

3

t-

~

3

~

o

o

iii

a;

'"

ttl

.l:

.l:

Q. o

Q.

'"

o'"

4

Passive At Rest State 5

4

Normalized Pressure Intensity. PA(zl/y.ft

Fig. 6.10. Change in distribution of static active earth pressure as the result of gradual wall yielding.

o

('it

a

5

'it.)

10

15

Normalized Pressure Intensity. Pp (zl/y, ft

Fig. 6.11. Change in distribution of static passive earth pressure as the result of gradual wall yielding.

257

256 be seen from the finite-element results shown in Figs. 6.12 and 6.13 (Potts and Fourie, 1986). The lateral pressure distribution changes considerably from the atrest distribution as the wall displacement increases, until the strength of the backfill is fully mobilized at the moving end for the case of rotation about the top.

Thereafter, it is expected that the lateral pressure distribution will gradually approach the straight line representing the translational wall movement case if the wall is allowed to move further. This is because the limiting state will propagate upward as the result of pressure redistribution. Consequently, the soil near the point of rota-

Ql

o

ACTIVE SIDE

o ~

::J

Curve Load Factor

CD ®

1Il

,.,~ /

0.20

~63

t

1'00~' I

@

1 i

I

.

,I@

'(j)

.I J

I

PASSIVE SIDE <.I:

1,"

1

"C

2

~ 21 ~

§

0

3]l

3

1 a 4r

t

--L.~Ka +@

~~

Ql

Cl

--2--'0'--0---''---100

(3)

.

l

100

® ®

,,:--..,....

'''-'' \.'\

~, ~

.

200

200

0.5 1.0

'~~ "'........,...

Ko"

KA "" "' .....

5'---'-

''''-..

® ...... '-Q2~P'';t', .........

'-------'-----"-200 100

100

' , .........-1;;.,.

300

400

Horizontal effective stress (k Pal (a) EQUAL TRANSLATION

(a) EQUAL TRANSLATION

o

0-.~

,"'-.

Horizontal effective stress (kPa)

Curve Load Factor

0.1 0.51 1. 00

~ ~\•.

I

4

cr:,

•

1.00

Curve Load Factor (1;, 0.1

Curve Load Factor

0.46

.~\~' ®

PASSIVE SIDE

ACTIVE SIDE

O~C. urve .~oad Factor

Curve Load Factor

(1" ®

(1)

tID ®

0.54 1.00

3

K 4-

I

5L---'100

4

.....

"-'--"~c=

100

(b) ROTATION ABOUT TOP

Curve Load Factor (j)

tID @

100

200

400

(bl ROTATION ABOUT TOP

Curve Load Factor

CD tID ®

0.08 0.5 1.00

Curve Load Factor

0.1 0.61 1.00

0.1 0.5 1.00

~

=-_......L..,'Lt=... = 200

sL.:!:_--'100

400

100

200

300

400

(c) ROTATION ABOUT BOTTOM

Fig. 6.12. Finite-element solutions of active and passive pressure distributions for smooth wall (' 25°, {, = 0, fJ = i = 0) (Potts and Fourie. 1986).

Fig. 6.13. Finite-element solutions of active and passive pressure distributions for rough wall (. = 25°. {, = '. fJ = i = 0) (Potts and Fourie. 1986).

259

258 tion will be brought to the ultimate state of 1/; = rf> and result in a uniform distribution of 1/;(z), if the soil is considered perfectly plastic, as is done in all limiting equilibrium methods of analysis, including limit equilibrium and limit analysis. The lateral pressure distributions as shown in Figs. 6.10 to 6.13 lead to a very interesting finding. Note that at the stage of full mobilization of strength at the moving end, the pressure distribution for the case of rotation about the top approaches the ultimate translational line with its corresponding resultant pressure the same as that for the translational case. On the other hand, the pressure distributions at the end of wall yielding for the rotation about the toe are not very different from the at-rest lines with their corresponding resultant pressures the same as the active and passive at-rest pressures in the active and passive cases, respectively. These tend to indicate that the resultant lateral pressure is dependent only on the mobilized ¢value at the toe, although the shape of the pressure distribution is surely affected by the assumed distribution of 1/;(z) as shown earlier. This finding explains why the lateral pressure distribution for the case of the rotation about the toe, in which the 1/;-value at the toe,1/;= 1/;a' remains unchanged upon wall yielding, does not change as much as that for the case of rotation about the top in which the 1/; value, or 1/;b-value, changes with the amount of wall displacement. Also, this helps to explain why rotation about the bottom gives higher KA-values and lower Kp-values than rotation about the top and translation do. The above also explains why the differences in pressure distributions for the rotation about the top, based on the Dubrova and the recommended distributions, do not differ as much as those for the rotation about the toe. This is illustrated in Figs. 6.6 and 6.7. In fact, the consequence ofKA = K p = 1.0 for the case of rotation about the toe based on the Dubrova distribution can be explained by the abovementioned finding. The 1/;-value is assumed by Dubrova as zero at the toe, and consequently the backfill behaves like a liquid or undrained clay with c/> = 0 and K = I, although the pressure distribution is not linear. The importance of the above-mentioned finding is particularly important when the shape of the assumed distribution of 1/;(z) is considered. After the two extremes of a 1/;(z)-distribution, 1/;a and 1/;b' are selected based on consideration of the state of stress at different stages of wall movement, the only problem remains to be solved is how to vary the 1/;-values between the two extremes. Perhaps, it has to be relied mainly on experimental observations. The linear distribution of 1/;(z) adopted in this chapter is based on limited experimental results available. Fortunately, the above suggests that the assumption of a linear variation of 1/;(z) between the two extremes, 1/;a and 1/;b' is justified. Based on Figs. 6.6 and 6.7, it appears that although the magnitude of resultant lateral pressure is essentially controlled by the 1/;-value at the toe, 1/;/, the shape of the pressure distribution is greatly influenced by the 1/;-value at the top, 1/;u' or the difference between 1/;/ and 1/;u' The shape of the 1/;-distribution assumed seems to in-

fluence the pressure distribution to a much smaller extent than that of the assumed 1/;u-value. Hence, once the two extreme values of 1/;-distribution, 1/;u and 1/;/, are properly selected, the shape of. pressure distribution, or the point of action will be influenced by the assumed shape of 1/;-distribution to a relatively minor extent. As discussed before, the magnitude of resultant force is practically unaffected by the shape of the 1/;-distribution when 1/;u and 1/;1 are specified. Therefore, the suitability of selecting a linear 1/;-distribution is greatly enhanced. The conclusion made from the study of changes in lateral pressure distribution as the result of wall yielding as shown here by the modified Dubrova method appears reasonably justified. Nevertheless, the general validity of the finding may need further justification. Terzaghi (1936), interpreting from his experimental observations, indicated that for rotation about the toe, the total resultant pressure drops off immediately after wall starts yielding, even long before the actual slip in the backfill is observed. The close-to-at-rest pressure as obtained, based on the modified Dubrova method and the recommended distribution of 1/; and ow' is not consistent with the actual observations of free standing retaining walls. The validity of the modification of the Dubrova method for actual application therefore needs further justification at least for the case of rotation about the toe. For translation and rotation about the top, the preliminary results obtained so far based on the modified Dubrova method seems reasonable, although it has been observed that rotation about top gives higher resultant pressure (Terzaghi, 1941). For rotation about the toe, it is apparent that the pressure so obtained is not representative of the limiting active or passive pressure. However, the lateral earth pressure so obtained using the recommended 1/;-distribution corresponds reasonably to that when the soil near the top of the wall has just reached the limiting active or passive state while the soil below remains essentially at rest. Hence, the pressure distribution so obtained is at least representative of this particular state, which may resemble the behavior of basement waIls during an earthquake or the behavior of rigid retaining structure fixed at the bottom and somewhat restrained at the top. Hence it is still worthwhile to present some numerical results for all three modes of wall movement before giving an overall evaluation of the modified Dubrova method.

6.3.3 Point of action for static conditions It is generally considered that Rankine's theory is applicable for lateral earth pressure assessments when the wall is rigid, vertical, and perfectly smooth, and the backfill is horizontal. The theory gives a linearly distributed lateral pressure behind a perfectly smooth rigid wall for both outward and inward translational wall movements. The points of action, expressed in terms of the height from the bottom

260

261

of the wall, ha for the active case and h p for the passive case, can be taken as HI3(H = wall height) above its base, according to the theory. Some results of the modified Dubrova method are first compared with those of Rankine for i = (3 = 0 = 0. Figures at 6.14 and 6.15 show the lateral pressure distributions as obtained from the proposed method for the two rotational modes of wall movement. The translational lateral pressure which is linearly distributed with its magnitude equal to the Coulomb's solution, or Rankine's solution for this special case in which 0 = 0, according to the modified Dubrova method is also shown in the same figures. It is found that the pressure distribution for the three different modes of wall movement differs from one another. For both rotational cases, the distribution is not linear, in general. The same results can be found in the finite-element solutions shown in Figs. 6.12 and 6.13.

N

2

f3-1=00

¢ = 30°, 8 = 0°

g-

PpE

(z) = dP PE dz

3

t-

Static ( kh = 0

~

o

0

"

'"

- - - Seismic ( k h - 0.20)

lD

\

\

\

C. 4

c'"

\ ~\

\\\ \ \\

-N

\\ \\

\\

\\

2

\

\

0

0

0.

",

1\ \

,

,

\

\ \

0 Q)

-

\ \ \

~

.Q

\

\ \

~

.s::

0

\'\ ,

\

3

ep =30

\' \

::=

{3=i=Oo

\\

4

l'

AE

Q)

0

6 0::-----:-5--.l---.lLJ.l0=----...Jl..J5L---.:L~2LJ.0~~--2J5

,, --Static ,, ----- Seismic ,, , ,

0

5

8 =0

dPAE (z)=-dz

\

0.

•

J"

,"'/ ... ...

... ,

( kh=O) (kh=0.20)

::r ... ,

,,

6 0

2

3

" ,,

4

Normalized Pressure Intensity

PpE (z)1

r.

ft

Fig. 6.15. Distributions of static and seismic passive earth pressures for a perfectly smooth wall.

... 5

Normalized Pressure Intensity .1' AEI z Vy. ft Fig. 6.14. Distributions of static and seismic active earth pressures for a perfectly smooth wall.

.The points of action h a and h p ' corresponding to each pressure distribution in FIgs. 6.14 and 6.15 are listed in Tables 6.1 and 6.2, except for translation in which h a = h p = l is found. For rotations, they are found to be different from the Rankine value: h a = h p = H13. The difference is expected, since in the modified Dubrova method, the 1/t-value is allowed to vary along the height of the wall while 1/t = if> is assumed in the Rankine theory. The h - and h -values are found to be highly dependent on the mode of wall movement p For the case of outward rotation about the toe, the state of stress in the soil near the top of the wall is expected to change from the at-rest condition to the active condition while that very close to the toe remains at-rest. The concave upward distribu-

263

262 tion curve as obtained by the proposed method is not unreasonable, although Terzaghi (1936), based on his large retaining wall tests (Terzaghi, 1934), claimed that h = H/3 for this case. It should be noted that the point of rotation in all the tiiting tests he conducted was not right at the toe. On the other hand, for rotation about the top, some arching effect is expected near the upper portion of the wall. The parabolic-like distribution as shown in Fig. 6.14 seems reasonable, although it is flatter than that reported by Terzaghi (1941). The nonlinear distribution of passive pressures for inward rotational cases as shown in Fig. 6.15 are also reasonable, since passive resistance increases as the degree of strength mobilization increases, in general. For the case of inward rotation about the top, the mobilized strength parameter if; increases as depth increases. The concave upward distribution may therefore be expected. The pressure intensity is supposed to decrease as depth increases for the case of rotation about the bottom

in which the degree of strength mobilization varies from the highest value at the top to the lowest value at the bottom. However, the effective overburden pressure increases as depth increases. This -may compensate partly the effect due to a reduction in if;-value with depth. As shown in Fig. 6.15, the change in pressure intensity with depth is, therefore, not as sharp as that if a material of less weight is considered. Some results on active and passive pressure distributions for a fairly rough wall are also obtained based on the modified Dubrova method. They are presented in Figs. 6.16 and 6.17. The distributions are found to be similar to those for perfectly smooth walls. The comparison with available information is presented as follows. There exists a very limited theoretical work in the open literature that deals with O . - - - - - . - - - - - - r - - - -.....- - - - - , - - - - - - ,

o.----.,-----,---,-----r----, 8

~,

-

t

khW

,/ ,/'"

~/~y,¢)

.....

-N

o

a.

N

=30 0

•

8=/2

'0 c-

o

I-

dz

\

£0

.r:

~

..

o CD

::a.

,,\

a;

k=O

Q.

I

\.

4

0

4

p () _ dPPE PE z --dz-

,~

q~

3

~

0

~

.c

\ \ \

~

dP AE 'PAE(z)=--

3

\

Oi

f3=i=OO

2

/3 - I - 0° ¢ -30 0 ,8-¢/2

2

~\

CD

o

-kh-O

\\~ \ \'

0.10 0.20

\~

\\~'\

J=

\ 5

6'--

6

o

4

5

Normalized Pressure Intensity. PAE( Z )/y. ft Fig. 6.16. Distributions of active earth pressure for earthquakes of different magnitude.

o

L..:L..L-

10

...L...

20

'----""----'"----""--

30

40

-'

50

Normalized Pressure Intensity, PpE (zl/y, ft

Fig. 6.17. Distributions of passive earth pressure for earthquakes of different magnitude.

264

265

lateral earth pressure distributions. Here, the work of Prakash and Basavanna (1969) which makes a distinction between a sliding (translational) wall and a tilting wall, a wall rotates about its toe, is selected for comparison with the modified Dubrova method for the active case. In their method, they take into consideration not only the force equilibrium on the triangular sliding wedge, but also the moment equilibrium of the rotating wedge. By optimizing the overturning moment or the active force for the determination of the most critical sliding surface, they were able to make the distinction between a tilting wall that occurs when the overturning moment is maximum, and a sliding wall that occurs when the active force is a maximum. Basavanna (1970) tried to improve the solution of Prakash and Basavanna (1969) for the sliding case based on the same framework. However, the improvement is very limited.

... ...

0

1.8.------,-------r------,------, \

.~

...

~

~

Q)

E

=c-

~

\

\ \ \ \

r<)

\

:I:

\

,

4

1'=30°, 8=7.50° dPAE 'PAE(zl=-dz

-'=

"

rt)

1'=30°,8=7.50°

1.4

{3=i=Oo

"-

:I: >.

Method

"'0 Q)

1.2

N

~

c

- - Rotation obout the Top _ . - Rotatian abaut the Tae

.!2

-

"-

Moditied Dubrova

~ Q)

{3=i=Oo

.c

0 0.

.c

1.6

"-

\ \ \ \ \

N

-

\

.~

2

A typical comparison of the result of Prakash and Basavanna (1969) and that of the modified Dubrova method is shown in Fig. 6.18. It is found that although Prakash and Basavanna (19.69) tried to differentiate sliding walls from tilting walls, they obtained identical results from the two different wall movements for the static case. The modified Dubrova's method, however, gives a different pressure distribution for a different wall movement. The distribution of translational pressure obtained by the modified Dubrova method, which is not shown in Fig. 6.18, is linear. It is close to the distribution curve for the rotation about the top, as shown in Fig. 6.14. It is therefore clear that the distribution of translational pressure obtained by Prakash and Basavanna (1969), although not linear, does not deviate much from that obtained by the modified

6

...E 0

Z

-'=

0. Q)

.~.

0

Static ( kh=Ol

8

Prakosh and Basavanna (J969 1

\

-0

c:

{kh=O.IO • ' \

'\

2

4

0.8 M.D.M.=Modified Dubrova

"0 a..

\.

0

0

+= 0


---- Rotation obout the Toe and Translation

Seismic'~

10

c:

0.6 L 10

Normalized Pressure Intensity, 'PAE( Z l/y, meter Fig. 6.18. Distributions of static and seismic active earth pressures by the modified Dubrova method and Prakash and Basavanna's solution.

o

-'-

0.1

Method

.--,;L-

-'-

--J

0.2

0.3

0.4

Horizontal Seismic Coefficient, k h Fig. 6.19. Comparison of h.-values obtained by the modified Dubrova method with solutions of Prakash and Basavanna (1969).

266 Dubrova method, which is linear. However, the distribution of active pressure for a tilting wall as obtained by them seems close to that for a wall rotating about the top rather than that for a wall rotating about the toe based on the modified Dubrova method. According to the modified Dubrova method, the distribution of the active earth pressure for a tilting wall about the toe is very close to the at-rest line. The points of action, ha, corresponding to those distributions in Fig. 6.18 are shown in Fig. 6.19. It is found that for the static case, the ha-value obtained by Prakash and Basavanna (1969) for both tilting and sliding walls is higher than those for the translational wall, ha = H13, and for walls rotating about the toe as obtained by the modified Dubrova method. For the reasons given earlier, it seems that the distribution curve should be concave upward for the case of rotation about the bottom. This tends to suggest that ha < HI3 as obtained by the modified Dubrova method is more reasonable. It should be noted that both 1/;- and ow-values are assumed to vary with depth in the modified Dubrova method, while they are assumed constant with depth and equal to their corresponding maximum values, c/> and 0, in the Prakash and Basavanna's formulation. This is probably partially responsible for the difference in pressure distributions as obtained by the two methods. In summary, it can be concluded that the distribution of lateral earth pressure, or the resultant pressure and its point of action, is highly dependent on the mode of wall movement. The current practice of assuming ha = h p = H13, regardless of the type of wall movement, appears to be very much in error. The distribution of static lateral pressure is found to be nonlinear for rotational wall movements, -regardless of the soil- wall interface roughness.

267 responding points of action ha, hp for the distributions shown in Figs. 6.14 and 6.15 are summarized in Tables 6.5 and 6.6. Figures 6. 16 and 6.17 show the gradual change in lateral pressure distribution for rotational wall movements as the result of a gradual increase in the magnitude of earthquake, which is represented by the seismic coefficient, k or its horizontal component k h · The corresponding ha- and hp-values are plotted against the kh-value as shown in Fig. 6.20. It is found that the ha- and hp-values are practically unaffected or only decreases gradually as the kh-value increases for the case of rotation about the top. For the case of rotation about the toe the ha-value gently increases up to k h = 0.1 and decreases thereafter, while the hp-value increases as the kh-value increases for this case. The comparison with available information in active cases is presented as follows. 0J45r----r------,-----,------.,

0:40

1-------

:x:: .....

c. s:.

6.3.4 Point of action for earthquake condition It is generally reported that the point of action of resultant lateral pressure for

the seismic case is higher than that for the static case. To see if this is the case, distributions of seismic lateral pressures for different wall movements, as obtained by the modified Dubrova method are shown together with the static pressure distributions in Figs. 6.14 and 6.15 for a perfectly smooth wall. It is found that the translational lateral pressure, which is the same as the Mononobe-Okabe or the modified Coulomb solution, is linearly distributed. The distribution of rotational lateral pressures are nonlinear, however, the shape of the distribution of seismic lateral earth pressure is found to differ from that of the static lateral pressure. In general, the active pressure intensity increases and the passive pressure intensity decreases when subject to the action of an earthquake. Consequently, K AE increases and K pE decreases. However, since the shape of the distribution is not much altered by the presence of earthquakes, the point of action of the resultant lateral pressure changes only slightly. The changes in K AE , K pE and their cor-

c co

Q 3 0 1 : : - - - - - - - - - -_ _

+:

....~co

1;;

~ 025 .."

rp= 300, 8 = rpl2 i O· ,( a=900) ,B=0·,H=6ft

Gl

.!:!

=

;;;

e<-

co

Z

0.20L...-

o

..L.-

0.1

...1--

0.2

-'-

.......J

Q3

0.4

Horizontal Seismic Coefficient,

Fig.

6.20.

kh

Effect of earthquake forces on point of action for different wall movements.

269

268 The point of action of resultant active earth pressure during earthquakes was first extensively investigated by Ishii et al. (1960) experimentally. Their model tests on gravity walls reflect that for the case of translational wall movement, the point of action h a increases gradually from a value of about HI3 at kh = 0 to a value of about 0.4 H when kh = 0.4. For the case of rotation about the toe, they found that the ha-value is only slightly increased as the kh-value is increased. Their observations seem to agree with the results of the modified Dubrova method for rotation about the toe, on one hand, and disagree for translation, on the other hand. However, it should always be recognized that all the model tests may distort the actual soil behavior because of scale effect. Results of model test may, therefore, be unreliable. For comparing the results of the modified Dubrova method with other theoretical solutions, the work of Prakash and Basavanna (1969) as described earlier is referred. A typical result of the active pressure distribution for the case of k h = 0.10 obtained by them is compared with that obtained by the modified Dubrova method as shown in Fig. 6.18. As in the static case, it is found that for the case of k h = 0.10, the pressure distributions for a sliding wall and that for a tilting wall are practically identical. Hence, only the average distribution is shown in the figure. Again, the seismic pressure distribution as obtained by them is close to that for the case of rotation about the top as obtained by the modified Dubrova method. The distribution of seismic active earth pressure for the case of rotation about the toe, as obtained by the modified Dubrova method, is found to be much flatter. Figure 6.19 shows the variation of h a with k h for results obtained by the two different methods. The ha-values as given by Prakash and Basavanna (1969) are found to increase rapidly as the kh-value increases. The distinction between the ha-value for a sliding wall and that for a tilting wall is clear only when the kh-value is high. They also show the vertical acceleration component, kvg, has some effects on the ha-values for both the sliding and tilting cases. The results of the modified Dubrova method, however, indicate that the point of action, ha, is practically unaffected by the kh-value for all three modes of wall movement. Also, the h a-value is found unaffected by the present of k v ' For the case of rotation about the top, ha > HI3 is obtained. For the case of rotation about the toe, h a < HI3 is found. The translational movement, similar to the modified Coulomb solution, gives h a = H13, which is completely independent of kh-value. However, based on Prakash and Basavanna (1969), the ha-value is found always higher than HI3 for either sliding or tilting wall. This finding may need further justification at least for the case of tilting wall about the toe for which a concave upward distribution is probably more representative of the actual distribution. In order to see how the earthquake alone affects the active earth pressure, seismic increments of active pressures on a perfectly smooth wall are calculated for different wall movements. The resulting distributions of the seismic increments are shown in

Fig. 6.21. It is found that, based on the modified Dubrova method, the distribution is linear only for the translational case. The point of action corresponding to !JJ(AE' h~ = HI3 (h~ is the ha-value corresponding to the seismic increment of the active earth pressure) as found by Nandahumaran and Joshi (1973) for the case of o = 0, is only true for the translational wall movement. It is interesting to note that the distributions of seismic increments of active earth pressures as shown in Fig. 6.21 are almost identical in shape to their corresponding resulting active pressure distributions for different modes of wall movement. Also it is found that K AE and MAE for the rotation about the top are the same as those for the translation case. The h~-value is larger than HI3 for the case of rotation about the top, while h~ < HI3 is found for the case of rotation about the toe.

° ¢ _30 0 8 _0 0

K"E -P"E /.5 Y H 2

6.

2

K"E -K"E -K"

k h -0 _0.20

3

'0

6.

K"E -0.15

0.

~

hi-O.333H

it a

Qi

.0

4

J:

a.III

o

5

6.

K"E -0.15

hi-

.361H

6

°

0.5

1.0

1.5

2.0

Seismic Increment of Pressure Intenslty,6. P"E (zl/y, ft

Fig. 6.21. Distributions of seismic increments of active earth pressure for different wall movements.

270

271 It is found that the h~-value is only slightly affected by the khyalue. This is also different from that found by Nandahumaran and Joshi (1973) who reported that h~ increased linearly with -the -kh-value. Nevertheless, due to the unreasonable assumption made by them, Nandahumaran and Joshi's findings are not necessarily justified. The comparison with available information in passive cases is presented as follows. Since passive earth pressure problems involve extensive interaction and progressive failure phenomena, the problem is more complicated than the active earth pressure case. Furthermore, the mQst critical sliding surface, unlike the active case, is generally curved for the passive case. For these reasons, studies on the seismic passive earth pressure are more difficult than those of the seismic active pressure.

0.70

ep=300 , 8= epl2 :I:

___ §:'L=_O~ ______ L~e~r~t~~!~~!...hi=!~3H_ __

......

"

.c

lIJ

0.60

- - - Modified Dubrova

Method

~cr.


-g

----- Oth.. '"II,bI.

o

',',m,H"

1

l

Common Proctlce

t il

.5

0.50

c.

----~~~~~~:~--_!!~!~~--- --

III

......

G)

"r--

U

S

:;:

.:£

Prakash (1981), h

0.40

/

o

"E

&

£ ,..-

or--------,------,--------,-----¢ - 30°

Hydrostotic

{] _ 0°

\

"'0

.~ c

'= 0.45 H

e ------------------------------

o

0.30

\f

...E o

Z

0.20 0.10

0.15

0.20

\r 0.25

K PE -- I'PE

/.1. 2 Y

H2

2

N

kh - 0

0.30

-

0.20

'0 Horizontal Seismic Coefficient, k h

n

/;:, K PE - 0.85

3

o

I-

Fig. 6.22. Point of action of seismic increments of active earth pressure.

h p ' - 0.333H

'it

o iii m

Figure 6.22 shows the change of h~-value with k h for different modes of wall movements as obtained by the modified Dubrova method for a fairly rough wall. The h~-values so obtained are found not much different from the hydrostatic value of H13. They are much lower than the theoretical value of 2H13, the value found by Wood (1975), and the value as suggested by Prakash (1981). However, according to the general wedge theory (Terzaghi, 1941), the point of action will rise when the sliding surface becomes curved, if both ¢ and 0 are assumed constant. In fact, Prakash and Basavanna (1969) also noted that h~ = HI3 for the case of 0 = O. Hence, it is believed that the high h~-value as obtained by many investigators for rough walls is partly the direct consequence of assuming if; = ¢ and Ow = all the way along the height of the wall, as is usually done in conventional earth pressure theories.

°

.s::

4

n

OJ

o

5 h p'

=

0.293H

6

o

2

4

6

8

Seismic Increment of Pressure Intensity. 6 ppE(zl/Y , It Fig. 6.23. Distributions of seismic increments of passive earth pressure for different wall movements.

I !

272

273

There is even less information available on the point of resultant seismic passive earth pressure, h p • As shown in Fig. 6.20 the hp-value for a translational wall movement is H13. The rotation about the top gives h p < H13, while the case of rotation about the toe gives hp > H13, which are the reverse of those for the active case. These results are consistent with the finding of Narain et al. (1969) based on model tests in sand. The deviation of h p from the hydrostatic value HI3 is found larger which indicates that the distribution is more curved for the passive pressure case than for the active pressure case. In order to compare results of the modified Dubrova method with the only available information on the distribution of seismic passive pressure, as given by Ghahramani and Clemence (1980), the seismic increment of passive earth pressure is considered. Figure 6.23 shows the distributions of seismic increments of passive pressure for different wall movements as obtained by the modified Dubrova method. It is found that the shapes of these distributions are almost identical to those of seismic passive pressure. By using the theory of zero-extension line, Ghahramani and Clemence (1980) developed a method of assessing the coefficient of passive earth pressure for the increment of pressure as the result of a seismic acceleration. Also, based on the work

of Sabzevari and Ghahramani (1979), in which walls were allowed to move independently at a maximum acceleration amax = kg in an incremental manner, they suggested a distribution of seismic acceleration for each mode of wall movement. These are shown in Fig. 6.24. By incorporating these distributions into their proposed method of analysis, they found that the distributions of passive pressure increments as the result of different wall movements are those shown in Fig. 6.24. Their shapes are independent of amax and o. Comparing Fig. 6.23 with Fig. 6.24, it is interesting to note that, although based on completely different concepts, the two methods give identical distribution for seismic increment of passive pressure in the case of translational wall movement. The distributions for the case of wall rotating about the top are practically similar, although the corresponding hp-value obtained by the modified Dubrova method is 0.70r - - - - , . . . - - - - , . . . - - - - , . . . - - - - - - - , 1

:r: ,0.

..c

0.60

w a..

Modified Dubrova

Method

Ghahramani & Clemence ( 1980)

~

<]

... 0

J= -=f

JF a

Pp

Pp

Pp

~=H/3 ( pP)mox

=HI2

en

0

()

v

c 0 -.;:; u

< ....

0!40

... 0

"

.5 0

a.

.... ClI

.!::!

;;;

...E

Pp

lThr~HI' ~ ( pp)mox

hp

0 0.

ao

lao"

0.50

.........

a

a

Da~.

l

0>

c '6 c

h~ = H/4

0

z

~-----------

020,L--pPp

h;=H/2

j..j

1/4(Pp)mox

Fig. 6.24. Distributions of acceleration assumed and passive pressure increments obtained by Ghahramani and Clemence (1980) for different wall movements.

0.10

..J-=...:.-_-'0.15 0.20

-L.

0.25

Horizontal Seismic Coefficient, k h Fig. 6.25. Point of action of seismic increments of passive earth pressure.

-.J

0.'30

II 275

274

6.3.5 Effects of strength parameters and geometry of soil-wall system on point of action

a little higher. The distributions for the case of rotating about the bottom, although very different, all indicate that h p > H13. The h '-values obtained by the modified Dubrova method for different wall moveme~ts for a fairly. rough wall are shown together with those reported by Ghahramani and Clemence (1980) in Fig. 6.25. It is found that they both follow the same general trend. More importantly, they both indicate that the h~-values are practically independent of the seismic coefficient. This is in contrast to the active case in which many investigators found that h~ increases as the kh-value increases. Although similar to the active case, the modified Dubrova method also shows that h~ is practically independent of k or k h'

In addition to the mode of wall movement that affects the lateral pressure distribution and the point of action of resultant lateral pressure, the ha- and h p values are also dependent on the strength parameters ¢ and a and the geometry of soil-wall systems, represented by the angle of backfill slope 13 and the angle of wall repose a or wall tilt angle i = 90° - a. The modified Dubrova method is adopted for studying the effects of these factors on the ha- and hp-values. The effects of strength parameters ¢ and a are first investigated. Figures 6.26 and 6.27 show the variations of ha and h p , respectively, with the ¢- and a-parameters 0.7.--------,,-------,------,------,

I- 0

0.6.--------,------,------,-----,

0

f3 - ¢

0.6

(a

_90° )

/2

i=O· (a=90·)

ta=¢/4, a = ep/2

-" I

- - Static Ckh -0)

O.S

~

.c

c.

.

c: 0

'';:; u

~

0

,r-

c

~o

Practi cally unaffecte~ by a-values

0.4

-

.....

.c

-

----

- - Selsmlc(kh -0.2)

0°, ¢/2, 2'1'/3,

¢

-o c

kh-values

'0

Q.

0

0.. '0 4J

-

8-

«

-------- -----

Translation,for all a-&

c:

0.5

't> a>

!! a;

0.3

N

C

Practically unaffected by a-values

E

'-

0 Z

E (;

~ ~

Z

Translatlon,for all

8-

&,k h - values

0.3t:===--------,--__-J

0.2

Seismic (k h = 0.20) 0.1'--

30·

--'

3S·

---1.

40·

--'-

--'

4S·

Angle of Internal Friction,

ep

Fig. 6.26. Variation of h a with cp- and a-parameters for different wall movements.

Angle of Internal Friction, ¢ Fig. 6.27. Variation of h p with cp- and a-parameters for different wall movements.

276

277

for different wall movements. It is found that the a-value has practically no effect on the ha-values for all modes of wall movement. As found earlier, the ha-value for the case investigated is also practically unaffected by the presence of earthquakes. Consequently, it can be concluded that the h~-value, is also practically unaffected by the a-value. This is different from that reported by Nandakumaran and Joshi (1973) who found h~ increases parabolically with increasing a-value. However, it should be noted that Nandakumaran and Joshi's work is based on a questionable assumption that the static-failure wedge is the same for the seismic condition. Their finding is therefore subjected to question. In the active case, it is also found that ha is only slightly affected by the ¢parameter for rotational wall movements. For the translation case, h a = HI3 is ob-

tained. It is not only independent of the kh-value and a-value, but also independent of ¢. In the .passive case, although as shown before, hp-values are practically unaffected by the kh-value for the cases studied, it does depend on both ¢ and aexcept for the translation in which h a = HI3 for all a- and ¢-values. As shown in Fig. 6.27, the effects of ¢ and aon h p are of the same order for the two basic types of rotational movements. Nevertheless, the effect of ¢ is larger when ais smaller. In general, h p increases as ¢ increases for a given a-value for the rotation about the bottom case. On the other hand, for rotation about the top, h a decreases as ¢ increases for a given a-value. The effects of geometrical factors {3 and i on the h a- and hp-values are presented 1.0 r-----,-----,----.....,----.,.-------,

0.6r------r----.,......------y-----,-------,

1_90° -

a

¢-400 1-

0.5

8 - ¢ 12

30~

J:

.,

0°,

~~~~;:;;~;:9~;:~"F==::==~~~:::=~

0.3!=:

- - - Selsmlc(k h -0.2)

.-..-

....., ---=~

0.4

0

Translatlon,for all 1- & k h - values

0.

~

- - Static (kh -0)

- - - --

;; «: '0

C

f

-30°

.l:!

Translation, for all i -& kh -values

c:

"tl

"tl

Ql

1l

.!:!

~

"iii E (;

1-

(;

z

130°,

0.6

.c_ c:

. 6

-

a

¢-35? 8- ¢/2

0.4

~'0

1_90° -

0.8

z

0.2

---

0.2

1-

- - Static (k h -0)

-30°,

0°,

300~

- - - Seismic (k h -0.20) 0.1'--

o

-L

0.1

.l-

0.2

Normalized Slope Angle

----'-

0.3

-'-

0.4

---'

0.5

,13 I ¢

Fig. 6.28. Variation of ha with geometry of soil-wall system for different wall movements.

0.0 0

0.1

0.2

0.3

0.4

0.5

Normalized Slope of Backfill ,PI

Fig. 6.29. Variation of hp with geometry of soil-wall system for different wall movements.

279

278 in Figs. 6.28 and 6.29. Again, the figures show that for translating walls, the haand hp-values are equal to H/3 and are independent of the geometry. In the active case, it is interesting to note that the ha-value is not always independent of the seismic acceleration k h . When i = 30° in the case of rotation about the top and i = - 30°, rotation about the bottom, ha is found to be different for both the static and earthquake cases. This finding is different from that reported earlier based on i = 0°. The conclusion that ha is practically independent of k h is therefore not always true and it depends on the geometry of soil-wall systems. Figure 6.28 also shows that ha decreases as the i-value decreases (or the a-value increases) for both types of rotation. Also, it is found that, in general, the ha-value for both the static and the earthquake cases, decreases as {3 increases when the wall rotates about the top. On the other hand, for wall rotating about the toe, it is found that ha increases slightly as {3 increases in the static case and ha has a tendency to decrease with {3 as {3 === e/>/4 in the seismic case. In the passive case, the following points are of interest: (a) h p is practically independent of k h for any soil-wall geometry; (b) when i > 0°, h increases with {3 for the cases of rotation about the bottom for the rotation aboulthe top; (c) when i :s 0°, h p decreases very slightly with {3 for the cases of rotation about the bottom and for the rotation about the top;. and (d) the effects of i and (3 on hp are of the same order for both types of rotatIOn. 6.4 Evaluation of the modified Dubrova method The modified Dubrova method developed in this chapter is mainly for the purpose of locating the point of action of the resultant lateral earth pressure. Since it gives a complete pressure distribution for a specified wall movement, the resultant force corresponding to a distribution or a type of wall movement can also be calculated. In order to study the validity of the method for actual applications, not only the point of action but also the magnitude of the resultant lateral pressure, as obtained by the method, have to be compared with actual field observations. All the previous results and numerical results for static and seismic cases obtained above based on the modified Dubrova method and the assumed ,p- and 0distributions for the case of rotation about the toe tend to suggest that the active and passive pressures so obtained are close to the active at-rest and passive at-rest pressures, respectively. However, as pointed out earlier, the full-scale free-standing wall tilting tests conducted by Terzaghi (1934) clearly showed that the total resultant pressure acting on the wall drops off long before an actual slip in the backfill occurs. The validity of the modified Dubrova method for actual applications, therefore, needs a careful evaluation.

6.4.1 Basic assumptions oj the modified Dubrova method The Dubrova method was Jirst developed by Dubrova (1963) for the case of vertical wall and horizontal cohesionless backfill by assuming the validity of the Coulomb solution and the existence of an infinite number of quasi-rupture planes on which the mobilized e/>-parameter is directly dependent on the amount of wall movement. The modified Dubrova method developed in this chapter is a direct extension of this method. The Mononobe-Okabe formula, or modified Coulomb's equations, for the seismic active and passive resultant forces are used as the framework for the development of the method. The Dubrova concept of pressure redistribution is used as the basis for taking the wall movement into consideration. The wall friction angle is allowed to vary with depth. Detailed investigation of the methodology proposed by Dubrova (1963) reflects that the basic assumptions of the modified Dubrova method can be summarized as follows. 1. The Mononobe-Okabe solution is assumed to be valid. It assumes that the shear strength is fully mobilized simultaneously along a planar sliding surface passing through the toe of a wall at the instant when the displacement of the wall is just sufficient to bring the soil mass behind the wall to a limiting active or passive state. The soil behind the wall behaves as a rigid body and the seismic accelerations are uniform throughout the entire soil mass so that the effect of earthquake force can be represented by two inertial forces corresponding to the effects of the vertical and horizontal components of the seismic acceleration. 2. There exists an infinite number of quasi-rupture planes on which the average mobilized e/>-parameter, represented by ,p(z) (where z is the distance from the top of the wall to the point where the assumed quasi-rupture plane of concern intersects the wall), is dependent on, but not proportional to, the amount of wall movement. 3. The mobilized wall friction 0w(z) corresponding to each ,p(z) is proportional to ,p(z) such that 0w(z) = m ,p(z), m = Ole/> :S 1. 4. The accumulative resultant active and passive earth pressure at depth z can be obtained by replacing H with z, e/> with ,p(z), and with 0w(z) in the MononobeOkabe equations, Eqs. 6.1 and 6.5, in which e/> and are assumed constant with depth, even though both ,p(z) and 0w(z) may be variables and functions of the depth z. 5. The functions ,p(z) and 0w(z) and their derivatives are continuous. It should be noted that this assumption is necessary for avoiding unrealistic jump in the calculated pressure distribution.

°

°°

280

6.4.2 Failure mechanisms for free-standing rigid retaining walls In evaluating the validity of Coulomb's solution, Terzaghi (1936) pointed out that in the case of outward wall movement, the lateral yield of retaining walls is also 'always' sufficient to mobilize at least the major part of the shearing resistance along the sliding plane which separates the sliding wedge from the soil mass. There is no doubt about this for the case of translation and rotation about the top. This is because a distinct sliding surface passing through the toe is always developed at the early stage of tests for these two modes of movement (James and Bransby, 1970), although it should be recognized that the sliding surface is generally curved for the case of rotation about the top, even in the active pressure case. For the case of passive rotation about the toe, James and Bransby's model test results (James and Bransby, 1970) clearly showed that rupture starts near the top and propagates downward as the amount of rotation increases. A distinct rupture surface is only found after a considerable rotation when the soil mass behind the wall reaches a limiting, impending collapse state. Their results showed that this distinct rupture surface intersected the wall at about mid-height rather than passing through the toe. The findings of James and Bransby (1970) indicate that rupture surfaces are first observed at the top of the wall, not near the toe, as would be anticipated from conventional earth pressure theories. For this reason, they indicated that the simple failure mechanisms, as adopted in most conventional methods of earth pressure analysis, do not occur for the case of rotation about the toe. The same mode of wall movement for the active case has been studied by Kezdi (1958). His experimental results on a model wall continuously rotating about the toe clearly show the progress of failure. Among the important findings observed by him are: 1. A continuous surface of sliding develops after a certain amount of rotation only. 2. The distinct sliding surface does not go through the toe and instead it intersects the wall at a height ho about the toe, in which h o is dependent on soil characteristics and the amount of rotation. 3. Definite slides rather than uniform deformation is found in the sliding mass. These findings clearly indicate that rupture pattern as observed for the active rotation about the toe case is similar to that postulated based on the passive test results (Fig. 6.2). He has pointed out that it is obviously wrong to consider a surface on which there is no displacement at all, as a surface of sliding. The sliding surface passing through the toe as assumed in conventional limit equilibrium solutions based on wedge theory is, in fact, a surface on which the force of reactions is corresponding to the 'earth pressure at rest' condition. Based on the above considerations, use of conventional analytical method for determining lateral earth pressures is therefore improper for the case of rotation

281

about the toe. It should be kept in mind that all the limit equilibrium and limit analysis methods are generally subjected to the limitation that the validity of the solution is highly dependent on 'whether the failure mechanism assumed is proper or not.

6.4.3 Characteristics of the modified Dubrova method The discussion of the preceding section tends to suggest that the modified Dubrova method, which is based on the assumption that the Mononobe-Okabe equations are valid for determining the resultant active and passive earth pressures, is applicable to the cases of translation and rotation about the top. The implication of this can be clearly seen from Eq. (6.2) or Eq. (6.6), which is the consequence of Assumption (4). Based on Eq. (6.2), the accumulative resultant pressure at depth z can be obtained by inputting the 1/;- and ow-values corresponding to that depth into the equation. For both the translation and rotation about the top, the 1/;-distributions with 1/;, = ~ are assumed. Also based on Assumption (3), Ow = m 1/;, = is assumed at the toe in both cases. Consequently, both types of movement give exactly the same expression for the accumulative resultant active force, which is actually the original Mononobe-Okabe equation, Eq. (6.1). This explains why the KA-values as obtained on the basis of the modified Dubrova method and the recommended 1/;distribution as shown in Section 6.3 are exactly the same for the cases of translation and rotation about the top. On the other hand, for the cases of rotation about the toe, the 1/;-distribution with 1/;, = 1/;a' in which 1/;a = ~o for the case of (3 = 0, is generally assumed. Also Ow m 1/;a at the toe. Consequently, at depth z = H, for the case of i = 90° and (3 = Ow = Ok = k y = 0, Eq. (6.2) becomes:

°

-YH

P

2(

cos~o )

= -2- 1 + sin~(;

2

-yH2 2

(1 - Sin¢o) 1

+ sin¢o

(6.20)

This is exactly the active at-rest pressure. Similarly, for the passive case, Eq. (6.6) gives: p

= -yH 2( cos~o ) 2

1 - sm¢o .

2

= -yH2 2

(11 +- sm¢o s~n¢o)

(6.21)

This is exactly the passive at-rest pressure. The reason why, for rotation about the toe, the resultant lateral force obtained based on the modified Dubrova method and the recommended 1/;-distribution equals to the at-rest pressure is therefore apparent.

282 In fact, the finding in Section 6.3.2 that the resultant lateral force obtained based on the modified Dubrova method depends only on the 1ftrvalue is simply the direct consequence of Assumption (4). However, this unique outcome is not actually observed. It is generally recognized that rotation about the top gives slightly higher K A-value than translation does; whereas, the modified Dubrova method gives the same KA-values for both cases. Its actual applicability therefore, needs further justification. 6.4.4 Validity of the modified Dubrova method in practical applications Based on the discussion given above, it appears that the modified Dubrova method is strictly applicable only to the cases of translation and rotation about the top, in which a distinct sliding surface passing through the toe actually develops at the limiting state. The limiting active and passive pressures obtained based on the method and the properly assumed 1ft- and a-distributions are, however, justified only if the actual sliding surface is essentially planar, which is seldom true for rotation about the top (Terzaghi, 1941). The method does not appear applicable to the case of rotation about the toe for determining the limiting lateral earth pressures, since the assumed failure mechanism is inconsistent with actual observations. Moreover, although a 1ftdistribution more realistic than that recommended can be assumed based on the work of Kezdi (1958), Eq. (6.2) tends to indicate that the total resultant pressure as obtained will not be affected once 1ft/ = 1fta is selected. This is also clearly reflected in Figs. 6.6 and 6.7. Changes of the 4-value atthe top, ¥tu ' and the shape of the 1ft-distribution seem to affect only the shape of the resulting lateral pressure distribution, and consequently the point of action of the resultant force. Hence, it seems that the lateral earth pressure corresponding to the limiting state for the case of rotation about the toe cannot be simply solved by improving the assumed 1ftdistribution. The method can therefore be applied to this type of movement only if the actual wall displacement is somewhat restricted at the top. In general, the point of action for the resultant lateral pressure is only directly influenced by how the 1ft-distribution assumed differs from the uniform distribution. If a 1ft-distribution is properly assumed based on actual field observations, the point of action as obtained by the modified Dubrova method could be a reasonable estimate. The work of Kezdi (1958) tends to indicate that for retaining walls tilting about the toe, a 1ft-distribution (should be continuous according to Assumption (5) before) with steeper slope at the upper part and gentle slope at the lower part, is more realistic. The results of James and Bransby (1970) also tend to indicate that similar kind of 1ft-distribution is more realistic for passive rotation about the toe. Based on Figs. 6.6 and 6.7, the ha-value obtained by using this kind of distribution is smaller

283 than that obtained by using a linear 1ft-distribution. The reverse is found for the hpvalue. Hence, use of the ha- and hp-values as shown in Section 6.3 will result in a slightly larger overturning moment and slightly smaller resisting moment than if a close-to-real 1ft-distribution is assumed. These ha- and hp-values could be useful if the actual wall displacement is close to what is assumed. The active and passive earth pressures as obtained by the modified Dubrova method based on the recommended linear 1ft-distribution, although not representative of the limiting active state, correspond to the pressure when the outward and inward rotations, respectively, are just enough to bring the soil right near the top of the wall to limiting active and passive states. At shallow depths, where the confining pressures are very small, the shear strength needs only a very small amount of displacement to reach its full mobilization, which can be seen from conventional triaxial test results. This is especially the case for failure due to lateral extension, which represents the mode of active failure. It is therefore expected that major portion of the soil is still essentially under the at-rest condition while the soil right near the top first reaches a limiting state. Assuming that there is a quasi-sliding surface, on which the mobilized -value is equal to 1fta, passing through the toe appears reasonable in this case. The results are, therefore, representative of the condition of wall movement just described. Although, the distributions of lateral pressure are different from those corresponding to the completely at-rest states, the resultant lateral pressures, which are exactly the same as the at-rest pressures, also seem reasonable. In fact, based on the assumption that there is a quasi-sliding surface, on which the mobilized -value is equal to 1fta (passing through the toe), the long time unsolved problems of evaluating the increase in at-rest pressure as the result of an earthquake can be easily handled by using the modified Dubrova method. By assuming a uniform 1ft-distribution with 1ft = 1fta, or 0 for the case of {3 = 0, both the static and the seismic at-rest pressures corresponding to either the active or the passive condition can be calculated. This could be of practical value for the case in which the retaining structures are rigidly connected and they move together with the surrounding soil mass during earthquake. In short, the modified Dubrova method, if properly used, could be a useful tool in actual design applications. However, its limitations should always be recognized in order to gain benefits from it. To sum up, the study of the point of action of the resultant lateral pressure shows that the point of action is highly dependent on the mode of wall movement for both static and earthquake cases. Results of analysis based on the proposed modified Dubrova method show that the point of action for both active and passive translational movements is always at H/3. The distribution of translation pressure is always linear whether there is an earthquake or not, if the ultimate state is reached. For the rotation about the top of wall, ha > H/3 and h p < H/3 are found for the active and the passive cases, respectively. On the other hand, for the rotation

284

285

about the toe of the wall, h a < H/3 and h p > H/3 are obtained. The ha-value is found practically independent of the seismic acceleration when i :$ 0° for the rotation about the top and i;::: 0° for the rotation about the toe. The hp-value, however, is found practically unaffected by the presence of earthquakes. Comparison of results obtained by the modified Dubrova method with others shows that the h a- and hp-values obtained by the proposed method are, in general, closer to the hydrostatic value of H/3, than the others for all types of wall movement. This is probably the consequence of the fact that the strength parameters adopted in the proposed method, if;(z) and 0w(z), are allowed to vary with depth according to the nature of the wall movement; whereas, if; = ¢ and Ow = a are generally assumed by the others. The modified Dubrova method, which allows the distribution of lateral earth pressure corresponding to each stage of wall yielding to be calculated, shows that the resultant lateral pressure is governed only by the assumed if;-value at the toe. This special characteristic of the method can be clearly seen from Eqs. (6.2) to (6.5). Of secondary importance are the difference between the two extreme if;-values and the shape of the if;-distribution assumed. For the case of rotation about the toe, in which if; = if;a is assumed at the toe, the active and passive pressure distributions do not shift considerably from the atrest pressure distribution curve. On the other hand, for the case of rotation about the top, for which if; = ¢ is assumed at the toe, both the active and passive pressure distributions shift gradually from the active and passive at-rest distribution curves, respectively, to those close to the ultimate translational pressure distribution curves. It is therefore concluded that once the two extremes of the if;-distribution, especiaIly that near the toe, are properly selected based on the consideration of the change in stress state in the backfill during wall yielding, the lateral earth pressure distribution corresponding to the condition of the displacement specified can be reasonably assessed by the modified Dubrova method. Parametric studies show that 1. ha is practically unaffected by ep and 0; 2. h a is affected by i and is moderately affected by {3 only when i is much larger than zero in the case of rotation about the top and when i is much smaller than zero in the case of rotation about the toe; 3. ep, and i, {3 all affect the hp-value for the case of rotation about the toe as much as that for the case of rotation about the top; 4. h p is practically unaffected by {3 when i :$ 0° for the case of rotation about the toe and when i ~ 0° for the case of rotation about the top. Overall evaluation of the modified Dubrova method reflects that the method, although it has its limitations, can be applied to several earth pressure problems. The resultant lateral earth pressure and its point of action for the cases of translation and rotation about the top can be reasonably assessed by the method based on

°

the recommended linear if;-distribution, if the actual sliding surface is planar. For rotation about the toe, the method is not strictly applicable for determining the lateral earth pressures corresponding'to the limiting active and passive states. The method, however, could be used for determining the seismic at-rest pressure acting on a rigidly connected box structures, if there is no relative displacement between the structures and the surrounding soils during earthquake. In actual applications, the limitations of the modified Dubrova method, nevertheless, should always be recognized. A computer program developed for calculating the lateral earth pressures based on the modified Dubrova method is given in the thesis by Chang (1981). 6.5 Effects of wall movement on lateral earth pressures Although the mode of wall movement is seldom specified in the development of most classical earth pressure theories, it has been reported that both the magnitude of resultant lateral pressure and its point of action are highly dependent on the wall movement. The effect of wall movement on the point of action of the resultant lateral force has been treated in Sections 6.1 to 6.4. This section intends to explore its effect on the resultant lateral pressures, which are generally expressed as the dimensionless coefficients K A and K p for the static case or K AE and K pE for the seismic case. Results of the limit analysis which represent the resultant lateral pressure for translational wall movement and results obtained by the modified Dubrova method as described in the preceding section are selected for comparison with those obtained by others for which wall movement has been considered. The modified Dubrova method, theoretically, can yield the lateral earth pressure corresponding to any specified mode of wall movement if the distribution of mobilized strength corresponding to the specified wall movement can be reasonably assumed. However, only results for purely active and purely passive translational and rotational wall movements are considered here.

6.5.1 Effects of wall movement on static and seismic active earth pressures The modified Dubrova method is versatile in that the effect of wall movement, which affects the distribution of lateral earth pressure, can be incorporated into the analysis. However, it should be kept in mind that the method is developed based on the assumption that the Mononobe-Okabe, or the modified Coulomb solution is valid for assessing the resultant seismic lateral forces. For the solutions of the modified Dubrova method to be justified, the assumed mechanism of failure has to be close to the reality in the first place. For the cases of rotation about the top and translation, it has been experimentally observed that

286

287

a distinct failure surface does run through the toe as was assumed by Coulomb. Hence, it is expected that the K A - and KAE-values obtained by the modified Dubrova method could be practically acceptable, if the actual failure surface is essentially planar, and the distributions of mobilized cp- and o-values, i.e., if; and Ow can be reasonably assumed. For the case of rotation about the toe, it has been reported by Kezdi (1958) that it is obviously wrong to consider that there is a planar failure surface running through the toe. The most critical sliding plane as assumed in Coulomb wedge analysis is actually a plane corresponding to the earth pressure at rest condition since 1.2

1.0

¢- - 30°

0.8

I - 0°,

8 -¢-/2 (a _

90 ° )

,a-00,H-61t

'": r

>-. ~IC\I 0.6 w

0..<

,

w <

lll:

0.2

M.D.M. - Modified Dubrova's Method

OL...--

o

L-

0.1

L-

0.2

L-

0.3

...J

0.4

there is no displacement occurring on this plane at all. He found that even in the ultimate stage of the rotation, the actual sliding plane last developed runs into the wall at certain height ho aboye_ the toe, in which ho is function of soil characteristics and the amount of rotation. Hence, it is expected that the results obtained by the modified Dubrova method and the recommended distributions of if; and Ow are only corresponding to the state at which the backfill near the top just reaches limiting equilibrium. The K A - and KAE-values so obtained are, therefore, close to the atrest values. This can be seen from Table 6.1, in which K A "" K o = I - sin cp = 0.50 is shown for this case. They are not representative of the actual active states. Therefore, they should not be used for design purpose, unless the wall movement expected is consistent with that assumed. The upper bound limit analysis, such as that developed earlier in previous chapters is based on a more flexible log-sandwich mechanism of failure. However, at present, it has been formulated for the case of translation wall movement only. Hence, it is expected that the results are applicable only when the wall movement is predominantly translational, theoretically speaking. Figure 6.30 shows the results on K AE for three different modes of wall movement as obtained by the modified Dubrova method for earthquakes of different magnitude. The translation KAE-values as obtained by the upper bound limit analysis are also shown in the figure. It is noted that both the translation and the rotation about the top give identical KAE-values based on the modified Dubrova method. The K AE for rotation about the toe, which is more representative of the at-rest condition, is much higher. The translational K AE as obtained by the limit analysis is found practically the same as that based on the modified Dubrova method. The active earth pressure problem has also been studied by Prakash and Basavanna (1969). Their method of analysis is based on the consideration of equilibrium of both force and moment on an assumed triangular failure wedge. They tried to differentiate sliding wall from tilting wall by optimization with respect to active force and with respect to overturning moment, respectively. Results obtained by them are compared with those by the modified Dubrova method in Fig. 6.31. It is found that for both sliding and tilting walls, Prakash and Basavanna's solution is very close to the solutions of the modified Dubrova method for the rotation about the top and the translation cases. The KAE-values for the rotation about the toe case as obtained by them are found close to the translational KAE-values, which are much more close to the reality than the solution of the modified Dubrova method does.

6.5.2 Effects of wall movement on static and seismic passive earth pressures Horizontal Seismic Coefficient, k h

Fig. 6.30. Comparison of KAE-values obtained by the modified Dubrova method with the translational KAE-values obtained by limit analysis.

In the passive earth pressure case, the actual failure surface is generally curved, especially when the a-angle, and 1>-, o-values are high. Hence, similar to Coulomb's,

289

288 1.0,------,-------,--------.-------,

¢ - 30°, 8= 7.5° /3=i=Oo

HI 0.8

analysis solutions, which are based on a more reasonable log-sandwich failure mechanism, is therefore worthwhile. The comparison is shown in Fig. 6.32. Similar to the active case, translation wall movement and rotation about the top give practically identical KpE-values based on the modified Dubrova method. The case of rotation about the bottom, however, gives the lowest KpE-values, which as explained before, are more representative of the passive at-rest values (see Table 6.2). Since planar quasi-sliding surfaces are assumed in the modified Dubrova method, while a more reasonable log-sandwich mechanism of failure is assumed in the limit analysis method, the KpE-value obtained by the limit analysis is therefore, as ex-

0.6

6r------,--------r------,------...,

'"J: ~

~IN

w 0.<

"w

<

--"--

~

M.D.M. -Modified Dubrova Method Prakash and Basavanna (1969) 0.2

- - Translation

.............. ~- ......~

4

_ - - Rotation About the Top

.......

'"J :

Limit Analysis

>..

~IN

0'--

o

-'0.1

s = <1>/2

...... -~ .... .....

-'-

--'-

-'

0.2

0.3

0.4

Solution

"

"

w

i-00,
......-

""

(La.

"w

_

.

......

3

= 6ft

'-8 ...........

""

...... .....

.....

0. ~

Horizontal Seismic Coefficient. k h Zero - Extension Fig. 6.31. Comparison of KAE-values obtained by the modified Dubrova method with solutions of Prakash and Basavanna (1969).

Line Theory 2

the K p - and KpE-values obtained by the modified Dubrova method are generally too high if the most critical sliding surface is far from being planar. For this reason, it is suggested that Dubrova's concept can be more reasonably applied to the passive pressure problem, if the basic framework of the formulation is based on equations developed by assuming a nonplanar sliding mechanism, such as that shown in Chapter 5, rather on the modified Coulomb or the Mononobe-Okabe equation. Nevertheless, the modified Dubrova method gives KpE-values for three different kinds of wall movement. When r/J and 0 are small and ex :5 90°, the overestimation of KpE-values may not be too serious. Comparing these results with the limit

M.D.M. - Modified Dubrova Method

o

0.1

0.2

0.3

Horizontal Seismic Coefficient, k h Fig. 6.32. Effect of wall movement on static and seismic passive earth pressures.

0.4

290

pected, lower than that of the modified Dubrova method for the translation case to which the limit analysis method is applicable. A study of seismic passive earth pressure was recently published by Ghahramani and Clemence (1980). Based on the zero-extension line theory, a method was developed by them for evaluating the seismic increment of passive earth pressure. They suggested that their work can be combined with Habibagahi and Ghahramani (1979) for obtaining the resultant seismic passive earth pressure. Since in the passive earth pressure case, a reduction rather than an increase in passive resistance of a soil- wall system is of concern to us, instead of adding the seismic increment to the static passive earth pressure as that was done by Ghahramani and Clemence (1980), it is suggested that the seismic increment should be substracted from the static pressure to give the resultant seismic passive pressure. Some results so obtained for the translation case are shown in Fig. 6.32 for comparing with solutions of the modified Dubrova method and the limit analysis method. It is found that the zero-extension line solution is lower than the limit analysis solution as well as the solution of the modified Dubrova method. The zero-extension line solution is based on a log-sandwich failure mechanism as the limit analysis method does. However, limiting equilibrium of force, rather than equilibrium of energy as adopted in the limit analysis, is considered in both Habibagahi and Ghahramani (1979) and Ghahramani and Clemence (1980). This difference may be somewhat responsible for the discrepancy between the two solutions. However, in general, most stability solutions consideririg overall limiting equilibrium are still considered as upper bounds to the exact solution. The fact that the zero-extension line solution for the translational mode of wall movement is lower than the limit analysis solution and the solution of the modified Dubrova method for the same mode of wall movement tends to indicate that the zeroextension line theory gives the best result at least for the particular case studied. By assuming different distributions of seismic acceleration for different modes of wall movement, Ghahramani and Clemence (1980) based on their proposed method found that for the same maximum kh-value, translation wall movement produces the maximum seismic increment of passive earth pressure. They also found that rotation about the top creates about j of the translation force, and rotation about the bottom creates t of the translation force. Based on their findings, KpE-values for the case of c/> = 30°, li = 15°, and (3 = i = 0° are calculated. They are shown together with those obtained by the modified Dubrova method in Fig. 6.33. It is found that the ..:lKpE-yalues obtained by both methods are practically the same for translational wall movement. The ..:lKpE-values for both rotational wall movements as obtained by the modified Dubrova method are much higher than those based on Ghahramani and Clemence (1980). The fact that the KpE-value obtained by the modified Dubrova method is close to the passive at-rest value, instead of the limiting passive state value, is probably partially responsible for the high ..:lKpE-value obtained for the case of rotation about the toe.

291 2.5.-------,-----...,------,.------,

ep=30',6=15'

i = 0'. f3 =0'

2.0

--

(IX=

90')

Modified Dubrovo Method

'"J: h.

~

1.5

w

n."
II ::t' 1.0 >::
0.5

- -- -=--

0.1

0.2

0.3

0,4

Horizontal Seismic Coefficient. k h

Fig. 6.33. Comparison of dR'pE-values obtained by the modified Dubrova method and the zeroextension line theory.

In summary, the mode of wall movement affects not only the point of action of resultant lateral earth pressure, but also the magnitude of the lateral pressure. It is found that the K AE- and KpE-values for the case of rotation about the top are practically identical to those for the case of translation. Based on the modified Dubrova method and the recommended distribution of if; and ow' the rotation about the toe case gives the highest KAE-value and the lowest KpE-value, which are in reality close to the active and passive at-rest coefficients of earth pressure, respectively. Comparison of the results of the modified Dubrova method with other solutions shows that for the case of translation, the KAE-value obtained by the method is practically the same as the limit analysis solution and the solution of Prakash and Basavanna (1969). The translation KpE-value obtained by this method is, however, higher than the limit analysis solution and the solution of Ghahramani and Clemence (1980). It seems clear that the zero-extension line theory gives the best result in this case.

293

292

In the active case, the KAE-value for the case of rotation about the toe as found by Prakash and Basavanna (1969), which is close to the solution for the translation case, seems reasonable. The KAE-value for this case as obtained by the modified Dubrova is more representative of the at-rest pressure. By comparing the b.KpE-values to different wall movements obtained by the modified Dubrova method and those calculated based on Ghahramani and Clemence (1980), it seems that the values are the same for the translation case. For rotational cases, the b.KpE-values as obtained by the modified Dubrova method are higher. 6.6 Earth pressure theories for design applications in seismic environments Since earthquake forces are of oscillatory nature, design of retaining structures in seismic-active zone based on displacement consideration seems to be more realistic. While a step-by-step dynamic analysis seems more rigorous, it requires accurate earthquake data, advanced constitutive models of soils, and sophisticated numerical techniques. A simplified displacement analysis as that described by Richards and Elms (1979) may be more promising. Nevertheless, the conventionally adopted design procedure based on ultimate load and empirical factors of safety still has the advantage of being simple and having high acceptability. For designing rigid retaining structures in seismic environments, the first step is to obtain required earthquake data. If a quasi-static approach is to be adopted in the analysis of seismic lateral earth pressure, the seismic coefficients can be assessed by methods such as that suggested by the Japanese Society of Civil Engineers (980), that recommended by Prakash (1981), and that summarized by Emery and Thompson (1976). Once the input earthquake information is ready, evaluation of seismic lateral earth pressure becomes the primary step to ultimate-load-based design of retaining structures. In displacement-based design, however, assessment of seismic lateral earth pressure is also absolutely necessary. This information is required when evaluating the yield acceleration, which is the acceleration over which the safety factor against the specified failure mechanism is less than one and permanent displacement is bound to occur. Hence, no matter which design method is used, seismic lateral earth pressure should be carefully evaluated. Most earth pressure theories are developed on the basis of different assumptions. Results of analysis based on different theories usually differ from one another. Furthermore, they are all subjected to certain limitations and may be suitable only for a certain type of wall movement. For these reasons, in selecting earth pressure theories for practical applications, the expected field conditions should be carefully considered. The assumptions behind each earth pressure theory and its limitations should be fully understood. No matter which design method is to be adopted, selec-

tion of a proper earth pressure theory or analytical method is of great importance. This section attempts to compare some well-known earth pressure theories or analytical methods that are suitable for assessment of seismic lateral earth pressures. For practical reasons, only those methods essentially based on the perfect plasticity or the limiting state equilibrium, which are practically applicable to rigid retaining wall problems, are included. Methods based on elastic wave theory and approaches based on elasto-plastic and nonlinear theory are not considered here.

6.6.1 Analytical methods for determining seismic active earth pressure The first well-known method for seismic earth pressure analysis, the MononobeOkabe (M-O) method was developed fifty years ago based on the Coulomb's wedge theory (Okabe, 1926; Mononobe and Matsuo, 1929). The analysis method has been widely used in current practice of retaining wall design. Basic assumptions of the M-O analysis, which is applicable to a soil- wall system with general configuration, dry cohesionless backfill, and zero surcharge, are: 1. The failure surface is planar and passes through the toe. 2. The backfill material is perfectly plastic and the wall displacement is sufficient to produce a limiting equilibrium state so that shear strength along the failure plane is fully mobilized simultaneously. 3. The soil wedge acts as a rigid body so that the vertical and horizontal seismic accelerations in the wedge are uniform and have the same magnitude as the base of the wall. 4. The lateral earth pressure increases hydrostatically with depth. For the active earth pressure case in which the most critical sliding surface is essentially planar in most cases and the soil- wall interaction effect or/and progressive failure phenomena is not very significant, use of the M-O analysis is practically desirable. The analysis has been shown to be reasonable by several experimental works. Probably for these reasons, the method is commonly used in determining the seismic active earth pressure in ultimate-load-based design of retaining structures. Although, boundary deformation condition is not considered in the formulation of the M-O method, it seems that the solution is applicable to the cases of translation movement and rotation about the top in which a distinguished failure surface passing through the toe is actually developed as found by James and Bransby (1970) in a passive pressure model study. However, the point of action of the resultant force as suggested by Mononobe and Okabe may need further revision by considering the wall movement effect and the earthquake magnitude. The M-O method has been modified and simplified by Seed and Whitman (1970). Suggestions have been given by them for actual design applications. They suggested that for practical purpose, K AE can be estimated as K A + ~ kh . In other words, the seismic active earth pressure can be calculated as:

295

294 (6.22)

In particular, they suggested that the value ha (H/3) obtained by Prakash and Basavanna (1969) can be used for locating the point of action of the resultant active force. Or for practical purpose, they suggested that the seismic increment of pressure can be assumed to act at 0.6H above the bottom and the static active pressure at ha = H/3. Based on essentially the same concept and assumptions as the M-O analysis, Prakash and Saran (1966) developed dimensionless earth pressure factors for active earth pressure analysis and assumed that the principle of superposition is valid. The earth pressure factors they developed are applicable to cohesive horizontal backfill with tension cracks considered. It seems that the method is a great improvement if the backfill does contain a large amount of cohesive material. However again, the method has the same limitations as the M-O analysis does. The point of action, ha, is taken as H/3, which is similar to the M-O analysis. Determination of the point of action, which is the center of dispute of the applicability of the M-O analysis, has been attempted by Prakash and Basavanna (1969). By making an assumption regarding the distribution of vertical pressure on any surface parallel to the ground surface, they were able to consider force and moment equilibrium of a triangular sliding wedge. Again, by making the assumptions similar to the M-O analysis, they developed a method for calculating the active moment, active force, and the point of action, ha. By optimizing the active moment and the active force, respectively, they were able to make distinction between wall tilting (rotating about the toe) and walL sliding (translating). The method, which is applicable to the case of cohesionless backfill with zero surcharge, gives a complete information on the distribution of active earth pressure. It is very promising. However, based on this method, the difference in pressure distribution for the two types of wall movement is very small, especially when the seismic coefficient is small (k s 0.15). In fact, it is understandable that the pressure distribution for wall translating and wall rotating about the toe can give significantly different pressure distributions even in the static case. The validity of differentiating the two different types of wall movement by optimization with respect to different physical components is therefore somewhat in doubt. The limit analysis method as developed in previous chapters based on the upperbound technique of perfect plasticity is capable of solving the lateral earth pressure problems for a general soil - wall system in earthquake environments. It has been described in detail in previous chapters. By using the concept of virtual work, or energy equilibrium, the limit analysis method is theoretically developed based on a translational wall movement. However, the log-sandwich mechanism assumed in the formulation is well representative of the actual failure condition for the case of translation and fairly represen-

tative of that for the case of rotation about the top (James and Bransby, 1970). It is suggested that the estimated lateral earth pressure is suitable for the above two types of wall movement. Similar to most limit equilibrium methods, the limit analysis method gives no information on the point of action. For practical applications, it is suggested that earth pressure tables developed in Chapter 5 based on the limit analysis method can be adopted for determining the KAE-values for both cases of translation and rotation about the top of wall movement, while, the modified Dubrova method as described in this chapter is recommended for assessing the corresponding point of action. The modified Dubrova method developed based on Dubrova's idea (Dubrova, 1963), which is applicable only to cohesionless soils, is theoretically capable of assessing both the magnitude of lateral earth pressure and the point of action of the resultant force, or the lateral pressure distribution, for different modes of wall movement. This modified method is based on the framework of the M-O equations. The assumptions are therefore similar to those in the M-O analysis, except that strength in the backfill is assumed to mobilize to that somewhat in proportion to the amount of wall displacement. Detailed formulation and description of the method have been given in Chapter 5. Comparison of those analytical methods as described above are tabulated in Table 6.7. For comparison of numerical results as obtained by these methods, two examples are given. The results are listed in Tables 6.8a and 6.8b. In the tables, K A , K AE , and MAE are listed. The corresponding points of action, static ha, seismic ha, and increment h~ are also listed in the tables. Dimensionless normalized overturning moment, M o = KA(ha/H), is also calculated for each analytical method and used as a basis for comparison. From Tables 6.8a and 6.8b, it seems that for the cases of translation and rotation about the top, all methods given essentially the same K A - and KAE-values, since in the active case the actual failure surface is practically planar. The major difference among them is on the ha-value. Based on the modified Dubrova method, the ha-value is found higher for the case of rotation about the top than for the translation case. Consequently, as expected, the Mo-value is larger for the case of rotation about the top. For the translational wall movement, the h~-value suggested by Seed and Whitman (1970), h~ = 0.6 H, and that calculated by Prakash and Basavanna (1969) for the case of ex = 90°, {3 = 0°, cf> = 30°,0 = 7.5° and kh = 0.15, h~ = 0.64 H, are much larger than those based on other methods that give h~ = 0.33 H. Consequently, the overturning moments as obtained by them are much larger than the others. However, based on Prakash and Basavanna (1969), the ha-value for the static translation case, of which the distribution is generally recognized as practically

N

\0 0'1

TABLE 6.7 Comparison of analytical methods for seismic active earth pressure analysis Methods

Basis

Mode of movement

Failure mechanism

Analytical techniques

Applicable soilwall conditions

Results given

Remarks

M-O analysis and Seed and Whitman (1970)

Coulomb's wedge theory

Not specified

Planar

Limiting force equilibrium, quasi-static

General soil-wall configuration, q,-soils, no surcharge

K AE

h~ =

Prakash and Saran (1966)

Coulomb's wedge theory

Not specified

Planar

Limiting force equilibrium, quasi-static

General soil-wall system with horizontal backfill

K AE

Considers tension cracks

Prakash and Basavanna (1969)

Couloumb's wedge theory

Translation and rotation about toe

Planar

Force and moment equilibrium, quasistatic

General soil-wall configurations, q,- soils, no surcharge

K AE , h a

Limit analysis

Upper-bound limit theorem

Translation

Log-sandwich

Energy equiIibrium, quasistatic

General soil-wall system

K AE

Translation and two types of rotation

Planar

Limiting force equilibrium, quasi-static

General soil-wall configurations, q,-soils, no surcharge

K AE , h a

Modified DuPressure rebrova method distribution (M.D.M.)

TABLE 6.8a Numerical comparison of solutions of various analytical methods for seismic active earth pressure analysis = 0 and 0.15) Analytical method

Static

Seismic

Increment

KA

haiR

K AE

haiR

M-O analysis

0.325

0.333

0.440

0.333

Seed and Whitman (1970)

0.325

0.333

0.438

-

Prakash and Saran (1966) Limit analysis

Moilifioo Dubrova method (M.D.M.)

~ Translation

!

Rotation about top

T ro"l,fi'" RotatIon

abo~t top RotatIOn about toe

(O!

tlKI\E

0.6 H is recommended by Seed and Whitman (1970)

= 75°,

{3

Considers wall movement effects on mobilized strength

= 0°, q, = 40°, 0

Normalized overturning h~1H

= (j)q" k"

Remarks

moment, M o

0.147 0.113

0.600

0.176

0.325

0.333

0.442

0.333

-

-

0.147

0.325

0.333

0.440

0.333

-

-

0.147

0.325

0.333

0.440

0.357

-

-

0.157

0.325

0.333

0.440

0.333

0.115

0.333

0.147

0.325

0.361

0.440

0.357

0.115

0.344

0.157

0.445

0.309

0.577

0.31 I

0.115

0.321

0.179

Suggested for O! = 90° only

The ha,value is based on M.D.M.

N

~

298

299

o

o

'"II -e. o

o OON\ClOO M f' I:"-r---

II

en.

o

0

0

0

00 '"

'" V"\

ci

ci

00 '"

o

'" V"\

0

V"\ 00

o

u

'~

00

::is

'iij Cf.l

o

\0

\0

o

M M M

("f)

M

d

0

0

~ M

M

lr'l

M

M

~ M

f""1

M M

o

ei

V'l

0

t.n

0

linear, is found higher than H/3. Whether the high ha-value as obtained by the same method is reasonable or not is difficult to justify. Hence, the high Me·values obtained by Prakash and Basavanna (1969) and Seed and Whitman (1970), whose recommended h~ = 0.6 H was mainly based on the work of Prakash and Basavanna (1969), need further justification. Based on the modified Dubrova method, ha = H/3 is found for all kh-values for the case of translation movement, which is the same as that assumed in the M-O analysis. Although the ha-value for the translation case may need further explorations, it appears that the KAE-value can be obtained by any of the methods as listed in Table 6.7. However, the limit analysis method, which is applicable to general soil- wall systems, is highly recommended, especially when surcharge, q, and cohesion, c, are present. In this case, both the active pressure resulting from the surcharge and the cohesion can be assumed to act at the middle height of the wall. Unless strong evidence showing that ha > H/3 is available, it is reasonable that the active pressure for q = C = 0 case be assumed to act at h a = H/3 for practical purposes, if the movement is found essentially translational and the amount of the movement is sufficiently large. For the case of rotation about the top, it is recommended that the modified Dubrova method which gives both the K AE-value and its corresponding ha-value can be used for the case of no surcharge (q = 0) and zero cohesion (c = 0). However, if surcharge and/or cohesion are present, the limit anaysis is highly recommended for determining the KAE-value. The value can be easily estimated from the tables and charts provided in Appendices A and B of Chapter 5. While the active pressure as contributed by the surcharge and cohesion can again assume to act at the mid-height, the ha-value for the case of q = c = 0 is recommended to obtain from the modified Dubrova method. For the case of rotation about the toe, it is found that the K A - and KAE-values obtained by the modified Dubrova method, which are close to the at-rest values, are much larger than those based on Prakash and Basavanna (1969) for the same mode of movement. On the other hand, based on Prakash and Basavanna (1969), the havalue for the case investigated (O! = 90°, (3 = 0°, ¢ = 30°,0 = 7.5°, and k h = 0.15) is as high as 0.433 H. As explained earlier, the distribution of active pressure for the case of rotation about the toe may be concave upward, and consequently h a < H/3 is expected. The result based on the modified Dubrova method, which gives ha < H/3, is consistent with this reasoning as far as the ha-value is concerned. However, considering the results from Prakash and Basavanna's method and the absence of results from full-scale tests, further study of this question is justified. Although both Prakash and Basavanna (1969) and the modified Dubrova method coincidentally give almost the same overturning moments for the case of rotation about the toe, for the reasons just discussed, it seems that both methods are not strictly suitable for assessing the overturning ,moment Me corresponding to the

301

300

limiting active state. The modified Dubrova method, however, can be used for estimating the seismic overturning moment for the case in which the retaining structure is allowed to rotate outward about the bottom for a very limited amount of rotation, such as the case of basement wall. If no better theories are available, it is recommended that for the case of rotation about the toe, the KAE-value, which is supposed to be close to the translational K AE-value if the limiting state is allowed to develop, can be taken from that obtained based on the limit analysis method. Due to the assumption inherent in the use of seismic coefficients the actual value of ha is uncertain. Accordingly, further study of this question is highly recommended. 6.6.2 Analytical methods for determining seismic passive earth pressure The well-known M-O analysis generally adopted for seismic active earth pressure analysis has also been used for calculating seismic passive earth pressure. However, in the passive case, there are serious interactions between the wall and the backfill material, progressive failure phenomena is of great significance. Consequently, the failure surface is generally curved and the actual average mobilized strength parameter can fall well below the peak value. For these reasons, the method is not highly desirable unless that the actual failure is practically planar and a reasonable assessed average 1>-parameter is considered. A completely new approach of solving passive earth pressure problems in static and seismic cases has been presented recently by Habibagahi and Ghahramani (1979) and Ghahramani and Clemence (1980), respectively. In this method, a simple zero-extension line field is first developed based on associated fields of stress and displacement. The limit equilibrium technique is then applied for determining the resultant passive earth pressure. In the static case, it appears that the theory is applicable only for the case of translational wall movement. No point of action has been mentioned by Habibagahi and Ghahramani (1979). In developing the approach for assessing seismic increments of passive earth pressure, Ghahramani and Clemence (1980) tried to incorporate the seismic acceleration effect in an incremental manner based on the work of Sabzevari and Ghahramani (1974). However, on the basis of the incremental approach, it appears that only the relative acceleration between soil above and soil below the toe is considered. The foundation soil is assumed to remain static. The suitability of this assumption is somewhat questionable. Nevertheless, based on a simple zero-extension line field, Ghahramani and Clemence (1980) developed a chart for obtaining the 'dynamic' seismic passive earth pressure coefficient for the case of translational wall movement. Also, by assuming different distributions of seismic acceleration, they showed that the LlKpE-values for the cases of rotation about the top and about the toe are j and t of the transla-

tional ~KpE-value, respectively. They further showed that the corresponding point of action hp equals to HI3 for the translation case, HI2 for the case of rotation about the toe, and HI4 for Jhe _case of rotation about the top. Ghahramani and Clemence (1980) based on their study, suggested that the resulting seismic passive earth pressure can be taken as the combination of the static passive pressure obtained based on Habibagahi and Ghahramani (1979) and the 'dynamic' increment as calculated by their suggested method. The limit analysis method and the modified Dubrova method, which have been briefly described earlier in this chapter, are applicable not only to the active earth pressure case, but also to the passive earth pressure case. Use of the limit analysis method, in which the log-sandwich mechanism of failure is assumed, in the determination of passive earth pressure is extremely beneficial. On the contrary, like the M-O analysis, the application of the modified Dubrova method to this case has to be carefully considered, since the planar failure mechanism as assumed in the method is quite often unrealistic. Consequently, the K p - and KpE-values may be seriously overestimated by this method. Nevertheless, due to lack of theoretically sound methods for assessing the point of action of resultant passive pressure, it is suggested that the modified Dubrova method can be used for estimating the hp-values. The hp -value so assessed can then . b e Incorporated with the KpE-values calculated based on the limit analysis method for obtaining a reasonable estimation of the normalized resisting moment , M r = K pE (h/H).

The above-mentioned analytical methods have been compared and summarized in Table 6.9. To compare the suitability of these analytical methods for practical design applications, an example has been worked out. Comparison of the solution based on these methods is given in Table 6.10. It should be pointed out here that in presenting the solutions of Ghahramani and Clemence (1980), it is assumed that for all types of wall movement the static Kp-value is the same. Also, negative seismic increments of pressure are considered. The static hp-value is taken as HI3 for all cases of wall movement. It is clear from Table 6.10 that both the M-O analysis and the modified Dubrova method seriously overestimate the K p - and KpE-values for the cases of translation and rotation about the top. The solutions of Ghahramani and Clemence (1980) and the limit analysis method, which are both based on the log-sandwich failure mechanism, are more reasonable. The zero-extension line theory appears to give the best result on the assessment of the K p - and KpE-values. For the case of rotation about the toe, the modified Dubrova method gives much lower K p - and KpE-values than the zero-extension line theory does. This is because the K p - and KpE-values so obtained based on the recommended 1/;- and 0distributions are more representative of the at-rest values.

w

13

TABLE 6.9 Comparisons of analytical methods for seismic passive earth pressure analysis Methods

Basic idea

Mode of movement

Failure mechanism

Analytical techniques

Applicable soiIwall conditions

Information given

MononobeOkabe analysis

Coulomb's wedge theory

Not specified

Planar

Limiting force equilibrium, quasi-static

General soil-wall configuration, cP-soil, no surcharge

k PE

Ghahramani Zero-extension line theory and Clemence· (1980)

Translation Log-sandwich and two types of rotation

Limiting force equilibrium, incremental approach, quasi-static

Vertical wall and horizontal cohesionless backfill, no surcharge

K pE , h~

Limit analysis

Upper-bound limit theorem

Translation

Energy equilibrium, quasistatic

General soil-wall system

K pE

Modified Dubrova method (M.D.M.)

Pressure redistribution

Translation and Planar two types of rotation

Limiting force equilibrium, quasi-static

General soil-wall configuration, q,-soil, no surcharge

K pE , h p

Log-sandwich

Remarks

Wall movement specified for seismic case only

Consider waIlmoment effects of mobilized strength

TABLE 6.10 Numerical comparison of solutions of various analytical methods for seismic passive earth pressure analysis (ex = 90°, {3 = 0°, cP = 40°, = 0 and 0.15) Analytical method

M-O analysis

f'~"btiOO

Ghahramani Rotation Clemence about top (1980) Rotation about toe Limit analysis Modified Dubrova method (M.D.M.)

Static

Seismic

Increment

Kp

hpiH

K pE

hpiH

f!J(PE

h~1H

Normalized resisting moment, Mr

18.72

0.333

16.42

0.333

-

-

5.47

11.90

0.333

10.16

0.333

-1.74

0.333

3.37

11.90

0.333

10.74

0.342

- 1.16

0.250

3.68

11.90

0.333

11.32

0.325

-0.58

0.500

3.68

13.09

0.333

1l.88

0.333

-

-

3.96

Rotation about top

13.09

0.237

11.88

0.236

-

-

3.09

T,~,""oo

18.72

0.333

16.43

0.333

-2.29

0.333

5.47

18.72

0.237

16.43

0.236

-2.29

0.214

3.89

5.27

0.447

4.53

0.450

-0.74

0.429

2.04

~ Translation

l

Rotation about top Rotation about toe

/j = (~)cP,

kh

Remarks

Translational K p and static h p are used for all cases The hp-value is based on M.D.M.

w w

o

305

304 It is interesting to note that the h~-values obtained based on the zero-extension line theory and the modified Dubrova method are the same or very close in all cases of wall movement. However, comparison of hp corresponding to K pE shows that h > HI3 for the case of rotation about the top and h p < HI3 for the case of rotati~n about the toe are obtained based on the zero-extension line theory. This contradicts to the solutions of the modified Dubrova method. This is because h p = HI3 is assumed for the static case in the zero-extension line solution. According to the modified Dubrova method, the relative magnitudes of h p and h~ for different wall movements are very close. That is, h p < HI3 for the case of rotation about the top and hp > HI3 for the case of rotation about the toe as found on the basis of the modified Dubrova method seem more reasonable. As far as the resisting moment is concerned, it appears that for both cases of the translation and the rotation about the top, both the zero-extension line theory and the limit analysis method with h -values based on the results of the modified Dubrova method, give very close ~d reasonable Mr-values. However, it should be noted that the larger Mr-value for the translation case than for the case of rotation about the top as found by the modified Dubrova is more reasonable for the reason just discussed. In fact, if a distinction between the Kp-value and the hp-value for different wall movements for the static case is possible based on the zero-extension line theory, the theory will give the best results on Mr' Nevertheless, the zero-extension line theory has so far been applied only to the very special case of vertical wall and horizontal cohesionless backfill with zero surcharge. The actual application of the method is, therefore,. very limited. In this regard, it apPears that the limit analysis method in combination with the h -value based on the modified Dubrova method could probably be a preferable ap;oach for assessing the passive resisting moment for actual design applications in the case that the mode of wall movement is expected to be mainly translational or rotating about the top. Appendices A and B of Chapter 5 which show KpE-tables and charts required for estimating KpE-values corresponding to the case of q =1= 0, C =1= 0, 8 =1= (nonhorizontal seismic acceleration) are recommended for determining the KpE value. If surcharge and cohesion are present, they can be assumed to act at the midheight of the wall. For the case of rotating about the toe, the resisting moments obtained by the zeroextension line theory and the modified Dubrova method are much different, although the corresponding h I -values obtained are fairly close. On the one hand, in the approach of Ghahrama~i and Clemence (1980), assuming that Kp and h a are the same for all types of wall movement for the static case is not justified. On the other hand, use of the modified Dubrova method, which is based on a planar failure mechanism, for assessing passive resistance is not necessarily proper either. Furthermore, the results obtained by the modified Dubrova method and the recommended 1/;- and a-distributions are not representative of the limiting passive state, as pointed

°

out previously in Section 6.4. Hence, it appears that further investigation on this mode of wall movement is urgently needed. Perhaps, one of the possible ways of solving this problem is to use .the equations developed based on the limit analysis method as the framework for developing the modified Dubrova method. At the same time, the 1/;- and o-distributions have to be assumed based on more extensive field observations. Its feasibility, however, needs further investigation. The difficulty with the modified Dubrova method in regard to the fact that the resultant lateral force may not solely be dependent on the mobilized ¢-value near the toe should be overcome first. Alternatively, the finite-element method may be used to obtain a more realistic solution (Potts and Fourie, 1986). Again, if no better theories are available the KpE-value obtained by the limit analysis method, which is representative of the ultimate translation value, could be used. With regard to the point of action, h p = HI3 could be adopted, although it is quite possible that h p > H13, as given by the modified Dubrova method, may be true for the case of rotation about the toe. 6.7 Design recommendations Evaluation of seismic lateral earth pressures is important in both ultimate-loadbased design and displacement-based design of retaining structures in seismic zones. For reasonable estimation of seismic lateral earth pressures and points of action of the resultant forces, the expected field conditions, especially the type and amount of wall movement, should be considered in the first place. Based on the comparison of some well-known earth pressure theories with the limit analysis method and the modified Dubrova method proposed herein, proper analytical methods can then be selected for reasonable assessment of the magnitude and the point of action of the resultant active and passive earth pressures for actual design applications. As described by Nazarian and Hadjian (1979), most reported damage of retaining walls by earthquake can be attributed to increased lateral pressures inducing sliding and tilting of the structures. Nandakumaran and Prakash (1970) also reported that outward wall movement and settlement of backfill are generally found on retaining walls damaged by earthquake. It, therefore, appears that tilting and translational wall movements are predominant in most earthquakes for the case of free-standing rigid retaining walls. The study of this chapter tends to indicate that all the earth pressure theories investigated are applicable to the determination of resultant seismic active earth pressure if the wall movement is essentially translational. However, the limit analysis method, which is applicable to general soil-wall systems, is highly recommended. Appendices A and B of Chapter 5 presenting earth pressure tables and charts for application of the earth pressures have been developed based on the limit analysis method. They can be used for actual design applications. In regard to the point of

307

306 action, h a > HI3 should be adopted since there are evidences showing ha > HI3 in the seismic case, although at the ultimate state h a = HI3 as given by the modified Dubrova method may be true. In case that tilting movement is predominant, the KAE-value estimated from Prakash and Basavanna (1969) or the limit analysis method is recommended if no better theories are available for this mode of movement. At present, the point of action, h a, should be taken as greater than H13, although, in this case, the distribution curve for the active earth pressure may be concave upward such that ha < H13. Quite possibly, most free-standing walls may move in a way combining tilt and translation. Such kind of wall movement, although not treated in this chapter, could be solved by the finite-element approach. It could also be reasonably handled by the modified Dubrova method by considering the point of rotation to locate at certain points below the bottom of the wall. The modified Dubrova method certainly has a great potential for treating problems of such kind. Nevertheless, its limitations should always be recognized. For the case of bridge abutment, in which lateral movement is restricted at the top, rotation about the top is the most likely mode of movement. In this case, the limit analysis method is recommended for assessing the KAE-value, if surcharge and cohesion are present. For the case of no surcharge and zero cohesion, the modified Dubrova method or the Mononobe-Okabe analysis can also be used for obtaining the KAE-value. However, it is suggested that h a should be obtained by the modified Dubrova method. According to the modified Dubrova method, the K AEvalue as predicted is the same .as the Mononobe-Okabe solution. However, based on this method, h a is found higher than that was assumed in the Mononobe-Okabe analysis, i.e., ha = H13. Consequently, the resulting driving moment obtained based on the modified Dubrova method is larger than that calculated by using the Mononobe-Okabe method. This may be one cause of the damage to bridge abutment in New Zealand in the 1968 earthquake, which was reported in Richards and Elms (1979). Foundation or basement walls of structures, although presenting more interesting design and construction problems than free-standing retaining walls, have received less attention in the literature. Estimation of seismic increment of at-rest pressure as the result of earthquake forces generally follows empirical rules. A simple method has recently been proposed by Hall (1978) for evaluating the 'dynamic' at-rest pressure by interpolating between the 'dynamic' active and passive pressures. However, it appears that the modified Dubrova method, which can theoretically be applied to any stage of wall yielding if the movement is essentially translation or rotation about the top, is capable of solving this problem directly. By assuming a uniform 1,l--distribution with its magnitude equal to the at-rest 1,l--value, 1,l-a' the static and seismic at-rest pressures can be easily obtained by the method. The seismic increment of the pressure can also be calculated. It is apparent that both the seismic

at-rest pressure and the increment of the pressure are higher than the actual values. Although the way retaining structures move under lateral pressures is seldom documented, purely active and virtually no passive states are not always predominant. Also, it has been pointed out by Nazarian and Hadjian (1979) that the center of rotation for the passive case could move up and down the wall depending on the interaction characteristics of the externally applied force, the wall, and the backfill. It appears that a combination of active and passive states, quite often, can occur in reality. If this is the case, the modified Dubrova method could be a useful analytical tool. In short, the earth pressure theories should be carefully selected in actual design of rigid retaining structures in seismic environments. This must be done by considering not only the assumptions, the limitation, and the applicability of the analytical method but also the actual conditions, such as the type and amount of wall movement, that might be expected in the field. It should also be kept in mind that the construction method, the loading and strain conditions, the soil-structure interface friction, and the progressive failure and scale effect as discussed earlier should always be taken into consideration in actual applications of all earth pressure theories. No earth pressure theories can be blindly applied to actual designs without these considerations. References Basavanna, B.M., 1970. Dynamic earth pressure distribution behind retaining walls. Proc. 4th Symp. on Earthquake Eng., uliIv. 'of Roorkee, India, Vol. I, pp. 311- 320. Chang, M.F., 1981. Static and seismic lateral earth pressures on rigid retaining structures. Ph.D. Thesis, School of Civil Engineering, Purdue University, West Lafayette, IN, 466 pp. Davis, E.H., 1968. Theories of plasticity and the failure of soil masses. In: I.K. Lee (Editor), Soil Mechanics: Selected Topics. Butterworths, London, pp. 341 - 380. Dubrova, G.A., 1963. Interaction of Soil and Structures. Rehnoy Transport, Moscow, U.S.S.R. Emery, J.J. and Thompson, C.D., 1976. Seismic design considerations for gravity retaining structures. Can. J. Civ. Eng., 3: 248-264. Ghahramani, A. and Clemence, S.P., 1980. Zero extension line theory of dynamic passive pressure. J. Geotech. Eng. Div., ASCE, 106 (GT6): 631-644. Habibagahi, K., and Ghahramani, A., 1979. Zero extension line theory of earth pressure. J. Geotech. Eng. Div., ASCE, 105 (GD): 881 - 896. Hall, J.R., 1978. Comments on Lateral Forces-Active and Passive. ASCE Specialty Conference on Earthquake Engineering and Soil Dynamics, Pasadena, CA, Vol. 3, pp. 1436 - 1441. Harr, M.E., 1966. Foundation of Theoretical Soil Mechanics. McGraw-Hill, New York, NY, 381 pp. Harr, M.E., 1977. Mechanics of Particulate Media - A Probabilistic Approach. McGraw-Hill, New York, NY. Ishii, Y., Arai, H. and Tsuchida, H., 1960. Lateral earth pressure in an earthquake. Proc. 2nd World Conf. on Earthquake Engineering, Tokyo, Vol. I, pp. 211- 230. Jaky, J., 1948. Pressures in soils. Proc. 2nd ICSMFE, 1: 103-107. James, R.G. and Bransby, P.L., 1970. Experimental and theoretical investigations of a passive earth pressure problem. Geotechnique, 20 (1): 17 - 37.

309

308 James, R.G. and Bransby, P.L., 1971. A velocity field for some passive earth pressure problems. Geotechnique, 21 (I): 61 - 83. Kedi, A., 1958. Earth pressure on retaining wall, tilting about the toe. Proc. Brussels Conf. on Earth Pressure Problems, Brussels, Vol. I, pp. 116 - 132. Lee, I.K. and Herington, J .R., 1972. A theoretical study of the pressure acting on a rigid wall by a sloping earth or rockfill. Goetechnique, 22 (I): 1-26. Matsuo, H., 1941. Experimental study on the distribution of earth pressure acting on a vertical wall during an earthquake. J. Jpn. Soc. Civ. Eng. 27 (2). Matsuo, H. and Ohara, S., 1960. Lateral earth pressures and stability of quay walls during earthquakes. Proc. 2nd World Conf. on Earthquake Engineering, Tokyo, Vol. I, pp. 165 ~ 181. Mononobe, N. and Matsuo, H., 1929. On determination of earth pressure during earthquakes. Proc. World Engineering Congress, Tokyo, Vol. 9, pp. 275. Murphy, V.A., 1960. The effect of ground characteristics on the aseismic design of structures. Proc. 2nd World Conf. on Earthquake Engineering, Tokyo, Vol. I, pp. 231 -248. Nandakumaran, P. and Prakash, S., 1970. The problem of retaining walls in seismic zones. Proc. 4th Symp. on Earthquake Eng., Univ. of Roorkee, India, Vol. I, pp. 307 -310. Nandakumaran, P. and Joshi, V.H., 1973. Static and dynamic active earth pressures behind retaining walls. Paper No. 136, Bull., I.E.S.T., 10 (3): pp. II3 - 123. Narain, J., Saran, S. and Nandakumaran, P., 1969. Model study of passive pressure in sand. J. Soil Mech. Found. Div., ASCE, 95 (SM4): %9-984. Nazarian, H. and Hadjian, A.H., 1979. Earthquake induced lateral soil pressures on structures. J. Geo!. Eng. Div., ASCE, 105 (GT7): 1049- 1066. Okabe, S., 1926. General theory on earth pressure and seismic stability of retaining walls and dams. J. Jpn. Soc. Civ. Eng., 12, (I). Potts, D.M. and Fourie, A.B., 1986. A numerical study of the effects of wall deformation on earth pressures. Int. J. Numer. Anal. Methods, Geomech., 10: 383-405. Prakash, S., 1981. Soil Dynamics. McGraw-Hill, New York, NY, 426 pp. Prakash, S. and Saran, S., 1%6. Static and dynamic earth pressures behind retaining walls. Proc. 3rd Sym. on Earthquake Eng., Univ.of Roorkee, India, pp. 277-288. Prakash, S. and Basavanna, B.M., 1969. Earth pressure distribution behind retaining walls during earthquake. Proc. 4th World Conf. on Earthquake Eng., Chile, pp. 133 -148. Richards, R. and Elms, D.G., 1979. Seismic behavior of gravity retaining walls. 1. Geotech. Eng. Div., ASCE, 105 (GT4): 449-464. Sabzevari, A. and Ghahramani, A., 1974. Dynamic passive earth pressure problem. J. Geotech. Eng. Div., ASCE, 100 (GTl): 15-30. Saran, S. and Prakash, S., 1977. Effect of wall movement on lateral earth pressure. Proc. 6th World Conf. on Earthquake Engineering, India, pp. 2371- 2372. Seed, H.B. and Whitman, R.V., 1970. Design of earth retaining structures for dynamic loads. ASCE Specialty Conf. on Lateral Stresses in the Ground and Design of Earth-Retaining Structures, Cornell Univ., Ithaca, NY, pp. 103-147. Terzaghi, K., 1934. Large retaining-wall tests. Eng. News Rec., Vol. 112, May. Terzaghi, K., 1936. A fundamental fallacy in earth pressure computation. J. Boston Soc. Civ. Eng., 23 (2): 71-88. Terzaghi, K., 1941. General wedge theory of earth pressure. ASCE Trans., 106: 68 - 80. The Japanese Society of Civil Engineers, 1980. Earthquake Resistant Design for Civil Engineering Structures, Earth Structures, and Foundations in Japan. Tokyo. Tschaebotarioff, G.P., 1962. Retaining structures. In: G.A. Leonards (Editor), Foundation Engineering. McGraw-Hill, New York NY, pp. 438-524. Wood, J.H., 1975. Earthquake induced pressures on a rigid wall structure. Bull. N.Z. Soc. Earthquake Eng., 18 (3): 175 - 186.

Chapter 7

BEARING CAPACITY OF STRIP FOOTING ON ANISOTROPIC AND NONHOMOGENEOUS SOILS 7.1 Introduction The bearing capacity of strip footings on homogeneous and isotropic soils has been extensively studied by several investigators. The fundamental theory and practical applications of strip as well as square, rectangular and circular footings have been described in details in Chen's book (1975). However, it has been recognized that natural soil deposits are nonhomogeneous and anisotropic with respect to shear strength (Casagrande and Carillo, 1944; Lo, 1965; Bishop 1966; Livneh and Komornik 1967). Considerable work has been done with regard to the influence of anisotropy and nonhomogeneity on the bearing capacity of clays, in which c/J = 0 (Raymond, 1967; Reddy and Srinivasan, 1967, 1971; Davis and Christian, 1971; Davis and Brooker, 1973; Livneh and Greenstein, 1973; Salencon, 1974a, b). Most of these studies found that the anisotropy and nonhomogeneity have a considerable influence on the bearing capacity of clays. On the other hand, however, very few attempts have been made for studying the effect of anisotropy and nonhomogeneity on the bearing capacity of c-c/J soils. Reddy and Srinivasan (1970) studied the effect of anisotropy and nonhomogeneity on the bearing capacity of c-c/J soils including c/J = 0 condition of soils. In their study, they used the method of characteristics to obtain the bearing capacity. Meyerhof (1978) obtained the bearing capacity for soils exhibiting anisotropy in friction by the conventional Terzaghi's type approach using two extreme values of c/J for the outer zones and an equivalent c/J for the radial shear zone. Shklarsky and Livneh (1961a) found from tests made on asphalt paving mixture that the ratio of strengths with load parallel and perpendicular to the direction of compaction was greater than one. It was shown that such paving mixture has anisotropic cohesion and isotropic angle of friction. Shklarsky and Livneh (1961b) also made an analysis of splitting tests for asphalt mixture and Shklarsky and Livneh (1962) presented equations for slip lines in anisotropic cohesion medium. It is generally believed that for compacted soils it is highly probable that, analogous to asphalt, the cohesion is dependent on the direction of compaction. Salencon (1974b) and Salencon et ai. (1976) presented analysis for the bearing capacity of c¢-'Y soil taking a linear variation of cohesion with depth. Limit analysis, particularly the upper bound analysis, has been found to be a convenient tool for solving such complex problems involving anisotropy and

~JI

311

310 nonhomogeneity of soil. Chen (1975) extensively dealt with the applications of limit analysis to soil engineering problems. He has applied the limit analysis for the bearing capacity of footings on a single layer and two-layered soils, exhibiting both the anisotropy and nonhomogeneity. Chen's analysis was again based on the assumption of a circular failure surface and hence agreed with the solution by Reddy and Srinivasan (1967). However, for soils exhibiting anisotropy and nonhomogeneity, the assumption of a circular failure surface is found to considerably overestimate the bearing capacity (Davis and Brooker, 1973; Salencon, 1974a). The assumption of a circular failure surface by many investigators was due to the fact that for anisotropic and nonhomogeneous soils, solutions become very difficult. Besides, Purushothama et al. (1972) obtained the bearing capacity factors of homogeneous and isotropic soil by treating the boundary wedge angles as variables. In 1982, Reddy and Venkatakrishna, by adopting a Prandtl type of failure mechanism, applied the limit analysis (upper bound) to obtain the bearing capacity of a strip footing on c-¢ soils exhibiting anisotropy and nonhomogeneity in cohesion. In this chapter, we shall introduce the Reddy and Venkatakrishna method as an extension and improvement of Chen's work (1975). The results are presented in a nondimensional form of charts for various parameters considered. 7.2 Analysis

The problem is treated here as a two-dimensional problem and is based on Limit Theorem II in Section 2.8 (Drucker and Prager, 1952). The following assumptions are made. (a) The Prandtl type of failure mechanism is valid (Fig. 7.1). In the experiments reported by Ko and Davidson (1973), it is observed that the Prandtl type of mechanism used in predicting the bearing capacity of rough strip footings is justified. In this failure mechanism, wedge abc is translating vertically as a rigid body with the same initial downward velocity vI as the footing. The downward movement of the footing and wedge is accommodated by the lateral movement of .the adjacent soils as indicated by the radial shear zone bcd and zone bdef. The

r-

"""''''"--------.---,

8

----I

r-.'--------:=n;;m<

C

hs

angles ~ and 1J 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. 7.1. (b) Based on earlier studies (Casagrande and Carillo, 1944; Shklarsky and Livneh, 1961a, b, 1962; Lo, 1965; Livneh and Komornik, 1967), the variation of cohesion ci' with the major principal stress inclined at an angle i with the vertical, is assumed to be given by: (7.1) where Cj is the cohesion corresponding to the inclination i of the major principal stress with the vertical direction. ch is the cohesion corresponding to the horizontal direction of major principal stress, and Cv is the cohesion corresponding to the vertical direction ofthe major principal stress (see Fig. 9.1b). The variation of cohesion with depth is assumed to be linear. Thus, the cohesion at depth h from the surface is given by (Salencon, 1974b; Salencon et aI., 1976): (7.2) where chs is the cohesion in the horizontal direction at h = 0, and {3 is the variation of the cohesion with depth. The ratio cv/ch is found to be constant for a given soil (Lo, 1965) and is denoted by k. Referring to Fig. 7.1, the angle (31 is obtained from .the geometry by the expression: 2(D/B) e(1r -

-

cos(¢ + 1J) cos¢

(7.3)

'1)tanq,

where ¢ is the angle of internal friction of soil. If Vo is the initial velocity along the line bc, VI is the downward velocity of the footing and v r is the relative velocity inclined at an angle ep with the line of velocity discontinuity bc, then VI and vr are given by (Fig. 7.2):

a~., b ep

C

Fig. 7.1. Assumed failure mechanism.

cos~ ~

v,

V

0

VI

Fig. 7.2. Velocity diagram.

312

313 Vo cosc/J/cos(~

VI =

(7.4)

- c/J)

(7.5)

The rate of external work done by the foundation load considering the symmetry is:

Wq

qBv cosc/J

o !---,-,-----,-

=

cos(~

(7.6)

- c/J)

, = V

=

tan¢

(7.13)

voeo tan¢

(7.14)

'0 eO

where the angle i is given by: i = !7r - [(!7r + c/J + I-t - ~) - 0] = !7r - (x - 0)

=h +

in which X

(7.15)

c/J + I-t-~

The rate of energy dissipation along the velocity discontinuity bc is: and

D bc

=

(average

Cj

along bc) (length bc) vr cosc/J

(7.7) (7.16)

The average

Cj

along the line bc is taken as the average of Cj at band

Cj

at c. At b: Substituting these values into Eq. (7.11), we obtain: (7.8)

D cd

At c: Cj

I

OJ [

Chs

(3B

+ (3D + ! -

cos~

o

=

+ (3D + ! (3B tan~]

[Chs

[1

+ (k - 1) cos 2 i]

(3B

[

hs

c.

o

[

2

tan¢

dO

eO

tan¢ sin(O +

]

cos~

(7.17)

After further simplifications, Eq. (7.17) can be reduced to:

Substituting the values of average

=

BV e20

(7.9)

where the angle i is given by:

D bc

=

]

+ (3D + -4 tan~

[1

Cj'

vr and the length bc into Eq. (7.7), we have:

+ (k - 1)sin2 (~ +

I-t)]

Bvo tan~ cosc/J

2

cos(~

- c/J)

(7.10)

where Ii' 12 , 13, and 14 are integrals given by:

I

OJ

II =

The rate of energy dissipation along the log-spiral cd is:

e 20

tan¢dO

(7.19)

tan¢ sin2(x - 0) dO

(7.20)

tan¢ sin(O + ~) dO

(7.21)

o

I

OJ

D cd

=

Cj' V

(7.11)

dO

o where Cj =

Cj

[Chs

where

=

+ (3D +

(3, sin(O

+

m[1

+ (k - 1)

20

o

on the log-spiral with an angle 0 is given by: cos 2 i]

fe OJ

12

(7.12)

13

=

fe OJ

30

o and 14

=

I

OJ

o

e30 tan¢ sin2 (x - 0) sin(O + ~) dO

(7.22)

314 At d:

The rate of energy dissipation in the radial shear zone bcd is given by:

f

0,

D bed =

(average

Cj;v

along any radial line)"r v dO

o where ciav' the average ci along any radial line, is taken as the average of ci at b given by Eq. (7.8) and ci given by Eq. (7.12), but with i = 7r/2 - (0 + ~ + p,). Substituting the values of average Cjav and other quantities into Eq. (7.23), we have:

D bed

f

1

0

=

[

Chs

+ (3D + -{3B4

o [1

+

cos~

(k - 1) sin2(0

+

eO tan¢

~

sin(O + ~)]

Bvo

+ p,)] - - e20 tan¢ dO 2

cos~

(7.32)

(7.23)

where the angle i = ! 7r - {32 + p" v3 = voe o, tan¢, (32 is the angle between the line de with the horizontal and the length de is given by the geometric relation: B e

de =

01 tan¢

.

Sill1) 2 cos~ cos(et> + 1)

Substituting these quantities into Eq. (7.30), we obtain: (7.24) D

de

=

[hS C

cos~

(3B

+ -4

After further simplifications, Eq. (7.24) reduces to:

Bvo e201 [2

=

01

J e20

tan¢

(7.26)

dO

o lb

=

f

01

e20 tan¢ sin2(0 + ~ + p,) dO

(7.27)

o

fe 01

Ie =

30 tan¢ sin(O

+ ~) dO

tan¢

sin1) coset> ] cos(et> + 1)

(7.34)

=i

-yB2 vO sin~ coset>

-~----

CoS~ CoS(~ - et»

(7.35)

where -y is the unit weight of soil. The rate of work done by the weight of soil in the radial shear zone bcd is given by: UV'Y)bed

=~

01

-y

f

r 2v cos(O

+ ~) dO

(7.36)

o

and

=

. (W ) 'Y abc

(7.28)

o ld

cos~

. {3 2] e01 tan¢. Sill1) Sill [1 cos(et> + 1)

Rate of work done by the soil weight in the triangle area abc, taking only one half, is:

where la' lb' Ie and ld are integrals given by: la

(7.33)

Equation (7.36) can be integrated and simplified to:

fe 01

30 tan¢ sin2(O

+ ~ + p,) sin(O +

n dO

(7.29)

o The rate of energy dissipation along the discontinuous line de is:

D de

=

(average

cj

along de) (length de) v3 coset>

(7.37)

(7.30) The rate of work done by the weight of soil in the area bdef is given by:

At e: (7.31)

316

317 TABLE 7.1 Comparison of values of N c for G = "(Blc h, = 0.0, k = cJch = I, and

COS

«(3 I

-

3(11" + fJ 1

T} ) e

-

~ -

TJ) tan¢

(7.38)

Equations (7.37) and (7.38) are the same as those given previously by Chen (1975). Equating the rates of external work done by the footing (Eq. 7.6) and by the weight of the soil (Eqs. 7.35, 7.37, and 7.38) to the rates of internal energy dissipation (Eqs. 7.10, 7.18, 7.25, and 7.34), and introducing the nondimensional parameters: d' 11

=

=

9.30 16.00 32.00 82.00

9.30 17.00 37.00 97.50

8.00 14.50 31.00 73.00

(3)

10 20 30 40

8.34 14.83 30.14 75.32

8.34 14.80 30.10 75.30

-

Limit analysis Reddy and Srinivasan

-'-'(1970)

140

Ca)

1/ =

PS Ic hs =0

100

0, and:

80

k = cvlch

60

~/

20 00

10

20

160 Nc

140

40

_.-

Limit analysis Reddy and

200 50

Srlnlvasan(1970)

180 (cl

degree

1/=

1.2

160

_ Limit analysis -· .... Reddy and Srinivasan (1970)

140

k=0.8

k=O.~

Cbl 1/ = 0.4

""

120

120

(7.43) (7.44)

60

li7

Values of the normalized bearing capacity pressure, q' = qlcvs ' have been obtained for rf> = 10, 20, 30, and 40°, DIB = 0.0, and 1.0, G = "fBI chs = 0.0 and 2.0, k = cvlch = 0.8, 1.0, 1.2, 1.6, and 2.0, and 11 = (3Blchs = 0.0, 0.40, 0.80, and 1.2.

20.J" 30 'I'

j)JI/

40 20 I - - -

~V 10

.'iii

60

~V

7.3 Results and discussions

40

50

~

10

20.J,,30 't'

degree

Fig. 7.3. Comparison of N c values for G

riJ

2~

80

VII

40 20

100

~.

1.0

80 2.0,

The minimum value of q' can be obtained by any optimization program.

i!

~i

1.0",

100

a

,

30

¢

220 Nc

~~

(7.42)

=a

//) W'~

the expression for q' is obtained as:

where the bearing capacity factors N c and N 7q are functions of the angles ~ and T} for given values of parameters, d' , 11, k, G and p,. For the minimum value of q' , the angles ~ and 'YJ must satisfy the conditions:

=0.8~

1.0 ,

40

aq'laT} =

8.34 14.80 30.10 75.31

(2)

(7.41)

oq' lo~

8.35 14.84 30.15 75.34

(I)

(7.41)

qlcvs

where cyS is the cohesion in the vertical direction at h

Meyerhof (1978) (8)

Salencon (I 974b)

120

q'

Terzaghi (1943) (7)

Sokolovskii (1965)

(7.40)

= (3Blchs

(4)

Reddy and Srinivasan (1970) (6)

Present analysis

0

= fJBlch' = 0.0

Chen (1975) (5)

¢ (degrees)

(7.39)

DIB

11

=

"(Blc h ,

=

0.0.

40

degree

50

319

318 TABLE 7.2 Comparison of ratios of Nc (nonhomogeneous) to Nc (homogeneous) for k = "(Blchs = 0.0, and v = {3Blc hs = 1.2

cJ ch

(I)

Present analysis (2)

Salencon (1974b) (3)

10 20 30 40

1.527 1.699 1.965 3.437

1.45 1.60 1.85 3.25


(degrees)

q' •

=

I, G =

Table 7.1 compares the values of the bearing capacity factor N c for an isotropic and homogeneous soil, obtained by the present limit analysis with those of others reported previously. The values of N c obtained by the present method are seen practically the same as those obtained by Sokolovskii (1965), Salencon (1974b) and Chen (1975). Table 7.2 compares the ratios of N c nonhomogeneous to N c homogeneous, obtained by the present analysis and those of Salencon (l974b). The ratios obtained by the present analysis are slightly higher than those of Salencon. Values of the pre-

141---I-......j.--I-

.2l._ c vs 24

_ _ Limit

q' 60

analysis

_.-ft~~gr and Srinivasan

20

_

Limit analysis _'_I~~~gr and Srinivasan

121--+---6,.£...j

50

L..-:::::

40 30 20

...,.-,:

'""'~-::..

~

.

~~

.=-- ~\

~ p\-'

6 ~==-+--+--+--I--j

l.:::::::':"= P2.0 1.0 k=0.8

41--+--J.--I--I-~

41--+--+--t--t---1 (a)

00

0.4

(a) D/B=O.O ,

10 (b)

16 =10~G/k.4

0.8

1.2

1.6

v. ,BB/chs

0.4

2.0

16.20., G/k=4

0.8

1.2

1.1

1,6

G=O.O

2 0 0.25 0.50 0.75 1.00 1.25 V=,BB/Chs

2.0

q' 600 ~~-,---,,--;-:-;::----,

q'

36-~-~-~-~~

_ Limit analysis _0_ Reddy and Srinivasan

q'

190--------,-----.,

550 1-----:1c!:19!!.7.!!.O;..)-

.....~"'l--

5001--I---I---,,AJ--;r"t--

3001-:::'-+C:--I---I--1---j 2501--+-+-+--+-" (c) 0/>= 30~G/k=4

50 0

0.4

0.8

1,2

1,6

1.1

Fig. 7.4. Comparison of q' values.

2.0

200 0

0.4

0.8

1,2 1.1

1,6

(c) D/B=1.0 ,

G=O.O

4 0 0.25 0.50 0.75 1.00 1.25 tJ

Fig. 7.5. Values of q' for


=

10 degrees.

(b) D/B=O.O , G=2.0

4 0 0.25 0.50 0.75 1,00 1.25 1.1

320

321

sent analysis are about 5.3% higher for cP = 10° and 8.3070 higher for cP = 10°. Figure 7.3 compares the values of N c obtained for G = 0.0 and II = 0, 0.4 and 1.2, by the present analysis, with those of Reddy and Srinivasan (1970). It can be seen from these figures that, the values of N c by the present analysis are slightly lower for II = 0.0 case and are slightly higher for II 0 case. Figure 7.4 compares the values of q' obtained by the present analysis with those obtained by Reddy and Srinivasan (1970) for the combination of cP = 10°,20°,30° and 40°, and G/k = 4. It can be seen from these figures, that the present analysis gives values higher than

*"

those obtained by Reddy and Srinivasan (1970). The difference in the values of q' obtained by the two methods, increases with an increase in the friction angle cPo Figures 7.5 to 7.8 show th~ q' variation with II = {3B/chs for several combinations of the parameters cP, D/ Band G considered. These figures reveal that the variation of q' with II is almost linear for the range of parameters considered. The values of q' increase with increasing II, and decreases with increasing k. Table 7.3 shows the percentage change in the bearing capacity pressure q' with

q.=....'L

q'

c vs 90,---,.---,.--,-------.----,

105 r----r--,----.----,,-----,

G=O.O 801---j-----r---r---I'--j

(bl 0/8=0.0, G=2.0 95 f---j---;---::t-::>"-if---j

10 '---,--'-_--'-_-'----'-_.....1 o 0.25 0.50 0.75 1.00 1.25

25 0 0.25 0.50 0.751.00 1.25

(a)

0/8=0.0

I

321--+-+---+--I-----l

121----+---+--+---+---1 (bl 0/8=0.0, G=2.0

(al 0/8=0.0, G=O.O

a

80

0.25 0.50 0.75 1.00 1.25

v

0.25

v

0.50 0.75 1.00 1.25 V

q'

V

q'170

q' (c) 0/8=1.0 , G=O.O

1501---+--j--+--If--i

(cl 0/8=1.0, G=O.O

80. 0.25 0.50 0.75 1.00 1.25

v

Fig. 7.6. Values of q' for 1>

=

20 degrees.

100 0.25 0.50 0.75 1.00 1.25

v

Fig. 7.7. Values of q' for 1> = 30 degrees.

0.25 0.50 0.75 1.00 1.25 V

322

323

·respect to the isotropic and homogeneous values for the extreme values of parameters considered. The change in q' , when compared to its values for k = I and 1J = 0, is seen to vary from - 48070 to + 305%. In summary, for the range of parameters considered, numerical values for the bearing capacity of anisotropic and nonhomogeneous soils have been presented in the nondimensional form of charts, which can be used easily in practice. The results clearly show that the anisotropy and nonhomogeneity of soils do have a considerable influence on their load-carrying capacity.

TABLE 7.3 Percentage change in values of q' = qlcvs with respect to homogeneous and isotropic values

(degrees) d'

=

DIB G

=

"fBlch; k' p

(I)

10

=

cv/ch

= (JBlchs

= 0.8 = 0.0

k = 2.0 p

= 1.2

p

=

0.0

k = 1.0 p

= 1.2

p

=

1.2

(2)

(3)

(4)

(5)

(6)

(7)

(8)

0.0

0.0 2.0

+ 14.33 + 15.83

+ 74.91 + 65.89

-28.72 -31.71

+ 8.00 - 8.44

+ 52.69 + 43.66

1.0

0.0 2.0

+ 15.81 + 19.33

+ 197.00 + 123.50

-32.33 - 39.50

+74.92 +22.32

+ 156.80 + 90.10

------------------------------.-------------------------------.--------------------------------------------

q

0.0

0.0 2.0

+ 16.41 + 18.93

+ 98.02 + 73.54

-32.82 -37.89

- 6.52

+ 69.91 + 46.86

0.0 2.0

+ 18.08 +21.91

+ 183.50 + 110.20

-36.30 - 43.46

+73.60 + 5.25

+ 169.10 + 75.22

0.0

0.0 2.0

+ 18.41 +23.52

+ 132.60 + 78.60

-36.92 -43.09

+24.15 - 12.43

+ 96.47 + 48.25

1.0

0.0 2.0

+ 19.81 +23.27

+248.40 + 99.06

- 39.69 -46.54

+ 79.13

- 6.99

+ 192.00 + 63.72

0.0

0.0 2.0

+20.11 +23.33

+ 191.08 + 79.08

-40.58 -46.73

+46.94 -18.22

+ 143.68 + 46.67

1.0

0.0 2.0

+21.05 +24.08

+305.20 + 88.24

-42.37 -47.18

+98.56 -16.29

+235.50 + 55.43

20 160

1.0

+ 13.69

-----------------------------------------------------------------------------------------------------------

140. 30

-----------------------------------.----------------------------------------------------------------------40 (bl 0/8=0.0, G=2.0

80 0 0.25 0.500.75 1.00 1.25

0.50 0.75 1.00 1.25

tI

tI

q'

370.--.----r--,--.----,,, (e)

0/8-1.0

,G-O.o

q'

840'--'-"--'--'-'" 7601--+-+--t--+-'"

29 0 I--+-+--I-¥-++-l

6801--+---1--+-7'9-----1

2501--+-+-+h~--r..,

6001--+--.J'''''--+--+-'''''''''

2101---+-.,.q..----rh-l--7-I

0.25 0.50 0.75 1.00 1.25 tI

Fig. 7.8. Values of q' for

=

References

(dl 0/8=1.0 , G=2.0

3301-----t----j--j--o;1><--..,

40 degrees.

1.00 1.25

Bishop, A.W., 1966. The strength of soils as engineering materials. Geotechnique, 16(1): 85 - 128. Casagrande, A. and Carillo, N., 1944. Shear failure of anisotropic materials. Boston Soc. Civ. Eng. Reprinted in Contributions to Soil Mechanics, 1941 - 1953, Boston Society of Civil Engineers. 1953. pp. 122 - 135. Chen, W.F., 1975. Limit Analysis and Soil Plasticity. Elsevier, Amsterdam, 638 pp. Davis. E.H. and Brooker. J.R., 1973. The effect of increasing depth on the bearing capacity of clays. Geotechnique, 23(4): 551 -563. Davis, E.H. and Christian, J.T., 1971. Bearing capacity of anisotropic cohesive soil. J. Soil Mech. Found. Div., ASCE, 97(SM5): 753 -769. Drucker, D.C. and Prager, W., 1952. Soil mechanics and plastic analysis or limit design. Q. Appl. Math., 10(2): 157 -164. Ko, H.Y. and Davidson, L.W., 1973. Bearing capacity of footings in plane strain. J. Soil Mech. Found. Div. ASCE, 99 (SMI): 1-23. Livneh, M. and Greenstein, J., 1973. The bearing capacity of footings on nonhomogeneous clays. Proc. 8th Int. Conf. on Soil Mechanics and Foundation Engineering, Moscow, USSR, Vol. 1, pp. 151 - 153.

324 Livneh, M. and Komornik, A., 1967. Anisotropic strength of compacted clays. Proc., 3rd Asian Regional Conference on Soil Mechanics and Foundation Engineering, Haifa, Israel, Vol. I, pp. 298 -304. Lo, K.Y., 1965. Stability of slopes in anisotropic soils. J. Soil Mech. Found. Div., ASCE, 31(SM4): 85 -106. Meyerhof, G.G., 1978. Bearing capacity of anisotropic cohesionless soils. Can. Geotech. J., 15(4): 592- 595. Purushothama Raj, P. Ramiah, B.K. and Venkatakrishna Rao, K.N., 1972. Limit analysis on bearing capacity of shallow foundations. Symposium on Strength Deformation Behavior of Soils, Bangalore, India, pp. 191 -196. Raymond, G.P., 1967. The bearing capacity of large footings and embankments on clays. Geotechnique, 17: 1-10. Reddy, A.S. and Srinivasan, R.J., 1967. Bearing capacity of footings on layered clays. J. Soil. Mech. Found. Div., ASCE, 93(SM2): 83-99. Reddy, A.S. and Srinivasan, R.J., 1970. Bearing capacity of footings on anisotropic soils. J. Soil Mech. Found. Div., ASCE, 96(SM6): 1967 -1986. Reddy, A.S. and Srinivasan, R.J., 1971. Bearing capacity of footings on clays. Soils Found., 11(3): 51-64. Reddy, A.S. and Venkatakrishna Rao, K.N., 1982. Bearing capacity of strip footing on c- soils exhibiting anisotropy and nonhomogeneity in cohesion. Soils Found., Jpn. Soc. Soil Mech. Found., Eng. 22(1): 49 - 60. Salencon, J., 1974a. Bearing capacity of a footing on a= 0 soil with linearly varying shear strength. Geotechnique, 24(3): 443 - 446. Salencon, J., 1974b. Discussion on the paper 'The effect of increasing depth on the bearing capacity of clays by Davis and Brooker. Geotechnique, 24(3): 449 - 451. Salencon, J., Florentin, P. and Gabriel, Y., 1976. Capacite portante globalendune Foundation sur un sol Nonhomogene. Geotechnique, 26(2): 351 - 370. Shklarsky, E. and Livneh, M., 1961a. The anisotropic strength of asphalt paving mixtures. Bull. Res. Council Israel, 9C(4): 183 -192. Shklarsky, E. and Livneh, M., 1961b. Theoretical analysis of the splitting test for asphalt specimens. Bull. Res. Council Israel, 9C(4): 202. Shklarsky, E. and Livneh, M., 1962. Equations of slip lines in anisotropic cohesive medium. Bull. Res. Council Israel, Vol. C(4): pp. 159 - 170. Sokolovskii, V.V., 1965. Statics of Granular Media. Pergamon Press, New York, NY, 237 pp. Terzaghi, K., 1943. Theoretical Soil Mechanics. John Wiley, New York, NY, 510 pp.

325

Chapter 8

EARTHQUAKE-INDUCED SLOPE FAILURE AND LANDSLIDES * 8.1 Introduction Slope failures and landslides occur extensively in all parts of the world and frequently result in a tremendous toll of death and destruction of properties. It is therefore of prime importance to devise the means and capability necessary to achieve a satisfactory assessment of the danger of slope failures and landslides. The term landslide refers to a sudden rupture of a mass of rock or soil and its movement downslope by the force of gravity. Slides on the slopes of man-made cuts or earth structures are generally referred to as slope failures. Slides on natural slopes are commonly referred to as landslides. On steep slopes, among the many types of landslides that may occur, falls, slides, and avalanches of rock and soil are most frequent during rainstorm or earthquakes (Chen, 1977). In this chapter, for the sake of convenience of discussion, the term landslide, refers to a sudden slip of a mass of soil along a well-defined slip surface for both natural slopes and slopes of man-made cuts. It is true that, in many instances, the most severe destructions and greatest number of casualties result from earthquake-induced landslides. Records show that this kind of earthquake-induced landslides occur most frequently on sloping earthmasses. They are observed on the slopes of dams, embankments, and other manmade cuts; on the banks of rivers, lakes, reservoirs, and along coasts as well as on mountain slopes (Koh and Chen, 1978). For simplicity, such sloping earth masses will be referred to as 'earth slopes' throughout this chapter. Because of the potential threats associated with these landslides, there is an urgent need to advance the state of the art to develop more effective methods for the assessment of such dangers. Common practice in such analysis involves two points: the first is the neglecting of the more complex soil behaviors and properties; the second is the simplification of the seismic forces as being a constant. This chapter focuses on the improvement of the analysis through the incorporation of nonconstant seismic forces throughout the slope height. In the following chapter this procedure will be extended and incorporates the nonhomogeneity of some soil properties.

* This chapter is based on the M.S. thesis by S.W. Chan (1980) and the paper by Chen, Chan en Koh (1984).

326

327

Thus, a more realistic estimation on the earthslope stability can be made with relative ease (Chen and Koh, 1978; Chen, 1980). The extension of the limit analysis method for the static case to the case of seismic loadings is a logical step forward. The existing approach involves essentially adding to the deadweight of the potential collapse mass a pseudo-static force simulating the seismic load. This force acting through the center of gravity of the soil mass is expressed as the product of the soil mass and a seismic coefficient as defined previously in Chapter 5. As shown in Fig. 8.1, m'Y is the soil mass and k h is the seismic coefficient in the horizontal direction. The criticisms on this rather crude approach are: Firstly, the seismic coefficient of a slope is not a constant value during any instant of earthquake. Owing to the nonrigidity of the soil layers, the reactive forces developed throughout the height of the slope as a result of the movement of the ground always vary. Secondly, because of the granular nature of the soil, it is reasonable to expect the density and strength properties to vary along the height as the consequence of different degrees of water saturation inside the intergranular voids of the soil. To get around the first problem, a more recent effort by Chen et al. (1978) was made to incorporate zones of different seismic coefficients along the height to approximate the actual variation (Fig. 8.2). While such approach is a definite improvement over the earlier ones, there are also restrictions to its application. A conceivable difficulty is the case where the variation of the seismic coefficient is so sharp that a large number of thin zones have to be used. In the presence of secon-

~-Tf2'r8-0

I f

-----k-;;;-

=

R - - - - - - k -;- -

~_"'"""'_ C -=--=--=E Contribution of

==- ==~~-= h

k hn

'\actual arbitrary seismic profile equivalent seismic zones

on BFGB=contribution of k hn on BEDB -contribution of khn on FEDGF

Cal Seismic zoning

f

OAQ..~f

____

_ _ _ _ _ _ _O~~

________ O~Q.. __--..J<--~-

Q,~

resulting gentle profile

Cbl Result of imposing 4 seismic zones of equal thickness on all slopes rigidly

---

Fig. 8.2. Schematic of the seismic coefficient zoning approach.

potential slip surface

Fig. 8.1. Schematic of a pseudo-static constant seismic profile approach.

dary slope as shown in Fig. 8.2 with {3 > 0, the analysis will necessitate the handling of two different slope geometries. In this case, rather than considering the original slope with one knee (section B-E-D-B in Fig. 8.2), one will have to consider a fictitious slope with two knees (F-E-D-G-F). With the need to prepare a map of coefficient zones for each profile of seismic forces at each instant, it would be quite timeconsuming when investigating the hazard of the slope at intervals within the duration of the earthquake. The next logical step towards a better analysis of the earthslope is to develop formulations to account for a more accurate seismic profile. The seismic coefficient as a function of the elevation about the ground level must be recognized. The vertical component of the seismic force, neglected in earlier works, must be included. Through appropriate interpretation, the vertical seismic profile can also be used to represent the variation of density along the height as well. With more rigorous and general formulations, further implications of the possibility of incorporating the

329

328 profiles of strength parameters into the model can be made. All these are undertaken in this chapter, following the work of Chen et al. (1984). Since every mathematical model has its own limitations, it is the wise application of a model to different situations that determines its usefulness and efficiency. The slight changes of the model for the analysis of the location of the most critical slip surface in a given slope are demonstrated. Further directIons for the applications of the model, as well as improvement for the future, are also discussed. 8.2 Failure surface

In the analysis of slope stability, the determination of the critical height of the slope, the height at which the slope is at the verge of collapse, yields an important criterion. According to the upper-bound theorem, failure can occur if a compatible failure mechanism exists in the body. A convenient way to approach the problem is to assume that a single, well-defined slip surface exists for the slope. For example, in Section 2.10, Chapter 2, the straight and logspiral surfaces were chosen as the translational and rotational mechanisms respectively to determine the critical height of the slope. The arguments were essentially based on the geometric compatibilities. The question that remains is that of these two types of mechanisms, which one would be more critical? That is, which one would be developed with the least input of external work? For the earthslope under the static loads, the ¢-logspiral has been shown to be the most critical (Chen, 1975). This can be seen in the example problem of Section 2.10, Chapter 2. However, it is still necessary to find out, for more complicated loading situations, whether this type of failure mechanism is still the most likely to occur. Since different shapes of failure surface will result in earthslopes of different soil masses, the stress state along the slip surface is mechanism-dependent. The process of determining the most critical surface is thus controlled by the consideration of the shape junction and the stress-distribution junction. The shape of the mechanism and the resulting stresses along the failure surface must be chosen in such a way that the external loads from the soil mass above this surface will be just balanced by the stresses developed at the surface. At the incipience of any collapse, when the stress state satisfies a certain criterion, the flow of the velocity discontinuity will then carry the block it supports along. Of all the possible shapes that satisfy the above requirement, the one that needs the minimum of applied load would be the most critical. This then is the criterion of optimization. Using the techniques of variational calculus, the applied load on a potential slip surface can be defined by a function (Fig. 8.3):

w

=

f H

[(dFx )2

+ (dFy )2]i

(8.1)

where dFx and dFy are the orthogonal force components of an infinitesimal soil layer at an arbitrary elevation. They are defined as: (8.2) and (8.3) Here, 'Y is the unit weight of soil; k x and k y are the seismic coefficients in X-, ydirections, respectively; lis the length of the infinitesimal soil layer, and dh is its thickness. For the static case involving the gravitational force only, kx = 0 and k y = 1. For seismic loading, both k x and k y are nonzero. The present formulation

h

H

/

L

-------------".-----"

~

Fig. 8.3. Arbitrary potential failure surface.

330

331

permits the consideration of cases where the seismic load has a vertical component; for this case, ky ¢ 1. To account for the variations of loadings along the vertical and horizontal distances from the toe of the slope, k x and ky are allowed to be any functions of rand 0, the reference polar coordinates (Fig. 8.3), as:

Eq. (8.6) is so general that it is true for any other slope-surface combinations (Figs. 8.5 and 8.6). Therefore, by transformation back to polar coordinates, I becomes: I

=

l(a,(3,r,O)

(8.7)

From the consideration of the geometry along the slip line (Fig. 8.3): (8.4) dh

=

ds cos(O - 5")

(8.8)

and (8.5) The function I depends on the shape of the slip surface, the geometry of the slope, as well as their relative positions with respect to each other. That is, the I-value depends on the slip surface 1/;J and the perimeter function 1/;2 of the slope (Fig. 8.4). Recoursing to the rectangular coordinates for the time being, we have:

where r is the angle between the perpendicular to the radius and the surface element ds, which is:

r dO ds = - cos r Therefore: dh = f(r,O,5")dO

I

=

1/;J (y) - 1/;2(a,(3,y)

=

l(a,(3,y)

(8.6)

Note that the actual form of 1/;2 is immaterial here. It can be any complicated function or even a Fourier series to account for the kink at the knee. Also notice that

y Fig. 8.5. The equivaJency of 1 for a raised sagging slip surface.

Fig. 8.4. The equivalency of I, the horizontal slice length, for a toe-surface of failure.

(8.9)

Fig. 8.6. The equivalency of 1 for a stretched slip surface.

(8.10)

333

332 With respect to the polar coordinates, and with 00 and 0h being the initial and final angles of the slip surface, the functional W may be expressed as:

o

.

h

2

J 0

W =

+

'Y l(r,O,O'.,(3) [kx (r,O)

2

I

ky (r,O)j2 f(r,O,ndO

0

=0,

f [7 coso -

I:.Fy

= 0, -

J[7 sino +

u sino] cis -

coso] ds +

u

f

'Y k x 1 dh

Jk 'Y

y

=0

1 dh

B 3 = 'Y I(O,r,O'.,(3) [(r sinO) kx(O,r) =

(8.11)

(8.12)

= 0

+

[r cosO -

! 1(0, r, 0'.,(3)] kyCO,r)] (8.23)

B 3 (O,r,O'.,(3,'Y)

where the R's are the reaction forces from the stress state of the slip surface, and the B's are the applied forces contributed by the soil block supported by the surface. In minimizing the functional:

The additional equations are the equations of equilibrium: I:.Fx

(8.22)

(8.13)

W =

J°h P(O,r,O'.,(3,ndO

(8.24)

° 0

subjected to the three constraints, it is necessary to make use of the Lagrangian multipliers:

and I:Mo = 0,

J[ra sinS- +

rr cost] cis +

J'Y [kxrl sinO

kyl (r cosO -

! I)] dh

=

0

where 0 = 7r - 0 - 1/;. These three equations can be simplified from the geometry. Furthermore, if we assume that the Coulomb's criterion (Eq. 2.32) is satisfied everywhere along the slip surface, then, they become:

J°h (R

I:.Fx =

aI ar

a2I ar' ao

a2I ar' ar

a2I = 0 ar' ar'

BI)dO = 0

l -

(8.15)

(8.26)

and

J°h (R

=

where AI' A2' and A3 are constants. By the Euler-Lagrange differential equation for multi-variable variational calculus, we get: _ - - - - r' - - - r " - -

00 I:Fy

(8.25)

(8.14)

2

+

B 2)dO

=

0

(8.16)

(8.27)

00 I:.Mo

=

f°h

(R 3

+

B 3 )dO

=

0

(8.17)

00

= -

R2

=

a [(r cosO)' tanc/>

aB I

aB2

aB3

ap

aa

au

aa

aa

-=-=-=-=0

where Rl

From the fact that P, B l , B 2 , and B 3 are independent of u, it is obvious that:

+

(r sinO)'] - c(r cosO)'

a [(r cosO)' - (r sinO)' tanc/>] - c(r sinO)'

=

=

R l (O,r,r ',c/>,c,a)

R 2 (O,r,r' ,c/>,c,u)

(8.18)

With these, and the condition that:

(8.19)

~= aa'

(8.20) (8.21)

0

(8.28)

(8.29)

335

334

Knowing that:

Equation (8.27) becomes:

(8.38)

(8.30)

Substituting the expressions of Rl' R 2, and R 3 in Eqs. (8.18), (8.19), and (8.20) into Eq. (8.30), we obtain:

we obtain:

Al [- [(r'cose - r sine)tan + (r'sine + r cose)]] + A2 [(r'cose - r sine) - tan (r'sine + r cose)] + A3 (rr' - r 2tan
Al [- tan -

To convert the coordinates to the cartesian system, the following transformation identities are used: X = {y

r cose

(8.32)

= r sine

-

tan~]

[x + y: - (x: - y)] = tan

0

2 Y A dx + (xA +2 A ) + tan [(y - AI) A - (X + A A3 )ddx ] = 0 ( Y - AI)dY 3 3 3

t o

dy dx 2 x--y-=r de de

-T-

--x

II'V Bo

(8.34)

r cose

I

I

/ I

",

x

(8.35)

and dx de

dy de

x-+y-=rr

,

(8.36)

Equation (8.31) is then reduced to: Y dY ] Al [ - tan -dx - -d ] + A2 [dx - - tande de de de

+ A3 [ X -dx + de

(ddeY- Y -dx)] =0 de

Y -dy - tan x -

de

(8.39)

By collecting terms, it becomes:

(8.33)

dx = r' cose - r sine de

~ = r'sine +

+ A3

~] + A2 [1

(8.37) /

&

Fig. 8.7. Transformation of the x-y (1'-0) coordinates into the X-Y (1'- 0) coordinates.

(8.40)

337

336 Translating the origin (0,0) of the x,y-system to (- A2/A3' Aj /A3) of the new X, Ysystem (Fig. 8.7), the differential equation evolves into:

or In T

(8.41)

In

+

(Ytan¢

+

X) dX = 0

T

Thus, Eq. (8.41) becomes:

+ X) dX + (v X2 - X2 tan¢) dv

0

(8.45)

Integrating Eq. (8.45), we obtain: dX = tan¢

In X

Co

8

=

f~ - f~ + 1+ v 1 + v 2

2

(8.46)

Co

is a constant. That is:

= tan¢ [arc tan v] - i

Since v

=

In (1

+ v2 ) +

Co

(8.47)

Y/X, then from Fig. 8.7 it is obvious:

(8.48)

arc tan (v)

(8.49) Therefore: =

80 tan¢

(8.53)

=

TO exp [(8 - ( 0) tan¢]

(8.54)

8 tan¢ - ~ In (TI X) 2 +

Co

This is the equation for a ¢-Iogspiral. The ¢-Iogspiral is then the most critical slip surface. From the solution ofEq. (8.27), it is obvious that the process of solving Eq. (8.26) will be even more tedious. The solution will be very complicated, and will result in the profile of the normal stress distribution along the slip surface. Since the resultant of normal and shear stresses along the entire surface is directed toward the center of rotation, they have no contribution to the internal work for our rotational mechanism. It is sufficient to know from the examination ofEq. (8.26) that the normal stress distribution along the slip surface varies with different loadings. Note that by means of similar derivations, the following statement can be obtained. For the case where body forces as well as soil nonhomogeneity and anisotropy in cohesion are considered only, the most critical failure surface, if the Coulomb's criterion is used, is still the ¢-Iog-spiral. 8.3 Determination of the critical height for seismic stability

and

In X

In TO -

(8.52)

(8.44)

v dX + X dv

where

=

Co

Substituting Eq. (8.53) back into Eq. (8.51), we finally obtain:

where v will be explained later, then we have:

fX

= 80 tan¢ +

(8.43)

Y=vX

(v 2 X

ro

or Co

=

(8.51)

(8.42)

Assuming:

dY

Co

From the boundary condition, where TO corresponds to 80 for the initiation of the slip surface curve:

or (Y - Xtan¢) dY

8 tan¢ +

(8.50)

As shown in Section 8.2, the ¢-Iogspiral surface may be used as the failure mechanism in the stability analysis of an earthslope under seismic loading situations. Following traditional approach in slope stability analysis, we predict the critical height of the slope rather than the critical load itself. Such a prediction is an upper bound on the actual value. It is quite useful in providing insight to the evaluation of the slope stability as well as guidelines for the design of earthprojects. Pertinent to the analysis of the stability of an earthslope under seismic loads is the development of a suitable representation for the loads. In conventional works

339

338 (Prater, 1979), the seismic load is considered constant and in the horizontal direction only. Further studies (Newmark, 1965; Seed, 1966, 1967; Ambraseys and Sarma, 1967) considered the seismic load to increase with height. In addition, studies have taken into consideration the vertical component of seismic load. More realistically, the seismic profile is nonlinear. A convenient representation is to treat profiles of the vertical and horizontal components as polynomials of the elevation, as follows:

and Y

= r sinO = ro exp [(0 - 0o)tan¢] sinO

(8.59)

with 00 =:; (J =:; (Jh' Substituting Eq. (8.57), into Eq. (8.55) for kx(h), we obtain: (8.60)

(8.55) which expands to: and ky(h)

= bo +

bjh

+

b2 h 2

+ ... +

bnh n

= .~

j=o

bjh j

+

kx
a jl1 - ajy

+

(8.56)

+

However, to conform to the polar coordinates used to describe the failure spiral surface, these must be expressed in terms of 0 and r. If the spiral is to pass through the toe of the slope, then the height of any horizontal layer of soil is (Fig. 8.8):

3a31Jy 2 -

a3y3

a2 11 2 - 2a211Y

+ ... +

a2y 2

+ +

am1Jm

a

+

a311 3 -

m E

[m ] 1J(m-k) yk k

m k=j

3a31J2y

(8.61)

Equation (8.61) may be rewritten as: kx(Y)

=

(8.57) where

(ao

+

aj1J

+

4a41J

+

am

3

a211

2

+ ... +

G] 1Jm-2)y2

+ ( ... +

(8.58)

+

+

a31J 3

+ ... +

j am [7] 1J m - ) y

+

- (a 3

am [~] 1Jm-m)

4a41J

am1Jm) -

+

(a 2

+

+ ... +

ym

(a j

+

2a21J

3a31J

+

6a41J2

+

3a31J2

+ ...

am [~] 1Jm-3)y3

+ (8.62)

If we set: Loading profiles (seismic)

p. j

mi· .

= E [.] a'1JI-j( - 1)j i=j

j

(8.63)

I

Equation (8.62) can be expressed as: (8.64) Similarly, for ky(Y) with: (8.65) we have: n

ky(y)

Fig. 8.8. Relation of the loading profiles to the geometry of the slope.

= .E

j=O

.

Il-jyj

n·

= .E

j=O

.

Il-j(r sinO)l

=

ky(r,O)

(8.66)

341

340

8.3.1 The critical height of toe-spiral A spiral that begins somewhere in the ,6-portion and terminates at the base of the a-portion of the slope shall be referred to as a toe-spiral. With the toe-spiral prescribed as the failure surface, together with the seismic profiles specified along the slope, the critical height of the slope may be derived. The critical height of a toe-spiral is the height of the slope at which such a failure mechanism can be developed so that the soil mass resting upon the failure surface will be carried down in the fashion of pure rotation. Yielding impends when the loads from the rotating mass perform external work at a rate equal to the internal rate of energy dissipation in the mechanism. It is then only necessary to impose a virtually small rotational velocity to the rotational block, and require that the energy rate equilibrium be observed. Equating the external work rate to the internal dissipation rate will provide the basis for the calculation of the critical height. By means of superposition, the rate of external work done contributed by the rotating soil mass D-B-E-D (Fig. 8.9) can be found as the rate of work done by A-BE-F-A (the gross work rate), minus the work rate by A-B-D-C-A and C-D-E-F-C (the fictitious work rate). Considering the region of A-B-E-F-A first, we have (Fig. 8.10): Fig. 8.10. Logspiral slip surface for seismic loading, calculation for the fictitious external work rate.

(8.67) r cosO, and y

with dFx = ')'kx(Y) dA, dFy Noting that: dy

[(dr/dO) sinO

+

r sinO.

r cosO] dO

[sinO d[r o exp [(0 - ( 0 ) tan¢>J]!dO + r cosOl dO (sinO tan¢> + cosO) r dO

(8.68)

and dA

jY! /

Fig. 8.9. Logspiral slip surface for seismic loading, calculation of the gross external work rate.

=x

(8.69)

dy

we have:

Jd W1x J°h ')'0 kx(Y) r [sin 0 cosO tan¢> + sinO cos 0] dO 3

=

and

°0

2

2

(8.70)

342

343

°h

Jd W1y = i J ky + cos30] dO 'YO

(8.71)

=

For region A-B-D-C-A, we have:

J

J

=

and

Yo

'YO kiY) [yx2 dy]

YB

+

J!

'YO ky
[x~dY]

YB

Yo

J (!)

=

JdW3y

The above six equations Eqs. (8.70), (8.71), (8.75), (8.76), (8.80), and (8.81), can be expanded by substituting the expressions for kx(Y) and ky
From the geometry, the expression of x 2 can be derived from: (y - YB)/(xB - xz)

=

(8.73)

f°h

JdW1X =

=

(xB

+ YB/tan(3)

- y/tan{3

=

/2 - y/tan{3

(8.74)

JdWl

J

0

'YO.f: [/t:J.r j + 3 (sinjO cos30 j=O

0

Thus:

(8.82)

j=O

f°h

=!

Y

E [p/j+3 (sinj+10 cos20 + sin j + 20 cosO tan
'YO .

00

so that, for YB :5 Y :5 YD: X2

(8.81)

'YO ky
YE

(8.72)

tan{3

(8.80)

y!taIJ.a)y dy

YE

Yo

+ dW2y

Yo

Jd W3x J 'YO kx(Y) (13 -

°0

W2 =' JdW2x

Thus:

+ sinj+10 cos20 tan
(8.83)

Yo

Yo

d W2x

=

J

'YO kiY) (12 - y/tan(3)y dy

J 'YO .E [Pj(l2 yj + 1 -

=

JdW2x

YB

(8.75)

yj+2/tan(3)] dy

(8.84)

j=O

YB

and

Yo

Yo

JdW2y =

J (!)

'YOky
dW2y =

!

(8.76)

YB

.

J (!) YE

Jd

YE

W3x

'YO.f: [/tj

YB

j=o

(I~ yj

- 2/2 yj+l/tan{3

+ yj+2/tan2(3)] dy

(8.85)

YE

JdW3x

Similarly, for region C-D-E-F-C: W3 =

J

+

Jd W3y = YoJ 'YOkx(Y) [yx3dy] +

2 'YOky
=

J 'YO .E [Pj (13 yj+l Yo

yj+2/tana)] dy

(8.86)

j=O

and (8.87) (8.77)

Yo

Equations (8.82) and (8.83) are expressed into more consistent forms with the application of the following identities:

where from: (8.88) (8.78)

and for YD :5 Y :5 YE:

and

r = (8.79)

e exp(O tan
(8.89)

344

345

where

<0 8

=

exp( - 00 tanet»

Q = TO

A2

(8.90)

+ B2

A2

A sinO - (B - 2) cosO A2 + (B - 2)2

8h

I'D

= Ix

B (B -

1) sinB - 30

+ B2

1) (B - 2) (B -

3)sinB - 50

--------=---~-+-----''---------'-----'-------'--'-------'---

Thus, we have:

JdW

+ B (B -

{sinB-IO (A sinO - B cosO)

E [v. Qj+3 0J e 8(i+3)tan1> (sinj+IO

j=o

J

- sin j + 30

A __ si_nO_-_...:.(B_-_4.:-)_c_os_O A2 + (B - 4)2

0

+ cosO sin j + 20 tan
(A2 + B2) [A2 + (B - 2)2]

(8.91)

' + . .. }+"

(8.95)

where A is the last term of the series. The expression for A depends on whether B is even or odd. If B is even, then A = Ae :

and

JdW

Iy

=

l.vD 2

I

E [p.. Qj+3 Jf°h e 0(i+3)tan1> (cosO sinjO

'-0

J

J-

- cosO sin j + 20

00

+ sinj+IO tanet> - sin j + 30 tan
(8.92)

Each of these two equations involves two integral forms, namely eM sinBO cosO dO, and

Je

AO

eA°

. BO

sm

sinBO dO. Their solutions are:

cos

0 dO

e

=

AO

sinBO [A cosO

+

A2 -

k =

J

AO

(8.93)

I

'e = "

Je

e

~

sm

e

sinB - 10 (A sinO - B cosO)

= ------'-----------'A2

+

If B is

+ B2

J

B (B - 1) e AO sm . B- 20 dO A2 + B2

(8.94)

We expand Eq. (8.94) iteratively as:

J

e AO sinBO dO =

I

A2

+ B2

A2

+

+

1 (B -

0

(8.96)

l)(B - 2)(B - 3)

[B - (B -

1)]

(8.97)

[A2 + [B - (B - 2)]2J

AO

=

od~,

~

+ B2) [A2 + (B - 2)2] ... (A2 + 22) A2 ~ (1)

0

if A

=1= 0

(8.98) if A = 0

(2)2

then A = AO:

kO~AO sinO dO = k o [A

sinO - cosO eAOJ A2 + 1

(8.99)

with:

[e AO sinB-IO (A

+ B(B - 1)(

=

B(B - 1) ... (1) A

?(A2

B 2(B - 2)2

. BO dO

if A

(A2 + B2) [A2 + (B - 2)2]

B(B - 1)

A8

0

or

and AO

B(B -

(B+ 1)2

AB e AO sinB-IO dO A2 + (B+ 1)2

=1=

with:

e

+ (B + 1) sinO]

if A

2)Z

sinO - B cosO)

__ B(.:.-B_-_l:.-)_ . ._.-=.[B_-_(.:....B_-_2.:.::..)]_ ko = _ [A2 + B2] ... [A2 + [B - (B - 3)]2J

[e AO sinB - 30 [A sinO - (B - 2) cosO]

or

(B - 2) (B - 3) eAO sinB - 40 dOl)]

(8.100)

347

346 Ao

= eAO [ B(B -

1) (A2 + B2)

Then, similarly, after some manipulation, Eq. (8.93) can be expressed as:

(2) (A sinO - cosO) ] (A2 + 32) (A2 + 12)

(8.101)

JAO e

Now, since:

[

B] 18

= (B) (B

-1) ... (B - 2s

+ 1)

. Bn sm u

n dn

cOSu

u

=

e AO sipBO JA cosO + (B + 1) sinO]

--:..-.::---'-=--'-----'---~----.::

A2 + (B + 1)2

AB - - - - I[A, B-1] = J[A, B] A2 + (B + 1)2

(8.102)

(18)!

(8.107)

Ultimately, Eqs. (8.91) and (8.92) can be reduced to the final form:

and

(8.108) and then, Eq. (8.95) can be expressed as: (8.109)

. (B)

JeAO sinBO dO = eAO

mt ');,2

• B 2s {[B](18)! [A smO - (B - 18)cosO] sin - 5=0 18 5 n [A 2 + (B - 2/)2]

0}

1

1=0

(8.103)

where

f\X(TO,OO,Oh)

-

where int(B/2) = the integer part of (B/2). Note that the last term, Ae or Ao is included. For, if B is even, then int (B/2) = B/2, and the last term in the sum of Eq. (8.103) is:

he

= B(B -

1) ... [B - 2(B/2) + 1] (sinO)-l (A sinO) eAO (A2 + B2) ... (A2 + [B - (B - 2)]2) [A2]

(8.104)

If B is odd, then int (B/2) = (B - 1)/2 and the last term in the sum of Eq. (8.103) is:

o .r:

=T

3

)=0

vj(}1+ 3 {I

1\

B(B - 1) ... [B - 2(B - 1)/2 + 1] (sinO)o (A sinO - cosO) e AO (A2 + B2) ... (A2 + 32) (A2 + 12)

I]Oh00 (8.110)

and

o .£

f1/TO,OO,Oh) = T 3

)=0

P-je j + 3

IJ:: -

(J[U+3)tanq"j]Oh - J[U+3)tanq"j+2]Oh 00

tanq, I [ U+ 3)tanq" j + 31::)

00

(8.111)

(8.105) . Next, the coordinates of the points B, D, and E of the soil mass D-B-E-D are seen to be (Fig. 8.10):

If Eq. (8.103) is written in the simple form:

fe AO sinBO dO = I[A, B]

3)tanq" j+

I[U+3)tan~,j+3]:: + tanq,J[U+ 3)tanq"j+2]::}

+ tanq, I [ U + 3)tanq" j + '0 =

[U+

(8.112) (8.106) YD

= TO

sinOo + L sinl3

(8.113)

348

349

and

(8.124) (8.114)

where

Therefore, from Eqs. (8.74) and (8.79):

f 2x ('o,OO,Oh) =

,-3

j=o

0

(8.115)

(8.116) The variables L, like H, is the geometric parameter describing the rotating soil mass D-B-E-D. From the geometrical configuration of the slope shown in Fig. 8.8, we have the following relations: '0

cosoo - 'h cosOh - H/tana - L cos{3

=0

-3,", Itj

='0

=0

°

f (

0)

3y '0' 0' h

j~O Pj

j

Eo

+ 2 j

[2 Y j

yi +

j

JIY

+ 3

2

2

J.

+

U + 2)tan{3 yj + 3

2

(8.125)

U + 3) tan{3 Y: 2/ y j +

+ I

j + 1

[/ 3

y

3

y +

J!YO(8.126)

U + 3)tan2(3 YB

JYE

(8.127)

+ 2 - U + 3)tano: Iyo

2 n [/ yj+1 -3 1: 3 ='0 j=O Itj j + 1

21 yj+2

-

3

U + 2)tano:

+

yj+3

JI YE(8.128)

U + 3)tan20: Yo

The total rate of external work done is now expressed as:

- IdW3X

(8.118)

The solution of these two simultaneous equations gives explicit expressions for both Land H:

J=O

-3 m

WE = WI - W2

'h sinOh - '0 sinOo - H - L sin{3

....

f 3x ('o,OO,Oh) ='0

(8.117)

and

J

n

( II ll) f 2y'O,VO,vh

and

E p.[/2yj + 2 _

-

-

W3 = !dWlx + JdWly IdW3y

-

IdW2x - IdW2y (8.129)

or (8.130)

L = ['0 sin(Oo

+

0:) -

'h sin(Oh + o:)]Isin(o: - (3)

(8.119) The next step is to calculate the rate of internal energy dissipation along the velocity discontinuity surface BE, where yielding occurs. Similar as Eq. (2.33) (Fig. 2.14), this dissipation rate for an infinitesimal surface element is:

and H = ['h sin(Oh + (3) - '0 sin(Oo + (3)] sino:/sin(o: - (3)

(8.120) d.D

Equations (8.84) to (8.87) are all in integrable forms, and can be expressed formalIyas:

(T -

(J

tan..p) l' dv

(8.131)

where

(8.121)

l'

(8.122)

in which t is the extremely small thickness of the velocity transition zone resulting from the dilatation. The negative sign for the second terms is necessary because (J represents the compressive normal stress while l' tanlj> stands for the outwards dilatation. Since the Coulomb's criterion (Eq. 2.32) must be satisfied, we have - (J tanlj> = C - T. Substituting it into Eq. (8.131) and integrating over the entire region of the mechanism results in:

(8.123) and

=

= ,0 coslj>/t

(8.132)

351

350

JdD = J(7 - u tanrf>H' dv = v·

v = cO

~osrf> Jr s

1

II

H* ::;; ~N*

(cr 0 cO;rf>)d.i ds

sO

d.i ds

(8.133)

with

0

(8.140)

Here, d.i is the differential thickness of an element. The extremely small thickness of the transition zone is constant throughout. Noting that ds = r dO/cosrf>, we have for the total internal rate of work (or the total energy dissipation rate) expression:

f

f

110

110

. clJ Bh Z I1h WI = D = t r t dO = clJ fro exp[(O Z

(8.139)

'Y

cOr o [exp[2(Oh -

°

0)

°

0)

such that r

o' °0and OJ; satisfy the conditions of: (8.141)

tanrf>]Jz dO

tanrf>] - 1)/(2 tanrf»

(8.134)

The dimensionless number N* is the seismic stability factor of the earthslope. The value of N* is a pure number, and is dependent on rf>, a, fJ, kiy) and kyCY). Note that when the loading force is a constant (Le. zero-th degree polynomial in Y), the function F becomes dependent on 0 and 0h only, and:

°

Equating the external work rate to the internal rate of dissipation: (8.135)

(8.142)

F(kx = constant, k y = constant) = F(OO,Oh)

8.3.2 Earthslopes of purely cohesive soil

we have: exp[2(Oh -

c

ro = 'Y 2 tanrf>

°

0) tanrf>] -

1

---~---:---,-------=---:---,---~--,-

[(fix - f 2x - f 3x )

+

~ (fly - fzy - f 3y )]

(8.136)

By the upper-bound theorem, this means that any toe-spiral satisfying the above equation will be a surface along which yielding impends. Substitution ofEq. (8.136) into Eq. (8.120) gives an expanded expression for H, the vertical distance of the knee D above the ground, or the height of the slope: (8.137)

A purely cohesive soil is one in which there is no internal friction (rf> = 0). It is also called the Tresca material. It is observed from Eq. (8.138) that: F

=

g(rf» q(rf»

(8.143)

where g(rf»

=

sina [e(8h

=

2 tan<1> sin(a - fJ) [fIx - f zx - f 3x +

-

Bo>tan¢ sin(Oh

+

fJ) - sin(Oo

+

fJ)][eZ(B h

-

Bo>tan¢ -

1] (8.144)

and

where

q(rf»

sina[e(8h

-

80}tan¢ sin(Oh + fJ) - sin(Oo + fJ)] [e2(Oh - 0O>tan¢ - 1]

(8.138)

2 tanrf> sin(a - fJ) [fIx - f 2x - f 3x + ~ (fly - fzy - f 3y )] The critical height of instability is then the minimum value of H attainable for a combination of <1>, a, and fJ, as well as kiY) and kyCY). It may be written as:

For rf>

=

!Cily

-

fzy - f 3y )]

(8.145)

0, function F becomes:

F(rf> = 0) = g(<1> = 0) = ~ q(rf> = 0) 0

(8.146)

352

353

By the L'Hopital rule, we have:

= 0) = lim q,-o

F(
g(
=

g' (0) q'(O)

(8.147)

This requires that the spiral be started out in the (3-zone of the slope. Thus, the second and third constraints assure the condition that the spiral traverses both zones of the slope under investigation_. (8.154)

Differentiating functions g(
= 2(Oh -

°

0)

[sin(Oh + (3) - sin(Oo + (3)]sina

(8.148)

[exp[(Oh - 0o)tanet>] sin(Oh + (3) - sin(Oo + (3)J sina/sin(a - (3) > 0

(8.149)

A close examination of the equation for the critical height as formally stated in Eq. (8.138) reveals that the value for the critical height can still be illusively positive yet physically unrealistic. This is the case when both the numerator and the denominator expressions are negative-valued. In order to rule out such a possibility, the constraint (d) is introduced. As it may seen quite redundant to use both expressions as constraints instead of just either one of these, it must be pointed out that using both can safeguard the function from assuming negative values. This is extremely important as far as the optimization process is concerned.

Accordingly:

(Oh - ( 0 ) [sin(Oh + (3) - sin(Oo + (3)] sina

(8.150)

(e) LlH

8.3.3 Physical ranges and constraints

Since the problem concerned has been associated with certain geometries, it is necessary to identify the physical constraints corresponding to the geometrical restrictions. Applicability of the analysis to physical situations is discussed in this section. A total of eleven constraints, stemming from physical considerations, can be identified. These are:

o

(8.151)

This is the only equality constraint. It is the same as Eq. (8.136), which must be satisfied for spiral failure mechanism. (b) rh sinOh - ro sinOo - L sin{3 > 0

(8.152)

This is similar to the second of the two simultaneous equations for the slope geometry, Eq. (8.118). Its inclusion in this list imposes the restriction that the spiral must terminate in the a-portion of the slope. (c) L

> 0

and

(8.153)

> 0.1

(8.155)

(8.156)

This essentially requires that the spiral not be skewed towards and along the height of the slope. The most part of it lies in the a-zone (Fig. 8.11). (f) H/L

> 0.1

(8.157)

It specifies that a spiral skewing out of proportions towards and along the top of the slope, i.e., the most part of it lies in the {3-zone (Fig. 8.11), is not acceptable. Such skewing tendencies are observable when the slope angle {3 is equal or close to the internal friction angle et>, in addition to a small a angle. The presence of these skewness usually results in critical height values that are very low. Two reasons are given to dispel such skewing spirals. The first being that for such spirals, the geometry is quite different from the ideal picture on which the derivations are based. So, results obtained may be questionable. Secondly, even if these skewed spirals are perfectly all right, the degree of hazard associated with them may not be as great as the less-skewed ones. Based on these considerations, the ratios are set as shown. Of course, they are subjected to relaxations or further restrictions, according to the judgements of the investigators. (8.158)

355

354

sin(Oo + ex) - exp[(Oh - ( 0) tancf>l sin(Oh + ex)

~-rn1-80---

> 0

(8.161)

Since: skewed along the height

f3

(8.162)

then:

l/H
sin(Oo + ex) - sin(Oh + ex) > 0

(8.163)

°

°

This can only be satisfied if 0h > hr - ex and 0 + ex > !11" or 0 + ex < !11". The result is then for the first case: 0h > 0 which is reflected in the constraint (g). For the second case, it is 0 + ex > 11" - (Oh + ex) which is constraint (i). Constraint (h), the expression for the slope height H, Eq. (8.120) is used. In order that it is positive, the following must be true:

°

°

exp[(Oh - 0o)tancf>l sin(Oh + fJ) - sin(Oo + fJ)

skewed along the top H/l
> 0

(8.164)

If the slope under investigation is composed of purely cohesive soil, then, Eq. (8.164) becomes:

> sin(Oo + fJ)

sin(Oh + fJ)

(8.165)

or Fig. 8.11. Skewed spirals.

IOh

>H

(8.159)

This constraint is for the determination of the location of the most critical spiral for a slope of given height only.

°° ><

(i) 0 0

!11"

I< !

11" -

(° 0 + fJ)

°

This assures that the spiral does not go backward. (h) H s

+ fJ -

°

This gives 0 < 11" - 2fJ - 0h and 0 < 0h' which are constraints (i) and (g). While much have been said of the constraints, the importance of the ranges of the independent variables ro' 0 and 0h must not be overlooked. Although no specific statement has been made in the derivations, the validity of these formulations can be easily seen to rest on the following implied variable ranges:

°

o< 2ex - 0h 11" - 2fJ - 0h' for purely cohesive soil only

(8.166)

ro <

00

(8.167)

7l" -

(8.160)

°

This constraint is related to the physical ranges of the spiral angles 0 and 0h' The first of these two is more general. It is derived from a consideration of the expression for the length L, Eq. (8.119). For the length to be greater than zero, the following must be true:

(8.168)

and (8.169)

356

357

However, to provide greater insight into the applicability of these formulations as well as to expedite the optimization process, better refining of these ranges is necessary. These narrowing down of the ranges can be achieved by geometric and algebraic considerations. The geometry of the model requires that the spiral be confined within the slope by the perimeter of the slope. This results in the upper and lower limit for (Jo and (Jh' respectively (Fig. 8.12):

and the implied and established limits for (Jo' It is obvious that: sin«(Jh + (3) > 0

(8.174)

or (8.175)

(8.170) or (8.171) - {3

The upper limit for (Jh can be further refined by next considering the expression for the slope height again, Eq. (8.120). To satisfy the fact that H is positive, the expression is reduced to: (8.172)

< (Jh <

7['

-

(8.176)

{3

Accounting for the above refinements, the ranges now become:

o < ro <

(8.177)

00

(8.178) with: and (8.173) ~7['

+ ¢ - ex < (Jh <

7['

-

{3

(8.179)

These restrictions should further reduce the efforts needed in the optimization. 8.4 Special spiral-slope configurations

The discussions presented in Section 8.3 pertain essentially to failure mechanisms with the ending at the toe of the slope. However, for special cases, it is possible that the spiral may terminate at some distance vertically above the toe, or even stretched horizontally away from the toe. 8.4.1 Sagging spiral

1/217+

ep< 8h + a or

8h >1/217 + ep-a lower limit for

8h

Fig. 8.12. Partial limits for 00 and 0h'

Before discussing the special cases mentioned above, it is worth noting yet another possibility, the case of a sagging spiral (Fig. 8.13). A spiral will be termed 'sagging' if its point vertically farthest away from the origin (M in Fig. 8.13) is not its endpoint E. This point of the largest vertical distance is a stationary point in the spiral: y

=

r sin(J = ro exp[«(J - (Jo) tan¢] sin(J

(8.180)

This point which corresponds to the maximum of y, is determined by solving the equation:

358

359 evaluation of the area inside two curves 1/;1(Y) and 1/;2(Y) (Section 8.2). In the case of the ordinary spiral, the external work rate is calculated formally: WI =

seismic profile

JdWlx + JdWly

for YB

<

<

Y

YE ' with boundary 1/;1(Y)

W2 = JdW2x + JdW2y

for YB < Y < Yo

~3

for Yo

.

= d ~3X + d W3y

J

J

<

Y

<

I

with boundary 1/;lY)

(8.183)

(8.184)

YE

For the case of a sagging spiral (Fig. 8.13), it is convenient to truncate the portion of 1/;1 (y) at point M, and add the remaining portion of the curve to 1/;2(Y)' such that:

constant below ground

Fig. 8.13. Sagging spiral.

dO

ro

= JdWI

for YB

~ Y ~ Ym '

with boundary 1/;1(Y)

(WI)2

=

JdWI

for YE

~ Y ~ Ym '

with boundary 1/;2(Y)

with

y

dy

(WI)I

(8.185)

WI = (WI)I

exp[(O - 00) tan¢] (tan¢ sinO + cosO) = 0

(8.181)

W2 = JdW2 W3 = dW3

f

for YB < Y < Yo ' with boundary 1/;lY) for Yo < Y < YE ' with boundary 1/;lY)

(8.186)

Therefore: The solution is:

f

YD

(8.182)

YB

From this, a criterion can be set to determine whether a given spiral is sagging or not. Clearly, we have: ordinary spiral: 0h

~

f

YE.

dW2

-

dW3

YD

(8.187)

Om

sagging spiral: 0h > Om In view of the possibility of having a sagging spiral failure surface, it is important that the analytical procedure developed for ordinary spiral failure surfaces be reexamined to determine its applicability to the case of sagging spirals. That the procedure is equally applicable to both cases is easily demonstrated. We note that the evaluation of the external work rate contributed by the soil block B-M-E-D-B (Fig. 8.13), defined by the spiral and part of the perimeter of the slope, is equivalent to the

Formally, this is the same as Eq. (8.129). Thus, the same formula may be treated for both the case of the sagging spiral and the ordinary spiral. In the latter case, the formula is applicable only when the entire length of the spiral is above the ground level. The exception taken in the last statement is justified by the constant seismic force beneath the ground level; in contrary to the variation of the seismic coefficient above the ground, with the elevation. This essentially divides the seismic coefficients into two regions (Fig. 8.13):

361

360

for h

~

8.4.2 Raised spiral

0 (8.188)

for h > 0 for h

~

0 (8.189)

for h > 0

A spiral which has its end.at an elevation above the toe of the slope is hereby refer c red to as a raised spiral. Typical slopes are shown in Fig. 8.14, (i.b), (i.c) and (ii). For such a spiral, the two simultaneous equations, Eqs. (8.117) and (8.118), governing the dimensions of the rotating block are unchanged. In fact, only minor modifications of the formulations need be made. The modified expression for 'Y/ is:

(8.192)

or for y

2: 'Y/

(8.190) for y <

where H T is the height of the raised spiral terminal. Corresponding changes in the expressions (8.137) to (8.141) for Hare:

YJ

H for y

c

=-

'Y

(8.193)

F(ro,8 0 ,8h ,HT )

2: 'Y/

(8.191) for y <

YJ

A close examination of the geometry of the earthslope and the possible combinations of relative position between the slope and the spiral failure surface reveals that there are basically four major categories of spiral failure mechanism. These are illustrated in Fig. 8.14 and are categorized as follows: (a) The spiral terminates at the toe and there is no sagging (8 h ~ 8m , YJ = rh sin8 h)· (b) The spiral ends some elevations above the ground and there is no sagging (8 h ~ 8m, 'Y/ > rh sin8 h). (c) The spiral is sagging, but its end is raised and it has no point below the ground (8h > 8m , YJ > r m sin8m)· (ii) Partially sunken: The spiral is sagging; despite the elevation of its end above the ground, part of its length is below the ground (8 h > 8m , rh sin8 h < YJ < rm sin8m )· (iii) Sunken: The spiral is sagging, ends at the toe, and the portion between the end and a certain point is completely below the ground (8 h > 8m , rh sin8 h = 'Y/ < rm sin8 m , d = 0). (iv) Stretched: The spiral is sagging, ends some horizontal distance d away from the toe, and the portion between the end and a certain point is completely grounded (8 h > 8m , d > 0, r h sin8h = YJ < r m sin8 m )·

(i.a) normal spiral

7)= rh sinh

"J"ILL-

(i) Normal:

OK 80

8h ~

T"'--

7)

7)

> rh sin 8 h

8h ~

r h sin 8 h < 7)

< rmsin 8m

8m

(i.b) normal spiral

8h

"

(ii) partially sunken spiral

(iii) sunken spiral r h sin 8 h ~ 7)

< r m sin 8 m

7)

8m d=O

(i.c) normal spiral 7) >r msin8 m

(iv) stretched spiral

7)

8h

>

8m

7)

r h sin 8 h = 7) < r m sin 8m

d >0 Fig. 8.14. Four major categories of spirals.

363

362 with

F(ro,Oo,(Jh,HT )

.

= H T 1: + c

(8.200)

sina[exp[(Oh-0O>tanep] sin(Oh+13) - sin(Oo+,B)J[e2(Oh-oO>tancf>-I]

2 tanep sin(a - ,B) [fIx - f 2x - hx + HfIy - f 2y - f 3y )] (8.194) such that:

H*::s; ':'N*

(8.195)

(8.201)

'Y

where (8.196) In addition,

(8.202)

ro' 00, 01';, H'T must satisfy the conditions: (8.197)

These modifications are sufficiently general and would include the toe spiral as a special case (HT = 0). When the raised spiral qualifies for the first category as a normal spiral, no modification is needed. For the spiral of the third category (Fig. 8.14.iii), the sunken spiral, the raised height is zero, but the spiral cuts through the ground level once. Referring to the angle corresponding to the ground level point G (Fig. 8.13) of the spiral as 0g' the external work rate (gross) can be modified as: (8.198)

Thus, the earlier equations for WI' Eqs. (8.108) and (8.109) can still be used, as long as the Eqs. (8.110) and (8.111) are modified as: (8.203) (8.204) The ground point angle Og can be found from the following equation derived from the geometry: G1

=

ro e(Oh

-

0O>tancf> . II (Og - 0O>tancf> 'nll 0 smuh - ro e S1 v g =

(8.205)

The elevation is to be carried out with the Newton's iterative root finding method: where

O(n

g

(8.199)

I

GI

f°h.d W

f -yOaor3 [sin 0 cosO tanep + sinO cos20]dO +

Og

Og

lB =

~

2

+

=

1) = O(n) -

g

dGI -d Og

= ro

G

I

/0'I

(0 -

e g .

O.\tan OJ

(8.206) . cf> (tancl> smOg + cosOgJ

(8.207)

The superscripts stand for the iteration number. For the initiation of the iteration process, or at the zero-th iteration, O~~) can be estimated by assuming that: (8.208)

365

364 Their safety ranges for convergency are:

So (8.209)

(8.218)

Since the possibility of divergency exists in Newton's method, the following bounds will assure that such possibility will be eliminated:

(8.219)

(8.210) For the spiral of the second category, the partially sunken spiral (Fig. 8. 14,ii), the raised height is non-zero and the spiral cuts through the ground level twice. Referring to the angles corresponding to the ground level points G I and Gz as 0gl and 0gZ' respectively, we have the following expressions for the functions Ilx and II associated with the gross external work rate: Y

8.4.3 Stretched spiral

When the end of a spiral is stretched a horizontal distance d away from the toe, the two simultaneous equations for geometry are changed to:

ro cosOo - rh cosOh - H/tanOl - d - L cos{3 = 0

(8.220)

r h sinOh - ro sinOo - H - L sin{3

(8.221)

=

0

Solving these equations simultaneously, we obtain: L

=

[ro sin(Oo

+ 01) - r h sin(Oh + 01) - d sinOl]/sin(OI -

[rh sin(Oh

+

(8.222)

(3)

and The angles 0gl and OgZ can be found from the following equation of geometric consideration:

H

=

(3) - r o sin(Oo

+

(3)

+ d sin{3 sinOl] sinOl/sin(OI -

(3)

with: (8.213)

o :5

d < 00;

Om < 0h <

71" -

{3

(8.223)

As before, the evaluation formula is Newton's iterative formula: gl,Z+ I) --

O(n

°

(n)

gl,Z

-

Gz/G z'

(8.214)

Accordingly, for the formulation of xE used in Eq. (8.116), modification is necessary: (8.224)

where G 2 = G{

(8.215)

The initial estimation for 0gl and OgZ are made in a similar procedure as before:

where it is noted that rh cosOh is negative because 0h is larger than 71"/2. Also, since the spiral is sagging, the formulation for WI must be modified as in Eqs. (8.198) to (8.204) for the sunken spiral. Eq. (8.137) now becomes: (8.225)

(8.216) and

where (8.217)

366

367

F( . 8 ro,

(} d)· _ d sin{3 sin(x /' .. , -. ( - + sma-I"R) ·c

7J

0' h'

sina [e(Oh - 0O>tanq, sin(f)h + (3) - sin(80 + (3)] [e2}"~ 9Jtanq, -1] 2 tant/> sin(a - (3) [fIx -

f 2x

-

f 3x + ! (fly

-

hy

-

f 3y )]

(8.226)

= ro sinf)o + L sin{3 + if

(8.231)

where His the height of the given slope and L is as defined in Eq. (8.119). In addition, H T , the elevation of the end point of the spiral above the ground, is now no longer an independent variable, but is given as: (8.232)

The critical height H* of the stretched slope is: H* :5 .: N*

(8.227)

/'

where (8.228)

H~

such that r(;, 8(;, 8j;, d* satisfy the conditions: 0,

0,

0,

With these changes introduced, the rest of the formulations for the critical height of a toe-spiral can be used as discussed in the preceding section. As for H*, it is now defined as the vertical distance between the knee of the slope and the end of the most critical spiral. It can thus be used to specify the dimension of the most critical rotating block of soil mass. In addition to the modifications to the formulations, an additional physical constraint must be recognized, namely:

(8.229)

In addition, since a stretched spiral is necessarily a sagging spiral, the range for 8h must be restricted as: (8.230)

The other ranges and constraints for the simple toe spiral still apply.

8.4.4 The most critical slip surface for a given earths/ope The determination of the critical height for a slope of given geometry and soil properties is useful in that it provides valuable criteria for the safety design of earthslope structures. However, for an existing earthslope, it would be more vital to be able to predict the most critical failure surface under a given seismic load. Investigations of the cumulative soil mass displacement of a slope during an earthquake, similar to those suggested by Newmark (1965) and Seed (1967) may be carried out. This will be presented in Chapter 10. To accommodate the analysis of the critical slip surface, in particular to determine the location of the probable failure of the existing slope, only a few modifications have to be made to the analysis presented in the preceding sections. Foremost, we have to set:

H*

(8.233)

Adding this extra constraint to the original constraints assures that the height of the potential failure surface is not higher than the physical height of the slope. 8.5 Calculated results and discussions In order that the formulations developed in this study can be readily applicable to related investigations, computer coding has been implemented. A listing of the computer program and some selected sample outputs are given in the appendices of the paper by Chen et al. (1984). An in-house optimization subroutine BIASLIB, developed by the Purdue University School of Mechanical Engineering (Root and Ragsdell, 1977) was used in the program. The program itself has been subjected to testings and debuggings, and should contains a minimum of residual errors. A total of nine cases were investigated and presented in the forthcoming. Their results are tabulated.

8.5.1 Static case The first two of these cases deal with a static situation, with gravity as the sole influence force. The stability factors associated with dead-weight induced collapse were calculated. For the static case, the loading profiles for the vertical and horizontal components are: (8.234)

368

369 , ..

TABLE 8.1 Stability factor N* = H* (oylc) for dead-weight induced failure, through non-stretched spirals. Loading profiles: k x = 0, ky = I

q, 0

5

10

{3

0

0

0

10

15

0

10

20

0

'"

20

25

0

10

20

30

0 10

N*,a

"1 L' c

30 60 90

6.43 5.25 3.83

30 60 90

9.14 6.16 4.19

( (

30 60 90 30 60 90

"1 ro c

00

0;

6.51) 5.25) ( 3.83)

6.54 4.37 3.50

11.70 7.73 10.02

0.288 0.327 0.479

2.156 1.581 1.003

0.986 1.561

9.13) 6.16) ( 4.19)

5.75 4.17 3.47

14.98 8.41 10.77

0.427 0.386 0.529

2.048 1.563 1.026

1.259

13.50 7.26 4.58 12.99 6.99 4.47

( 13.50) ( 7.26) ( 4.58) ( 12.89) ( 6.99) ( 4.47)

5.58 4.04 3.44 11.89 5.17 3.92

21.31 9.25 11.56 27.67 9.96 10.45

0.571 0.447 0.579 0.671 0.445 0.521

1.986 1.561 1.062 i.942 1.562 1.105

1.497

30 60 90 30 60 90

21.67 8.63 5.02 21.16 8.38 4.91

( 21.69) ( 8.63) ( 5.02) ( 21.14) ( 8.38) ( 4.91)

5.77 3.96 3.42 9.26 4.87 3.82

34.93 10.31 12.35 36.60 10.86 11.19

0.732 0.512 0.630 0.730 0.506 0.577

1.939 1.568 1.081 1.945 1.580 1.122

30

30 60 90 30 60 90

41.22 10.39 5.51 40.69 10.16 5.40 38.81 9.79 5.24

( ( ( ( (

41.22) 10.39) 5.50) 40.69) 10.16) 5.40) 38.64) 9.74) 5.24)

6.46 3.91 3.40 8.93 4.68 3.75 18.63 7.31 4.38

73.65 11.68 13.30 73.61 12.13 12.01 76.15 16.01 11.18

0.917 0.581 0.684 0.904 0.573 0.634 0.887 0.669 0.586

1.891 \;580 1.110 1.896 1.587 1.144 1.909 1.570 1.188

30 60 90 30 60 90 30 60 90 60 90 60 90

120.64 12.74 6.06 119.33 12.52 5.95 117.28 12.14 5.80 16.04 6.69 15.82 6.59

(119.9 ) ( 12.74) ( 6.06) (119.4 ) ( 12.52) ( 5.95) (117.4 ) ( 12.14) ( 5.80) ( 16.04) ( 6.69) ( 15.82) ( 6.59)

11.74 3.89 3.39 12.02 4.57 3.68 22.88 5.91 4.22 3.90 3.37 4.50 3.63

300.15 13.55 14.30 286.17 13.92 12.91 294.68 14.74 11.95 16.24 15.42 16.54 13.99

1.160 0.655 0.739 1.139 0.647 0.692 1.138 0.646 0.647 0.735 0.794 0.727 0.754

1.820 1.595 1.141 1.824 1.599 1.169 1.823 1.605 1.203 1.612 1.173 1.614 1.195

(

( ( (

q,

0'g

( (

60 90 10

N'

TABLE 8.1 (continued)

35

{3

'"

20

60

30

90 60

0 10 20 30

1.544 40

0 10

1.725

"1 L' c

1 c

ro

00

0;

90

( 15.47) 6.44) ( 14.78) ( 6.22)

5.64 4.09 8.41 4.95

17.20 12.86 19.09 12.20

0.724 0.711 0.737 0.672

1.617 1.222 1.623 1.258

60 90 60 90 60 90 60 90

20.94 7.42 20.73 7.32 20.40 7.19 19.78 6.98

( 20.94) 7.42) ( 20.73) ( 7.32) ( 20.40) ( 7.19) ( 19.78) ( 6.99)

3.94 3.36 4.49 3.58 5.51 4.04 7.67 4.70

20.43 16.70 20.69 15.26 21.22 15.27 22.57 13.08

0.822 0.851 0.815 0.816 0.811 0.800 0.814 0.739

1.630 1.205 1.631 1.222 1.633 1.234 1.635 1.272

60 90 60 90

28.92 8.29 28.71 8.19 28.39 8.06 27.82 7.87 26.46 7.56

( 28.91) 8.29) ( 28.71) ( 8.19) ( 28.39) ( 8.06) ( 27.82) ( 7.87) ( 26.45) ( 7.56)

4.03 3.33 4.56 3.54 5.50 3.90 7.35 4.55 12.05 5.73

27.69 17.84 27.91 16.58 28.37 15.41 29.40 15.42 31.82 13.69

0.918 0.906 0.913 0.878 0.908 0.847 0.907 0.830 0.908 0.779

1.647 1.239 1.648 1.252 1.649 1.268 1.650 1.279 1.654 1.320

20

60

30

60 90 60 90

40

N!·a-

15.47 6.44 14.78 6.22

90

1.718

N'

(

(

(

0'g

q" {3 and", in degrees; 0;, O~ and 0; in radians. a Data published by Chen et at. (1969) and Chen (1975).

kx(h)

= 0.0

(8.235)

Data for this simple loading are in abundance. The purpose here is to provide an indication on how good results from the new model agree with the existing ones. Such a comparison is possible because the present model is quite general and that the dead-weight collapse is but one special case. As shown in Tables 8.1 and 8.2, the stability factors are in good agreement with the published values (Chen, 1975), both for the nonstretched and the stretched spirals. Also listed in these two tables are the coordinates and dimensional parameters of the spirals. These parameters are useful because they provide valuable insight into the estimation of the coordinates of the most critical spiral for a slope of similar geometry and properties. The estimation corresponds to the choice of a feasible star-

370

371

ting point for the optimization process. This optimization process can be quite sensitive to the choice taken.

TABLE 8.2 Stability factor N* = hlc)H* for dead-weight induced failure, through stretched spirals. Loading profiles: k x = 0, k y = I. (3

IX

0

0

15 30 45

5.60 5.56 5.53

5

0

15 30 45 15 30 45

14.38 9.13 7.35 13.71 8.83 7.18

5

-y ,

8.5.2 Cases of constant and linear pseudo-seismic profiles

'1. L' c

- ro c

00

°h

0'g

'1. d'

( 5.53) ( 5.53) ( 5.53)

42.85 32.81 51.48

56.64 39.77 57.75

0.349 0.332 0.352

2.688 2.657 2.685

0.456 0.484 0.456

40.34 30.36 49.05

(14.38) ( 9.13) ( 7.35) (13.71) ( 8.83) ( 7.18)

10.15 6.09 4.69 18.49 8.76 5.62

45.67 15.21 9.97 50.97 16.93 10.52

0.627 0.402 0.372 0.646 0.407 0.379

2.242 2.119 1.817 2.239 2.174 1.825

1.053 1.184 1.498 1.056 1.126 1.490

5.90 1.30 0.00 7.42 2.80 0.00

N*,3

N'


c

Tables 8.3 and 8.4 present the results for the cases of constant and linear pseudoseismic profiles. The constant seismic profile is taken so that:

a

0,

TABLE 8.3 Stability factor N* 0.325, k y = 1
(3

10

0

20

0

'"

= (-yIc)H*

N'

for constant seismic horizontal component. Loading profiles: k x

N*,a

'1. L'

-y ,

C

- ro c

00

0'h

0'g

30 60 90

4.98 4.32 3.22

( 4.98) ( 4.32) ( 3.22)

12.80 5.26 3.56

16.86 9.40 13.57

0.882 0.781 0.839

2.107 1.669 1.178

1.367

30 60 90

8.83 5.63 3.65

( 8.83) ( 5.63) ( 3.65)

6.94 4.53 3.35

17.32 10.26 17.53

0.949 0.885 0.962

2.058 1.665 1.217

1.776

30

0

60 90

7.44 4.13

( 7.44) ( 4.13)

4.20 3.14

11.78 18.65

1.008 1.058

1.697 1.284

40

0

60 90

10.25 4.66

(10.25) ( 4.65)

4.01 2.93

14.19 21.31

1.142 1.166

1.743 1.349

=

a

fJ and IX in degrees; 00, 0h and 0; in radians. Data from Chen et al. (1978).

0.325

(8.237)

TABLE 8.4 Stability factor N* bJh, k y = 1

= {-ylc)H*

'"

b~

N'

0

30 60 90

0.0388 0.0468 0.0635

5.24 4.25 3.13

20

0

30 60 90

0.0221 0.0362 0.0562

30

0

60 90

40

0

60 90


(3

10

for linear seismic horizontal profile. Loading profiles: k x

N"b

'1. L'

-y ,

= 0.225 +

c

- ro c

00

°h

0'g

'1. d'

(5.16) (4.27) (3.15)

10.96 5.50 3.59

18.26 10.91 16.39

0.941 0.856 0.888

2.005 1.635 1.168

1.478

0.0

9.09 5.50 3.54

(9.06) (5.53) (3.56)

8.25 4.83 3.35

22.49 12.29 18.95

1.064 0.970 0.990

1.998 1.638 1.222

1.840

0.0

0.0275 0.0500

7.22 3.98

(7.28) (4.00)

4.55 3.12

14.67 19.47

1.101 1.080

1.673 1.292

0.0201 0.0447

9.87 4.47

(9.97) (4.47)

4.44 2.88

18.65 18.42

1.243 1.166

1.720 1.369

c

0, 0h and 0; in radians. Imposing four equi-thickness zones of different coefficients (0.25, 0.30, 0.35, 0.40) indiscriminate of slope heights in effect results in the variation of seismic profile with the size of the slope. b Data from Chen et al. (l978).

(8.236)

Again, the stability factors obtained from the present analysis are in good agreement with the published values (Chen et aI., 1978). It should be noted that for the linear profile, the profile itself is not the same for each slope. Instead, shorter slopes have steeper profiles as tall slopes have more gentle ones. This is due to the fact that the first published data (Chen et aI., 1978) for a linear profile were obtained by imposing on the slopes a maximum of four zones of equal thickness. These zones are of different seismic coefficient values

fJ and IX in degrees; 0 0; and 0; in radians. Data from Chen (1975).


1.0

372 (kx1 = 0.25, k X2 = 0.30, k X3 = 0.35, k X4 = 0.40) to approximate the original profile of linear variation (Fig. 8.2b). Such a zone-restricting technique, which distributes the seismic load linearly through the slope height, tends to subject shorter slopes to heavier seismic loadings. While it is necessary to ascertain the validity of the philosophy underlying this technique, we dispense with the philosophical arguments and stilI use the published data to check the results of the present study. For this comparative study, we first identify the equation of the seismic profile for the slope configuration for a published critical height. Thus, if the seismic coefficients for the slope are of the form:

373 kx(h)

= 0.0057 + 0.0084

h - 0.000076 h2

+ 0.00000032 h3

(8.244)

which is ,sufficiently accurateJQr the slope configurations studied. The data obtained are shown to be larger (from 1.5 to 2.5 times) than those for the constant profile of 0.325 in Table 8.3. On the other hand, they are less than those for the static case, as expected.

374

375 h

TABLE 8.6 Stability factor N* = (ylc)H* for the general profile oriented at a direction of arctan (0.1) with the horizon. Loading profiles: k x = 0.0057 + 0.0084 h - 0.000076 h2 + 0.00000032 h3 • ky = 1. + 0.1 k x

300

250

c/>

k x (hl=0.0057+0.00B4 h-0.000076 h

N*

1J/

3

o 30

0

90 150

o 30

20

90 20 30 90

100

50

'!- L' c

Y , - ro c

8'0

8;'

8'g

d = '!- d'

1.019 0.495

d

= 32.73

h

= 0.773

c

or

Ii = '!c

Hi:

2

+0.00000032 h -0.00000000041 h 4

200

{3

o 60

40

k x (h)-o,0057+0.00B4 h-O'o00076 h2 +0.00000032 h3

90 40 60 90

~i::.0~-0"'T2-L-0'.4--0'.6--""'0.-B--1.r-0--12r---------kx

C/>. {3 and

Fig. 8.15. The general average seismic profile (horizontal).

1J/

6.10 5.15 3.74

6.97 35.20 3.53

11.98 42.34 10.33

0.349 0.362 0.505

2.123 2.646 1.008

23.51 5.27 9.27 4.99

7.93 3.43 100.8 4.54

47.94 14.57 136.42 11.74

0.982 0.732 1.168 0.632

1.905 1.112 1.891 1.202

20.45 7.60 7.61 6.81

4.51 3.36 76.08 6.41

24.84 22.34 79.81 15.15

1.032 0.982 1.195 0.856

1.656 1.238 1.821 1.350

in degrees; 80, 8;' and

Ii = 0.0021

0; in radians.

Tables 8.6 and 8.7 exhibit data corresponding, respectively, to the following loading profiles: ky(h)

=

1.0 + O.lkAh)

(8.245)

kx(h)

=

0.0057 + 0.0084 h - 0.000076 h2 + 0.00000032 h 3

(8.246)

and

c/>

(8.247) kx(h)

TABLE 8.7 Stability factor N* = (y/c)H* for the general profile oriented at a direction of arctan (0.5) with the horizon. Loading profiles: k x = 0.0057 + 0.0084 h - 0.‫סס‬OO76 h2 + 0.00000032 h 3• ky = I. + 0.5 k x

= 0.0057 + 0.0084 h

- 0.000076 h 2

+

0.00000032 h 3

0

N*

1J/

o 30 90

(8.248) 20

These calculations were made to understand the effect of a weak vertical seismic component on the critical height. The results indicate that while there are decreases in the values for an increase of the vertical component, these decreases are generally not too significant. This insignificant effect can be observed for a vertical component as strong as half the magnitude of the horizontal one. There are also relatively no significant change in the spiral coordinates. These small changes may be due to the fact that the two component profiles were assumed to be similar. Therefore, any statements extracted from these two tables may not be general enough to warrant

{3

o 30 90 20 30 90

40

o 60 90 40 60 90

C/>. {3 and

1J/

'!- L' c

'!c

ro

80

8;'

0'g

d='!-cforh='!-H;'

1.018 0.502

d=

28.47

Ii

0.011

6.02 5.14 3.70

6.89 31.13 3.50

11.82 37.74 10.33

0.349 0.352 0.508

2.124 2.639 1.006

22.91 5.20 9.02 4.92

7.49 3.38 98.15 4.48

45.53 14.40 122.05 11.61

0.970 0.731 1.129 0.633

1.911 1.111 1.920 1.202

19.97 7.48 7.53 6.69

4.32 3.31 75.46 6.33

23.73 21.91 78.23 14.96

1.024 0.981 1.192 0.858

1.658 1.238 1.823 1.351

in degrees; 80, 8;' and 8; in radians.

c

=

c

376

377

TABLE 8.8 Location of the most critical slip surface for a slope of 30 feet in height. Loading profiles: kx 0.0057 + 0.0084 h - 0.000076 h2 + 0.00000032 h 3, ky = I. Height of the slope: H = 30 ft. (3

cf>

IH'

C<

c

IH y

Ie c

1'0 c

00

oJ:

c

0

0

30 90

3.10 3.20

26.90 26.80

27.87 3.73

21.74 11.30

0.602 0.643

2.353 1.081

20

0

30 90 30 90

17.17 4.32 4.89 4.03

12.83 25.68 25.11 25.97

7.05 3.41 49.09 4.99

32.34 16.99 43.78 11.20

0.943 0.872 0.966 0.741

1.960 1.162 2.046 1.308

60 90 60 90

16.22 5.86 4.65 5.05

13.78 24.14 25.35 24.95

4.30 3.19 46.24 8.10

19.85 24.79 34.82 13.97

1.055 1.097 1.084 0.959

1.686 1.288 1.894 1.487

20 40

0 40

0*g

0,

cf>, (3 and Ci in degrees; 0 OJ: and 0; in radians. * Values calculated for a (cl-y) ratio of I.

=

the omission of the vertical component profile in future works, as they might be quite different in real life. More detailed investigations concerning the vertical loading should be made in the future when such profiles are available. Tables 8.8 and 8.9 are the tabulations of the locations of the most critical slip surface in slopes of different configurations. The height of the slope was given as 30 feet in Table 8.8 and 50 feet in Table 8.9. Under loadings specified by Eqs. (8.243) and (8.244), the soil near the top of the slope experiences the worst conditions and is the most likely place for a spiral to develop. The two tables reflect this fact and also the reductions in spiral heights as a result of the more intense loading of a taller slope. In all these tables, some more or less common features can be observed. They are: 1. Stretched spirals are present only in slopes with low angle of internal friction, cp, and small slope angle a; 2. Sagging spirals are also found only when cp and a are small, but their ranges are usually larger than those of the stretched ones; 3. Partially sunken spirals have not been studied completely so far. A close examination reveals a relatively general pattern for the variations of the spiral coordinates for similar configurations and loading conditions. This may be helpful in choosing the feasible starting points for future analyses with the computer coding. 8.6 Concluding remarks

TABLE 8.9 . Location of the most critical slip surface for a slope of 50 feet in height. Loading profiles: kx = 0.0057 + 0.0084 h - 0.000076 h2 + 0.00000032 h 3, ky = I. Height of the slope: H = 50 ft.

(3

Ci

IH* c

IH y

Ie c

1'0 c

00

oJ:

c

0

0

30 90

2.03 2.96

47.97 47.04

21.49 3.82

14.92 11.13

0.491 0.693

2.489 1.131

20

0

30 90 30 90

11.18 3.90 3.44 3.60

38.82 46.10 46.56 46.40

6.78 3.39 34.03 5.42

21.06 17.70 26.29 10.74

0.936 0.931 0.878 0.793

2.023 1.193 2.111 1.386

60 90 60 90

12.22 5.09 3.49 2.95

37.78 44.91 46.51 47.05

4.12 3.06 34.77 29.46

16.09 26.95 22.26 26.06

1.107 1.159 1.020 1.118

1.719 1.317 1.933 1.812

20 40

0 40

cf>. (3 and

* Values

Ci in degrees; 00, OJ: and 0; in radians. calculated for a (cl-y) ratio of I.

0*g

This chapter discusses and develops a more general and consistent mathematical model for the analysis of the instability of earthslopes under seismic loads. By recognizing the disadvantages of existing models and with a better understanding of the nature of an earthquake's influences, such a more involved model has been successfully formulated. The treatment of several possibilities has been categorized such that a better insight into the influence of the slope geometry on the spiral failure mechanism can be gained. By looking into the possible changes seismic loads may have on the shape of the most critical slip surface, far-reaching conclusions may be drawn. For instance, in the derivation of the slip surface equation, it could be observed that the inhomogeneity and anisotropy in the cohesion of the soil has no effect on the critical shape. In fact, the only controlling factor on the shape of the slip surface is the internal friction angle cp. Most important of all is the flexibility inherent in the present formulation to account for the variations of the seismic forces along the height. In considering both vertical and horizontal components for the seismic loads, not only is the variation of magnitude of the seismic force with height, but also the variation of the direction of the seismic force accountable. Such loading profiles, if interpreted wisely and

379

378 "

, ,

with c~re: can also be used to allow the ~ariation of the spe~ifk weight of the soil along the height. To facilitate the adaption of the present model to future analyses related to' seismic-infirmed earthslopes, computer coding of the tedious formulations has been implemented, and results of fairly simple cases have been studied. These data constitute two main functions: to provide indicators of agreement between results of the present model and previous established models, and to provide further information relating to the seismic loadings that were not available previously. Of importance are the tables of the spiral coordinates for the different cases studied. They not only show the general patterns of the variation of the coordinates, but also pro-. vide good indications for estimating the initial coordinates for the spiral optimization process. Thus, the iterative algorithm of the optimization subroutine (used in the computer program) can be initiated in the right direction, resulting in the expedited analysis, cutting down on run time and cost, as well as preventing convergence onto local minimum. The ability to predict and estimate the relative location of the most critical failure surface in an earthslope of given property, geometry, and height is even more significant. It allows for the cumulative displacement analysis proposed by Newmark (1965) and Seed (1967) to be developed. All in all, the computer model developed in this chapter is a step forward in recent efforts to understand better the seismic effects on the stability of earthslopes. It is, nevertheless, quite idealized with respect to the actual time variation feature of the seismic forces, and the changes in soil properties. The changes in properties are results of compaction, pore water .pressure variation, liquefaction, seepage forces, nonlinear post-elastic responses, hysteretic strain behaviors, etc., caused by the loadings. In the following chapter, the present formulation will be extended to account for the inhomogeneity and anisotropy of the soil cohesion. It is obvious from the present study that the varying cohesion has no effect on the spiral equation, and it does not enter in the equations of external work rate. Thus, all is needed is a modification of the internal energy dissipation rate equation to account for two cohesion profiles, one with respect to the elevation, the other with respect to the orientation. Such modification is similar to that obtained previously by Chen (1975) and will not induce much changes in the formulations or the computer coding. In Chapter 10, an extensive computer program to incorporate the hazards of a slope at different time intervals during the occurrence of an earthquake will be attempted. Such coding will include the examination of the seismic profiles at different intervals, to see if displacements along a well defined slip surface are inflicted or not. The displacements at the end of each interval will be integrated to determine the total displacement after the quake. This is essentially the new approach to the assessment of seismic slope hazards proposed by Newmark (1965) and Seed (1967).

However, no specific method of identifying the slip surface was mentioned in these earlier articles. The present study has provided part of the answer. The method developed for identifying the. most critical slip surface for a slope of given height (such as those in Tables 8.8 and 8.9) can be used to determine the progressive development and movement of the failure surface in the course of the quake. References Ambraseys, N.N. and Sarma, S.K., 1967. The response of earth dams to strong earthquakes. Geotechnique, 17: 181- 213. Chen, W.F., 1975. Limit Analysis and Soil Plasticity. Elsevier, Amsterdam, The Netherlands, 638 pp. Chen, W.F., 1977. Mechanics of slope failure and landslides. Prec. Advisory Meeting on Earthquake Engineering and Landslides, U.S-Republic of China Cooperative Science Program, Taipei, Taiwan, August 29-Sept. 2, pp. 219-232. Chen, W.F., 1980. Plasticity in soil mechanics and landslides. J. Eng. Mech. Div., ASCE, 106(EM3): 443-464. Chen, W.F. and Koh, S.L., 1978. Earthquake-induced landslide problems. Proc. Central Am. Conf. on Earthquake Eng., San Salvador, EI Salvador, January 9-14, Envo, PA, pp. 665 -685. Chen, W.F., Giger, M. and Fang, H.Y., 1969. On the limit analysis of stability of slopes, soil and foundations. Jpn. Soc. Soil Mech. Found. Eng., 9(4): 23-32. Chen, W.F., Chang, C.J. and Yao, J.T.P., 1978. Limit analysis of earthquake-induced slope failure. Prec. 15th Annual Meeting, Soc. Eng. ScL, R.L. Sierakowski (Ed.), December 4-6, Gainesville, Florida, pp. 533 - 538. Chen, W.F., Chan, S.W. and Koh, S.L., 1984. Upper bound limit analysis of the stability of a seismicinfirmed earthslope. In: A.S. Balasubramaniam, S. Chandra and D.T. Bergado (Editors), Proc. Symp. Geotechnical Aspects of Mass and Material Transportation, Bangkok, 1984. Balkema Publishers, Rotterdam, The Netherlands, 1987, pp. 373 -428. Koh, S.L. and Chen, W.F., 1978. The prevention and control of earthquakes. Prec. U.S.-S.E. Asia Symp. on Eng. for Nat'J. Hazards Protection, Manila, Philippines, Sept. 1977; Univ. Illinois Press. Newmark, N.M., 1965. Effects of earthquakes of dams and embankments. Geotechnique, 15(2): 137 -160. Prater, E.G., 1979. Yield acceleration for seismic stability of slopes. J. Geotech. Eng. Div., ASCE, 105(GT5): 682 - 687. Root, R.R. and Ragsdell, K.M., 1977. BIAS: A nonlinear programming code in Fortran IV. User's Manual, Design Group, School of Mech. Eng., Purdue Univ., West Lafayette, IN. Seed, H.B., 1967. Slope stability during earthquakes. J. Soil Mech. Found. Div., ASCE, 93(SM4): 299-323. Seed, H.B. and Martin, G.R., 1966. The seismic coefficient in earth dam design. J. Soil Mech. Found. Div., ASCE, 92(SM3): 25-58.

381

Chapter 9

SEISMIC STABILITY OF SLOPES IN NONHOMOGENEOUS, ANISOTROPIC SOILS AND GENERAL DISCUSSIONS 9.1 Introduction

As mentioned in the preceding chapter, the conventional method for evaluating the effect of an earthquake load on the stability of a slope uses the so-called 'pseudo-static method'. In this method, the inertia force is treated as an equivalent concentrated horizontal force, i.e., the pseudo-static force, at some critical point (usually the center of gravity) of the critical sliding mass. The inadequacies of the method for slope stability analysis have been discussed in various papers by many authors (e.g., Chen, 1982). Despite these criticisms, the pseudo-static method continues to be used by consulting geotechnical engineers because it is required by the building codes, it is easier and less costly to apply, and satisfactory results have been obtained since 1933. This method will continue to be popular until an alternative method can be shown to be a more reasonable approach. In the first part of this chapter, we shall extend the upper bound technique of limit analysis for the seismic stability of slopes to account for the nonhomogeneity and anisotropy of the soil cohesion. The Mohr-Coulomb yield condition is described by two parameters, cohesion c and friction angle c/>. Here, as in Chapter 7, we shall assume that only the parameter c is nonhomogeneous and anisotropic. The friction angle c/> is assumed to remain homogeneous and isotropic throughout the calculations, i.e., a constant value for a given type of slope. The term 'nonhomogeneity' of cohesion implies a variation of c with respect to depth z and the term 'anisotropy' of cohesion implies a variation of c with respect to direction at a particular point. It has been shown from the preceding chapter that the variation of cohesion has no effect on the spiral mechanism, and it will not enter in the equation for the external rate of work. Thus, all we need here is a modification of the equation for the rate of internal energy dissipation. For simplicity, we assume in this chapter a uniform distribution of lateral acceleration with respect to depth, and we use the Mohr-Coulomb failure criterion with constant c/> but variable c. In the following, a brief description of the two terms 'nonhomogeneity' and 'anisotropy' follows a rigorous analysis capable of dealing with a general slope. Extensive numerical studies of slopes made using this analysis are then presented. Details of this development are given in the paper by Chen and Sawada (1983).

382

383

In the later part of this chapter, the assessment of seismic stability of slopes will be discussed. This discussion includes the mechanics of earthquake-induced landslides, the applicability of some dynamic properties of soils to earthquake loadings, and the selection of appropriate analysis methods. The salient features of the most widely used methods proposed by Newmark (1965) and Seed (1966) for the analysis and design of the slides of dams and embankments induced by earthquake motions are briefly introduced. This introduction provides the necessary background information for the assessment of seismic displacement of slopes described in Chapter

Cohesion,

C

10. 9.2 Log-spiral failure mechanism for a nonhomogeneous and anistropic slope

The term 'nonhomogeneous soil' implies only the cohesion strength, c, which is assumed to vary linearly with depth (Fig. 9.lc). Figure 9.2 summarizes diagrammatically some of the simple cutting in normally consolidated clays with several forms of cohesion strength distribution being considered previously by several investigators. (Taylor, 1948; Gibson and Mogenstern, 1962; Odenstad, 1963; Reddy and Srinivasan, 1967). The term 'anisotropic soil' implies here the variation of the cohesion strength, c,

C

v

M-Anlsotropy

' ' \ (cv>ctl

,

Depth,z (a) Taylor 1948 <0=/3)

(b) Gibson and Morgenstern 1962

(e) Odenstad 1963 (Q./3 )

(d) Reddy and Srinivasan 1967

(a=/l)

(a=/l)

Fig. 9.2. Some types of linear variations of cohesion with depth.

with direction at a particular point. The anisotropy with respect to cohesion strength, C of the soil has been studied by several investigators. (Taylor, 1948; Odenstad, 1963; Lo, 1965; Reddy and Srinivasan, 1967). It is found that the variation of cohesion strength, c, with direction approximates to the curve shown in Fig. 9.1b. In this section, the variation of the apparent friction angle ¢ is not considered with respect to either the nonhomogeneity or the anisotropy. In the following we assume that the cohesion strength Cj' with its major principal stress inclined at an angle i with the vertical direction, is given by the same equation as Eq. 7.1: (9.1)

lb) Anisotropy with Direction

o N

(a) Log-spiral Failure Mechainlsm (clNonhamogenelty with Depth

Fig. 9.1. Failure mechanism for a nonhomogeneous and anisotropic slope.

where ch and Cv are the cohesion strength in the horizontal and vertical directions, respectively. These cohesion strengths may be termed as 'principal cohesion strengths' (Lo, 1965). For example, the vertical cohesion strength, Cv can be obtained by taking vertical soil samples at any position and being investigated with the major principal stress applied in the same direction. The ratio of the principal cohesion strengths ch lev' denoted by )(, is assumed to be the same at all points in the medium. ci = c h = Cv (or)( = 1.0) is an isotropic material. In Fig. 9.1a, the angle m is the angle between the failure plane and the plane which is normal to the direction of the major principle cohesion strength kept at an angle i with the vertical direction. This angle, according to Lo's test (1965), is found to be independent of the angle of rotation of the major principal stress. The geometrical relationsL/ro, H/ro and N/ro in Fig. 9.1a can be expressed in the forms:

384

385

L

D

H

'0

'0

cosOO - cosO h exp[(Oh - O~tanc/>l - - - - (a l cotfJ

H

+ a2 cotn!)

sinOh exp[(Oh - O~tanc/>l - sinOo

(9.2)

where aI' a2 , D and N are defined in Fig. 9.1 a. The rate of external work done by t~e regi<;lll A-A' -C-B '.-B-A can be obtained from the algebraic summation of WI - W2 - W3 - W4 - Ws' Herein, WI' W2, W3 , W4 and Ws represent the rates of external work done by the soil weight in the region O-A-B O-B'-B O-C-B', O-A' -C an O-A-A', respectively. " These expressions are:

'Y'~O [ ~

+

1 [(3 tanc/> cosOh 9 tan2c/»

3 tanc/> cosOo - sinOoJJ

· = 'Y'o" W 2

3(1

30 l '

= 'Y'~O '0

:y,~O

Os

(9.9)

Similarly, the rate of external work done by the inertia force on the soil weight can be found by a simple summation of W6 - W7 - Wg - W9 - WIO' Herein, W6 , W7 , Wg, W9 and WIO represent the rates of of external work done by the inertia force due to sliding soil weight in regions O-A-B, O-B' -B, O-C-B' , O-A' -C and O-A-A' , respectively. These expressions are as follows:

+ sinOh) exp[3(Oh - 0o)tanC/>l

°

1

L{2 cosOo - -L} = 'Y'0002 3

6 smOo '0

0t ~ sinOh[{2 cosOh exp[2(Oh - O~tanc/>l

exp[(Oh - Oo)tanC/>l] ~

(9.4)

=

'Y'6

(9.3)

N

WI

Ws =

(9.5)

(9.6)

(9.7)

a2 H . D a2 H ) . 2 + ( -D cotn! smO h + - - smO h cot n! + 2 - cosOh + - - cotn! cosO '0 2 '0 ,o 2 ,0 h

(9.8)

(9.14)

387

386 - in the region Om and 0h

The rate of external work due to the surcharge boundary pressure p and· its associated inertia force are found to be:

(Cj)n

- due to the surcharge load pL

{L

2 P ron -

ro

cosOo - -

L} =

2ro

p

2 ronf p

{I + C~ ,,') COS iJ c {n, + n2N~ronl (sinO exp[(O 2

=

- sinOm exp[(Om -

(9.15)

(0)

(0)

tan¢]

tan¢]) }

where x = ch/cv' i = 0+ , = - (11"/2 + ¢ - m) and no, n 1 and n2 are defined in Fig. 9.1c. After integration and some simplifications, Eq. (9.17) reduces to:

- due to the inertia force of the surcharge load pL (9.16)

where the factor kh = Xkh is the seismic coefficient for the surcharge load pL. The inertia response of the surcharge load pL caused by the earthquake is represented by khPL. The total rates of internal energy dissipation along the discontinuous logspiral failure surface AB is found by multiplying the differential area r dO/cos¢ by Cj times the discontinuity in velocity, V cos¢, across the surface anti integrating over the whole surface AB. Since the layered clays possess different values of Cj' the integration is therefore carried out into two parts:

rdO J Cj (V COS¢) COS¢ =f 8h

8m

80

(Cj)I rOVO exp[2(0 - (0) tan¢]dO

(9.21)

in which Cis the cohesion of soil at the level of the toe (see Fig. 9.1), and: (9.22) The functions Ql' and Q2 and Q3 are: Q,

=:

+ Q2 =

80

f

8h

(C)n rO Vo exp[2(0 - (0) tan¢]dO

(9.17)

eXP(2~0 tan¢) IB+ C~ x)E I:~ + eXP(2:0'ran¢) IB+ C~ x)E I:: 1 - no

.

(~)eXP(300 tan
8m

-x)

The log-spiral angle, Om at the level of the toe, and the logspiral angle, 0h at the exit of the spiral, are related from the geometric configuration shown in Fig. 9.1a as

sinO m exp[Om tan¢]

=

(9.18)

sinOh exp[Oh tan¢]

+ ( -1 x Q3

= (N) -

Referring to Eq. (9.1) and the geometry of Fig. 9.1, ed as:

(c~I

ro

and (Cj)n can be express-

+

- in the region 00 and Om (Cj)I

(9.20)

= (1 + ~ COS2 C (no + 1 - no (sinO exp[(O - 00> tan¢] - sin ( 0») x H/ro

i)

(9.19)

IA

(9.23)

- B sinOo exp[Oo tan¢]

C - E sinOo exp[Oo tan¢] 188~

n2 - nl

IA -

(9.24)

B sinO m exp[Om tan¢]

exp(300 tan
C~ X)c -

E sinOm exp[Om tan¢ll::

(9.25)

in which: A

B

(3 tan¢ sinO - cosO) exp[30 tan¢] 1 + 9 tan2

= exp[28 tan¢l 2 tan¢

(9.26)

(9.27)

388

389

= exp[30 tanepl [3 tanep sinO - cosO 2 1 + 9 tan 2ep

C

+

cos2

tanep sin30 - cos30 { 6(1 + tan2
ty. Details of the program are given elsewhere by Chen and Sawada (1982). The optimization technique reported by Sigel (1975) was used to minimize the function of Eq. (9.32) without calculating- the derivatives. The results are summarized in Tables 9.1 to 9.7 and Fig. 9.3. Some of the solutions are compared in Tables 9.1 and 9.2 with the existing limit equilibrium solutions. Table 9.1 compares the critical heights, H c ' obtained by the limit equilibrium method with those obtained by the present limit analysis for anistropic slopes with constant shear strength (Lo, 1965). Here, as in Lo's work, the value of m is taken to be 55° and the values of friction angle and acceleration k h are put nearly equal to zero so that the statical log-spiral failure surface reduces to the circular one. Generally speaking, both results are in a good agreement. Table 9.2 compares the cases of anisotropic slopes with shear strength increasing linearly with depth (Fig. 9.2b). In this way, the critical heights, H c' can be compared with those obtained previously by Lo (1965) using the limit equilibrium method. A good agreement is again observed.

cosO - 3 tanO sinO) + ----------:-2(1 + 9 tan2
. {sinO + 3 tan cosO - ------'---:--sin30 + tan COS30)] - sm2 2 2(1 + 9 tan
(9.28)

= exp[20 tanl [_1_

E

2

2 tan

+ cos2<1> (tan cos20 + sin20) - sin2<1> (tan sin20 - COS20)] 2(1 + tan2ep)

(9.29) . I

By equating the total rates of external work, Eqs. (9.5) to (9.16) to the total rate of internal energy dissipation, Eq. (9.21) and assuming kh = X k h , we obtain: kh

=

TABLE 9.1 Comparison of critical height: He for anisotropic soil with constant shear strength Curved failure surface

F(OO,Oh,D/ro) c(QI

+ Q2 +

Q3) - 'Y rO(OI - 02 - 03 - 04 - 05) -pfp

'Y rO(06 -

07 - 08 - 09 - 010)

+ xpfq

Slope angle (degree) Anisotropy factor 01=(3 x

aF = 0, -aF - - 0 aD/ro

, aeh

Limit analysis Log-spiral, (2)

Ratio (1)/(2)

(9.30) 90

1.0 0.9 0.8 0.7 0.6 0.5

95.75

110.57

0.870

70

1.0 0.9 0.8 0.7 0.6 0.5

119.75 118.00 116.25 114.50 112.25 110.25

136.62 132.36 128.14 123.89 119.12 114.92

0.877 0.892 0.907 0.924 0.942 0.960

50

1.0 0.9 0.8 0.7 0.6 0.5

142.00 138.50 133.75 129.75 127.25 121.25

142.00 137.50 129.40 125.50 120.75 116.50

1.000 1.007 1.034 1.054 1.054 1.041

The function F(OO,Oh,D/ro) has a minimum value and, thus, indicates a least upper bound, when 00,Oh' and D/ro satisfy the following conditions:

o

Limit equilibrium ¢ circle", (I)

(9.31)

Thus, the yield acceleration factor, k c is denoted as: (9.32) 9.3 Numerical results and discussions

9.3.1 Calculated results Extensive numerical results for the yield acceleration factor k c were reported by Chen and Sawada (1983) using a computer program developed at Purdue Universi-

" Lo (1965).

.(

390

391

TABLE 9.2 Comparison of critical height: He for anisotropic soil with shear strength increasing linearly with depth Curved failure surface Slope angle (degree) Anisotropy factor O!

= {3

Limit equilibrium circle", (1)

<1>

}{

Limit analysis Log-spiral, (2)

Ratio (1)/(2)

90

1.0 0.9 0.8 0.7 0.6 0.5

50.00 50.00 50.00 50.00 50.00 50.00

60.97 60.45 60.30 59.40 58.85 58.35

0.820 0.827 0.829 0.842 0.850 0.857

70

1.0 0.9 0.8 0.7 0.6 0.5

69.25 68.25 67.25 66.25 65.25 62.50

72.10 72.06 70.77 70.40 70.20 68.68

0.961 0.947 0.950 0.941 0.930 0.910

50

1.0 0.9 0.8 0.7 0.6 0.5

94.50 91.50 89.00 86.25 82.75 79.25

103.70 100.50 98.00 95.40 92.40 89.50

0.911 0.911 0.908 0.904 0.896 0.886

30

1.0 0.9 0.8 0.7 0.6 0.5

137.50

135.50

1.015

125.00

127.00

0.984

Figure 9.3 illustrates graphically the kc-values of anisotropy case normalized by the corresponding kc-value of isotropy case with C-anisotropy type (or x > 1) and M-anisotropy type (or x < ·1} as shown in Fig: 9.1b (Lo, 1965). Some typical results for the yield acceleration k c corresponding to the general case of nonhomogeneous and anisotropic soil are tabulated in Tables 9.3 to 9.7. Table 9.3 gives the yield acceleration factor k c with constant stability number N s and surcharge p. The others consider (see Fig. 9.2) Taylor's model (1948), Gibson and Mogenstern's model (1962), Odenstad's model (1963), and Reddy and Srinivasan's model (1967), in which the angle m between the failure plane and the major principal plane as shown in Fig. 9.1a is taken to be 71'/4 + cPl2. 9.3.2 General remarks A practical approach to obtain effectively the stability solutions of nonhomogeneous and anisotropic slopes under an earthquake loading has been established (Fig. 9.3). Herein, the upper-bound techniques of limit analysis have been applied to obtain the yield acceleration factor for different cohesion strength distributions. The formulation of the problem is seen to be rather straightforward and simple. The numerical results are found to be in a good agreement with the existing limit equilibrium solutions. It can therefore be concluded that the upperbound techniques of limit analysis provide a convenient and effective method for the analysis for seismic stability of nonhomogeneous and anisotropic slopes.

1.1

104.50

114.00

0.917

kg- Anisotropy kC-lsotropy

" Lo (1965). TABLE 9.3 Yield acceleration factor k e with constant stability number N s and surcharge p Anisotropy factor }{

1.0 0.9 0.8 0.7 0.6 0.5

p = 5.75 kPa.

p=o k{,/kh = 0 0.477 0.457 0.437 0.416 0.396 0.377

p = 120 psf k{,/k h = 0

p = 120 psf k{,/kh = 0.5

fan~fg:ropy

1.2

K-l.3 _ _ _ _ _ _ _ _ _- - - - - . K - l . 2

}

C-Anlsotropy

K -1.1

1,0 -I==~------------===.!.:.!. ---------------,K-0.9 0.9

mO.7

0.8

K-0.5

0.7

a =30

0

H -50 feet

0.455 0.436 0.417 0.399 0.380 0.361

0.450 0.431 0.413 0.394 0.376 0.357

o 5 10 15 20 25 30 35 40 m- (45 0 ) (47.5 0 )150°) (52.5°)(55°)(57.5°) (60°) (62.SOX65°)

ep Fig. 9.3. Variation of

Y -120

P -0 K'h- O

(degree)

kc-anisotrop/kc-isOlrOPY

with

<1>.

@ _30 0

Ib./cu.tt.

I I

393

392 TABLE 9.4 Anisotropic but homogeneous soil: Taylor model (1948) with constant c, (4) = constant, 1.0, nl = 1.0, nz = 1.0 Friction angle 4> (0)

C-anisotropy type

\.,.~;'O (m = 45°)

M-anisotropy type

Anisotropy factor x 1.3

1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5

Ci

= (3),

no

=

Yield acceleration factor: k c Ci

= 30°

0.290 0.275 0.260 0.246 0.231 0.216 0.201 0.187 0.172

Ci

= 50°

0.285 0.270 0.256 0.241 0.227 0.212 0.198 0.184 0.169

Ci

= 70°

0.282 0.268 0.253 0.239 0.225 0.211 0.196 0.182 0.168

Ci

= 90°

0.280 0.266 0.252 0.237 0.223 0.209 0.195 0.181 0.167

_______________ M _____________________________________________________________________________________________________________________

0.356 0.354 0.352 1.3 0.361 0.342 0.337 1.2 0.346 0.339 0.327 0.323 0.331 0.325 1.1 5.0 0.316 0.312 0.310 0.309 1.0 0.294 (m = 47.5°) 0.297 0.296 0.9 0.301 0.283 0.281 0.280 0.286 0.8 M-anisotropy 0.268 0.267 0.266 0.7 0.271 type 0.252 0.251 0.256 0.253 0.6 0.5 0.240 0.239 0.238 0.237 -----------------------------------------------------------------------------------------------------------------------------------0.449 C-anisotropy 0.459 0.454 0.451 1.3 type 0.434 1.2 0.435 0.433 0.438 0.422 0.420 0.418 1.1 0.432 10.0 0.407 0.405 0.403 1.0 0.411 (m = 50°) 0.391 0.389 0.388 0.9 0.394 0.374 0.373 0.8 0.378 0.375 M-anisotropy 0.359 0.358 0.357 0.7 0.362 type 0.343 0.342 0.346 0.344 0.6 0.327 0.326 0.5 0.329 0.328 -------------------------------------------------------------------------------------------------------------------------------------0.721 C-anisotropy 0.725 1.3 0.739 0.729 type 0.707 0.704 0.701 1.2 0.717 0.683 0.680 0.695 0.686 1.1 0.660 20.0 0.666 0.662 1.0 0.673 0.639 (m = 60°) 0.645 0.641 0.9 0.651 0.619 0.623 0.621 0.8 0.629 0.598 M-anisotropy 0.602 0.600 0.7 0.607 0.578 type 0.6 0.585 0.581 0.579 0.557 0.560 0.558 0.5 0.562 C-anisotropy type

Before we proceed to describe the methods for a valid assessment of the seismic stability of slopes in the following chapter, it is necessary to mention here that the success in developing a practical method for a valid assessment of the hazard of slope failure and landslides requires the considerations of the structure of the slope and its water saturation as well as the physical properties of the soils comprising the slope. Supplemental data on the local earthquake activity, climatic and hydrologic conditions of the region, and the history of local landslides should also be collected. Estimates of the slope stability must be based on the results of these investigations (Chen, 1977). The effects of these factors on the stability of slopes from the viewpoint of soil mechanics will be described in the forthcoming. TABLE 9.5 Anisotropic and nonhomogeneous soil: Gibson and Morgenstern Model (1962) for c, increasing linearly with depth no = 0, n1 = 1.0 Friction angle 4>

n

0.0 (m = 45°)

5.0 (m

=

47.5°)

10.0 (m = 50°)

20.0 (m

=

60°)

Anisotropy factor x

Ci

1.0 0.9 0.8 0.7 0.6 0.5

0.231 0.216 0.201 0.186 0.172 0.157

0.227 0.213 0.199 0.184 0.170 0.155

0.226 0.211 0.197 0.183 0.169 0.155

0.224 0.210 0.196 0.182 0.168 0.154

1.0 0.9 0.8 0.7 0.6 0.5

0.303 0.288 0.273 0.258 0.243 0.228

'0.300 0.286 0.271 0.256 0.242 0.227

0.299 0.284 0.270 0.255 0.241 0.227

0.298 0.283 0.269 0.255 0.241 0.226

1.0 0.9 0.8 0.7 0.6 0.5

0.399 0.383 0.367 0.350 0.344 0.318

0.396 0.380 0.365 0.349 0.333 0.318

0.394 0.379 0.364 0.348 0.332 0.317

0.393 0.378 0.363 0.348 0.332 0.317

1.0 0.9 0.8 0.7 0.6 0.5

0.659

0.653 0.632 0.611 0.590 0.569 0.548

0.650 0.629 0.609 0.588 0.568 0.547

0.648 0.628 0.608 0.587 0.567 0.547

Yield acceleration factor: k c = 30°

Ci

= 50°

Ci

= 70°

Ci

= 90°

394

395

TABLE 9.6 Anisotropic and nonhomogeneous soil: Odenstad Model (1963) for cy increasing linearly with depth (q, = constant, Ci *- (J), no = 1.0, nl = 1.0, n2 = 1..5. Friction angle q,

Anisotropy factor

Yield acceleration factor: k c

(0)

"

Ci

1.0 0.9 0.8 0.7 0.6 0.5

0.329 0.311 0.293 0.274 0.255 0.237

0.323 0.305 0.287 0.269 0.250 0.232

0.320 0.302 0.284 0.266 0.248 0.230

0.318 0.300 0.282 0.264 0.246 0.228

1.0 0.9 0.8 0.7 0.6 0.5

0.395 0.376 0.356 0.337 0.318 0.299

0.389 0.370 0.351 0.332 0.314 0.295

0.386 0.367 0.349 0.330 0.311 0.293

0.383 0.365 0.347 0.328 0.310 0.292

1.0 0.9 0.8 0.7 0.6 0.5

0.486 0.465 0.445 0.424 0.403 0.382

0.480 0.460 0.439 0.419 0.399 0.379

0.477 0.457 0.437 0.416 0.396 0.377

0.474 0.455 0.435 0.415 0.395 0.376

0.0 (m = 45°)

5.0 (m = 47.5°)

10.0 (m = 50°)

_..; _ _ w . .' -

20.0 (m = 60°)

..;

-:.

.._ _. .

1.0 0.9 0.8 0.7 0.6 0.5

..

~

,_-_....

.._

= 30°

..... _ _ ..;;.;.;;'_ _ ,;_.:._..;__

0.741 0.713 0.686 0.658 0.631 0.603

Ci

= 50°

... .:._;._:.._'__ _ :...

:...;~.;...;.:..:

0.732 0.705 0.678 0.652 0.625 0.598

Ci

= 70°

:..

0.727 0.701 0.675 0.648 0.622 0.595

Ci

= 90°

. ; _ ~

..; _ _ ...'

the landslide movements will be very extensive and the motion is termed af/ow slide. The action of the pore-water pressure, u, in a landslide can be compared to that of a hydraulic jack. The greater ·the pore-water pressure, the greater is the part of the total weight of sliding mass which is carried by the water, and as soon as the porewater pressure becomes equal to the normal pressure, the sliding mass 'floats' and a flow slide occurs. In many cases a liquefied zone may not be extensive so that sliding occurs partly through liquefied soil and partly through non-liquefied soil along the slip surface. In such cases, the strength of the nonliquefied soil may be sufficient to prevent sliding as soon as the inertia forces due to earthquake ground motions stop. The TABLE 9.7 Anisotropic and nonhomogeneous soil: Reddy and Srinivasan Model (1967) for cy increasing linearly with depth (q, = constant, Ci = (J), no = 0.5, nl = 0.7, n2 = 1.5 Friction angle c/>

Anisotropy factor

Yield acceleration factor: kc

(0)

"

Ci

1.0 0.9 0.8 0.7 0.6 0.5

0.304 0.286 0.268 0.250 0.232 0.213

0.298 0.281 0.263 0.245 0.228 0.210

0.295 0.278 0.261 0.243 0.226 0.209

0.293 0.276 0.259 0.242 0.225 0.207

1.0 0.9 0.8 0.7 0.6 0.5

0.372 0.354 0.335 0.317 0.298 0.280

0.368 0.350 0.331 0.313 0.295 0.277

0.364 0.347 0.329 0.311 0.293 0.276

0.362 0.345 0.327 0.310 0.292 0.275

1.0 0.9 0.8 0.7 0.6 0.5

0.467 0.447 0.428 0.409 0.390 0.369

0.461 0.442 0.423 0.404 0.385 0.365

0.459 0.440 0.421 0.402 0.383 0.364

0.456 0.438 0.419 0.401 0.382 0.363

1.0 0.9 0.8 0.7 0.6 0.5

0.739 0.712 0.685 0.658 0.631 0.605

0.730 0.704 0.678 0.652 0.626 0.600

0.725 0.700 0.674 0.649 0.624 0.598

0.722 0.697 0.672 0.647 0.622 0.597

0.0 (m = 45°)

..:

=

30°

Ci

=

50°

Ci

=

70°

Ci

=

._

0.724 0.698 0.672 0.646 0.620 0.593

9.4 Mechanics of earthquake-induced slope failure

5.0 (m = 47.so)

10.0 (m = 50°)

Earthquake accelerations cause a temporary increase in shearing stress, 7, within a soil mass and at the same time significantly decrease the shearing strength, s, in some materials such as water saturated sand. If the existing slope is already in a near-critical stress condition, the earthquake will trigger the landslide. If, as a result of cyclic loading due to accelerations, the pore-water pressure in a mass of undrained cohesionless soil increases to the point where it is equal to the externally applied pressures, the soil loses completely its shearing strength and the soil is said to liquefY· If liquefaction occurs along a large portion of the potential sliding surface,

20.0 (m = 60°)

90°

397

396

movements involved in such cases are relatively small compared to the size of the slide mass (Seed, 1968). This is usually referred to as landslide due to liquefaction. Liquefaction can only persist as long as high pore-water pressures persist in a soil. For some materials such as a medium-dense cohesionless soil, liquefaction may be developed initially over a small deformation range but the material stiffens rapidly as a result of the property of dilation at large deformation. This behavior of dilation reduces the pore-water pressure and results in a self-stabilizing effect against large movements. Similarly, if drainage can occur rapidly in the soil, large displacements of landslides during earthquake due to soil liquefaction can not develop. Some case studies of landslides due to liquefaction and flow slides are reviewed by Seed (1968), among many others. No slide can take place unless the factor of safety or the ratio's/ T' between the average shearing resistance, S, of the ground and the average shearing stress, T, on the potential surface of sliding has previously decreased from an initial value greater thaI1- one to unity at the instant of the slide. For most slides, the change of the ratio is rather gradual, which, in turn, results in a gradual progressive type of deformation process for material located above the potential surface of sliding and some downward movements of all points located on the surface of the potential sliding mass. The differential downward movements of the slope which proceeds the slide may be detectable by a careful measurement. Further, this relative displacement is likely to cause tension cracks along the upper boundary of the slide area. This phenomenon may be used as an indication of a possible disaster of slope failure or landslides. Sometimes, these effects are such that they are conspieuous enoughto attract the attention of animals (Terzaghi, 1950). The only slope failures and landslides which occur suddenly as a result of an almost instantaneous decrease of shearing strength, s, and instantaneous increase of shearing stress, T, are those due to earthquakes by liquefaction. Earthquakes alone without extensive liquefaction can produce only a cumulative finite displacement as the magnitude of each cycle of the acceleration increases, decreases and reverses. Since the inertia forces are developed in such short periods of time, the factor of safety of a slope may drop below unity several times during an earthquake with a cumulative finite displacement resulting, but the earthquake produces no 'failure' in the conventional sense of a major collapse or change in configuration. However, for a certain soil, if an earthquake leads to a simultaneous soil liquefaction along the potential surface of sliding and this liquefied zone extends to a free surface, the extensive lateral movements and shape change of a slope, characteristic of flow slides, will be inevitable and the slope fails or collapses completely. The conventional method for evaluating the effect of an earthquake shock on the stability of a slope is to determine its minimum dynamic factor of safety against sliding using the rigid-block-type of motion. In the analysis the inertia force is

treated as an equivalent concentrated horizontal force, P, which is expressed as the product of a seismic coefficient, k, and the weight, W, of the potential sliding mass. The magnitude of the dynami.c ~hear stress, T, is determined by simple statics such as the slice method (Morgenstern and Price, 1965). The values of the dynamic factor of safety against sliding is then determined by comparing the average shearing stress, T, along the potential surface of sliding to that of the average shearing resistance, S, of the material mobilized, with due regard to the considerable reduction in shearing strength of the material owing to the dynamic effects on the porewater pressures. The numerical value of the seismic coefficient, k, depends on the intensity of the earthquake. This empirical value is in the range of 0.05 - 0.15 for most engineers designing earth dams in the United States, but somewhat higher values are used in Japan, ranging from 0.12 to 0.25, depending on the location of the dam, the type of foundation and the possible downstream effects of damage caused by an earthquake. Some analytical procedures in obtaining these values will be described later. The magnitude of the dynamic shearing resistance depends on the current state of initial stresses along the potential surface of sliding before the earthquake, the magnitude of increase and decrease in shearing stresses and the number of their stress cycles during the earthquake and slope drainage conditions. At the present time, there is no analytical material model available for determining the dynamic shearing stress resistance under the above-mentioned conditions, although analytical modeling to this problem of dynamic properties of soil is in a very active stage of research and development (Chen and Baladi, 1985). In such a situation, the only engineering recourse is to attempt to determine the soil strength characteristics in the laboratory for soil samples taken from the field and prepare them as closely as possible to the field conditions and test them under the conditions applicable in a given design earthquake. This procedure will have to be used until a more satisfactory strength and deformation model becomes available. A brief review of some dynamic properties of soil as applicable to earthquake loading is summarized in the following.

9.4.1 Dynamic shearing resistance of soils During an earthquake, the soil along a potential surface of sliding is subjected to a series of alternating shearing stresses which vary in magnitude in a rather random fashion, in addition to the initial shearing stresses due to the weight of the earth slope. The dynamic effects of earthquake motions on soil shearing resistance involve mainly the change of pore-water pressure, which, in turn, is a function of the volume change of the material. In general, it is the undrained shearing resistance that is of importance in an earthquake. For highly permeable materials, however, the drained shearing resistance may be appropriate.

398

399

At the present time little analytical work has been done concerning the mechanism of pore-water pressure build-up under given cyclic loading conditions except, that we know from laboratory tests that are markedly different from those developed under static loading conditions, depending on the type of soils considered (Chen, 1980). So long as the strains are relatively small, for many soils, the stress strain curve for a soil sample in a pulsating loading is the same as for a single application of stress as shown in Fig. 9.4. However, for some soils after a certain strain has been reached the stress may drop from the original virgin curve as the pore-water pressure builds up progressively during each cycle of loading, which in turn reduces the shearing resistance of the material. This is illustrated by the dotted line in Fig. 9.4. For the; case of saturated sand, the development of pore-water pressures can be equal to the applied confined pressure after some number of cycles. As soon as this happens, the effective stresses become zero and the soil 'floats'. The soil loses all of its strength. This is known as liquefaction. Conditions for the possible development of liquefaction for a soil under cyclic loading depend on the magnitude and number of stress cycles, the initial density and stresses, and drainage situation. In general, laboratory tests show that (Seed, 1968): 1. the larger the magnitude of the applied cyclic stresses, the fewer the number of cycles required to induce liquefaction; 2. the magnitude of the cyclic stresses required to induce liquefaction increases rapidly with increase in initial density; 3. the higher the confining pressure the greater the cyclic shearing stress required to induce liquefaction; and 4. the larger the ratio of initial shearing stress. to initial confining pressure acting on the same plane, the greater the cyclic shearing stress required to induce liquefaction in a given number of cycles. Experimentally determined relations between these variables for the conditions of liquefaction of a saturated sand under a cyclic loading are reported by Seed (1968), among many others. However, only limited data is available for clay types of soils,

STRAIN

Fig. 9.4. Stress-strain relations for ideal and real soils under pulsating soils.

and it appears to indicate that liquefaction problems are not likely to develop in plastic soils such as soft' clay, even for strain cycles with an amplitude as large as 2070.

9.4.2 Seismic coefficient

The seismic coefficient is a simple engineering means of designating the complex dynamic effects induced by earthquake motions by a single static equivalent force. The static equivalent force may be chosen to represent: (1) the maximum or average inertia force developed on the soil mass during the design earthquake; or (2) the maximum or average deformations of the embankment developed during the design earthquake. The value of the seismic coefficient depends of course on its precise meaning or definition. If a dam is assumed to behave as a rigid body, the accelerations will of course be uniform throughout the dam and equal at all times to the ground accelerations. Probable intensities of maximum acceleration for major earthquakes in California are about 0.3 - 0.5 g. Thus, it may be argued that the design seismic coefficient, k, should reflect this maximum ground acceleration, k = 0.3 - 0.5. However, this simplified approach assumes that the horizontal acceleration kg acts permanently on the slope material and in one direction only. This conception of earthquake effects on slopes is obviously not correct because, the reduction in stability of a slope only exists during the short period of time for which the unfavorable direction of inertia force is induced. As soon as the ground acceleration is reversed, the direction of the inertia forces is also reversed with a corresponding increase in stability of the slope. During the period of the unfavorable direction of inertia force, the factor of safety may drop below unity and some permanent displacements can occur but this movement will be arrested as soon as the magnitude of the acceleration decreases or is reversed. For most stable soils such as clays with a low degree of sensitivity, in a plastic state, or dense sand above the water table, the overall effect of a series of large but brief inertia forces causes only a cumulative displacement of the embankment but the slope will not fail even with a severe permanent displacement. The factor of safety of the section after the earthquake will be just about the same as it was before the earthquake. Hence the effects of earthquakes on embankment stability should be assessed also in terms of the displacements they produce. This was first proposed by Newmark (1965), and Seed (1966) subsequently presented improved methods of analysis based on this concept which requires the complete time history of the inertia forces acting on the embankment during the earthquake. This will be described later. Since soils are not rigid, the magnitude of the acceleration in a dam will vary depending on the material properties and damping characteristics of the dam as well

400

401

as the nature of the ground motions. To assess the effect of dam flexibility on the design seismic coefficient, Mononobe et al. (1936) assumed the dam consisting of a series of infinitely thin horizontal slices which are connected to each other by linearly elastic shear springs and viscous damping devices and obtained visco-elastic response solutions. Subsequent developments by others extend this analysis to include the variation of horizontal response over both the length and height of the dam, the dam resting on an elastic layer of finite thickness instead of a rigid foundation, and a variation of increasing shear modulus as the cube root of the depth. Application of these solutions leads to a distribution of seismic coefficient varying from a maximum value at the crest of the embankment to zero at the base. In the 1960's, Seed and Martin (1966) determined the average seismic coefficients based on the concept of visco-elastic response solutions which represent the entire time history of the inertia force to which the embankment has been subjected by any given ground motion. It was concluded in their evaluation that, in designating seismic coefficients for design purposes, it is important to differentiate between dams of different heights and different material characteristics, as well as different positions of the potential slide mass with the embankment section. For example, for a 100-ft-high embankment subjected to EL CENTRO earthquake (shear wave velocity = 1000 FPS, 20070 critical damping, 15 significant cycles of force, with a predominant frequency of 3.3 cycles per second), the following equivalent maximum seismic coefficients operative over different portions of the embankment are shown in Table 9.8 (Seed and Martin, 1966). Suggested values for the seismic coefficients appropriate for other cases are also given in the same reference. 9.4.3 Rigid-plastic analysis

As mentioned previously, for some soils, their stress-strain behavior may be approximated by two straight lines as shown by the dashed lines in Fig. 9.4. A hypothetical material exhibiting this property of continuing plastic flow at constant stress is called ideally plastic or perfectly plastic material. The yield stress level used in an analysis may be chosen to represent the average stress in an appropriate range of strain applicable to an earthquake condition.

The simplest type of motion of slope failures or landslides associated with this perfectly plastic idealization is the rigid-block-type of sliding separated by narrow transitionJayer. The two mostimportant cases of rigid body sliding for a dam subjected to an earthquake as observed in the field are shown in Fig. 9.5 (Newmark, 1965). The two slip surfaces as indicated in Fig. 9.5, marked curve 'a' and curve 'b', are the result of successive ground motions, with the net result of a major settlement at the crest. This has been observed in several old dams which may not have been designed to have adequate earthquake resistance (Newmark, 1965). For some cases, natural soil strata can lose part or almost all of their shearing resistance under shock conditions, as the result of liquefaction. Under such conditions, motion of the dam as a rigid block along the base can occur, as indicated in Fig. 9.5, line 'c'. A typical example of this type of failure for natural embankments which slide major distances on sensitive clay strata or on loose sand layers has been observed in the Anchorage earthquake (Newmark, 1965; Seed, 1968). Once a failure mechanism has been assumed, values of the dynamic factor of safety against sliding can be determined by conventional analysis or by limit analysis with due consideration of inertia forces due to earthquake and dynamic shearing strengths of material whose values may be considerably reduced owing to the dynamic effects on the pore pressures. As pointed out by Newmark (1965), the effects of earthquakes on embankment stability should also be assessed in terms of the deformations they produce rather than depending mainly on the concept of minimum factor of safety. In estimating the magnitude of displacements, Newmark (1965) proposed a simple method which assumes that the slipping of a mass of soil along a failure surface is analogous to the slipping of a rigid block on an inclined plane. Thus, it is necessary to determine the yield acceleration at which slippage will just begin to occur, and the actual acceleration generated by the earthquake for the sliding mass. When the induced acceleration exceeds the yield acceleration, movement will occur and the magnitude

TABLE 9.8 Equivalent maximum seismic coefficients of embankment

Extent of potential slide mass

Equivalent maximum seismic coefficient

Upper quarter of embankment Upper half of embankment Upper three quarters of embankment Full height of embankment

0.4 0.35 0.30 0.25 Fig. 9.5. Possible mechanisms of failures of an earth dam in an earthquake.

403

402

of displacements may be evaluated by double integration' of that part of the acceleration history above the yield acceleration. The applicability of this procedure for coliesionless soil has been examined by Seed and his associates at Berkeley (see Seed, 1966). It was concluded that if the yield acceleration can be accurately evaluated and due allowance is made for a variation in yield acceleration with increasing slope displacement, Newmark method provides a reasonably good estimate of slope displacement induced by an earthquake. This appears to be the case for the analyses of displacements resulting from surface sliding of the downstream slope and the analyses of displacements of the upstream slope when the reservoir is empty (see Fig. 9.5). Under such condition.s, the main part of the sliding surface is developed essentially in dry or partly saturated cohesionless soils for which the pore pressures has negligible magnitude. However, for the analyses of displacements of the upstream slope with water in the reservoir, the determination of the yield acceleration in the saturated cohesionless soil requires an assessment of the progressive change in pore-water pressures during the period of the earthquake. At the present time, this is still not possible. The only engineering recourse is to estimate the pore-water pressures form the results of simulatedearthquake loading tests in the laboratory for the samples taken from the field. This latter procedure has been proposed by Seed (1966). In short, the slope stability of embankments during earthquakes should be assessed in terms of embankment displacements as well as the factor of safety against sliding. The method of displacement analysis proposed by Newmark is simple to apply for dry or partly saturated cohesionless soils for which rigid body motion occurred on a wellcdefined slip surface. In the case of saturated cohesionless soils in which pore pressure changes may occur during the earthquake, special types of laboratory tests in which soil samples are subjected to simulated field conditions are needed. This requires the consideration of the time history of inertia forces developed in a dam during an earthquake as well as the initial stresses on soil elements along any potential failure surface before the earthquake. A step-by-step analytical procedure in estimating these stresses along with the laboratory test procedure is given by Seed (1966). For cohesive soils for which large deformations approaching failure can occur under pulsating load conditions even when the maximum applied stress is somewhat less than the static strength of the soil, a broad shear zone within the embankment will contribute to the overall displacements of the embankment. At the present time, no means is available for integrating these deformations to determine the slope displacements and overall change in slope configuration. Nevertheless, Seed's procedure is still valid and can be applied to calculate the factor of safety against sliding for such slopes, once the shearing strength along the potential failure plane has been evaluated by laboratory tests appropriate to an earthquake condition.

References Chen, W.F" 1977. Mechanics of slope failure and landslides, Proc. Advisory Meeting on Earthquake Engineering and Landslides, US-ROC Cooperative Science Program, Taipei, Taiwan, R.O.C., August 29 - Sept. 2, pp. 219 - 232. Chen, W.F., 1980. Plasticity in soil mechanics and landslides. J. Eng. Mech. Div., ASCE, 106(EM3): 443-464. Chen, W.F., 1982. Soil mechanics, plasticity and landslides. In: R.T. Shield and G. Dvorak (Editors), Special Anniversary Volume on Mechanics of Material Behavior to Honor D.C. Drucker. Elsevier, Amsterdam, pp. 31-58. Chen, W.F. and Baladi, G.Y., 1985. Soil Plasticity: Theory and Implementation. Elsevier, Amsterdam, 230 pp. Chen, W.F. and Sawada, T., 1982. Seismic stability of slope in nonhomogeneous, anistropic soils. Structural Engineering Report No. CE-STR-82-25, School of Civil Engineering, Purdue University, West Lafayette, IN, 108 pp. Chen, W.E. and Sawada, T., 1983. Earthquake-induced slope failure in nonhomogeneous, anisotropic soils. Soils Foundations, Jpn. Soc. Soil Mech. Found. Eng., 23(2): 125 -139. Gibson, R.F. and Morgenstern, N.R., 1962. A note on the stability of cutting in normally consolidated clays. Geotechnique, 12(3): 212 - 216. Lo, K.Y., 1965. Stability of slopes in anisotropic soils. J. Soil Mech. Found. Div., ASCE, 91(SM4): 85-106. Mononobe, N., Takata, A. and Matumura, M., 1936. Seismic stability of the earth dam. Proc., 2nd Congress on Large Dams, Washington, D.C., Vol. IV. Morgenstern, N.R. and Price, V.E., 1965. The analysis of the stability of general slip surfaces. Geotechnique, 15(1): 79-93. Newmark, N.M., 1965. Effect of earthquakes on dams and embankments. Geotechnique, 15(2): 139-160. Odenstad, S., 1963. Correspondence. Geotechnique, 13(2): 166-170. Reddy, A.S. and Srinivasan, R.J., 1967. Bearing capacity of footing and layered clays. J. Soil Mech. Found. Div., ASeE, 93(SM2): 83-98. Seed, H.B., 1966. A method for earthquake-resistant design of earth dams. J. Soil Mech. Found. Div., ASCE, 92(SM1): 13-41. Seed, H.B., 1968. Landslides during earthquakes due to soil liquefaction. J. Soil Mech. Found. Div., ASCE, 94(SM5): 1055 -1122. Seed, H.B. and Martin, G.R., 1966. The seismic coefficient in earth dam design. J. Soil Mech. Found. Div., ASCE, 92(SM3): 25-58. Sigel, R.A., 1975. STABL User Manual, Joint Highway Research Project, JHRP-75-9, School of Civil Engineering, Purdue University, West Lafayette, IN, 112 pp. Taylor, D.W., 1948. Fundamental of Soil Mechanics. John Wiley and Sons, New York, NY, 700 pp. Terzaghi, K., 1950. Mechanics of Landslides, Engineering Geology Volume, The Geological Society of America, November, pp. 83 -123.

405

Chapter 10

ASSESSMENT OF SEISMIC DISPLACEMENT OF SLOPES*

10.1 Introduction

In current seismic stability analysis of slopes, there are two basic approaches, one is the conventional pseudo-static analysis and the other is the stress-strain analysis. With the advent of the finite-element technique and knowing the material properties, the cross-section of the slope, and the time history of acceleration of a design earthquake, it is possible to analyze the section for deformation and safety by computing the stresses and strains in the structure. However, a complete progressive failure analysis of stress and strain in a soil mechanics problem by using the stressstrain approach is too complicated for practical applications. The most important feature of seismic stability analysis is the estimation of the seismic loads which will cause slippage of the soil mass and the overall movements of the sliding soil mass throughout an earthquake. However, the traditional pseudostatic approach is too crude to predict the behavior of a slope under earthquake loading condition. In this traditional analysis, the pseudo-static inertia force is applied as an equivalent permanent concentrated horizontal force, Le., the pseudo c static force, acting at some critical point (usually the center of gravity) of the critical sliding mass (Chen, 1980). Earthslopes are then analyzed with the calculation of the factor of safety when this inertia force is considered. If the factor of safety is less than unity, the slope is considered to be unsafe. In reality, however, the reduction in the stability of the slope exists only during the short period of time for which the inertia force is acting. Thus, during the earthquake, the factor of safety may drop below unity a number of times which will induce some movements of the failure section of a slope, but this may not cause the collapse of a slope. Thus, the stability of slopes should depend on the cumulative displacements developed during an earthquake. Newmark (1965) first proposed the important concept that the seismic stability of slopes should be evaluated in terms of the displacements rather than the traditional concept of minimum factor of safety. After carefully investigating the influence of the earthquake on the stability of the slope and upon the awareness of the drawbacks and deficiences of currently available methods, an effective, practical and relatively accurate analytical pro-

* This chapter is based on the Ph.D. thesis by C.l. Chang (1981) and the paper by Chang, Chen, and Yao (1984).

407

406

in Chapter 9, the computation of the yield acceleration factor by using the upperbound limit analysis is based on the following conditions: plane strain condition; pseudo-static earthquake loading; uniform horizontal distribution of lateral acceleration; Mohr-Coulomb criterion for failure with constant c and cP, homogeneous and isotropic slope.

cedure which extends the present pseudo-static method, is proposed herein for the analysis of earthslope stability subjected to earthquake loading. The necessary steps are outlined in the forthcoming. In estimating the magnitude of displacement, we assume that the slipping of a mass of soil along a failure surface is analogous to the slipping of a rigid block on an inclined plane. Thus, it is necessary to determine the yield acceleration at which slippage will just begin to occur, and to compare it with actual acceleration generated by the earthquake for the sliding mass. When the induced acceleration exceeds the yield acceleration, rigid-body-type of slope movement will occur and the magnitude of displacements can be evaluated by double integration of that part of the acceleration history above the yield acceleration. The applicability of this procedure for cohesionless soil has been examined by Seed and his associates at Berkeley (1966). It was concluded that if the yield acceleration can be accurately evaluated and due allowance is made for a variation in yield acceleration with increasing slope displacement, the proposed method provides a reasonably good estimate of slope displacement induced by an earthquake. It should be noted that the Newmark's analytical procedure can not account for the dynamic effects of pore-water pressures built-up and the possible loss of shear strength of soil owing to liquefaction during an earthquake shaking, but it will probably provide a satisfactory result, whenever the major portion of the sliding surface, that develops the resistance to sliding in a slope, is made of either clays with a low degree of sensitivity, dense sand either above or below the water table, or loose sand above the water table. In this chapter, the yield acceleration of slopes is first computed by using the upper-bound techniques of pseudo-static limit analysis method. Based on the calculated yield acceleration and its associated failure mechanism, Newmark's concept is then used to evaluate the displacements of earth slopes during an earthquake.

10.2.1 Infinite slope failure

Figure 10.1 shows an infinite slope. We assume that the slope is very wide in the direction normal to the cross-section, and consider only the stresses that act in the plane of the cross-section. Further, the slopes are relatively long and presumably homogeneous and isotropic. Thus, if the horizontal inertia force is uniformly distributed along the slope, the stresses on any vertical planes at same depth below the surface should be equal. Thus, sliding is likely to begin at any depth, depending on the properties of the soil, magnitude of inertia force and slope angle, 0:. Referring now to Fig. 10.1, the rate of external work done by soil weight and the inertia force, respectively, are:

W-y

= 'Y (d/coso:) V

and WE

= kh 'Y

(d/coso:) V cos(o: -

cP)

(10.2)

where 'Y is the unit weight of soil, cP is the internal friction angle of soil, V is the discontinuous velocity vector of sliding block, kh is the horizontal seismic coefficient of inertia force, and d is the perpendicular depth of soil stratum overlying bedrock (Fig. 10.1).

10.2 Failure mechanisms and yield acceleration The possible failure modes of slope include the infinite slope failure and the local slope failure. The former is the sliding parallel to the earth surface or the bedrock in a very wide range. The latter, however, is due to the surcharge boundary loads and any other conditions that may change the possible occurrence of parallel sliding. The critical mode of failure depends on the properties of soil, slope angle, magnitude and direction of inertia force, and thickness of the soil overlying the bedrock. The slope is considered unstable when either an infinite slope failure or a local slope failure occurs. This section is concerned with the calculation of the critical or yield horizontal inertia force corresponding to the yield acceleration factor, k c' at which a condition of incipient slope movement is possible along the potential sliding surface. Here, as

(l0.1)

sin(o: - cP)

/(

Fig. 10.1. Analysis of infinite slope.

408

409

The internal rate of dissipation of energy along the slip surface is: D

=

c (llcosa) V cos

(10.3)

Equating the sum of the external rates of work, Eqs. (10.1) and (10.2), to the rate of internal energy dissipation, Eq. (10.3), yields: k

=

c

k

=

h

c/(d'Y cosix) - tana + tan4> (1 + tan4> tana)

(10.4)

10.2.2 Plane failure mechanism of local slope failure

H sin(a - 8) sina sin8

= c (H/sin8) V cos

=

Wp

= p L V sin(1J -
WE''Y

4 'Y H L V sin(8 - 4»

=

WE,p

kh

! 'Y H

(10.7)

(10.8)

where k h is the seismic coefficient corresponding to the surcharge load p whose magnitude can be related to k h by: (10.11) kh/kh can be greater or less than unity. By equating the external and internal rates of work, we obtain the expression for k h as:

~=

(10.6)

(10.10)


cos4> . [ sina _ ~ sin(8- ~)-,-NsSin.(8-4»]. cos(8 - 2 cos Ns

px -+2 c

Fig. 10.2. Translational local slope failure mechanism.

.

(10.12)

where N s' the stability factor, is equal to 'YH/c. The yield acceleration factor kh has the minimum value k c' when (} satisfies the condition:

o H

(10.9)

L V cos(8 - 4»

= khp L V cos(8 -

(10.5)

The rate of internal energy dissipation along the surface AC is: D

W'Y

and

The construction of structures on an infinite slope changes the geometric and stress conditions. Thus, the local slope failure may be more critical than that of the parallel sliding. For local slope failure, the plane slip surface and the logarithmic spiral slip surface of angle 4> are the two popular surfaces of velocity discontinuity that are permitted in the limit analysis for a rigid-body motion relative to a fixed surface. Figure 10.2 shows the first of the two possible failure mechanisms - plane failure mechanism. In this figure, region ABC translates as a rigid body with the relative velocity V to the rigid body below the discontinuous surface AC. The failure mechanism is assumed to end at point A in Fig. 10.2 with height H. The assumed failure mechanism can be specified by two variables 8 and L, where 8 is the angle of slip surface with respect to the horizontal line and L is related toH by:. L

The external rate of work done by soil weight, surcharge loads, and the inertia forces, respectively, are:

(10.13)

Solving this equation and substituting the 8-value thus obtained into Eq. (10.12) yield a least upper-bound k c for the yield acceleration factor for plane failure mechanism. When no surcharge exists, Eq. (10.12) can be reduced to: k = h

cos4> [~ ~~ + sin(4) - 8)] cos(1J -

(10.14)

411

410

rates of work done by the soil weight in regions OAC, OBC and OAB, respectively:

10.2.3 Log-spiral failure mechanism of local slope failure The motion of plane failure mechanism is a translational type. In this section, we consider a rotational log-spiral failure mechanism as shown in Fig. 10.3. The region ABC rotates as a rigid body about the center of rotation 0 with the angular velocity relative to the materials below the logarithmic failure surface AC. As shown in Fig. 10.3, the parameters H, eo and eh are used to specify the failure mechanism. The geometrical relations of Hlro and Llro can be expressed as follows:

°

3

- 3 tane/> coseo - sineol = 'Yro 3

. W 2

!!.. = sineh exp[(eh

-

ro

eo> tane/>] - sineo

ro

W3 = (10.16)

For the failure mechanism passing through the toe, the internal dissipation o( energy occurred along the discontinuity surface AC is:

D =

° rL(

6

r de f!!h (cVcoSe/»cose/>

"I

rg °exp[(eh

(10.18)

l

.D..

L) .

2 coseo - - smeo ro o

(10.15)

sin(a + eo> - exp[(eh - eo> tane/>] sin(eh + a) sina

L

o

"I r- =-

°f

= 'Yro3/2

(10.19)

eO>tane/>] [Sin(eh - eO) - !::... Sine h] {coseo - !::...

-

~

6

+ coseh exp[(eh

-

eo>tane/>]} = "I

~

rg (} f 3

(10.20)

Similarly, the rate of external work done by the inertia force due to soil weight can be found by the summation W4 - Ws - W6, in which W4, Ws and W6 are the rates of work done by the inertia force due to soil weight in regions OAC, OBC and OAB, respectively:

80

2

~o

= - - (exp[2(eh 2 tane/>

- eo)tane/>] - 1J ...

2

= croOfc .

(10.17)

The rate of external work due to soil weight for region ABC can be found by a simple algebraic summation WI ..:. W2 - W3 , in which WI' W2 and W3 are the

- 3 tane/> sineo 3

k h "I ro Ws = 6 .

+

coseoJ = k h "I

°L 2 sm. eo 2

ro

3

= kh'Y ro

rg °f 4

(10.21)

°f s

(10.22)

(10.23) H

Fig. 10.3. Rotational local slope failure mechanism.

The external rates of work due to surcharge boundary loads and their inertia forces are:

pLO ( ro coseo - L) - = pro2 2

°-L ( coseo ~

-L) = pro2 2~

°f

p

(10.24)

412

413

and

tia effect including the consideration of saturation on the energy-balance equation. Computer programs developed at Purdue University using the aforementioned analytical procedures are listed at the end of this chapter. Some selected sample outputs have been obtained by this program. The results are tabulated in Table 10.1 and illustrated graphically in Figs. 10.4 and 10.5. From the results in Table 10.1 b, we can see that the log-spiral failure surface is more critical than the plane failure surface. It appears that the surcharge loading has little effect on the yield acceleration factor. This is due to the fact that the surcharge is only a small fragment when compared to the sliding mass in this case with height H = 100 ft. Table 10.2 presents a comparison of the limit equilibrium solutions with the upper-bound limit analysis solutions. The limit equilibrium solutions are run by a computer program STABL (Sigel, 1975) with which the simplified Janbu method of slices (Janbu, 1957) and circular failure surface are adopted to find the factor of safety of a slope. In the use of STABL program, only the factor of safety of slopes can be assessed. Besides, log-spiral failure surface is not included and Xis always assumed equal to zero. Thus, in order to compare the limit analysis solutions with the solutions obtained by STABL, we assume X equals to zero and use k c' obtained by the limit analysis method, as input data to find the corresponding factor of safety. The factor

(10.25)

Equating the rate of internal energy dissipation to the external rate of work gives:

(10.26)

where N s' the stability factor, is equal to 'YHIe and the functions 1's are defined as above, which are all functions of 0 , 0h' k h is the upper-bound solution of the yield acceleration factor of the logspiral failure mechanism. By taking the first derivatives of Eq. (10.26) with respect to 0 and 0h' respectively, and equating them to zero, Le.:

°

°

o

(10.27)

°

°

and solving Eq. (10.27), we obtain the critical values of 0 and 0h' Substituting 0 and 0h values so obtained to Eq. (10.26), we have'the least upper=bound for the yield acceleration factor, kc = min(kh), of the rotational log-spiral failure mechanism. When no surcharge exists, Eq. (10.26) can be reduced to:

TABLE 10.1 Yield acc~)eration factor, ke,of.infini,te slope flliIlJre a!ldloclll ,sl,<;>pe J;lHl!re withf =900 psf, '" = 40,°, and 'Y = 120 pcf (a) ke of infinite slope faiIure with d 15 0.593

=

30 0.293

50 feet 45 0.D28

60

75

• The slope has already failed before imposing any seismic force.

(10.28)

(b) ke of local slope failure with H = 100 ft, '" = 40°,

p=O

The yield acceleration factor of the log-spiral failure mechanism generally gives a lower value than that of the plane failure mechanism. Thus, the log-spiral failure mechanism generally controls the local slope failure. The value of 'Y in N s is the unit weight of sliding soil mass. If the soil is partially saturated or submerged, the yield acceleration factor, k h , can be evaluated by introducing an average unit weight of the sliding mass or directly calculating the iner-

C

= 1800 psf and 'Y

=

120 pcf

p = 120 psf, X = 0

p = 120 psf, X = 0.5

",0

Plane

Logspiral

Plane

Logspiral

Plane

Logspiral

15 30 45 60 75 90

1.111 0.951 0.759 0.560 0.353 0.124

0.926 0.819 0.677 0.516 0.333 0.116

1.126 0.961 0.764 0.562 0.350 0.116

0.928 0.820 0.677 0.514 0.329 0.108

1.115 0.951 0.757 0.556 0.347 0.115

0.925 0.817 0.674 0.511 0.326 0.107

414

-

TABLE 10.2 Factor of safety obtained by the limit equilibrium method with H 120 pcf, P = 120 psf and X = .a

c/yd =0.10

-0.15" =0.20 c : cohesion ep: Internal frictional angle

Factor of safety

= 100 ft, cf> = 40°, C = 1800 psf, 'Y =

15

30

45

60

75

90

0.920

0.885

0.891

0.933

1.006

1.025

of safety thus obtained is listed in Table 10.2. The factor of safety so obtained should be very close to unity as checked by the limit analysis. However, due to the difference in failure surfaces assumed, i.e., circular surface in the limit equilibrium method and the logspiral surface in the limit analysis method, the results are somewhat different.

0:3 0.2 0.1

10.3 Assessment of seismic displacement of slopes a

Slope Angle,

10.3.1 General description

Fig. 10.4. Yield acceleration factor, k c' of infinite slope failure as a function of slope angle a.

N s = 10

-- --

1.1

- 6.66 • 5

------- ... ............. .........

1.0 0.9

-"-40

0.8

--~ --30

0.7

----

....

c: cohesion frictional angle

ep: Internal 40~

"'

......

:::::-- 30

-

'.... ---~

k c 0.6 0.5

...

40

- - 2 0 ..

------20

--

........

~......................... -........:::~~............... --.....,

-.......

30

-

.............

-'"

".........

........." ....~ ............. ~;:,~

0.3

,

0.2

~ a "

H

":~

--..............

0.4

X =0.5 p=120psf

...........,

'"

,

....

In 1965, Newmark first proposed the basic elements of a procedure for evaluating the potential displacements of an embankment due to an earthquake shaking. Newmark envisaged that sliding would be imminent once the inertia forces on a potential failure block were large enough to overcome the yield resistance and that movement would stop when the initial forces were reversed (Newmark, 1965). In his analysis, a soil mass moving downward along a failure surface under inertia force due to earthquake shaking is considered to be analogous to a rigid block acted on by an external force sliding on an inclined plane as shown in Fig. 10.6. Thus, the movements of slope would begin to occur if the inertia force induced by earthquake on a potential slide mass exceeds the yield acceleration. The failure mechanism and corresponding yield acceleration must be determined first so that the analogous inclined plane and external force can be simulated. Subsequently, the overall displacements of a failure slope under earthquake loads can be assessed (Chen et

" '"" "

.............

''''','' '...... ........... ' .........',""" "~',""'-~ ........ ""', " " ......... , '" """',

"

........................:~

0.1

'"

0

O·

15°

30°

45°

60'

,

"""' .....

90°

Fig. 10.5. Yield acceleration factor, k c' of logspiral failure mechanism as a function of slope angle a.

Fig. 10.6. Rigid block on an inclined plane.

416 aI. 1978; Chen, 1980). This can be achieved in the following steps (Chang et aI. 1984): 1. Calculate the yield acceleration at which slippage will just begin to occur. 2. Apply various values of the pseudo-static force to the slope. These values are obtained from a discretized accelerogram of an actual or simulated earthquake. 3. According to the yield acceleration and accelerogram of an earthquake, the time history of velocity of the sliding soil mass of a slope can be calculated. The magnitude of displacements can be evaluated by integrating all the positive velocity. 4. Determine the 'stability' of the slope on the basis of this estimated total displacement by rigid body sliding. The computation of the yield acceleration by the upper bound techniques of limit analysis has been described in the preceding Sections. Based on this yield acceleration and its associated failure mechanism, the equation of motion for the estimation of displacements along the potential failure surface is developed in this Section. Newmark's concept implies that movements would stop when the inertia forces were reversed. Actually, the velocity could remain positive even if the inertia forces were reversed or the inertia forces were not reversed but less than the yield resistance on the potential failure surface. Positive velocity thereby causes sliding on the surface. On the other hand, the velocity could be negative even though the inertia forces were greater than the yield resistance. It all depends on the magnitude and direction of both the velocity and the inertia force. Besides, as also indicated by Newmark, the uphill resistance may be taken as infinitely large without serious error in the calculations. In this situation, ground motions in the direction of the downward slope tend to move the mass downhill, but ground motions in the upward direction along the slope leave the mass without relative additional motion except where the ground motions are extremely large in magnitude. Thus, the negative velocity or velocity heading uphill is not allowed in this analysis. Accordingly, by computing an acceleration at which the inertia forces become sufficiently high to cause yielding to begin and integrating the effective velocity on the sliding mass as a function of time, ultimately, displacements of the slope can be evaluated. Thus, the slope failure can be judged on the basis of the overall displacement caused throughout the earthquake.

417 rigid body with resistance mobilized along the sliding surface may proceed by the following steps:

Step 1 Select a design earthquake with time history of acceleration. A constant time interval may be chosen to designate the time and subsequently estimate all the corresponding accelerations. The acceleration within a time interval is assumed to be linear but not necessarily constant. . Step 2 The inertia forces, caused by the accelerations, tend to reduce the stability of the slope. Once the induced acceleration, khg, is greater than the yield acceleration, keg, the movement of the failure section will occur. Based on the computed yield acceleration and the accelerations obtained from the accelerogram of the design earthquake, calculate the motion acceleration, X, of the sliding block. The motion acceleration, X, acting on the sliding block for different failure mechanisms can be calculated, respectively, as follows: (a) Infinite slope failure Referring to Fig. 10.7, when Xl (10.29)

(10.30)

(10.31)

10.3.2 Numerical procedure The resistance to earthquake shock of a block of soil that slides along a surface is a function of the shearing resistance of the material under the conditions applicable in the earthquake (Newmark, 1965). However, in this section we shall assume that the resistance to sliding does not change during the earthquake. The calculation based on the assumption that the whole moving mass moves as a single

a

a (al

(b)

Fig. 10.7. (a). Equilibrium of forces on resting block (infinite slope). (b). Forces on sliding block (infinite slope).

419

418 WI sina

+

kh WI COsa - [C

+

(Nz - U-;) tanl = PI cos

(10.32)

where, WI is the weight of the sliding block of unit length, i.e., 'Y d/cosa, C is the total cohesion; N I , N z are the normal forces, UI' Uz are the pore-water forces for the cases mentioned above respectively, and PI is the force acting in the direction of motion. Assuming UI = Uz' from the cancellation of N I and N z from Eqs. (10.29) to (10.32), and using the relation PI = (WI/g) I' we obtain:

x

(10.33)

XI = (kh - kJ g cos( - a) (b) Local plane failure Referring to Fig. 10.8, when Xz Wz cosa - kc Wz sina Wz sina C

-

- [C

+

=

+ pL cosa

0: k h

=

(N4 -

U~

(10.37)

tanl = P z cos

where Wz (= 'YH LI2) is the weight of the sliding wedge of plane failure mechanism N 3, N 4 are the normal forces; U3' U4 are the pore-water forces for the cases mentioned above respectively, and P z is the force acting in the direction of motion. Assuming U3 = U4' from the cancellation of N 3 and N 4 from Eqs. (10.34) to (10.37), and using the relation P z = (Wz/g) z' we obtain:

x

h-

kJ g cos( - a)

(I + X:~)

(10.38)

k c;

- X k c pL sina = N 3

(10.34)

(c) Local log-spiral failure Referring to Fig. 10.9, when (j

kc'Yr~

=

(10.35)

U3) tan

+ pL cosa - X kh pL sina

=

0, k h

= k c:

15 - 16 ) + 'Yr~ (fl - I z - 13 ) + X kc p r~/q + p r~/p = z eh c role u l r Z tan dO eo which the I's are defined in the preceding sections. (f4 -

f

in Wz cosa - kh Wz sina N 4 _. P z sin

+ k h Wz cosa + pL sina + X k h pL cos a

Xz = (k

+ kc Wz cosa + pL sina + X kc pL cosa

+ (N3

Wz sina

(10.39)

=

(10.36)

L

(a)

(b)

Fig. 10.8. (a). Equilibrium of forces on resting block (plane failure surface). (b). Forces on sliding block (plane failure surface).

(a)

lb)

Fig. 10.9. (a). Equilibrium of forces on resting block (logspiral failure surface). (b). Forces on sliding block (logspiral failure surface).

421

420

3

k h 'Y ro (f4 2

C

role -

Is - 16 ) +

f°h

U2

3 'Y r o (fl -

r2 tan cjJ

dO

h - 13 ) +

2

Step 4 The motion velocity, xi at time ti can be calculated as:

2

x k h pro I q + prO I p =

+M

(10.44)

(10.40)

Knowing Xi' the motion velocity Xi + I at time ti + I can similarly be calculated as

°0 where up u2 are the pore-water pressure distributions along the failure surface for the cases mentioned above, respectively. Assuming u l = u2 and substituting Eqs. (10.39) and (10.40) into M = (W3 /g) (j p, we obtain: (10.41) in which W3

=

weight of the sliding block of log-spiral failure mechanism 'Y r~ = --.

{eXP [2(Oh - ( 0 ) tancjJl - 1

2

- [sin(Oh - ( 0 )

ro

2 tantf> -

L. - -

~ Sin{;lh] exp[(Oh

SIllOo

- (;Io) tancjJl}

(10.45) In this calculation, Xi + I is obtained from Eq. (10.33), (10.38) or (10.41). Using the same procedure, all the velocities corresponding to all the selected time instants can be calculated. Besides, as Newmark pointed out, the uphill resistance may be taken as infinitely large without causing serious errors in the calculations. In this situation, ground motions in the direction of the downward slope tend to move the mass downhill, but ground motions in the upward direction along the slope leave the mass without relative additional motion except where these are extremely large in magnitude. Thus, the sliding block can only move downhill with positive velocity, regardless of the direction of the acceleration. If the acceleration changes from negative to positive and the velocity changes from positive to negative during a time interval, by referring to Fig. 10.10, in addi-

(10.42) ~x

M = moment taken about center 0 (al

Step 3 Using the results of step (2), starting from the beginning of the earthquake, find the first positive motion acceleration Xi (or (j), with which the downhill movement will start to occur at time ti . If xi is the first positive motion acceleration, then xi _ I at t i _ I must be negative except in a special case in which Xi _ I = O. Thus, it needs to calculate time t, at which = 0 and motion velocity X will start to increase from zero. By linear interpolation, we obtain:

(bl

(10.43)

(e)

X

x

at time t, when the induced acceleration exceeds the yield acceleration, the motion velocity of the slide block increases from zero and the motion displacement x occurs.

t 1+1

X

Fig. 10.10. Motion of failure block on slope with a positive velocity at ti + 1 and negative velocity at ti + 2' (a). Negative acceleration at ti + 1 and positive acceleration at Ii + 2' (b). If negative velocity is permissible. (c). If negative velocity is not permissible.

422

423

tion to time In + I' time In + 2 also needs to be computed. the time In + I' at which becomes zero can be computed by using the following relationship:

x

(a)

t 1+1

I------='f"'!=-:--L--- t :

t

1+2

I I

+

x; + I (tn + I -

Solving for In +

1 -

I; +

I) = 0

(10.46)

I; + I' thus gives:

,,2 2(x; + 2 - X; + I)x; + IJ! ± [ X; + I - - - - - - - - - (I; + 2 - I; + I) (tn + I - I; + I) = - - - - - - - - - - - - - - - - - - -

(bl

X~_t X

1 I

I I

I I

(10.47)

(X; + 2 - X; + I)

(0)

(ti + 2 - I; + I)

Only one solution of the two obtained from Eq. (10.47) is reasonable, at which the velocity is zero and displacement ceases. Because, as mentioned earlier, the velocity cannot be negative, the velocity after time In + I is zero rather than negative. This zero velocity will remain until time In + 2' During In + I and I; + 2 whenever motion acceleration becomes positive the failure block will move again, and the time is In + 2' at which the failure block move downward again, In + 2 can be expressed as follo~s: ,

In + 2 =

- x; + I (t; + 2 -

Ii + I)

(x; + 2 - x; + I)

+

I; + I

(10.48)

Thus, during the time period from I; + I to Ii + 2' two nonconsecutive displacements can be calculated. When the acceleration changes from negative to positive and the velocities at times I; + I and I; + 2 are positive, as shown in Fig. 10.11, it is necessary to check the velocity at time In + 2 so that the velocity actually occurs as sketched in Fig. 11.11b or c. The velocity from I; + I to Ii + 2 must be all positive provided that n + 2 is positive, otherwise, a procedure similar to that described previously for finding In + I and In + 2 should be used. When the acceleration changes from positive at time I; + I' and the velocity at time Ii + 2 is positive, or the accelerations are all positive during the time interval from I; + I to I; + 2' we can calculate the deformation by proceeding to step (5) directly. If the velocity at I; + 2 is negative in the former case, as shown in Fig.

x

(d)

Fig. 10.11. Motion of failure block on a slope with positive velocity at both ti + 1 and ti + 2' (a). Negative acceleration at ti + I and positive acceleration at t i + 2' (b). Velocity is all positive. (c). Velocity is negative between In + 1 and In + 2' if negative velocity is permissible. (d). Velocity is zero between In + I and In + 2. if negative velocity is not permissible.

10.12, the time In + follows:

I'

X;

+

at which the velocity becomes zero should be calculated as

I + [X 7+ I -

2

Xi

+

2 - X;

Ii + 2 Ii + I

+

I; + I

I x; + IJ! (10.49)

X;+2- X;+1 Ii + 2 -

I; + I

Finally, when the accelerations at both I; + I and I; + 2 are negative, the time In + l' at which the velocity becomes zero can also be calculated in a similar manner as that of Eq. (10.49). Slep 5 Based on the accelerations and velocities between two times, e.g., I j and I; + I' the displacement of the failure section between I; and I; + I' thereby can be calculated as:

424

425

x

(10.50) Similar procedures will be performed to find the overall displacement until the end of the earthquake. 10.3.3 Numerical results

t;+1r--------"''''''''-::----------.-------t;+2

x

A computer program using the aforementioned analytical procedure has been developed at Purdue University and is listed at the end of this chapter. The problem as shown in Fig. 10.13 is considered. The design earthquake record selected is the Pasadena earthquake (1952) scaled to 0.15 g, as shown in Fig. 10.15. After solving the problem, the yield acceleration factor for the log-spiral failure mechanism is found to be 0.01764. For simplicity, we shall assume this yield acceleration factor to be constant during the earthquake. Two diagrams of displacements at the top of failure surface versus time are shown in Fig. 10.14. From the result, we can see that the displacement increases sharply during the range close to the positive peak acceleration, and that the displacement at the top in both directions are rather close. This is due to the fact that the value of 80 is 43.65°, that is close to 45°. Actually, the displacement in any point along the failure surface can be obtained by introducing the corresponding angle 8.

x

(0=239.24 It

X=0.5

(h = 301ft

p=120 pet

H=100 t1

Y=120 pet e=600 pst

Fig. 10.12. Motion of failure block on a slope. (a). Positive acceleration at Ii + I' negative acceleration at Ii + 2' (b). If negative velocity is permissible. (c). If negative velocity is not permissible.

tP = 10° k c=0.01764

Fig. 10.13. Example problem.

...

.~'

426

427

--.

2.00

(0 )

2.0

1.75 1.5 1.50

1.0

.:::

.

1.25

"

1.00

C

....

a Ol

Ii

'" i5

0.5

x

E

~

0.0

c:

0.75

0

.

-0.5

«""

-1.0

~

4i

0.50 0.25

I

-1.5

0.0

2

4

6

8

10

12

14

16

-2.0

0

2

Time ( Sec)

12

14

16

10.4 Summary

1.50

This Chapter attempts to develop a practical method to effectively analyze the stability of earth _s~pes under earthquake loading. The proposed approach is -an extension and modification of the pseudo-static method of slope stability analysis. It is a pseudo-static approach and not a dynamic analysis. Thus, it should be viewed only as a useful and practical computational tool. The corresponding computer program listing for the modified pseudo- static approach are given in Appendices. The following two parts of work are presented in this chapter: 1. The determination of failure mechanisms and corresponding yield accelerations. 2. The assessment of seismic displacement with Newmark's analytical procedure. In the first part of the work, the limit analysis method is used to determine the critical failure surface and its yield acceleration of earth slopes subjected to earthquake loads. In this analysis the acceleration is assumed to be uniform throughout the depth of the slope. However, because the slope is not rigid, acceleration is not really uniform throughout the slope. The distribution of seismic coefficient is therefore a function of the height of a slope. This is the subject of study of Chapter 8. Using the concept of superposition, the variation of the horizontal pseudo-static force throughout the depth of the slope can be incorporated. The critical state of a slope against collapse with nonuniform seismic coefficients thus can be determined.

1.25 1.00

Ii

'" i5

10

1.75

.:::

"

8

Fig. 10.15. Time history of acceleration of Pasadena earthquake.

2.00

..E ...

6

Time (Sec)

(b)

C

4

0.75 0.50 0.25

0

2

10

6

16

Time ( Sec)

Fig. 1O.14(a). Horizontal displacement at the top of the failure surface. (b). Vertical displacement at the top of the failure surface.

429

428 The concept of superposition can be used not only to handle the variation of acceleration throughout slopes but also to solve the multi-layer property of a slope. However, to apply the concept of superposition to multi-layer slopes the compatibilities of failure surfaces among all the layers have to be satisfied. According to the associated flow rule of limit analysis, the shape of the failure surface is directly related to the frictional angle cP and for a multi-layer slope, the angle cP is different for each layer. Thus, to find the critical state of nonhomogeneous slope, some studies have been made by Chen and Sawada (1983). The failure surface is assumed here to pass through the toe. In some cases, the failure surface may pass below the toe. For simplicity, we consider that the inclination on the upper part of slope is horizontal. If the slope has an inclined upper part and the failure surface may pass below the toe, the procedure described in this chapter for analyzing the seismic response of slope during earthquake can still be applied. Details of this general formulation are not given here. The related equations for a general slope without the seismic effect can be found in the book by Chen (1975). For seismic slope stability analysis, in addition to calculating the yield acceleration and failure mechanism of a slope after a given earthquake, the effect of earthquake on the displacements of a slope must also be assessed. Based on the calculated yield acceleration and its corresponding failure mechanism, Newmark's analytical procedure is used to assess the soil displacements of the earth slope which is subjected to a design earthquake. It is the magnitude of the displacement which forms the basis for assessing the stability and adequacy of the section and not the seismic factor of safety. This is the second part of the work as described in the later part of this Chapter. The method is particularly useful in cases where the yield resistance of the soil can be reliably determined and therefore for the analysis of cases where pore-water pressures do not change significantly as the earthquake motions continue or shear displacements occur. This leads to the conclusion that the pseudo-static analysis procedure provides an acceptable method of analysis for some types of soil, which do not build up large pore pressures or cause significant strength loss due to earthquake shaking and associated displacements. In fact, the Newmark analytical procedure for assessing the seismic displacement of a slope is based on the effective stress method. If the pore-water pressure does not change significantly or the induced pore-water pressure can be appropriately evaluated during an earthquake, this procedure can be effectively applied for analyzing the seismic displacement of a slope with saturated soils. Sarma (1975) attempted to use pore-pressure parameters to evaluate the seismic displacement of earth dams based on an effective stress analysis. In the evaluation of the seismic displacement of a slope during earthquake, the theory of perfect plasticity together with its associated flow rule of Mohr-Coulomb

II

material are assumed. This may overestimate the volume change for some soils such as sands. A similar procedure can be applied for different types of soil models. Furthermore, even if there exist someJimitations in the present procedure, we can extend and modify the present procedure to make it more suitable to the analysis of the related geotechnical engineering problems. References Chang, C.l., 1981. Seismic Safety Analysis of Slopes. Ph.D. Thesis, School of Civil Engineering, Purdue University, West Lafayette, IN, 125 pp. Chang, C.l., Chen, W.F. and Yao, l.T.P., 1984. Seismic displacements in slopes by limit analysis. l. Geotech. Eng., ASCE, 110(7): 860 - 874. Chang, C.l., Chen, W.F. and Yao, l.T.P., 1985. Evaluation of seismic factor of safety of a submarine slope by limit analysis, Proc. 1983 Symposium on Marine Geotechnology and Near Shore/Offshore Structures, Shanghai, China, Tongi University Press, pp. 262-295. Chen, W.F., 1975. Limit Analysis and Soil Plasticity. Elsevier Amsterdam, 630 pp. Chen, W.F., 1980. Plasticity in soil mechanics and landslides. l. Eng. Mech. Div., ASCE, 106(EM3): 443-464. Chen, W.F. and Sawada, T., 1983. Earthquake-induced slope failure in nonhomogeneous anisotropic soils. Soils Found., lpn. Soc. Civ. Eng., 23(2): 125 -139. Chen, W.F., Chang, C.l. and Yao, l.T.P., 1978. Limit analysis of earthquake-induced slope failure. In: R.L. Seirakowski (Editor), Proc. 15th Annual Meeting of the Society of Engineering Science. University of Florida, Gainesville, FL, pp. 533 - 538. lanbu, N., 1957. Earth pressure and bearing capacity by generalized procedure of slices. Proc. 4th Int. Conference on Soil Mechanics, 2, pp. 207 - 212. Newmark, N.W., 1965. Effects of earthquake on dams and embankments. The Fifth Rankine Lecture , of the. I!ritish .Geotechnical Society."Geotechnique, 15(2): 137 -160. Sarma, S.K., 1975. Seismic stability of earth dams and embankments. Geotechnique 25(4): 743 -761. Seed, H.B., 1966. A method for earthquake resistance design of earth dams. l. Soil Mech. Found. Div., ASCE, 92(SM1): 13 -41. Sigel, R.A., 1975. STABL User Manual, Joint Highway Research Project, JHRP-75-9. School of Civil Engineering, Purdue University, West Lafayette, IN, 112 pp.

APPENDICES: Program Listings Appendix 1:

Plane Failure Surface

c---------------------------------------------------------c This program can be used to obtain the yield acceleration c factor for plane failure surface c c

c c

h c. f a

height of slope (ft) cohesion of soil (psf) friction angle of soil (degree) slope angle (degree)

430 c

c c c c c c

c c c c

431 p uniform surcharge (psf) x seismic coe~ficient of surcharge xns: stability factor (= #rh/c) r average unit weight of soil xl,x2,dx minimum,maximum values and interval of angle respectively in the process of searching for the yiel~ acceleration factor 'Xmin critical value of yield acceleration factor xa : failure surface angle corresponding to xmin

c---------------------------------------------------------_ read*,h,c,f,a read*,p,x,r xns=r*h/c c---- 1st-round trial xl=0.05*a x2=0.95*a f=f/57.2958 a=a/57.2958 dx=l. call xx(xl,x2,dx,a,f,c,p,x,xns,xa,xmin) c---- 2nd-round trial xl=xa-l. x2=xa+l. dx=0.05 callxx(xl,x2,dx,a,f,c,p,x,xns,xa,xmin) c---- 3rd-round trial xl=xa-0.05 x2=xa+0.05 dx=O.OOI call xx(xl,x2,dx,a,f,c,p,x,xns,xa,xmin) print*,xa,xmin stop end c-----subroutine to search for the critical value subroutine xx(xl,x2,dx,a,f,c,p,x,xns,xa,xmin) xmin=3. xa=90. m=(x2-xl)/dx do 10 i=I,m+l bb=x 1+( i-I) *dx b=bb/ 57.2958 bl=cos(f)/cos(b-f) b2=sin(a)/sin(a-b) b3=p*sin(f-b)/(c*cos(f» b4=0.5*xns*sin(f-b)/cos(f) tl=bl*(b2+b3+b4) b5=0.5*xns+x*p/c xkc=tl/b5 if(xkc-xmin.lt.O.and.xkc.gt.O.) then xmin=xkc

xa=bb else go to 10 endif 10 continue return end Appendix 2:

Logspiral Failure Surface

c---------------------------------------------------------c This program can be used to obtain the yield acceleration c factor for logspiral failure surface c c

c c c c c c c c

c c c c c c c c c c c

c c c c c

height of slope (ft) h c cohesion of soil (psf) f friction angle of soil (degree) a slope angle (degree) sl the 1st-round trial value of theta 0, (60 - 120) p uniform surcharge (psf) x seismic coefficient of surcharge xns: stability factor (= flrh/c) r average unit weight of soil xl,x2 minimum, maximum values of theta 0 respectively in the process of searching for the yield acceleration factor yl,y2 minimum, maximum values of theta h respectively in the process of searching for the yield acceleration factor interval of theta 0 and theta h in the process da of searching for the yield accleration factor the distance between the rotation center and the rO top of failure surface ratio of surcharge length to rO lr smin critical value of yield acceleration factor xx : the value of theta 0 corresponding to smin yy : the value of theta h corresponding to smin

c---------------------------------------------------------real lr read*,h,c,f,a,sl read*,p,x,r,ss xns=r*h/c c---- 1st-round trial xl=-0.2*a x2=sl yl=90-0.6*a y2=175 da=I.0

432

a=a/57.2958 f=f 157. 2928 call vx(x1,x2,y1,y2,da,a,f,p,c,x,xns,xx,yy,smin,ss) c---- 2nd-round trial x1=xx-1 x2=xx+1 y1=yy-1 y2=yy+1 da=O.OS call vx(x1,x2,y1,y2,da,a,f,p,c,x,xns,xx,yy,smin,ss) xx1=xx/57.2958 yy1=yy/57.2958 tt=(yy1-xx1)*tan(f) rO=h/(sin(yy1)*exp(tt)-sin(xx1» lr=(sin(a+xx1)-exp(tt)*sin(a+yy1»/sin(a) print*,xx,yy,smin,rO,lr stop end c---- subroutine to search for the critical value subroutine vx(x1,x2,y1,y2,da,a,f,p,c,x,xns,xx,yy 1,smin,ss) real lr xx=90 yy=180 smin=3. m=(x2-x1)/da n=(y2-y1 )/da do 10 i=l,m aa 1=xl+(i-1 )*da a1=aal/57.2958 do 10 j =1, n' aa2=y l+(j -1 )*da a2=aa2/57.2958 t1=(a2-a1)*tan(f) t2=3.*(1.+9*tan(f)*tan(f» ck=sin(a2)*exp(t1)-sin(a1) if(ck.lt.O.O) go to 10 lr=(sin(a+a1)-exp(t1)*sin(a+a2»/sin(a) if(lr.lt.O.O) go to 10 fc=(exp(2.*t1)-1)/(2.*tan(f» f1=«3.*tan(f)*cos(a2)+sin(a2»*exp(3.*t1)-(3.*tan( If )*cos (a1 )+sin( 1a1»)/t2 f2=lr*(2.*cos(a1)-lr)*sin(a1)/6. f3=exp(t1)*(sin(a2-a1)-lr*sin(a2»*(cos(a1)-lr+cos( 1a2)*exp(t1»/6. f4=«3.*tan(f)*sin(a2)-cos(a2»*exp(3.*t1)-(3.*tan( 1f)*sin(a1)-cos(a1»)/t2 f5=lr*sin(a1)*sin(a1)/3. f6=exp(t1)*(sin(a2-a1)-lr*sin(a2»*(sin(a1)+sin(a2) 1*exp(t1»/6. fp=lr*(cos(a1)-0.S*lr)

433 fq=lr*sin(a1) b1=fc-xns*(f1-f2-f3)/(sin(a2)*exp(t1)-sin(a1»-p* 1fp/c b2=ss*xns*(f4-fS-~6)/(sin(a2)*exp(t1)-sin(a1»+x*

1p*fq/c xkc=b1/b2 if(xkc-smin.lt.O.and.xkc.gt.O.) then smin=xkc xx=aa 1 yy=aa2 else go to 10 endif 10 continue return end Appendix 3:

Limit Analysis During Earthquake

c---------------------------------------------------------

This program can be used to analyze the response of slopes by limit analysis method during earthquake (EQ) c c

c c c c

c c c c c c

$, J I

I I ~

j

i

c c c

c

c c c c c c

c c

c c c c

ug(i): acceleration for EQ record or motion of slope mass d(i) : accumulated rotation of slope mass during EQ v(i) : angular velocity of slope mass during EQ dx (i) : displacement in horizontal direction for some specified PClint a10Ilg f,ail~r!,!, surface dy(i): displacement in vertical direction for some specified point along failure surface m1 length of acceleration,no. of EQ record input mm no. of lines of EQ record input (mm=ml/8) dt time interval scale: scaling factor of EQ h height of slope (ft) c cohesion of soil (psf) friction of soil (degree) f slope angle (degree) a theta 0 (degree) a1 theta h (degree) a2 yield accel. factor xkc surcharge loading (psf) p seismic factor of surcharge x stability factor (= rh/c) xns average unit weight of soil (pcf) r the distance between the rotation center and rO the top of failure surface 'coefficient in Eq. 10.41 af,t,er the ,induced accleration factor substracted by yield acceleration factor

cc

435

434 c c c c

rx

the distance betwein the rotation center and the specified point along failure surface theta: the value of angle from the horizontal direction corresponding to rx

c-------~-------------------------------------------------

dimension ug(900),d(900),v(900) dimension dx(900),dy(900) read* ,m1 ,mm read*,dt,scale c---- input earthquake record do 10 i=l,mm read(5,20) (ug(j),j=i*8-7,i*8) 20 format(8f9.6) 10 continue

70 c---80

c--------------------------------------------------------c---- scaling earthquake record 30 40 c----

c----

c---c----

50 c---60 c----

ugmax=O. do 30 i=1,m1 if(abs(ug(i))-ugmax.gt.O.) ugmax=abs(ug(i)) continue ratio=scale/ugmax do 40 i=1,m1 ug(i)=ratio*ug(i) input the required data for slope geometry, etc. read*,h,c,f,a,a1,a2,rO read*,p,x,r,xkc read*,theta xns=r*h/c open files for saving the displacement open(8,file='t1',status='new') open(9,file='t2',status='new') b=b/57.2958 f=f/57.2928 a1=a1/57.2958 a2=a2/ 57.2958 1\ theta=theta/57.2958 ~ rx=rO*exp(tan(theta-a1)) find the value of cc (defined above) call vx(a1,a2,a,f,h,p,c,x,r,xns,cc) set initial values of followings as zero do 50 i=l,ml d(i)=O. v(i)=O. dx(i)=O. dy(i)=O. continue subs tract accel. of earthquake by yield accel. do 60 i=1,m1 ug(i)=(ug(i)-xkc)*cc continue search for time at which the positive movement starts do·70 i=1,m1

100 200

if(ug(i).gt.O.) then k=i go to 80 else go to 70 endif continuebr compute the accumulated angular rotation tx=-dt*ug(k-l)/(ug(k)-ug(k-1)) v(k)=0.5*ug(k)*(dt-tx) d(k)=ug(k)*(dt-tx)*(dt-tx)/6.0 mx=m1-1 do 90 j =k, mx v(j+l)=v(j)+0.5*(ug(j)+ug(j+l))*dt if(ug(j).gt.0.0.and.ug(j+1).gt.0.) go to 100 if(ug(j).gt.0.0.and.ug(j+1).le.0.) go to 200 if(ug(j).le.0.0.and.ug(j+1).gt.0.) go to 300 if(ug(j).le.0.0.and.ug(j+1).le.0.) go to 350 d(j+l)=d(j)+v(j)*dt+(2*ug(j)+ug(j+l))*dt*dt/6.0 go to 90 . if(v(j+1).ge.0) go to 100 b1=(ug(j+1)-ug(j))/dt tn=-(ug(j)+sqrt(ug(j)*ug(j)-2.~bl*v(j)))/bl

300

350

352 351 301

d(j+1)=d(j)+v(j)*tn+(2.*ug(j)+tn*b1)*tn*tn/6. v(j +1)=0. go to 90 if(v(j+1).le.O) go to :301 b1=(ug(j+l)-ug(j))/dt tn3=-ug(j )/b1 vm=v(j)+0.5*ug(j)*tn3 if(vm.le.O) go to 301 go to 100 if(v(j+l).ge.O.) go to 100 b1=(ug(j+l)-ug(j))/dt if(b1.lt.0.0001) then tn=-v(j )/ug(j) go to 351 else go to 352 endif b2=sqrt(ug(j)*ug(j)-2.*bl*v(j)) tn=(-ug(j)-b2)/b1 v(j+1)=0. d(j+1)=d(j)+v(j)*tn+(2.*ug(j)+tn*b1)*tn*tn/6.0 go to 90 bl=(ug(j+1)-ug(j))/dt b2=sqrt(ug(j )*ug(j )-2. *b l*v(j)) tn1=(-ug(j)+b2)/b1 tn2=(-ug(j)-b2)/bl if(tnl-tn2.gt.O) tn=tn2 tn=tnl d(j+1)=d(j)+v(j)*tn+(2.*ug(j)+tn*b1)*tn*tn/6.0

436 tn3=-ug(j)/bl v(j+l)=0.S*ug(j+l)*(dt-tn3) ~j +1 )=d (j +,1 )+ug(j +1 )*(dt-tn3)*(dt-tn3 )/6.0 90 corttinue c---- compute the accumulated displacement do 400 j =1 ,ml dx(j)=rx*d(j)*sin(theta) dy(j)=rx*d(j)*cos(theta) 400 continue do 500 j =1 ,ml tt=dt*(j -1) write(8,SOl) tt,dx(j) write(9,SOl) tt,dy(j) 500 continue 501 format(10x,2flO.7) stop end c---- subroutine to search for the value of cc subroutine vx(al,a2,b,f,h,p,c,x,r,xns,cc) real lr,l tl=(a2-al)*tan(f) t2=3.*(1.+9*tan(f)*tan(f» lr=(sin(b+al)-exp(tl)*sin(b+a2»/sin(b) fc=(exp(2.*tl)-1)/(2.*tan(f» fl=«3.*tan(f)*cos(a2)+sin(a2»*exp(3.*tl)-(3.*tan( If)*cos (al)+sin( lal) »/t2 f2=lr*(2.*cos(al)-lr)*sin(al)/6. f3=exp(tl)*(sin(a2-al)-lr*sin(a2»*(cos(al)-lr+cos( la2)*exp(tl»/6. f 4= ( ( 3. * tan ( f) * s in (a 2 ) - cos ( a 2 ) )'* ex p ( 3 • ,\ t 1 ) - ( 3 • * tan ( If)*sin(al)-cos( lal»)/t2 fS=lr*sin(al)*sin(al)/3. f6=exp(tl)*(sin(a2-al)-lr*sin(a2»*(sin(al)+sin(a2) 1*exp(tl)/6. fp=lr*(cos(al)-O.S*lr) fq=lr*sin(al) rO=h/(sin(a2)*exp(tl)-sin(al» xl=r*rO*rO*rO*(f4-fS-f6)+x*p*rO*rO*fq w=0.S*r*rO*rO*(0.S*(exp(2.*tl)-1)/tan(f)-lr*sin(al) 1-(sin(a2-al)-lr 1*sin(a2»*exp(tl» x2=r*rO*rO*rO*(fl-f2-f3) x3=r*rO*rO*rO*(f4-fS-f6) l=sqrt(x2*x2+x3*x3)/w cc=32.2*xl/(w*l*l) return end

437

Chapter 11

STABILITY ANALYSIS OF SLOPES WITH GENERALIZED FAILURE CRITERION 11.1 Introduction

In many practical problems, such as the frozen gravel embankments, concepts adopted recently in offshore arctic engineering, sufficient experimental data have shown that the frozen gravel follows a highly nonlinear failure criterion, but it is difficult to obtain experimental data on its deformations. As a result, finite-element methods cannot be applied directly to solve such problems because of a lack of specific information about the behavior of this material under in situ conditions. Fortunately, our main interest here is the overall stability of slopes, not the detailed history of stresses and deformations. Limit analysis methods, together with a nonlinear failure criterion, provide a powerful tool to engineers to obtain approximate solutions under this situation. All methods of stability analysis in soil 'mechanics are highly dependent on the particular failure mechanism chosen for the problem. The selection of a proper failure mechanism is therefore of great importance for properly assessing the collapse load. It has been shown by the variational calculus that the logarithmic spiral rotational failure mechanism utilized in the upper-bound limit analysis solution is the appropriate failure surface for a rigid-body type of rotational sliding mechanism (Chen and Snitbhan, 1975). However,this conclusion is true only for the material that follows the linear Mohr-Coulomb failure criterion. We cannot immediately apply the linear limit analysis method to nonlinear failure problems. It is necessary therefore to investigate the soil stability problems and to develop practical solution methods based upon a generalized failure criterion. This is described in the present Chapter. When the failure criterion in 0'-7 space is a straight line, the logarithmic spiral rotational failure mechanism utilized in both the limit analysis and limit equilibrium solutions can be obtained directly without the information about the normal stress distribution along the failure surface. When the failure criterion is nonlinear, however, the calculation of the internal dissipation of energy along the slip surface is influenced by the normal stress distribution. In this case, the variational calculus can be used to obtain solutions of stability problems in soil mechanics. In 1970, Chen suggested to use the variational calculus to obtain the normal stress distribution along the slip surface for stability problems in soil mechanics. Afterwards,

439

438

Chen and Srtitbhan (1975) applied the variational method to obtain the shape of the slip surface and its corresponding normal stress distribution of a vertical slope. Bakerand Garber (1977) suggested a similar variational approach to solve bearingcapacity problems. Baker (1981) extended the same approach to include both the tensile strength and tension cracks in slope stability problems. Baker and Frydman (1983) appear to be the first to discuss the effect of nonlinearity of a generalized failure criterion on the upper-bound solution procedure. They applied the variational calculus to formulate the bearing capacity of a strip footing on the upper surface of a slope under the nonlinear failure case. Two solution procedures are suggested. One is based on minimization with four unknown parameters. Another is based on solving a system of four simultaneous equations. Both solution procedures proposed by Baker and Frydman, however, are not realistic except in very special cases. Zhang and Chen (1987) adopted the same variational calculus, developed an inverse solution procedure, and obtained numerical results for the critical height of an embankment. Uu et al. (1989) recently developed a new effective solution procedure called the combined method, suitable for the slope stability problems with a generalized nonlinear failure criterion. By this method, they obtained extensive numerical results for the critical height of slopes and the bearing capacity of a strip footing on the upper surface of a slope, and extended the method to obtain solutions of the stability of layered frozen gravel embankments. In this Chapter, the variational calculus approach in the limit analysis of strip footings is first introduced. A solution procedure suitable for the analysis of slope stabilityproblems is then,eJ{plaiI!eci,and finally, a realistic engineering problem in~ volving a layered analysis with nonlinear failure c;iteri()n; is described. 11.2 Variational approach in limit analysis and the combined method

The solution procedure required in an upper-bound limit analysis consists of the following steps. First, a class of kinematically admissible collapse mechanisms is postulated in terms of some geometrical variables. Second, for a set of values of the geometrical variables, an estimate of the collapse load is obtained by equating the rate of external work to the rate of internal dissipation. The estimate of the collapse load is expressed in terms of the geometrical variables defining the collapse mechanism. Third, the lowest value of these estimates represents the least upperbound value for this class of collapse mechanisms. Probably, other classes of collapse mechanisms may be considered, and the least upper-bound obtained from all classes of assumed mechanisms is taken as the best estimate of the true collapse load. The application of the variational calculus approach to the upper-bound limit analysis provides a systematic procedure to find the lowest value of the collapse load

required in the third step. To this end, we shall develop a more rigorous approach and obtain both the failure surface and the critical stress distribution that will provide a more complete result ,of the stability problems in soil mechanics. To focus our attention on. the implication of the nonlinearity of the failure criterion on the stability analysis, we shall take here the simplest type of kinematically admissible discontinuous velocity field, that is, a single rigid-body rotation along a surface of velocity discontinuity. The soil is assumed to be isotropic and homogeneous and no pore-water pressure exists. The variational calculus approach developed in the forthcoming considers the general case of bearing capacity of a strip footing on the upper surface of a slope (Fig. 11.1). The failure criterion of the soil can generally be expressed as: T

= f(a)

(11.1)

where a and T are the normal and shear stresses on the failure surface respectively (Fig. 11.2). If the plastic normal strain ratei:P and the plastic shear strain rate .yp are superimposed on the a-T space, the plastic strain rate vector (i:P, .yP) should be normal to the yield curve at the yield stress state. In Fig. 11.2, the plastic strain rate vector is seen making an angle cPt' the tangential friction angle, with the plastic shear strain rate vector .yP, where: dT df(o) tan cP = - = - t da da

(11.2)

The upper·bound method requires that the rate of external work be equated to the rate of internal dissipation for all plastically deformed zones. This requirement may be expressed as: b P b 2

2

Fig. 11.1. A strip footing on a slope surface.

441

440 W(P, 1])

=

f D(1]i) d v v

(11.3) V

where W is the rate of external work, P is the applied load, 1]i are a set of geometrical parameters defining the failure mechanism, and D is the rate of dissipation of energy per unit volume. For the present purpose, it is more convenient to write Eq. (11.3) in the form: Q(P, 1])

=

JD(1]) d V v

= 0

7

M =

7

cosO)ds

+

J (y ~ y)

'Y dx

+P

(11.7)

~

J[(u cosO

7

sinO) y

+

(u sinO

+

7

cosO) x]ds

s

J 'Y (y

(11.4)

(11.5)

where M is the resultant moment about some reference point. H, Vare the resultant horizontal and vertical forces acting on the rigid body. 0 is the rate of virtual rotation of the rigid body. it and v are the rates of horizontal and vertical virtual displacements of the reference point. In the present case, the reference point is taken at point 0 in Fig. 11.1, through which the external load P is applied. Referring to Figs. ILl and 11.3, the resultant horizontal, vertical forces and moment with respect to 0 can be written as:

J(u cosO -

(u

-

y)

(11.8)

x dx

Xo

Q=itH+vV+OM=O

=

J sinO + s

oX;,

W(P, 1]i)

where Q may be defined as the total work of the system at collapse. Considering the equilibrium of the rigid body shown in Fig. 11.1, the total virtual work of the sliding mass may be written as:

H

= -

oX;,

sinfJ)ds

where y = y(x) is the equation describing the discontinuous surface, i.e., the slip surface along which the nonlinear yield condition (11.1) is satisfied. u = u(x) and 7 = 7(X) are the normal and shear stress distributions along the slip surface y(x), respectively (Fig. ILl). Xo and x n are the end points of y(x). s is the arc length along y(x), and the angle 0 satisfies:

dy

cot fJ

(11.9)

=-

dx

The equation soil.

y

y (x) describes the soil surface,

(11.6)

s

P

?1:('

Y d

I

COMPRESSION

y

Fig. 11.2. A nonlinear yield criterion and its associated flow rule.

Fig. 11.3. The kinematic constraints of the slope.

u

and 'Y is the unit weight of the

442

443

Re-arranging the integrations in Eqs. (11.6) to (11.8), we have: gz =

~

H

£(u: -

=

(11.10)

7)dx

g3

~

V =

I [- (u + + £[(u + :)x -

(y -

7 :)

7

Taking

(11.11)

(y -

Y)

~ x+ (u:

- 7)Y] dx

(11.12)

vas unity and defining:

(VI

+H

£I

+ M 0) =

0

Substituting Eqs. (11.10) to (11.12) into Eq. (11.14), and assuming 0 tain: ~

1

P= {[u+ 7:~- (y-

(11.14)

'* 0, we ob-

gin:

P G[y(x), u(x),dy/dx, U, 0] f g dx =

(11.19)

Xo

is stationary when its first variation vanishes. In Eq. (11.19), G is the functional relation between the load and the unknown functions £I, and O. From Eq. (11.14), we can conclude that G represents the work done by the body forces (soil weight) and the dissipated work along the slip surface. G is not related to the external load P. The least upper bound value P can therefore be obtained by minimizing this functional relation with respect to the parameters rJi defining the failure mechanism. This minimization process must also be subjected to the constraint conditions: the failure mechanism is kinematically admissible. This procedure can be summarized in the following equations:

y(x), u(x),dy/dx,

(11.20)

P min = Min [G)

rJi

YhJ- u(u:~- 7)+ o[-(u+ 7:~)X

(11.21) where P min is the minimum value of P, and K(rJi) represents the kinematic constraints. If the u and 0 have been chosen, then we can use the Euler equation to obtain the necessary conditions for the stationary requirement. One of these conditions is:

or ~

P =

(11.18)

jI)

P

Equation (11.5) may be solved for P and yields:

+

- x)(y -

(11.17)

~

(11.13)

P

=(~

x) - :~(~ + Y)

Now the problem becomes a typical problem in variational calculus. The integral

Y) ~ ] dx + P

~

M =

(~ -

f 0 (7 gi + u gz -

~ g3)dx

(11.15)

Xo

oG ag - =- =

(11.22)

0

au

au

where

a is the variational operator. Eq.

(11.22) can be rewritten as:

where: (11.16)

[(~ + y)+ *(i - x)J~ +[(i - x)- *(~ + y)]= 0 where f(u) is given by Eq. (11.1).

(11.23)

444

445 da

Using the polar coordinate transformation (Fig. 11.3):

dlJ + 27 - 'Y r coslJ = 0

x

= Xc -

r coslJ

(11.24)

Y

= Yc + r sinlJ

(11.25)

~he polar coordinate system rand IJ centers at point (xc' Yc)' By the definitions of 0, ii, and ii, we have:

(11.26) (- Yc)

0=

U

(11.27)

Le. (11.28)

(11.33)

Equations (11.30) and (11.33) are the two necessary conditions for a minimum solution of the problem. However, by the upper bound theorem of limit analysis, the existence of such a minimum solution is guaranteed. Thus, there is no need to study the nature of the stationary point using the second variation. Equations (11.30) and (11.33) constitute a pair of simultaneous differential equations for the determination of the functional forms of r(lJ) and a(IJ). Obviously, four boundary conditions are needed. According to the Peano's theorem, the solution of Eqs. (11.30) and (11.33) can always be written in the forms: r = r(1J I A, B, ii, Q)

(11.34)

a = a(1J I A, B, it,D)

(11.35)

Using Eqs. (11.24), (11.25), (11.28), and (11.29), Eq. (11.23) can be simplified to the form:

in which, A, B are the two integration constants and it, 0 define the origin of the polar coordinate system. Note that Eqs. (11.34) and (11.35) do not imply the existence of explicit forms of solutions for r(lJ) and a(lJ) along the failure plane. It only emphasizes the dependence of the solutions on the four unknown parameters A, B, it, O. By substituting Eqs. (11.34) and (11.35) into Eqs. (H.20) and (11.21), we can write the general solution in the form:

dr

Pmin =

= -

Yc

-=

dlJ

o

df(a)

r-da

(11.29)

(11.30)

Min A,B,

u,n

K(A, B, it, 0)

Similarly, the other necessary condition can be found by the condition: (11.31)

Le.: (11.32) and this equation can be simplified by the transformation to the polar coordinate shown in Fig. 11.3:

=

[O(A, B, ii, 0)]·

(11.36)

0

(11.37)

Thus, the solution procedure for the stability problem is based on the minimization of the functional 0 with respect to the parameters, A, B, it, and 0 satisfying the kinematic constraints (11.37). The parameters, A, B in fact depend on the boundary conditions. For example, as shown in Fig. 11.1, the slip surface starts at the footing edge, Le., Xo = - b/2 and Yo = O. For an assumed set of values of ao, it, and 0, the pair of simultaneous differential equations, Eqs. (11.30) and (11.33), can be converted to an initial value problem, and we can obtain the distributions of the functions r(lJ) and aClJ) by a formal numerical method without difficulty. According to Eq. (11.19), this minimization can also be achieved by considering the conditions aOlail = 0 and aOlao = O. From Eqs. (11.14) and (11.19), we have:

446

447

~

G~):~

H

M ( a~):;" an..

=

0

(11.38)

='0

(11.39)

.

Therefore, the minimization of G (or P) with respect to iI and 0 is equivalent to the satisfaction of the equilibrium conditions H = 0, M = O. Furthermore, from Eq. (11.5), we also have V = 0, it follows that the minimization of G (or P) is equivalent to enforce the equilibrium requirement. In other words, we can use the equilibrium conditions H = 0 and V = 0 to check the minimum value G (or P) so obtained in a numerical procedure. Introducing the equilibrium function: (11.40) The general procedure of the variational'calculus approach can now be summarized as follows (Wang and Liu, 1988):

Pmin =

Min

[G]

Fig. 11.4. Geometric significance of the plastic velocity vector d.

tan
(11.41)

80 , (10. U. iJ

which implies that the plastic velocity vector,
F=O

(11.45) ~,

at the slip surface acts at an angle

(11.42) 11.3 Stability analysis of slopes

T

= f(O')

K«()o,

0'0'

(11.43)

ii,

0) =

0

(11.44)

It is worth to note the following points:

(1) The proposed combined method is neither a conventional upper-bound method nor a conventionallower·bound method. It combines the upper-bound and lower-bound methods together. The assumed admissible velocity field not only satisfies the kinematic constraints and the yield criterion, but also satisfies the equilibrium equations. The combined method appears to lead to the best solution within the framework of rigid-body sliding and perfect plasticity for the material. (2) The kinematic collapse mechanism so obtained satisfies the associated flow rule. For example, the necessary condition (11.30) is in fact the normality requirement for a rigid-body rotation. As shown in Fig. 11.4, ~n and ~t are normal and tangential components of the plastic velocity vector ~ to the slip surface respectively. From Fig. 11.4 and Eqs. (11.2) and (11.30), we find:

In the preceding section, we have introduced the variational calculus approach in the limit analysis and the combined method by an illustrative example, Le., the bearing capacity of a strip footing on the upper surface of a slope (Fig. 11.1). In the following, we shall discuss the general solution procedure. Since it is possible that when P = 0, the soil mass can still slide under its own weight, the critical height of a slope must also be considered in the analysis. To find the critical height of a slope, we can use the same solution procedure as we did for the bearing capacity of a strip footing on the upper surface of a slope, but let P = 0 and let the starting point of the slip surface flexible.

11.3.1 The solution procedure for the bearing capacity of a strip footing on the upper surface of a slope As shown in Fig. 11.5, the solution procedure can be summarized as follows: Step 1: Input soil property data and known slope geometric parameters of the strip footing on a slope.

II

448

449 Step 8: Change the initial normal stress lTo, repeat Step 3 to Step 7 until P min is found. Step 9: Output the results .. _ _ 11.3.2 The solution procedure for the critical height of slopes

y Fig. 11.5. Bearing capacity problems with slope angle (3.

As shown in Fig. 11.6, there is no external load, and the length L is unknown. Thus, we have: Step 1: Input soil property data and the known slope geometric parameters. Step 2: Assume the length L. Step 3: Assume the initial normal stress lTo' Step 4: Assume the coordinates of the rotation center (xc' Ye>, and calculate the initial radius ro and the initial angle 80 , Step 5: Select the step size /l8. Using a numerical method to calculate the values of r(8) and IT(8) from Eqs. (11.30) and (11.33) until the slip surface reaches the slope surface. Step 6: Calculate the resultant horizontal and vertical forces Hand V and the load P by the numerical integration. Step 7: Use an optimization method to repeat Step 4 to Step 6, until the load P reaches a minimum value. Step 8: Calculate the equilibrium function F when P is a minimum value. Step 9: Change lTo, and repeat Step 3 to Step 8, until F is very close to zero. Step 10: Determine the coordinates of the exit point of the slip surface (xn' y n). Let the height of slope H o = YnStep 11: Change L, and repeat Step 2 to Step 10, until the minimum H (Hmin ) is found. Step 12: Output H cr = H min • 11.3.3 Numerical results

Step 2: Assume the initial normal stress lTo' Step 3: Assume the coordinates of the rotation center (xc' Yc)' and calculate the initial radius ro and the initial angle 80 , Step 4: Select the step size /l8. Using a numerical method to calculate the values of r(8) and IT(8) from Eqs. (11.30) and (11.33), until the slip surface reaches the slope surface. Step 5: Calculate the resultant horizontal and vertical forces H and V by the numerical integration. Step 6: Use an optimization method to repeat Step 3 to Step 5, untilF = .JH2 + V2 is very close to zero. Step 7: Calculate the external load P from Eq. (11.15) and the corresponding coordinates of the exit point of the slip surface (xn, y n ).

Define the stability factor N s as: (11.46) where c is the nominal cohesion of the nonlinear failure curve shown in Fig. 11.2. The height Her is the critical value of the slope as shown in Fig. 11.6. The following three-parameter failure criterion is used. A similar criterion has been used previously by Zhang and Chen (1987): (11.47)

451

.450.

where c is the nominal cohesion of soil. . Ij> is the nominal angle of internal friction of soil. m·is the nonlinear coefficient. It can be seen that when m = 1, Eq.(l1.47) reduces to the well-known linear Mohr-Coulomb failure criterion. Comparison of the present numerical results for the linear case with the existing limit analysis solutions(Chen, 1975) is given in Fig. 11.7 and Table 11.1. It is seen that the agreement is good and in most cases, the combined method gives lower solutions. From Table 11.1, if we change the values of c and 'Y, and keep 'Y/c constant,

-0.06

-0.07

a (YH I m3 0.25

Her

.=

e = 73 (a)

0.26

0

(b)

Chen (19 7 5)

Combined Method

Fig. 11.7. Comparison of the distributions of the normal stresses (f3

= 70°,

c/>

= 20°).

Ns y

m=1.0

14

Fig. 11.6. Critical height problem.

£

0.8

0

2 TABLE 11.1 Comparison of the Ns:values for the combined method with the linear limit analysis

I 2

(3

90°

80°

70°

60°

Linear limit analysis (Chen, 1975) Nonlinear method (a) c = 0.01 MPa, 'Y = 0.0016 kg/cm l Relative error (OJo) (b) c = 0.015 MPa, 'Y = 0.0024 kg/eml Relative error (0J0)

5.50

6.75

8.13

10.39 13.45

5.50 0.00 5.46 0.72

6.71 0.59 6.68 1.00

8.23 1.23 8.22 1.11

10.35 13.63 0.38 1.34 10.28 13.30 1.06 1.12

Note: The Mohr-Coulomb criterion with c/>

=

20° is used in both cases.

50°

10

>-

0.6

:0 '" C;;

0.4 6

2

9LO---..J80'------'70------'6-0----5L..O----~-

pO

Slope angle

Fig. 11.8. The relationship between N s and (3 (q,

=

20°, c

=

0.01 MPa, 'Y

=

0.0016 kg/eml ).

452

453

the combined method leads to different answers, while the analysis for a linear failure criterion leads to the same answer.

Effect of the nonlinear coefficient m on the stability factor N s Values of the stability factor N s corresponding to constant values of the nominal cohesion c and the nominal angle of internal friction cP with the nonlinear coefficient m varying from 0.4 to 1.0 are given in Fig. 11.8 (cP = 20°, c = 0.01 MPa, and

TABLE 11.2 Influence of the nonlinear coefficient m on the bearing capacity P (unit: kN/m)

100

m= 1.0

P=28.94 kN 1m

0.8

14.72 kN/m

0.6

0.61 kN/m

0.4

-10.56 kN 1m

a

300

400

y (em)

Fig. 11.10. The failure mechanisms for bearing capacity problems (fJ = 70°, H o = 400 em, b = 250 em).

Fig. 11.9. The relationship between P and m.

455

454 'Y = 0.0016 kg/cm 3). It is observed that the stability factor N s decreases significant-

ly with a decrease in the nonlinear coefficient m, especially in the range of less slope angle {J.

m=1.0

H cr = 508.3 em

0.8

449.9 em

0.6

399.6 em

0.4

356.2 em

Effect of the nonlinear coefficient m on the bearing capacity As shown in Fig. 11.5, Table 11.2 gives the relationship between the bearing capacity Pand the nonlinear coefficient m. According to Fig. 11.9 and Table 11.2, it can be seen that when {J = constant, the bearing capacity of slope decreases with a decrease in the nonlinear coefficient m. Effect of the nonlinear coefficient m on the failure mechanisms Figure 11.10 shows that when {J = 70°, H o = 400 cm, b = 250 cm, even when the bearing capacity has been reached, the effect of m on the failure mechanisms is still not obvious. From Fig. 11.11, it is found that when (J = 70°, the critical height of slope decreases with a decrease in the nonlinear coefficient m. Effect of the nonlinear coefficient m on the distribution of normal stresses Figures 11.12 and 11.13 give the distributions of normal stresses for both the critical height and the bearing capacity, respectively. In general, when m decreases, the normal compressive stress increases somewhat. For the bearing capacity problems, the initial normal compressive stress ao decreases with a decrease in the coefficient m.

x (em)

(xc'Yc)

..(,~ \

....

,

(xc'Yc )

;/-(}

........

\

\

....

I I

\ \ \

.... ,

\ x (em) \300

x (em) \ 300 200 I

E

E

"

I

o

o

I

'
\ I

II

I/')

\

II

I

u

I

-0.0051

200

\

\ \ c" o o

'

\ \ \

\

\ \ \

\

I \

\

\

I

Y (em)

Y (em)

0.0211

Y (em) Fig. lUI. The failure meehanisms for critical height of slopes (j3 = 70°).

(a) m=1.0 .fJ=700, L=235 em

(b) m=0.6,{J=700 , L=230 em

Fig. 11.12. Normal stress distributions of the critical height of slopes.

456

457

(xc 'Yc )

~

""

\

\ \ I

ds sinO

"

""

\

""

J(', \

""

P=28.94 kN

\

I

200

\

100

\

""

\ \

"

-

P= 0.61 kN

\

"

\

\300

(11.49)

dy

The horizontal force along ds is:

(xc 'Yc )

\

=

(J

ds sinO -

7

- (J tanO

200

dx sm . 0-

ds cosO

dx -

Tdx

(J - -

cosO

= -

dx cosO

7 --

cosO

((J :; + T) dx

(11.50)

0.0049 \

\ \

\

\ \

\


I

\ \

\

\

y(cm) (a) m=1.0.

f3

=70°, L=250cm

y(cm) (b) m=0.6,

---

f3 =70° , L=250cm

Fig. 11.13. Normal stress distributions of the bearing capacity of footing on slopes.

------..-- --------

11.4 Layered .analysis of embankments

---

Fig. 11.14. Typical layered embankment.

In reality, every mass of soil exhibits some anisotropy and some nonhomogeneity. The slope under consideration is usually made of several different soils. In practice, we may treat this as a composite slope made of different layers, and in each layer, the soil can be treated as an isotropic and homogeneous medium. In this section, we shall extend the combined method to the stability problem of layered slopes (Uu et al., 1989). As an illustrative example, we consider the particular cross section of embankments consisting of several layers (Fig. 11.14). The load includes the ice bump pressure p and the surcharge pressure q. In many cases, the ice sheet or flow will seriously damage the embankment surface and even cause the sliding failure of the shoulder. In the following discussion, we shall assume the ice pressure p causes the sliding failure of the slope. Referring to the rigid body sliding failure in Fig. 11.14 and neglecting the interface slips between different layers, we shall derive in the forthcoming the governing equation similar to that of Eq. (11.5). Referring to Figs. 11.15 and 11.19, we have the following geometrical relationships: ds cosO

= dx

(11.48)

Fig. 11.15. Calculation scheme of the embankment.

---- ---- ---

---

459

458

0 "*

Let find:

0 and

u=

1. Substituting Eqs. (11.52) to (11.54) into Eq. (11.5), we

y)(~ + X)}dX + ;1 qd3

+ I'(y -

(11.55)

Now the problem reduces to:

=

p

_ _ _ _---=-_----"'L"'----'-_ _---l O'

G [y(X),

a(x), :~, v, 0] =

~

j

g* dx

(11.56)

Xo

Fig. 11.16. Geometric relationship on the slip surface.

Similarly, introducing: (11.57)

Similarly, in the vertical direction, we obtain: [- 7

:~ + a -

(y -

the solution procedure becomes: (11.51)

y) l'] dx

P min

= Min

(11.58)

[G]

'1/;

Integrating from xB to xA' we obtain:

F=O

(11.59)

XA

H = -

f (7 + dxdy a) dx +

p d1

(11.52)

7

(11.60)

= f(a)

XB

1[(- :~

(11.61)

XA

V

=

7

+

a) - (y - 1'] y)

The moment equilibrium of the mass with respect to the point 0

Using the following coordinate transformation (Fig. 11.15):

(11.53)

dx - q d 3

I

v + r SIll17 .

X

=

Y

= -;- -

II

- -

o

(Fig. 11.15) is:

(11.62)

XA

M=

L [-(7:~- a)x- (y-y)I'X+(T+ a:~)yJdX

(11.54)

1

r cos

()

e

(11.63)

the necessary conditions for the determination of the minimum ice pressure pare: dr

dO

df(a)

r-da

(11.64)

461 ' . ,

460

,'~.

For layered soil, by superposition, Eq. (11.68) can be written as:

and

(11.70)

:; +

2r

+

"t ,

sin () = 0

(11.65)

In the present case, we shall assume the slip surface begins at Point A (Fig. 11.15). If '0 and 00 are assumed, the location of the Point 0 is known as:

where "'Ii and Si are the unit weight and the area moment of layer i, respectively. The calculation for Si in Eq. (11.70) is outlined in the following. Referring to Fig. 11.17, we first calculate (8)125' and (8)123765' and then use the following relationship:

(11.66) (8)23765 = (8)123765 -

Assume the normal stress ao at Point A. Thus, the necessary conditions (11.64) and (11.65) can be treated as a pair of simultaneous differential equations with initial values. Many numerical methods can be used to find the slip surface and the corresponding normal stress distribution by a step-by-step process. Then, the value P can be obtained from Eq. (11.55). To find the minimum ice pressure P, we reassume the initial values of '0' ()o' and ao and repeat the above procedure. The minimum p can usually be found by using an available optimization program. In the layered stability analysis, several different layers must be considered in Eq. (11.55). The following expressions may be used for the evaluation of Eq. (11.55) in such a case. We divide Eq. (11.55) into three parts as:

(11.71)

(8)125

to find (8)23765' In this way, Si values for different layers can be obtained. In what follows, we shall only consider a common case, in which the interfaces between different layers are approximately straight lines, and q = O. For convenience, the definition of the interface MN with the end points M and N is necessary. As shown in Fig. 11.18, starting from A(xo'yo)' following the surface of the em-

4 Fig. 11.17. Basic idea of layered analysis.

Y

(11.67)

(11.68) and E

(11.69) where PI is the part of P carried by the internal stresses along the slip surface. Pb is the part of ice pressure carried by the body force (the soil weight), and P q is the part by the surcharge load.

~......A_(XO , Yo )

C.. .

-----------------~D

Fig. 11.18. Definition of M and N.

462..

463

bankment counterclockwise, the first intersection point with all il'ltersurface is called N, the other one is called M. To calculate Si' computer program will be used, considering the triangular type (Fig. I1.19a) and sector type (Fig. I1.19b). For the triangular type, we can develop a subroutine to calculate the centroid location and the area by the coordinates of three vertexes. For the sector type, the numerical integration can be used, such as:

a

82

j

S012

f

r 3(8) sin 8 d8

(11.72)

81

c

D (b)

(a)

a

; ;

;

II

/I

;

I I I I

;

".".

/

F/

y

;'1

;

I

I

I

4

I I

, I

N 3

2

2 (a)

(b)

E

c'---------...:.:.~ D

x----..J o'

(d) (0)

Fig. 11.19. The triangular type and sector type.

C!----------'~D

(e)

Fig. 11.20. Type I.

Fig. 11.21. Six cases in Type I.

C!---------.:.:...~D

(f)

: r

464

465'

According to the different distribution of layers, three different locations of the failure surface end B(xn , Yn ) will be considered. For each location of B(xn , Y n ) there exist several cases depending on the location of MN. (1) Type I

~)Typell

. As shown in Fig. 11.22, point B is -located between E and F, i.e., on the right surface of the embankment. There are three cases exist. (a) N is located between EF (Fig. 11.23a):

As shown in Fig. 11.20, the point B is located between F and 0, i.e., on the upper surface of the embankment. Based on the locations ofM and N, there exist six cases. (a) M is located between BOCD, and N between FB (Fig. Il.2la):

(11.78)

(11.73)

1° /il

II/

(b) M is located between BOCD, and N between EF (Fig. l1.2lb):

I / / f / / / I / f /

(11. 74)

G::-_ _..:F:.,.S /

\(/~

/

(c) M is located between BaeD, and N between AE (Fig. l1.2lc):

M/(11.75)

2~

/

C (a)

(d) M is located between FB, and N between EF (Fig. 11.2ld): (11.76)

(e) M is located between FB, and N between AE (Fig. 11.2le): (11.77)

(f) There is no intersection between MN and the failure surface (Fig. 11.2lf).

c (b)

.

Q

G

c

/ /

(c)

Fig. 11.22. Type II. Fig. 11.23. Three cases in Type II.

D

1

466 (b) N is located between AE (Fig. 11.23b): (11. 79)

467 As an example, we consider a two-layered embankment (Fig. 11.24) in three cases: Case < 1 >: cl = 0.01 MPa, cPl := 20° 1'1 := 0.0016 kg/cm3, m l = 1.0

(3) Type III Point B is located between 0 and C, Le., on the left surface of the embankment. There exist ten cases. But in practice not all cases will occur, especially when FO (Fig. 11.18) or ex (Fig. 11.15) is not small. Details of this development will not be given here. The procedure for this program can be stated in the following steps: Step 1. Input soil property data and known embankment geometric parameters. Step 2. Assume the initial normal stress 0"0' Step 3. Assume the initial parameters of the rotation center Xc and y c' and calculate the initial radius ro and the initial angle (}o' Step 4. Choose step length !:.(), and calculate the values of r«(}) and O"«(}) of Eqs. (11.64) and (11.65) until the slip surface reaches the embankment surface by Runge Kutta method. Note that when the failure surface reaches a different layer the corresponding layer property data should be used and the relative position numbers should also be recorded. Step 5. Considering different layers, calculate Si' the resultant horizontal force, H, the vertical force V, and the external load p by a numerical integration. Check the equilibrium fl.:l,nction F. Step 6. Repeat Step 3 and Step 5 until the equilibrium function F is close to zero by an optimization method. Step 7. Record the p = p«(}O> when F is close to zero. Step 8. Change 0"0' and repeat Step 3 to Step 6 again until Pmin is satisfactory. Step 9. Output the calculation results.

1'2

= 0.01 MPa , cP2 = 20° = 0.0016 kg/cm3, m2 = 1.0

c2

(c) There is no intersection between MN and the failure surface (Fig. 11.23c). Case <2>:

cI

=

1'1

=

0.02 MPa, cPI = 10°, 0.0016 kg/cm 3, m l = 1.0

c2

= 0.01 MPa ,

2

0.0018 kg/cm 3,

1'2 =

= 20° m2 =

1.0

y (em)

8(1)

250

\

\ \ [2]

\ 200

\ \

\ \

\ M

Interface'

E

E u

[ 1]

0 0

C\J

50

x n

y 1000 em

"

x (om)

G-

250

I'

150

200

=

50

O

2.548 x 10 2 kN 1m

(1)

P

(2)

P = 2.405 x 102 kN 1m

(3)

P

(2) N

M

E (1)

x-------I

A

=

2,201

X

10 2 kN/m

C"""------------------------~D Fig. 11.24. A two-layered embankment example.

A

Fig. 11.25. Failure mechanisms of the layered embankment example.

469

468

References

c I = 0.02 MPa , cPI = 10°, 1'1 = 0.0016 kg/cm3, m l = 0.8

Case <3>:

c2 = 0.01 MPa , 1'2

= 0.0018

cP2 = 20°, kg/cm3 , m2 =

0.8

The results of this calculation are shown in Table 11.3 and Fig. 11.25. It can be seen that Case < 1 > is in fact a single-layered embankment, its failure surface is continuously smooth. In Case < 2 > or Case < 3 >, the reduction of cP causes a decrease of the bearing capacity, even though c and l' increase. From Fig. 11.25, it is seen that the failure surface can not remain smooth. Comparing Case < 3 > with Case < 2 > , the bearing capacity is found to decrease by an amount of about 8.50/0. 11.5 Summary (1) The linear limit analysis procedure is generally limited to the linear MohrCoulomb criterion. The extended nonlinear procedure which is called the combined method herein is valid for both linear and nonlinear failure criteria. This procedure yields not only the mechanism of rotational discontinuity, but also its associated stress distribution along the slip surface. (2) One of the necessary conditions used in the extended procedure is the normality condition. (3) The combination ofthe variational calculus and the optimization method provides an effective solution procedure, that is convenient for solving various stability problems in soil mechanics including the layered analysis of embankment. (4) For the special case of a linear failure criterion the present results reduce to those obtained previously by the linear limit analysis, but in addition, the associated normal stress distribution along the slip surface is also obtained.

TABLE II.3 Layered embankment example Case

<2> <3>

P (kN/m)

Uo

un (MPa)

(Xc,yJ

(Xn'Yn)

(MPa)

(cm)

(cm)

254.8 240.5 220.1

0.1050 0.1066 0.0998

0.0069 0.0083 0.0083

( -572,258) ( -397,262) ( -395,282)

(261,251) (224,229) (212,223)

Baker, R. and Garber, M., 1977. Variational approach to slope stability. Proc., 9th ICSMFE, Vol. 2, Tokyo, pp. 8 -12. Baker, R., 1981. Tensile strength, tension cracks, and stability of slopes. Soil Found. 21(2): 1-17. Baker, R. and Frydman, S., 1983. Upper bound limit analysis of soil with non-linear failure criterion. Soil Found., Jpn., 23(4): 34-42. Chen, W.F., 1970. Discussion on Circular and Logarithmic Spiral Slip Surfaces, by Eric Spencer. J. Soil Mech. Found. Div., ASCE, 96(SMl): 324-326. Chen, W.F. and Snitbhan, N., 1975. On slip surface and slope stability analysis. Soil Found., lpn., 15(3): 41- 49. Garber, M. and Baker, R., 1977. Bearing capacity by variational method. l. Geotech. Eng. Div., ASCE, 103(GT-ll): 1209- 1225. Chen, W.F., 1975. Limit Analysis and Soil Plasticity. Elsevier, Amsterdam, The Netherlands, 638 pp. Liil, X.L., Wang, Q.Y. and Chen, W.F., 1989. Layered Analysis with Generalized Failure Criterion. Computers Structures, 33(4): 1117 -1123. Wang, Q.Y. and Liu, X.L., 1988. Limit analysis based on variational method. Proc. Int. Conf. of Engineering Problems of Regional Soils, Beijing, China, pp. 469-472. Zhang, X.J. and Chen, W.F., 1987. Stability analysis of slopes with general non-linear failure criterion. Int. l. Numer. Anal. Methods Geomech., 11: 33-50.

471

SUBJECT INDEX

Active earth pressure, 105-108, 112-114, 122 -124, 155, 156 Adhesion, 143 -145 Angle of dilatation, 96, 113, 116, 137, 242 Anisotropic hardening model, 8 Anisotropy, 309, 310, 377, 381-383, 389-394

Drucker-Prager model, 6, 12, 14, 23, 86 - 89, 92 Dummy index, 29'(see also Index notation) Dynamic shearing strength, 397 - 399

Critical state, 69 Critical confining pressure, 69 Critical void ratio, 70

Earth pressure - - cohesion, 173-176, 177, 192-197 - - design, 305 - 307 - - earthquake, 165 -170, 183 -185, 195 -197, 242-245 - - friction angle, 161-164 geometry, 164, 165 resultant, 245, 246 seismic point of action, 266 - 278 sliding surface, 177 - 180 static point of action, 259 - 266, 275 - 278 - - strength parameter, 246 - 253 - - surcharge, 171-173, 176, 177, 185 -192, 195 -197 - - wall movement, 253 - 259, 285 - 292 Earth pressure tables, 148, 198-230 - - - active pressure KA , 198 - 210 - - - active pressure, NAc ' 223, 224 - - - design applications, 292 - 307 - - - passive pressure, Kp, 210 - 222 - - - passive pressure, Npc ' 224, 225 Earth slope, 325 Earthquake engineering, 150 Effective strength parameters, 73 Effective stress, 62 - 65 Elasticity problem, 1 Embankment, 456, 457, 461-468 Energy dissipation, 115-118,312-315

Dilatation, 97, 98, 396 Discontinuous stress field, 46, 58 Discontinuous velocity field, 46, 47 Drained condition, 65 - 72 Drucker's stability postulate (see Stability postulate)

Failure mechanism, 111-114,280,281,310, 406,407,410,417-419,455 Failure surface, 72 - 92, 328 - 337 Flow rule, 6, 31, 32 - - associated, 5, 16-23,27,28,32,93, 127, 441, 446

Bauschinger effect, 8 Bearing capacity, I, 309, 447 - - factor, 316-323 Cambridge model, 7 Cap Models, 7, 16, 22 - - elliptic cap, 16 - - plane cap, 16 Cohesion, 73, 83, 143-145,311,383 Cohesionless soils, 62, 96, 97 Cohesive soils, 62, 95, 96, 351 Collapse load (see Limit load) Combined method, 438-456 Compaction, 67, 68 Constant volume condition, 65 Constitutive relations, I, 7, 63 Continuum mechanics, 62 Convexity, 35, 39 Coulomb criterion (see Mohr-Coulomb model) Creep, 2 Critical height, 47-59, 337-351, 374, 389, 390, 449

1 i

473

472 - - non-associated, 6, 15, 16, 19-21,42-45, 48, 96, 101, 127 Flow slide, 395 Friction angle, 73 -75, 77, 83, 97, 98, 101, 115-118,133-143,234-241,253-259 Friction material, 42, 43, 48, 97 - 105, 132 Frictional theorems, 43 Hardening rules, 8 (see also Work-hardening) isotropic, 7 - - kinematic, 7, 8 - - mixed, 7, 8 Homogeneous field, 56 Hooke's law, 2 Index notation, 28, 29 - - dummy, 29 - - free, ,,9 Interface friction, 138, 139 In-situ condition, 437 K o value, 241, 242, 254

Kinematically admissible velocity field, 28 Lade model, 90-92 Lade-Duncan model, 80- 83, 92 Landslides, 325, 396 Layered analysis, 456 - 468 Limit analysis method, 3, 4, 6, 9 -11, 27, 28, 45,57,61, Ill, 127,286-291,293-305, 389, 390, 413, 415, 437, 451 Limit equilibrium method, 1, 9-11, 58, 61, 109, 127, 389, 390, 413, 415, 437 - Limid load, 38 Limit theorems, 5,10"27,28,38-42,57,58 - - non-associated flow rules, 42 - 45 Limiting state concept, 232 Liquefaction, 65, 66, 394, 396-399, 401, 402 Logspiral, 50, Ill, 120, 127,337, 338, 352-357,382,410,419,425,431 - raised spiral, 361- 365 - sagging spiral, 357 - 360 - stretched spiral, 365, 366 - toe spiral, 340 Log sandwich mechanism, 122 Lower-bound theorem, 27, 28, 40 Modified Dubrova method, 231 - 234, 251, 278 - 285, 293 - 305

Mohr-Coulomb model, 1,2,5,6,9, 12-14,48, 56,83-86,89,92,93, 101, 104, 108, 113, 138 Mononobe-Okabe solution, 157 -160, 231, 279, 281,293-305 Nonhomogeneity,381-383 Nonlinear failure criterion, 449 Normality condition, 5, 27, 32, 35, 39, 93, 95 - 97, 132 Pasadena earthquake, 425, 427 Passive earth pressure, 105 -108, 112 -114, 116-122, 150-155 Peano's theorem, 445 Perfect plasticity, 2, 7, 9-12, 29, 30, 93, 107, 400 Perimeter function, 330 Plane failure surface, 51, 408, 418, 429 Plane strain, 133 - 137 Plastic strain, 66, 71, 72 Pore water pressure, 62 - 66, 398 Principal cohesion strength, 383 Progressive failure problem, 2, 12, 139, 140 Progressive index, 140 Pseudo-static analysis, 320, 371, 372, 381, 405 Rankine solution, 247 - 249, 259 Relative density, 134 Rigid-plastic analysis, 400 - 402 Rowe's dilatancy equation, 97, 98 (see also Dilatation) Scale effect, 139, 140 Seismic coefficient, 147, 150, 161, 326, 329, 397, 399, 407 Seismic displacement, 425 - 428 Seismic earth pressure tables (see Earth pressure tables) Seismic stability factor, 351, 367 - 377 Shape function, 328 Shear zone, 402 Slip surface, 330, 366, 376 Slip-line method, 9, 10, 58, 125 Slope failure, 325 Soil plasticity, 4, 6, 8 Specific volume (see Void ratio) Spiral (see Logspiral) Stability factor, 103, 449 - 452

Stability of material, 5 Stability postulate, 31 - 35, 94 Stability problem, 1, 437, 447 Stable material, 31, 32, 36, 94 Statically admissible stress fiels, 27 Strain softening, 30 Stress characteristic, 112, 127, 138 Stress distribution function, 328 Stress-dilatancy relation, 133 Stress-strain analysis, 405 Stress-strain relations, 1, 5, 7, 10 Strip footings, 12 - 23, 309 - 323, 439 Summation convention, 29 (see Index notation) Surcharge, 143, 170, 171, 176, 185, 195 Toe spiral, 340 (see also Logspiral) Tresca model, 6, 78, 79, 92, 351 - - extended, 86 Undrained condition, 65, 66-72, 95 Unstable material, 33 Upper-bound theorem, 28, 40, 41, 45, 46, 446 Variational calculus, 328 - 337, 437 - 456 Velocity characteristic, Ill, 127, 138

Velocity fields, 16, 18,20-23 - - kinematically admissible, 28 - - Prandtl solution, 15, 16, 18-23 - - statically admissible, 27 - - Terzaghi solution, 15, 16, 18-23 Virtual work equation, 36 - 38 Visco-elastic, 2, 400 Void ratio, 69 (see also Critical void ratio) - - critical, 70 - - initial, 76 Volumetric strain, 66, 67 von Mises model, 79, 80, 92 - - extended, 6 (see Drucker-Prager model) Wall, 164, 285 Work-hardening plastic, 6, 7, 11, 12, 18 (see also Strain-hardening plasticity) Work-softening, 11 (see also Strain softening) Yield acceleration factor, 388, 390, 392-394, 401, 406 Yield criterion, 29, 30, 31 Yield surface, 6, 30, 31 Zero extension line theory, 113, 125, 272, 289-291,301-304

.

,

I

I

475

AUTHOR INDEX

Ambraseys, N.N., 338,379 Arai, H., 197, 268, 307 Baker, R., 438, 469 Baladi, G.Y., 6, 7,8, 12, 16,25, 397,403 Basavanna, B.M., 232, 264, 265, 266, 268, 270, 287, 288, 291, 292, 294, 295, 299, 306,307, 308 Bazant, Z.P., 3,24 Bassett, R.H., 109 Bishop, A.W., 64, 73, 74, 93, 97, 101, 108, 309, 323 Blight, G.E., 64, 108 Booker, J.R., 5,24, 309,310,323 Bransby, P.L., 112, 113,146,234,238,241, 280, 282, 293, 295, 307, 308 Brumund, W.F., 139, 146, 234, 308 Caquot, A., 127, 128, 129, 130, 131, 146 Carillo, N., 309, 311,323 Casagrande, A., 309, 311,323 Chan, S.W., 3, 24, 379 Chang, C.J., 3,24,379,416,429 Chang, M.F., 3,24, 61, 96, 108, 111, 145, 146, 148, 152, 156, 180, 181, 197, 231, 285, 307 Chen, W.F., 3, 6, 7, 8, 9, 12, 15, 16,24, 25, 57, 60, 61, 87, 89, 96, 103, 108, 109, 111, 113, 115, 116, 124, 143, 145, 146, 148, 149, 150, 152, 153, 180, 197, 309, 310, 316, 317, 319,323. 325, 326, 328, 368, 369, 370, 371, 378,379. 381,388,389,393,397,398,403. 405, 415, 416, 428, 429, 437, 438, 449, 451, 456,469 Christian, J.T., 309,323 Clemence, S.P., 233, 272, 274, 290, 291, 292, 300, 301, 304, 307 Cole, E.R.L., 109 Collins, I.F., 44, 60 Cornforth, D.H., 70, 109, 134. 135, 146

Davidson, L.W., 310,323 Davis, E.H., 5, 6,24, 97, 98, 100, 101, 103, 105, 109, 138, 146, 242, 307, 309, 310, 323 Dejong, D.J.D., 44, 60 Desai, C.S., 3, 24 Drescher, A., 44, 60 Drucker, D.C., 5, 6. 7, 9, 24, 40, 42, 49, 60. 86, 89, 109, 310, 323 Dubrova, G.A., 231, 233, 234, 239, 240, 245, 247, 249, 279, 295, 307 Duncan, J.M., 80, 81, 82, 109 Dvorak, G.J., 3, 24 Eggleston, H.G., 45, 60 Elms, D.G., 292, 306, 308 Emery, J.J., 292, 307 Fang, H.Y., 103, 109, 379 Fellenius, W.O., 1,24, 54, 60 Finn, W.O., 61, 109. 111, 146, 152, 197 Florentin, P., 324 Fourie, A.B., 106, 107, 109, 232,238. 242, 256, 257, 305, 308 Frydman, S., 438, 469 Gabriel, Y., 324 Gallagher, R.H., 3, 24 Garber, M., 438, 469 Ghahramani, A., 113, 125, 146. 180, 197. 233, 234, 238, 241, 272, 273, 274, 290, 291, 292, 300, 301, 304,30~ 308 Gibson, R.E., 6,25. 382, 383, 391, 393, 403 Giger, M.W., 103, 109. 379 Greenstein, J., 309, 323 Habibagahi, K., 113, 125, 146, 290, 300, 301, 307 Hadjian, A.H., 305, 307, 308 Hall, J .R., 306, 307

[ f.

476 Harr, M.E., 61, 109, 231, 233, 307 Hansen, B., 96, 98, 109, 129, 146 Henkel, D.J., 6, 25, 95, 109 Herington, J.R., 129, 130, 146, 242,308 Hettiaratchi, R.P., 112, 146 Hill, R., 5, 24 Hodge, P.G., 5, 25 Horne, M.R., 137, 146 Housner, G.W., 165, 197 Humpheson, C., 25 Hvorslev, M.J., 69, 109 IABSE, 9,24 I1'yushin, A.A., 42, 60 Ishii, Y., 157, 197, 232, 268, 307 JSCE, 292, 308 Jaky, J., 241, 307 James, R.G., 112, 113,146,234,238,241,280, 282, 293, 295, 307, 308 Janbu, N., 413, 429 Johnson, S.C., 141, 146 Joshi, V.H., 233, 269, 271, 276,308 Kerisel, J., 127, 128, 129, 130, 140, 146 Kezdi, A., 280, 282, 286, 308 Ko, H.Y., 3, 10,25,310 Koh, S.L., 325, 379, Komornik, A., 309, 311,324 Ladanyi, C.B., 106, 108, 109 Lade, P.V., 75, 80, 81, 82, 90, 91, 109 Lee, I.K., 129, 130, 242, 308 Lee, K.L., 70, 75, 109 Leonards, G.A., 139,146 Lewis, R.W., 25 Liu, X.L., 438, 456, 469 Livneh, M., 309, 311, 323, 324 Lo, K.Y., 309, 311,324, 383,389,390,391,403 Makhlouf, H.M., 72, 109 Marachi, D.M., 77, 109 Martin, G.R., 372, 379, 400, 403 Matsuo, H., 147, 157, 197, 231,232,234, 293, 308 Matumura, M., 403 McCarron, W.O., 3, 24 Meyerhof, G.G., 108, 109, 309, 317,324 Mizuno, E., 3, 12, 15, 24, 25

'477· Mogenstern, N.R., 382, 383, 391, 393, 397, 403 Mononobe, N., 147, 157, 197, 231, 234, 293, 302, 303, 308, 400, 403 Mroz, Z., 44, 60 Murphy, V.A., 180, 197, 233,308 Musante, H.M., 75, 91, 109 Nandakumaran, P., 233, 271, 272, 276, 305, 308 Narain, J., 246, 272,308 Nazarian, H., 305, 306, 307,308 Newmark, N.M., 338, 361, 366, 378,379, 382, 399, 401, 403, 405, 415, 416, 429 Odenstad, S., 382, 383, 391, 393, 403 Ohara, S., 232, 308 Okabe, S., 147, 197, 231, 234, 293, 302, 303, 308 Okamoto, S., 147, 150, 156, 197 Palmer, A.C., 6, 25, 44, 60 Parry, R.H.G., 25 Peaker, K., 140, 146 Peck, R.B., 96, 110, 129, 146 Potts, D.M., 106, 107,109, 232,238, 242, 256, 305,308 Prager, W., 5, 6, 10,24,25,49,60, 87, 89, 109, 3.10,323 • Prakash, S., 148, 193, 197, 232, 233, 239, 242, 247, 257, 264, 265, 266, 268, 270, 287, 288, 291, 292, 294, 295, 296,299, 305, 306,308 Prater, E.G., 338, 379 Price, V.E., 397, 403 Purushothama, Raj., P., 310,324 Radenkovic, D., 44, 60 Ragsdell, K.M., 367, 379 Ramiah, B.K., 324 Raymond, G.P., 309,324 Reddy, A.S., 309, 310, 317, 318, 320, 321, 324, 382, 383, 391, 395, 403 Reece, A.R., 112,146 Richards, R., 292, 306, 308 Root, R.R., 367, 379 Roscoe, K.H., 94, 109 Rosenfarb, J.L., 113, 124,146, 148, 149, 150, 197 Rowe, P.W., 97, 109, 133, 134, 135, 137, 138, 140, 141, 142, 146

r

l i

[ I I

1

i \

I'

Sabzevari, A., 180, 197, 233, 234, 238, 241, 271, 273, 300, 308 Sacchi, G., 44, 60 Saleeb, A.F., 3, 25, 87, 109 Salencon, J., 309, 310, 311, 317, 318, 319,324 Saran, S., 148, 193, 197, 233, 239, 242, 247, 272, 294, 299, 308 Sarma, S.K., 338, 379, 428, 429 Save, M.A., 44, 60 Sawada, T., 381, 388, 389,403,428,429 Scott, R.F., 113, 146 Seed, H.B., 70, 75, 109, 166, 197, 293,295, 299, 308, 338, 366, 372, 378,379, 382, 396, 398, 399, 400, 401, 402, 403, 405,406,429 Selig, E.T., 3, 10,25 Shield, R.T., 3,24, 49, 60 Shklarsky, E., 309, 310, 311,324 Sigel, R.A., 389, 403, 413,429 Snitbhan, N., 438, 469 Stewart, I.J., 72, 109 Sokolovskii, V.V., 5, 9, 10,25, 109, 125, 126, 129, 130, 146, 317, 319, 324 Srinivasan, R.J., 309, 310, 317, 318, 320, 321, 324, 382, 383, 391, 395, 403

Takata, A., 403 Taylor, D.W., 104, 109, 382, 383, 391, 392,403 Terzaghi, K., 1, 9, 10, 14,25, 50, 56, 60, 62, 64, 65, 96, 110, 129, 146, 259, 262, 270, 278, 280, 282, 308, 317,324, 396,403 Thompson, C.D., 292, 307 Tschaebotarioff, G.P., 141, 146, 232,308 Tsuchida, H., 197, 268, 307 Venkatakrishna Rao, K.N., 310, 324 Wang, Q.Y., 438, 456, 469 Whitman, R.V., 166, 197, 293,295,296, 299, 308 Wood, J.H., 270, 308 Worth, C.P., 96,110 Yao, J.T.P., 379, 429 Yong, R.N., 3, 10, 25 Zhang, X.J., 438, 449, 469 Zienkiewicz, O.C., 16, 25