An examination of some theories of earth pressure on shaft linings

Imtitrrre ofForoldntiorl Etzgitlcerirrg rrrlrl Soil Mecl~rrnics, Swiss Federn1 Itlstitrrte c ~ f ' T d ~ t ~ o l o g g . ETH - Hijnggerberg, CH - 5093 Ziiricll

Received April 14. 1976 Accepted October 14. 1976

Various theories for determining the earth pressure on shaft linings in cohesionless soils are discussed, and results are presented for a Coulomb-type analysis with a conical sliding surface. The assumed shape of the failure surface approximates closely the one given in published results obtained by the method of characteristics. The simplicity of the cone permits an investigation ofa number ofparameters. e.g. the earth pressure coefficient on radial planes, which turns out to be a decisive parameter in the analysis, and accounts for the widely differing published values for earth pressures on shaft linings. Certain theories c o ~ ~ l d lead, especially at greater depths. to rather conservative designs.

A similartheory is also presented for earth pressureson shafts in cohesive soils. In this case the possibility of base failure must be considered as well, and it is shown that this might be the deciding failure mechanism.

Cet article presente une disc~~ssion des diverses theories relatives a la determination de la poussee des terres sur les rev2tements de puits dans les sols pulverulents, et les resultats d'une analyse de type Coulomb avec surface de glissement conique. La forme supposee de la surface de rupture est une bonne approximation de celle d i j i publiee et obtenue par la methode des caracteristiques. La simplicit6 du c8ne permet I'Ctude d'une quantiti de paramttres, tel que le coefficient de poussee des terres sur des plans radiaux qui s'avkre ttre le paramktre dtcisif de l'analyse et expl iq~~e les grandes differences entre les valeurs publikes de poussCes des terres sur des rev2tements de puits. Certaines theories peuvent conduire a des designs particulierement conservateurs. specialement a grande profondeur.

Une theorie similaire est presentee pour les pressions des terres sus les revttements de puits en milieu coherent. Dans ce cas la possibilite de rupture par soulkvement du fond doit egalement Ztre considCree et on montre que $a peut 2tre le mecanisme qui controle la rupture. Can. Geotech. J., 14, 91 (1977)

Part I. Earth Pressure on Shaft Linings in Cohesionless Soils

Introduction The earth pressure of cohesionless soils on

shaft and caisson linings has been investigated theoretically by a number of authors. Some of these theories are reviewed here. None of the theories discussed considers the effects of a water table, or the rigidity of the lining.

All the theories described herein are based upon plasticity methods. The simple approx- imate method dealt with more fully in this paper considers only the statics of the forces acting on an assumed failure mass defined by a rupture surface. Stress-strain behaviour of the material has not been considered. To the author's knowledge it has not been considered anywhere else either. An axisymmetric finite element program with a Drucker-Prager or some other nonlinear material model could be used to investigate the lower bound yielding conditions, and the effects of specified struc- tural movements. Such results will probably be forthcoming in the next few years.

[Traduit par la revue]

Review of Previous Work The earlier theories of Westergaard (1940)

and Terzaghi (1943) are based upon a con- sideration, along the lines of the classical Rankine theory, of plastic equilibrium on hor- izontal planes passing through the shaft. Sim- plifying assumptions are made at the boundary separating plastic and elastic zones in order to smooth out the singularities in stresses occur- ring there that result from an assumption in the theoretical so1ution.l However, the earth pres- sure predicted by the Westergaard-Terzaghi theory is in reasonable agreement with other theoretical results, but the disadvantage of the

'A limiting condition is placed upon the three dimensional state of stress, in that in the plastic zone two of the principal stresses are equated, viz. U I I is set equal to cr, and it is further assumed that ut = uz ur, i.e. within the plastic zone the earth pressure coef- ficient h = 1, where h is the earth pressure coefficient on radial planes. In the elastic zone, on the other hand, X = K O (the coefficient of earth pressure a t rest) and ut = G., (ut p u,). Thus, theoretically, ab- rupt changes in ut and u, occur a t the elastic-plastic surface of transition.

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Y L CAN. GEO'lbCH. J . v v ~ . I + , 1 7 1 1

theory is its unrealistic prediction of the shape of the plastic zone, which according to Terzaghi increases in radius with increasing depth reaching a limiting value asymptotically. This suggests that the foundations of the theory are inadequate. Other investigators, Steinfeld (1958), Karafiath ( 1 9 5 3 ~ ; see also 1953b), have assunled a Coulomb-type failure surface, which for axisymmetric conditions leads to a sliding mass in the form of a cone. The unknown inclination of the sliding surface is determined mathematically by means of the conditions that the total earth pressure, i.e. the resultant force acting on the wall, has a maxi- mum value.

Unfortunately, the papers by Karafiath (1953a, 19.538) contain mistakes in the draughtsmanship, which have partly misled later investigators. Schultz (1970) has cor- rected one of these mistakes, but the other mistake involving a factor of 10 in values of h/r (depth/radius of shaft) has remained un- noticed. Figure 5 of Karafiath's German pub- lication is reproduced corrected and with some notational changes in Fig. 5 of this paper.

Steinfeld (1958) published a similar theory a few years after Karafiath ( 1 9 5 3 ~ ) but his theory contains an incorrect theoretical con- sideration. I-Ie assumes that the distribution of earth pressure may be obtained directly. His method is best understood by reference to Figs. 2 through 4 taken from his publication. It is well known, however, see for example Terzaghi ( l936) , that Coulomb's theory furnishes only the magnitude of the earth pressure force and not even its line of action. In the case of a smooth wall and plane strain conditions Stein- feld's method would fortuitously give the earth pressure distribution, which in this case is deducible from Rankine's theory. For rough walls or in the absence of plane strain condi- tions Steinfeld's method breaks down and ex- hibits obvious oversimplifications in consider- ing the forces acting on the thin cylindrical ele- ments: he assumes that

[I] 6P + 6F = 6W tan ( a - @) whereas it is only for the sliding mass as a whole that the equilibrium relation

[21 P + F = W tan (a - a) holds, where W is the weight of the complete

Failure surface

FIG. 1. Assumed rupture model for a shaft in co- hesionless soil with the forces acting on the sliding mass.

FIG. 2. Method of computation of earth pressure according to Steinfeld (1958). 6 F - T . 68, FP + 6F = 6W tan ( a - +).

sliding mass, F is the outward component of the forces acting on the sides of the element (Fig. I ) , and P is the total earth pressure. Schulz (1 970) has extended Steinfeld's theory to include the effects of elastic deformation of the sliding mass associated with a nonrigid shaft lining or a recessed lining (Fig. 6) . Schulz realized that the negative earth pres- sures predicted by Steinfeld's theory are un- realistic but failed to locate the error in Stein- feld's theory-he simply disregards the neg-

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PRATER

r f a c e

Sha f t diameter D = 3.55 m Depth h - 7.5 m

2 - 1 I

""I

E f f e c t i v e e s r t l ~ prossure

B I~educ:ion o f :he eartl: pressure due t o the s ide fo rces FIG. 3. Earth pressure distribution on a shaft lining (after Steinfeld 1958).

lo) Depth h Lrnl

Shaft dimeter 0 - 3.55 m A - 1 $ - 7 0 ' x-1.8 t/m3

FIG. 4. Variation of the earth pressure for different shaft depths (after Steinfeld 1958).

ative portion of the pressure distribution given and the ratio h/r, which derives from [2] not by Steinfeld's theory. What is less satisfactory, [I]. Schulz's results, as remarked by Walz however, is the fact that he revises a part of ( l973), do not lead in principle to a different Steinfeld's theory which is correct, that is, the distribution of earth pressure. relation between the angle of the slip surface It is assumed by Walz (1973) that Kara-

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94 CAN. GEOTECH. J. VOL. 14, 1977

FIG. 5. Earth pressure coefficient k , as a function of shaft geometry 11/r for different friction angles 9 (after Karafiath, earth pressure at rest - Hungarian standards).

FIG. 6. Caisson with recess.

fiath's and Steinfeld's results lead to the same distribution of earth pressure. This is not the case, however. Karafiath does attempt by means of a graphical construction to locate the point of application of the earth pressure force, but gives no direct information about the pres- sure distribution.

A further publication which has attained wide readership is by Lorenz (1966). He adapted the results of an earlier publication by Steinfeld (1952), in which Steinfeld investi- gated the passive earth pressure around a circular shaft. In this publication Steinfeld neglected the force F (see Fig. 1) acting on the element of sliding mass. Lorenz, in adapt- ing Steinfeld's theory to active earth pressure conditions, while being aware of this force, neglected it, thereby predicting earth pressures well on the conservative side.

Berezantzev ( 1958, and references therein) and Snarsky (1972) have applied the Sokolov- ski 'step-by-step' method of solution of the

limit equilibrium plane strain problem to prob- lems with axial -symmetry. Here again (cf. Terzaghi's theory 1943) some assumption is made with regard to the tangential stress crt in order to make the problem statically determin- ate. From the conditions of the problem U , is a principal stress, i.e. ut = Fir. The assumptions made are as follows: V , = u i r = rI for active earth pressure conditions, ot = a i r = VrII for passive earth pressure conditions.

Thus in the plastic zones an earth pressure coefficient (A) of unity is tacitly assumed for active earth pressure conditions. Should KO (the earth pressure coefficient at rest) be much smaller than unity, it is possible that h does not increase to unity for the conditions of limit equilibrium, in which case Berezantzev's theory would underestimate the earth pressure. The results of Berezantzev are show; in Fig. 7

It is interesting to note that the failure sur- faces predicted by Berezantzev's theory may be adequately approximated by the surface of a cone. Therefore a cone shaped sliding mass should be satisfactory in a coulomb-type anal- ysis. The advantage of a Coulomb-type anal- ysis is that the influence of the depth/radius relation for the shaft is readily elucidated an- alytically (also with different values of A) , whereas-by the Sokolovski-Berezantzev method a complete numerical (step-by-step) analysis to obtain the families of slip lines (the 'char- acteristics') is required for each set of shaft conditions, i.e. for each set of the following

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PRATER 95

H, which always acts in the outward direction. i-51 F = 2T sin (66/2) whereby F = TS8 for small values of 69.. The reaction Q acting on the sliding body is di- rected at an angle p to the normal, where p lies within the following limits: - @ < p < @. If p is expressed as a fraction of the friction angle, i.e. p = k @, k lies within the limits f 1. The values -1 and +1 represent active and passive earth pressure conditions respectively.

The magnitude of the force T is determined by an earth pressure coefficient A. Integration of the pressure over the depth leads to the following expression for T:

[61 T = hyh3/6 tan a Lorenz assumes that this force may be safely neglected, i.e. h = 0, and also F = 0.

Substitution of W and F in [2] results in the expression for the earth pressure acting on the shaft lining :

rh2 [7] P = ss - 2 tan a FIG. 7. Results according to Berezantzev: (a) the radial stress, (b ) the earth pressure force, as a func- tion of @ and the depth factor z / r . h

x [tan (E + B) (== + r ) - $1 parameters: c, @, 6 (wall friction), h, r, and surface loading q.

In the following analysis, which is an ex- tension of the work of Karafiath and Steinfeld, shaft-wall friction (6) and the surface loading q are neglected. Their inclusion could easily be handled, but this would not help greatly towards the main aim of this section of the paper, which is to investigate the role played by the coefficient A.

Earth pressure relationships are derived for the active case, with h set equal to 1 and KO = 1 - sin @.

The aspect ratio radius/depth (r/h) is now introduced into [7] and the earth pressure is integrated over the shaft circumference to give the total earth pressure force

To obtain the limiting value of the earth pres- sure dPt/da must be set equal to zero, ac- cording to the method of the calculus of vari- ations. This step results in a relationship be- tween the aspect ratio n and the inclination of the failure surface. Steinfeld presents this rela- tion as a cubic equation in tan a. An alterna- tive and somewhat simpler expression is as follows:

Method of Computation of the Earth Pressure The forces acting on the sliding body are

shown in Fig. 1 . For a sector of angle 819

where R = h/tan LY + r. Thus by eliminating R from [3], W may be

written q 2 - X - 3 tan u 3 tan(u + P) [9] n = (X - 1) [4] W = y 6 9 (h3 16 tan2 u -t- h2 1-12 tan a ) where The tangential force T has a radial component

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CAN. GEOTECH. J . VOL. 14, 1977

FIG. 8. Relationship between shaft aspect ratio rz and the inclination of the failure surface or for different friction angles + and = 1 - sin a.

X = sin 2a/sin 2(a + p) Equation 9 has been programmed on a small computer assigning the following values to the parameters k and A: k = -1, active earth pres- sure, for h = 1 and h = 1 - sin a.

Some results are presented graphically in Fig. 8. The earth pressure per unit length of the shaft circumference, PI = Pt/2rr, may be written in the form

[lo] PI = k, -yh2/2 analogous to the plane wall case. k, is the co- efficient of earth pressure for cylindrical shafts. It is a function of n and h:

1 [ll] k, = - n tan or

It may easily be shown that for active earth pressure ( p = -a) and zero depth (n = m )

tan (a - cD) (IcJ,,, = tan a and also from [9] for n = .o

sin 2a: = sin 2(a: - a )

Thus k, is seen to reduce to the simple Rankine expression tan2 (45 - @/2).

A further result may be obtained by an ex- amination of [I I], i.e. that k, may exhibit zero values, either when cot a: = 0 or when (using also [9]) [I21 sin2 (a: + p) = h sin2 a: Equation 12 is solved for values of CY which with the aid of Fig. 8 determine the corre- sponding values of aspect ratio nCrit. The con- clusion to be drawn from this result is that theoretically (i.e. for a conical failure surface) below a certain depth, i.e. depth ratio less than nCri,, no earth pressure is exerted on the shaft l in ing .Vhe complete relationships between k, and n are presented graphically in Fig. 9.

'Karafiath (1953) carried out a small scale model test to verify this theoretical conclusion. To maintain the stability of individual grains the 2 cm diameter hole of about 40 cm height was lined with thin blotting paper containing vertical slits. No caving-in of the hole was observed.

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PRATER 97

FIG. 9. Relationship between depth factor h / r and the earth pressure coefficient k, , for p = -+, h = 1 - sin % a n d A = 1.

TABLE 1. Earth pressure results (total force per metre circumference)

Earth pressure force (t/m) at depth:

Active Rankine Pressure at rest (plane wall KO) Terzaghi Berezantzev Lorenz, X = 0 Steinfeld, X = 1 Karafiath, X = 0 . 5 Steinfeld, X = 0 . 5 The author, X = 0 .5

Some results obtained by Karafiath were pre- sented in Fig. 5 . The influence of the coeffi- cient h is clearly seen in Fig. 9. It was men- tioned earlier in the paper that for h = K,,, the active earth pressure coefficient, k , is indepen- dent of the aspect ratio n. This is inferred from [I 11. By assuming that h = tan (a + p)/tan CY (= KJi for k = -1 and CY = 45' + (P/2), k , becomes tan (CY + p)/tan CY and thus the earth pressure exhibits a hydrostatic form. If a value of less than tan (CY + p)/tan CY is chosen, the earth pressure exceeds the active Rankine pres- sure. Lorenz (1966) set h equal to zero and obtained values of k , which are very probably much too conservative. The sensitivity of the

results to changes in the coefficient h is made clear by the results shown in Table 1, taken from Schulz's paper and modified. For all cases the following values are adopted: -y = 2.0 t/m3, (P = 30, and r = 2.0 m.

It is evident that for realistic results h should be chosen in the range KJi < h < KO. The same model is used by Steinfeld, Lorenz, Karafiath, and the author, but Lorenz by choosing h < KL1 obtains a result about 15 times greater than with h = 1 - sin a, and 3.5 times greater than the active Rankine pressure in the above ex- ample for a depth of 20 m. The deviation from the Rankine value increases more and more with depth, while theories with h > K.\ show

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CAN. GEOTECH. J. VOL. 14, 1977

I --- Shaft o f dialeter R n 1 / 1 1

FIG. 10. The earth pressure force at different depths for a shaft with diameters 8 and 16 m and h = 1 - sin 9.

a decrease in comparison with the active Ran- kine earth pressure.

The influence of X on the earth pressure is further brought out in Fig. 10 where the in- fluence of the radius, i.e. the arching action, is also to be seen. It should be noted that the force shown in Fig. 10 is the total force acting at each depth. The pressure distribution over the depth must be calculated in a step wise manner. The construction for a shaft of 16 m diameter is shown in Fig. 11, the smooth curves being interpolations of the steps in the pressure diagrams. A further worked example is given below. The active Rankine pressure lines are included for comparison purposes. A comparison between the results obtained here and those of Berezantzev (1958) and Lorenz (1966) is presented in Fig. 12. Worked Example

A guide to the use of the figures presented for Part 1 of this paper is now given. First of all it should be noted that the computations for the shaft must be made for various depths and the procedure is incremental.

Let us suppose we want to investigate a shaft of diameter 4 m and depth 20 m in a cohesionless soil characterized by = 2 t/m3 and @ = 30". The water table is assumed not to enter the problem. We shall assume that the coefficient x = KO = 0.5.

If we are interested in the extent of the failure zone we may resort to Fig. 8. For a depth h = 20 m (i.e. n = 0.1) the potential slip surface is inclined at approximately 70.2'. It can be shown by a graphical construction that this slip surface encloses all other poten- tial slip surfaces above it, and thus the assumed cone mechanism is valid for this depth.

Now using Fig. 9 the value of h/r at which theoretically the earth pressure force becomes zero is 12.75, i.e. at about 25.5 m depth. Fur- ther use of Fig. 9 shows that the earth pressure coefficient k , corresponding to 20 m depth is 0.07. Thus, using [ l o ] , the total force is 28 t/m circumference of shaft. If instead of the total force the maximum pressure is required, a graphical construction like the one shown in Fig. 11 must be performed. Working in incre- ments of (say) 2 m depth the force for each

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PRATER 99

FIG. 11. Distribution of earth pressure for a shaft of diameter 16 m, 7 = 2 Urn".

increment is found as follows. Beginning with the top 2 m the force is calculated as above and then divided by the depth increment to give the average pressure for the increment, and so on for each load difference between successive increments. The computational steps are outlined in Table 2.

The incremental procedure shows that the maximum force is not in fact 28 t/m, but about 30.5 t/m. The maximum pressure of about 2.7 t/m2 is reached at a depth of about 10 m, and zero pressure is obtained at about 17.5 m depth.

This example could be reworked using Fig. 12 which was constructed for the same param- eters as were used in the worked example. Though for other values of a, etc. Fig. 9 must be used.

Part 11. Earth Pressure on Shaft Linings

undrained, unconsolidated state, a state which may exist directly after loading. This case does not previously seem to have been dealt with in the literature, but an approximate solution could be obtained using one of the arching theories for trenches dug in purely cohesive soils (Prater 1973). The same mode of failure is assumed here as was previously assumed with non-cohesive soils. Method of Computation

The assumed mode of failure, with the forces acting on the sliding mass, is shown in Fig. 13. The following forces are known:

Weight of the sliding mass

Cohesive force (7

for Soil in the 'a = 0' Condition h2 Introduction

As an extension of the earth pressure the- The tangential force ories discussed in Part I, the case is treated here of soil in the = 0 condition, i.e. in the [15] F = TS9 = S8Xy h3/6 tan a

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CAN. GEOTECH. J. VOL. 14, 1977

-- active Rankine ------ Berazantzev

- - - Karafiath / Prster [ ~ - ~ ~ - l - ~ i n + )

FIG. 12. Total earth pressure force per metre of circumference according to different authors (as a function of the shaft aspect ratio).

The equilibrium equation at the point of incipient failure is

[la] C + cos a ( P + JO2' TSB)

- Wsina = 0 By substitution of the known forces into

[16] the unknown active earth pressure may be determined (per metre circumference of the shaft) 1171 p 2 =yil?[Ze) 3 tan a + 1

where n* = h/r, the aspect ratio of the shaft. As with Coulomb's earth pressure theory,

Failure surface d FIG. 13. Assumed rupture niodel for a shaft in

purely cohesive soils, with the forces acting on the sliding mass.

the condition for the maximum (or minimum) value of earth pressure, determined by dP/da = 0, defines the value of in terms of the geom- etry of the failure mass, i.e.

(1 - h)yh n* = (1 - tan2 .)/ (cot a - 6c

If A is set equal to one,3 the following simple relation is obtained

1 181 n* = tan (1 - tan2 a ) which is independent of the term c/yh. Sub- stituting for n* in [17] yields the active earth pressure (per unit length of shaft circum- ference) [19] P = (yh2/2) (1 - N/sin or cos3 or) where N is the stability number (after Taylor), N = ~ / ~ h . Equation 19 may be rewritten

where k , is the earth pressure coefficient for the circular shaft in soil in the = 0 con- dition.

The relationship between k , and n for differ- ent values of N is shown graphically in Fig. 14. In the same figure rz is also given as a function of or.

The critical depth of shaft (cf. with slopes, the expression 'critical height' is used), that is

3Here again we assume A =KO. For cohesive soil 0.6 < KO < 1, i.e. KO is generally higher than in sands, and may exceed 2 for greatly overconsolidated clays.

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PRATER

TABLE 2. Computational steps

Increment Depth (Ed, (@I), ei No. (9 h, (m) (hlr)i (kh (tlm) (t/m) (tlm2>

1 2 1 0.302 1.20 1.20 2

0.60 4 2 0.273 4.37 3.17

3 1.59

6 3 0.247 8.91 4.54 4

2.27 8 4 0.221 14.14 5.23

5 10 5 0.195 19.50 5.36 2.62

6 2.68

12 6 0.168 24.24 4.74 7 14 7 0.144 28.33 4.09

2.37

8 16 8 0.119 30.45 2.12 2.05

9 1.06

18 9 0.093 30.13 [-0.321 - 10 20 10 0.070 28.00 [-2.131 -

Aspect rstio n-. h/r

FIG. 14. Relationship between k , , d y h , a, and n:': for purely cohesive soils.

the depth to which theoretically no support h,, = ( c / ~ ) (l/sin a cos3 a ) is required as the total earth pressure is zero, may be found from [21], setting k, equal to For plane conditions (n* = 0, a = 45" ), zero. Thus at the critical depth hco = 4c/y

[221 N = = sin a C O S ~ a the well-known value h,. Equation 22 is only valid when no tension cracks exist behind the

or wall. According to Terzaghi (1943, p. 154),

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CAN. GEOTECH. J. VOL. 14, 1977

Aspect ratio ng. h/r

FIG. 15. Determination of the critical depth for a shaft in purely cohesive soil with differeat shaft diameters ( h = 1).

in the presence of tension cracks a reduction in the value of h, of one third is to be ex- pected.

The critical depth, on the assumption A = 1, may be estimated with the help of Fig. 15. For a specific shaft diameter the critical depth is reached when the curve for the depth, with parameter r, intersects the curves constructed on the basis of [22] with parameter c/y. The results may also be presented in a dimension- less form (Fig. 16). This figure is used in the following manner:

Step 1. Calculate 6,: 8, = arctan (ry / c). Step 2. Construct the straight line OA. Step 3. From the point of intersection with

the curve determine (hCoy/c) and thus he$ The procedure is illustrated by the following

example: r = 4 . 2 m c / y = 1 . 5

Thus ry/c = 2.8, = arctan (2.8) and hCoy/c = 9

hco = 13.5 m (cf. h, = 4c/y = 6.0)

Some further examples are given in Fig. 16. These examples show that hCo may be many times larger than h,, the critical depth for a long trench.

The evaluation of the above values for hco does not take into consideration the possibility of a base failure in the shaft. A simple check on base stability can be made by using the theory outlined below - cf. the plane strain theory of Terzaghi (1943, pp. 189-194). It could alternatively be assumed that the failure surfaces for active and passive conditions are identical - this would not, of course, be true for non-cohesive soils - and then utilize a formula for the bearing capacity of a pile in clay.

With reference to Fig. 17, the vertical pres- sure, level with the base of the shaft, i.e. on the surface cd is

[23] a = yh, - 2nr(l + $)hsc rr2([l + $1' - 1)

provided that a hard layer of soil or rock does

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PRATER

FIG. 16. Determination of the critical depth of a shaft in purely cohesive soil (dimensionless curves).

not impede the formation of the failure sur- face de.

For base failure

where N, is bearing capacity parameter. It is usual to assume that N , lies within the limits 6 < N , < 9 for round foundations.

It follows from [23] and [24] that

It is now possible to compare values of critical

depth based on side failure hco and on base failure h, for different values of shaft radius r and c / y values.

The results shown in Table 3 indicate that the critical depth obtained by considerations of base failure may be smaller than that for side collapse. The smaller value is obviously the deciding one. Usually, however, the differ- ence between the critical depth values is not significant.

Figures 18 and 19 illustrate the influence of the tangential stresses (for A = 1) using the cone type failure to estimate the active earth

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104 CAN. GEOTECH. J. VOL. 14, 1977

TABLE 3. Values of ciitical depth

F a i l u r e surface

FIG. 17. Rupture model for base failure of a shaft.

FIG. 19. Earth pressure coefficient k , as a function of depth / I for shafts of different diameter and c/y = 1 m (A = 1).

pressure on shaft linings. The figures give the t,. I I.r I-h earth pressure coefficient k,. as a function of

depth placing restrictions on the values of shaft diameter (Fig. 19) and of the soil prop- erty c/./ (Fig. 18).

Conclusions Some existing theories for the prediction of

earth pressures on shaft and caisson linings have been summarized in this paper. In par- ticular, attention was paid to the extension of Coulomb's theory to axially symmetric con- ditions. Coulomb's method is easy to apply and it has found most favour in the German literature. The results, however, are found to

, be very sensitive to the assumed earth pressure coefficient h acting on radial planes. h must be greater than the active Rankine value for realistic results and this explains Lorenz's high overestimation of the active pressures when he sets h equal to zero (Fig. 12) . However, if h is set equal to unity, as is tacitly assumed in

d,p,h (.) Terzaghi's or Berezantzev's theories respec- FIG. 18. Earth pressure coefficient k , as a function tively, Coulomb's method delivers very low

of depth h with different c/y values for a shaft of earth pressure. be On the safe diameter 4 m (X = 1) . side, therefore, h should be set equal to the

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PRATER 105

earth pressure coefficient at rest (KO) when using the Coulomb method.

The different forms of the pressure distri- bution diagrams of Steinfeld and Befezantzev, drawn attention to by Walz (1973) has also been satisfactorily explained. The inconsistency lies in the theory of Steinfeld (1958), wherein it is incorrectly assumed that the earth pres- sure distribution may be obtained directly using the Coulomb method. In other words, the method proposed by Steinfeld (illustrated in Fig. 2 of this paper) is not acceptable, since statically it is only possible to isolate the forces shown in Fig. 1. Steinfeld partially corrects his pressure distribution diagram when he comes to consider various depths of shaft (Fig. 4 ) . Strictly speaking, however, his diagrams are incompatible: Figs. 3 and 4 should both give the same pressure distribution for the same shaft depth, were it possible to obtain the dis- tribution directly (Fig. 3) .

The Coulomb method indicates that below a certain depth no earth pressure acts (Kara- fiath 1953a), whereas the limit equilibrium theory used by Berezantzev (1958) does not give this result. The earth pressure according to Berezantzev reaches a limiting value asymp- totically which albeit is much smaller than the active Rankine value at greater depths. In practice, there may in fact be a reduction of pressure at depths greater than some critical depth, due to arching action. A similar phe- nomenon is known to exist for retaining walls not fulfilling the plasticity deformation condi- tions (Terzaghi 1943, p. 66) and in slurry wall trenches (Prater 1973). For design purposes, however, it would be advisable to ignore the reduction of pressure and dimension the shaft lining for the maximum earth pressure (e.g. as in Fig. 1 1 ) , as would in any case be common practice. If the pressure reduction is neglected, Berezantzev's results and those obtained with the assumptions of Coulomb's method are comparable (Fig. 12).

Certain investigators have simplified Bere- zantzev's theory of limit equilibrium. Walz (1973) has assumed that the slip lines are straight, corresponding to Rankine's pattern of slip lines. In this case the simultaneous par- tial differential equations for the stresses in the plastic zone reduce to simple differential equa- tions, which may be easily solved. Costantino

and Longinow (1964) have considered the case of a rough wall in their investigation of the pressures acting on silos. They assume that the angle of wall friction is equal to the angle of internal friction of the soil, in which case the cylindrical surface becomes a characteristic surface and the differential equations reduce to a simple form on the cylindrical surface. If wall friction is neglectcd, however, the results lie on the conservative side, and the above theory (or that of Berezantzev) may be ap- plied. If linings to shafts and wells are driven then it is possible that passive pressures may be developed at the cutting edge. Only in these circumstances is it necessary to consider Lorenz's results, and then only for the cutting shoe itself if a recessed lining is used (Fig. 6 ) .

Final Remarks The present paper has dealt mainly with

theoretical aspects of earth pressures in shaft linings. Some preliminary results of small-scale laboratory model tests in sand are reported by Walz (1973) - work initiated by Prof. Lorenz.

Case records where field measurements have been made are very scant in the literature. Kany (1972) does give some results for a 30 m diameter cylinder 40 m high. He found that locally high pressures occurred when compact- ing the backfill material around the cylinder. These stresses, however, decreased during later functioning of the structure, and at the base of the cylinder the pressures were espe- cially small. This effect could be produced both by wall friction and the phenomenon of arching described in this paper.

If a shaft lining can deform to allow the arching action to become fully effective it is suggested that some account of reduced earth pressure be taken according to the theories referred to in this paper. But a considerable amount of engineering judgement is necessary and for uncertain ground conditions the active Rankine pressures should probably be used, or some theory applicable to rough retaining walls.

NOTE ADDED IN PROOF-A relevant paper by Walz appeared after the submission of this work. Only the reference (see Walz 1976) and no discussion can be given here.

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106 CAN. GEOTECH. J. VOL. 14, 1977

BEREZANTZEV, V. G. 1958. Earth pressure on the cylindri- cal retaining walls. Brussels Conf. on Earth Pressure Problems, 11, pp. 21-27.

COSTANTINO, C. J. and LONGINOW, A. 1964. The theory of limiting equilibrium for axisymmetric problems: a com- parison with experiment on silo skin friction. Proc. Symp. on Soil-Structure Interaction, Arizona, pp. 583-592.

KANY, M. 1972. Measurement of earth pressures on a cylinder 30 m in diameter (pump storage plant). 5th Eur. Conf. Soil Mech. Found. Eng., Madrid, pp. 535-542.

KARAFIATH, L. 19530. Erddruck auf Wande mit kreisfor- migem Querschnitt. Bauplanung und Bautechnik, Ber- lin, 7, pp. 319-320. - 19536. On some problems of earth pressure. Acta

Tech. Acad. Hung., Budapest, pp. 327-337. LORENZ, H. 1966. Offene Senkkasten. Grundbautaschen-

buch. W. Emst & Son, pp. 795-798. PRATER, E. G. 1973. Die Gewolbewirkung der Schlitz-

wande. Der Bauingenieur, 48, pp. 125-131. SCHULZ, M. 1970. Berechnungdes raumlichen Erddruckes

auf die Wandung kreiszylindrischer Kolper. Disserta- tion, University of Stuttgart.

SNARSKY, A. S. 1972. Design of an axisymmetric retaining

wall subject to the pressure of a cohesionless loose medium. 5th Eur. Conf. Soil Mech. Found. Eng., Ma- drid, pp. 95-99.

STEINFELD, K. 1952. Der raumliche Erdwiderstand von Kreiszylindem und seine Vergrosserung durch Riit- telung. Dissertation, Tech. Hochschule, Hannover.

1958. Uber den Erddruck auf Schacht- und Brun- nenwandungen. Contribution to the Foundation En- gineering Meeting, Hamburg. German Soc. of Soil Mech. Found. Eng., pp. 111-126.

TERZAGHI, K. 1936. A fundamental fallacy in earth pres- sure computations. 111 Contributions to Soil Mechanics: 1925-1940, Boston Soc. Civ. Eng., 1940, pp. 277-294.

1943. Theoretical soil mechanics. J. Wiley &Sons, NY. pp. 202-215.

WALZ, B. 1973. Apparatur zur Messung des raumlichen Erddruckes auf einen runden Modell-Senkkasten. Baumaschine und Bautechnik, 20, pp. 339-344.

1976. Active soil pressure on a cylindrical caisson compared with model measurements. (In German.) 6th Eur. Conf. Soil Mech. Found. Eng., Vienna, 3, pp. 669-672.

WESTERGAARD, H. M. 1940. Plastic state of stress arounda deep well. J. Boston Soc. Civ. Eng. 27, pp. 1-5.

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