On dimensioning the base of a traditional retaining wall

R. BUTTERFIELD*

When conventional retaining walls fail, they usually do so either by being transported bodily within alandslide of the material they are retaining or by a local failure of their foundation, particularly if it is arigid base at shallow depth. The purpose of this letter is to present a rational way of determining theoptimum geometry of such a base – for a specific wall height and relevant soil parameters – using athree-dimensional interaction diagram for a rigid base buried at shallow depth. The optimisationprocedure required is novel because the loads acting on the base are a function of their dimensions.Furthermore, if the soil strength is scaled down, as in a ‘limit state’ design, the lateral load on the wallis increased whilst the load capacity of the base is decreased. The analysis presented provides asimple means of exploring the consequences of complex interactions of this kind. The methodologydeveloped is independent of the method by which the resultant base loads (V, H, M/B) are decidedand the analytical form of the failure locus.

KEYWORDS: bearing capacity; design; failure; footings/foundations; retaining walls; soil/structure interaction

ICE Publishing: all rights reserved

INTRODUCTIONFigure 1 shows an idealisation of a common form ofretaining wall and the forces acting upon it. In thefollowing analysis the wall is represented by the boldoutline shown so that, for simplicity only, self-weight forcescan be ignored. The objective is to determine values for thebase dimensions (a, b, d), herein called optimal, for whichthe wall is on the point of failure under the applied loads.The breadth of the footing B 5 (a + b) and the depth of anydownstand D 5 dB.

The wall, of height h, is subjected to resultant designloads (V, H, M/B). The load capacity of a shallow, rigidfoundation subject to such force resultants is thereforecentral to the analysis. For illustration, simple, butplausible, values are used for the forces (calculated perunit length of wall). With unit soil weight c, these are asfollows.

N V 5 cah; the weight of the rectangular block of soiloverlying section a of B.

N H 5 Kach2/2; the conventional value of active earthpressure acting at h/3 above the surface of the base.

N M 5 Hh/3 – Vb/2; the moment contributions of V and Habout an axis through the point shown on the centrelineof the surface of the base.Since the supporting soil is confined by both the

downstand D and the backfill, it is assumed that the loadcapacity of the base will be enhanced to that of a shallow-depth, rigid, buried footing with cross-section B6D; inpractice, the protrusion of the downstand below theconcrete base will usually be small.

If a more elaborate assessment of the forces developedwere to be used to calculate (V, H, M/B) (by including theactual weight of the wall and its base, more precise valuesof the active earth pressures and water pressures acting onthem), their values may differ from those used here,

although they will still be expressible as algebraic functionsof (h, a, b, d) and the soil properties.

THE LOAD CAPACITY OF RIGID BASESAn ‘interaction diagram’ is a failure locus – in the presentcase a closed three-dimensional (3D) surface in (V, H, M/B)space – such that load points lying within the surface defineadmissible loads, those lying on it identify a failure stateand load points plotting outside it cannot be supported.Such diagrams are now a well-established means ofidentifying the load capacity of rigid pad foundations.

For a surface strip footing, the failure loci in (V, H) and(V, M/B) planes can be represented satisfactorily by simple,geometrically similar parabolas (Butterfield & Ticof, 1979)such as

8(H=Vmax)~8r(1{r)t (1a)

11(M=BVmax)~8r(1{r)t (1b)

in which t 5 0?5 and r 5 V/Vmax where Vmax is the verticalcentreline load capacity of the surface strip. Although morerecent research has elaborated these expressions, they areprobably precise enough for everyday practical use.

The corresponding 3D locus in (V, H, M/B) has beenestablished as a ‘cigar-shaped’ surface with slightly rotatedelliptic cross-sections in (H, M/B) planes (Butterfield &Gottardi, 1994)

(8H=Vmax)2z2b(88HM=BV2max)z

(11M=BVmax)2{½8rt(1{r)�2~0(2)

in which b 5 0?22 radians is the rotation and the positivesign on this term locates the quadrant, the weaker one (seeFig. 4 below), in which (H, M/B) points have to plot whenM, calculated conventionally (i.e. clockwise in Fig. 1), istaken to be positive. The rotation of the ellipse can beeliminated by setting b 5 0.

Butterfield (2006) proposed an extension of this model toinclude burial depths of at least 0¡d¡1 in which, with d 5B/D, the following assumptions apply.

Manuscript received 1 February 2012; first decision 15 March2012; accepted 30 March 2012.Published online at www.geotechniqueletters.com on 9 May2012.*School of Civil Engineering and the Environment, University ofSouthampton, Southampton, UK

Butterfield, R. (2012) Geotechnique Letters 2, 25–28 http://dx.doi.org/10.1680/geolett.12.00001

25

(a) The vertical centreline failure load becomes V* 5Vmax(2d + 1) (i.e. for d 5 {0, 0?5, 1?0}, V* 5 {1, 2,3}Vmax, values that are almost identical to those widelyused).

(b) The horizontal load capacity of the buried footing,when V 5 0, is the conventional lateral passiveresistance, equal to KpcD2/2.

(c) The failure loci in the (H, V) plane remain parabolicbut their peak H values increase with d according to H*5 V*/f with f 5 (8 2 2d) (i.e. for d 5 {0, 0?5, 1?0}, f 5{8, 7, 6}). These values are in line with f < 7 from anumerical analysis of a pad with d 5 0?5 (Gottardiet al., 2005) and more conservative than f < 8 2 4ddeduced by Butterfield & Ticof (1979) from tests usingbrass-rod analogue material, which generated failureloci much like those shown in Fig. 3.

(d) The failure loci in the (V, M/B) plane are assumed to begeometrically similar to those in the (H, V) plane whilemaintaining the scaling (8/11) of the two parabolasestablished for the surface footing, d 5 0, case.Although this is a reasonable assumption there is, asyet, no experimental evidence supporting it. Figure 4shows elliptical cross-sections of the yield locus, withcalculated load points at failure superimposed: the M/Bcomponents are seen to be very small. Figure 2 is anexemplar of such an (H, V) failure locus, whichnecessarily extends (by mV*) beyond the coordinateorigin at O to an alternative origin O9, in relation towhich V9 5 V + mV*, V*9 5 V*(1 + m), whereas (H, M)are unchanged by the transformation.

When referred to O9, the defining parabolic equationscorresponding to equations (1a) and (1b) become

8(H=V�’)~8r’(1{r’)t’ (3a)

11(M=BV�’)~8r’(1{r’)t’ (3b)

in which

r’~V ’=V�’~VzmV�

V�(1zm)

The equation for the failure locus of a buried footing(Butterfield, 2006), analogous to equation (2), is then

(8H=V�’)2z2b(88HM=B(V�’)2)z

(11M=BV�’)2{½8r’t’(1{r’)�2~0(4)

From assumption (c), H*5V*/f5V*9t9/4 using equation(3a), whence, substituting for f, V*/(822d)5V*(1+m)t9/4,from which

m~2=(4{d)t’{1 (5)

As a consequence of assumptions (a)–(d), t9 becomes afunction of d, and is the positive root of equation (6), asshown in Butterfield (2006)

D = dB

a/2

h/3

Vh

H

Ba b

Fig. 1. Loading on and dimensions of a typical retaining wall

V */f = V *′t/4

tan-1(t ′)

O′ O ( V, V ′)V *mV *

V *′

H

Fig. 2. Notation for a buried footing

0.5

0.4

0.3

0.2

0.1

0

-1 0

f = 6

f = 7

f = 8 d = 0

d = 0.5

d = 1.0H = V(0.538)

H = V(0.569)

H = V(0.500)

1V *0 V *0.5 V *1

V/V *′

H/V

*′

2 3

Fig. 3. Details of a set of failure loci for D/B 5 {0, 0?5, 1?0}

0.1

0.05

-0.1

-0.05

-0.08 0.08-0.04 0.040M/BV ′

H/V ′ 0

Fig. 4. Cross-sections of the 3D locus showing load points atfailure for case (i) (solid) and case (ii) (open)

26 Butterfield

2(4{d)(t’)2{4t’zd2

1z2d~0 (6)

Figure 3 shows a set of (V/V*, H/V*) buried-footingfailure loci for d 5 {0, 0?5, 1?0}. Using this information,together with equation (4), the equation for the 3D failurelocus can be expressed (using a computer package such asMathematica) in terms of the unknown quantities (a, b, d)embedded in (V9, V*9, m, t9).

It is important to appreciate that an expressioncorresponding to equation (4), formulated in terms of (a,b, d, h), can be deduced for any (V, H, M/B) failure locusthat can be expressed algebraically. Therefore, the solutionprocess presented below does not depend on the specificalgebraic form chosen to represent the locus.

OPTIMISING THE DIMENSIONS OF THE RETAININGWALL BASEThe following section demonstrates how this methodologycan be used to deduce optimum base dimensions for aretaining wall typified in Fig. 1. By optimum, we mean aset of values for (a, b, d) that will generate a (V, H, M/B)load point lying on the surface of the 3D failure locus whileminimising some function of (a, b, d) – for example, theminimum total length of the base (B + D) 5 (a + b)(1 + d).

This is a classical ‘minimisation’ problem for whichcomputer packages are available. Mathematica contains sucha procedure called Minimize, a command line for which is

Minimize azbð Þ½ 1zdð Þ, cigar~~f 0,

a1 ƒ a ƒ a2, b1 ƒ b ƒ b2, d1ƒ d ƒ d2g, a,b,df g�

This reads ‘find the values of (a, b, d) that minimise (B +D) whilst satisfying cigar 5 0 – where cigar represents theyield function, equation (4), expressed in terms of (a, b, d) –within the specified ranges of a1 # a # a2 , b1 # b # b2, d1# d # d2’.

If, for example, the strength of the retained material isscaled down, thereby increasing the active pressure on thewall while simultaneously reducing the passive resistance ofthe downstand, then the process generates a ‘limit state’design for the base.

Consider, as an illustrative example, a wall with h 5 8 m,retaining material with an effective c 5 18 kN/m3 and Q9 530u. If tan Q9 is scaled down by a partial factor of 1?3, say,the value of Q9 to be used in a limit state calculationbecomes 25u.

Table 1 shows the output obtained for six cases, in eachof which a different function of (a, b, d ) was minimisedusing the above procedure. The solutions for cases (i) to (vi)in Table 1 refer to the following.

(i) Minimising (B + D),with d 5 0; providing B 56?16 m for the idealised optimum surface footing.

(ii) Minimising (B + D), with a reduced arbitrarily to4 m; now B 5 5?94 m but (B + D) 5 6?69 m since adownstand D 5 0?75 m is needed.

(iii) Minimising (B + 1?5D), also with a reduced to 4 m.This can be thought of as a ‘minimum cost’ examplein which the cost per metre of the downstand isassumed to be 1?5 times that of a surface pad. Ddecreases to zero again and, although (B + D)increases to 6?78 m, it is a less expensive solutionthan (ii).

(iv) If a were to be reduced further, say to 3 m (perhapsdue to construction constraints), B 5 9 m but (B +D), at 10?4 m, increases dramatically due to the largevalues of b 5 6 m and the downstand D 5 1?4 m,although the D/B ratio is little different in cases (ii)and (iv).

In practice, d will not be zero; the solid base will alwayshave a finite thickness, with D/B 5 d 5 0?1 for example. Incases (v) and (vi), this ‘solid base’ condition has beenimposed, leading to

(v) a reduced, minimum value for B 5 5?80 m with D 50?58 m when Q9 5 25u, providing ‘limit state design’solution.

(vi) the much smaller value of B 5 4?45 m (with D 50?45 m) at which the wall base would fail were the fullsoil strength Q9 5 30u to be mobilised.

Figure 4 shows a cross-section of the 3D failure locuslocated at V/V*9 5 0?32 (case (i) in Table 1, d 5 0) onwhich the associated (H/V*9, M/BV*9) values are plotted(solid point) lying on the ellipse close to the M/BV*9 5 Oline. Comparable data for case (ii) at V/V*9 5 0?21 lies(open point) nearer the origin O9 and therefore on a smallerellipse. In both cases, the value of the H/V*9 load is seen tobe critical and the M/BV*9 contribution is very small, asreflected in the low eccentricity ratios of the resultant loadon the base, e/B 5 (0?05, 0?02).

CONCLUDING REMARKSThis letter presents an extension of the ‘interactiondiagram’ philosophy for pad foundations beyond that ofa simple, rigid, surface strip footing to a case in which thefooting may not only be buried at shallow depth but also,and more importantly, the loads acting on it are functionsof its dimensions (as in the base of a retaining wall).Determination of an ‘optimum’ set of dimensions for thefooting (i.e. those that minimise either its extent or itscost) then becomes analogous to the determination of‘minimum cost’ functions in linear programming forwhich computer packages exist. Solution examples usingsuch a package suggest that it may be a practically usefultool in limit state design as well as having interestingheuristic possibilities. In particular, it introduces amethodology for exploring, in a convenient manner, theconsequences of different assumptions about soil proper-ties, loading regimes, wall geometry and their influenceon both limit state design and wall–base dimensions atfailure.

Table 1. Output for six cases in each of which a different function of (a, b, d) was minimised

Case Function (a, b, d) to be minimised a: m b: m d B: m D: m

(i) (a + b)(1 + d), Q95 25u 5?08 1?08 0?00 6?16 0?00(ii) (a + b)(1 + d), a 5 4 4?00 1?94 0?14 5?94 0?75(iii) (a + b)(1 + 1?5d), a 5 4 4?00 2?78 0?00 6?78 0?00(iv) (a + b)(1 + d), a 5 3 3?00 6?00 0?15 9?00 1?40(v) (a + b), d 5 0?1, Q95 25u 4?86 0?94 0?10 5?80 0?58(vi) (a + b), d 5 0?1, Q95 30u 3?91 0?54 0?10 4?45 0?45

On dimensioning the base of a traditional retaining wall 27

REFERENCESButterfield, R. (2006). On shallow pad foundations for four-legged

platforms. Soils and Found. 46, No. 4, 427–435.Butterfield, R. & Gottardi, G. (1994). A complete three-

dimensional failure envelope for shallow footings on sand.Geotechnique 44, No. 1, 181–184.

Butterfield, R. & Ticof, J. (1979). The use of physical models indesign. Proc. 7th Euro. Conf. Soil Mech., Brighton 4, 7.28, 259–261.

Gottardi, G., Govoni, L. & Butterfield, R. (2005). Yield loci by‘swipe’ testing. Proc. Int. Symp. on Frontiers in OffshoreGeotech, Perth, pp. 469–475.

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