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Lower-bound calculations of the bearing capacityof eccentrically loaded footings in cohesionlesssoil

Sven Krabbenhoft, Lars Damkilde, and Kristian Krabbenhoft

Abstract: Lower-bound calculations based on the finite element method are used to determine the bearing capacity of astrip foundation subjected to a vertical, eccentric load on cohesionless soil with varying surcharges. The soil is assumed per-fectly plastic following the Mohr–Coulomb failure criterion. The results are reported as tables and graphs showing the bear-ing capacity as a function of the eccentricity and surcharge. Normalised interaction diagrams in the vertical force versusmoment plane have been produced. The results from the analysis are in reasonable agreement with existing methods forsmaller eccentricities, whereas for greater eccentricities (e > 0.25B–0.3B, where B is the width of the foundation), thelower-bound values in general — and especially for greater surcharges — are considerably smaller than the bearing capaci-ties predicted by existing methods. For the special case of no surcharge, the results are in very good agreement with resultsobtained by the effective-width approach originally proposed by Meyerhof.

Key words: Bearing capacity, footings, cohesionless, lower bound, finite elements, eccentricity.

Résumé : Des calculs de limite inférieure basés sur la méthode par éléments finis sont utilisés pour déterminer la capacitéportante d’une fondation linéaire soumise à une charge verticale et excentrique sur un sol sans cohésion et avec des surchar-ges variables. Le sol est considéré comme parfaitement plastique et suivant le critère de rupture de Mohr–Coulomb. Les ré-sultats sont présentés sous forme de tableaux et graphiques montrant la capacité portante en fonction de l’excentricité et dela surcharge. Des diagrammes d’interaction normalisés de la force verticale versus le plan du moment ont été produits. Lesrésultats de l’analyse concordent bien avec les méthodes existantes pour les faibles excentricités, tandis que pour les grandesexcentricités (e > 0,25B–0,3B, où B est la largeur de la fondation), les valeurs de frontière inférieure — et particulièrementpour les grandes surcharges — sont généralement considérablement plus faibles que les capacités portantes prédites à l’aidedes méthodes existantes. Dans le cas particulier où il n’y a pas de surcharge, les résultats correspondent très bien avec lesrésultats obtenus par l’approche de la largeur effective proposée initialement par Meyerhof.

Mots‐clés : capacité portante, semelles, sans cohésion, frontière inférieure, éléments finis, excentricité.

[Traduit par la Rédaction]

IntroductionIt is always desirable to design shallow foundations or

footings so that the resulting force passes through the cent-roid of the base of the foundation, to avoid subjecting thefooting to any moments, but in many cases this is not possi-ble because different load cases must be taken into account.In these circumstances — for example, in the design of foun-dations for lightweight buildings, masts, pylons, steel chim-neys, and wind turbines — eccentricities invariably willoccur, and probably the most commonly applied equationsfor bearing capacity in such cases are based on the work car-ried out by Meyerhof (1953) and Brinch Hansen (1961).Meyerhof proposed the following equation, which can befound in any geotechnical textbook, for the bearing capacityof a strip foundation in cohesionless soil

½1� qult ¼ qsNqiqdq þ 0:5gB0Ng igdg

where qult is the bearing capacity per unit area of the effec-tive area of the foundation, qs is the vertical, effective stressat the foundation level on either side of the foundation, Nqand Ng are bearing capacity factors, g is the effective soilunit weight, iq and ig are inclination factors, dq and dg aredepth factors, which are always greater than unity and veryoften they are assumed equal to unity, as will be done in thisstudy. Only vertical loads will be considered, so the inclina-tion factors are also equal to unity. The expression for Nq isgiven by Reissner (1924), and for the bearing capacity coeffi-cient Ng, several expressions have been suggested (e.g., Za-droga 1994 and Zhu 2000). B′ is the effective width of thefoundation, which can be taken equal to B – 2e, where B isthe width of the foundation and e is the eccentricity, found asthe moment, M, divided by the total vertical force, V (e = M/V). In this way, the vertical load and the moment is consid-ered as a vertical force applied at a distance e = M/V from

Received 8 June 2011. Accepted 9 November 2011. Published at www.nrcresearchpress.com/cgj on 15 February 2012.

S. Krabbenhoft and L. Damkilde. Department of Civil Engineering, Aalborg University, Niels Bohrs Vej 8, 6700 Esbjerg, Denmark.K. Krabbenhoft. Department of Civil, Surveying, and Environmental Engineering, The University of Newcastle, Australia.

Corresponding author: Lars Damkilde (e-mail: [email protected]).

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Can. Geotech. J. 49: 298–310 (2012) doi:10.1139/T11-103 Published by NRC Research Press

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the centre of the footing. The way B′ is found is often re-ferred to as “the effective width method”.The expression for the surcharge contribution on the right-

hand side of eq. [1], which now becomes qsNq, was derivedon the assumption that both the footing and the surchargewere placed symmetrically about the resulting vertical loadV. However in cases where moments occur, this assumptionis impossible to fulfill and ignoring this will always lead tounsafe design, which of course is an unwanted situation.The degree to which the surcharge contribution becomes

unsafe depends on the eccentricity ratio e/B, in that thegreater the e/B, the more unsafe the design becomes. A usualrequirement is that the load V must be kept within the middleone-third of the footing, that is e < (1/6)B (Day 2005). Onthe other hand, Eurocode 7 (CEN 2004) considers the magni-tude of the term qsNq valid for values of e < (1/3)B andstates that in cases where this value of e is exceeded, specialprecautions must be taken. Such precautions include carefulreview of the design loads and construction tolerances. Pra-kash and Saran (1971) analyzed the problem for verticallyloaded strip footings for eccentricities up to e = 0.4B. Theyassumed that the failure mechanism always develops to theside of the eccentricity, away from the footing, and theirwork resulted in proposals for bearing capacity factors Ngeand Nqe being dependent on the friction angle, 4, and the ec-centricity ratio, e/B.Purkayastha and Char (1977) also considered a one-sided fail-

ure mechanism like the above, and on that basis they introduceda reduction factor Rk defined as Rk = 1 – qu(e)/qu(e = 0), where qu(e)and qu(e = 0) are the bearing capacities at eccentricities e ande = 0, respectively. Rk can be found from the expression

½2� Rk ¼ ae

B

� �k

It was found that a and k depend on the magnitude of thesurcharge relative to the width B, but not on the width B noron the friction angle of the soil.Michalowski and You (1998) examined the effective width

rule in calculations of bearing capacity of shallow footingson both cohesive and cohesionless soil, and for the latterthey found that for a surface foundation, the effective widthrule — for an eccentricity ratio e/B of 0.25 — overestimatesthe bearing capacity by 35% and that this overestimation in-creases with increasing e/B. Loukidis et al. (2008) examinedthe bearing capacity of surface footings in sand subjected toeccentric and inclined loads using finite elements, and for aneccentricity e up to (1/3)B their results match the effectivewidth approach very closely. They also found that the failuremechanism — although not symmetrical — developed to ei-ther side of the eccentrically placed vertical force. Yamamotoand Hira (2009) also used finite elements to calculate the bearingcapacity of surface foundations on frictional soils under eccentricloadings, and for a friction angle of 35° and an eccentricity e =(1/3)B, they found a bearing capacity equal to about 45% of theone determined by the effective width approach.The bearing capacity eq. [1] assumes that the total bearing

capacity can be found as the sum of the two contributionsdue to the surcharge and the selfweight of the soil, and as re-ported by amongst others Zhu et al. (2003), this assumptionyields results that are conservative.

The aim of the present study is to evaluate the bearing ca-pacity of eccentrically loaded footings in cohesionless soil foreccentricities in the interval 0 to 0.5B with varying sur-charges, without using approximate superposition of the con-tributions from the selfweight of the soil and the surcharge.For this purpose, lower-bound limit analysis using finite ele-ments is applied. The results are reported as tables and chartsfor practical design, showing the bearing capacity as a func-tion of the friction angle of the soil, the eccentricity ratio,and the surcharge ratio. The results are compared with pre-vious studies and current codes.

Lower-bound limit analysis

Application of the lower-bound theoremThe lower-bound theorem assumes a perfectly plastic soil

model with an associated flow rule, and states that any stati-cally admissible stress field will provide a lower-bound valueof the correct load limit. In real soils the normality conditionis never fulfilled, as the angle of friction is always greaterthan the angle of dilation. As shown by, amongst others,Wang et al. (2001), Yin et al. (2001), and Zhao et al.(2011), ignoring this will lead to unsafe design, in that for aspecific value of the angle of friction, the smaller the angleof dilation, the smaller the bearing capacity. Therefore, whenusing the lower-bound theorem, a modified value of the fric-tion angle as suggested by Davis (1968) must be used.The lower-bound theorem, which in principle is simple,

very often becomes difficult to use in practical problems ifthe stress fields must be constructed manually, and this ap-plies especially if nonsymmetrical stress fields occur and theselfweight of the soil must be taken into account.In such cases, the finite element has proved to be a very

powerful tool and in using this method, the soil mass is dis-cretized into a large number of interconnected elements. Stat-ically admissible stress discontinuities are permitted to occurat the interfaces between adjacent elements and these condi-tions, together with the stress-boundary conditions, equili-brium conditions, and yield criterion, lead to an expressionfor the collapse load, which is maximized subject to a set oflinear and nonlinear constraints. In this study, the numericalformulation of the lower-bound theorem, using the Mohr–Coulomb yield criterion, has followed a method proposed bySloan (1988) and the formulation put forward by Poulsen andDamkilde (2000) has been applied. A finite element programhas been prepared and for the optimization process the softwareMosek (Andersen et al. 2003; Mosek ApS 2010) was applied.

The meshThe domain used in all analyses is shown in Fig. 1a and

the mesh in Fig. 1b. The dimensions of the domain arechosen large enough to ensure that the boundaries will haveno effect on the calculated results. The footing is assumed tobe rough, the vertical load V is applied at a distance e fromthe centreline, and a surcharge qs is placed at the foundationlevel. The mesh is unstructured and composed of three-nodedtriangular elements, and at each element there are nine un-known stresses. The mesh consists of 20 816 elements and10 691 nodes, and refinements have been made in the areaaround the footing edges to ensure results that are as accurateas possible. To validate the results, some preliminary analy-

Krabbenhoft et al. 299

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ses were carried out. These included the calculation of thefailure load (Ng and Nq) for both the g case (qs = 0) and thesurcharge case (g = 0) and were carried out for values of thefriction angle of 25°, 30°, and 35°. In all tests, the unitweight was taken as g = 20 kN/m3 and the width B =1.0 m. The results are given in Table 1 together with Ng val-ues found using the software package ABC (Martin 2004)and Nq values according to Reissner’s (1924) equation.As can be seen, the lower-bound values are in reasonable

agreement with both the Martin and Reissner values, and fur-thermore the lower-bound values are conservative, whichmust be regarded as an advantage in design. Therefore themesh shown in Fig. 1b is considered as being sufficiently fine.

Soil characteristicsAll the simulations were performed for friction angles 4 =

25°, 30°, and 35°, and because the normality condition mustbe fulfilled the dilation angle j was assumed equal to thefriction angle 4. The unit weight of the soil was taken asg = 20 kN/m3.

LoadingsAs shown in Fig. 1a, the vertical load is applied at an ec-

centricity e. The value of the eccentricity varied stepwisefrom 0.00 to 0.45 m, with the increment being either 0.05 or0.10 m. The surcharge is expressed by the surcharge ratio, l,defined by

Fig. 1. (a) Geometry and (b) mesh.

Table 1. Bearing capacity factors: computed (lower bound (LB)), Martin (2004), and Reissner (1924).

Ng Nq

4 (°) LB Martin Error (%) LB Reissner Error (%)25 6.35 6.49 2.2 10.56 10.66 0.930 14.27 14.75 3.3 18.17 18.40 1.335 32.70 34.48 5.2 31.66 33.30 4.9

300 Can. Geotech. J. Vol. 49, 2012


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½3� l ¼ qs

gB

where qs is the surcharge at the base of the footing, g is theeffective soil unit weight, and B is the width of the founda-tion. The analyses were carried out for values of l in the in-terval 0 to 1.0. The reason for introducing the surcharge ratiois that for footings having the same value of the surcharge ra-tio, the ultimate load can be normalized with respect to thequantities g and B. One series of the simulations was carriedout considering the soil being weightless.

Results from the analysisThe results from the analyses are shown in Table 2, and

Figs. 2 through 5 show failure mechanisms for various valuesof the eccentricity ratio. Figures 6 through 10 show thelower-bound values from the limit analysis together with val-ues calculated on the basis of the Eurocode 7 (CEN 2004)and the bearing capacities reported by Prakash and Saran(1971) and Purkayastha and Char (1977).From Table 2 it can be seen that in a few cases, for large

values of the eccentricity ratio, numerical problems occurredresulting in the optimal solution not being reached. For val-ues of l and e/B other than the ones given in Table 2, it issufficiently accurate to apply linear interpolation.In Figs. 2 through 5, the failure mechanisms for eccentric-

ities equal to 0.00, 0.20, 0.30, and 0.40 are shown for a fric-tion angle 4 = 30° and a surcharge ratio l = 1.0. It can beseen that the greater the eccentricity, the greater the part ofthe failure mechanism that develops under the unloaded partof the foundation. This supports the fact that the contributionfrom the surcharge to the total bearing capacity decreaseswith increasing eccentricity, and this trend is seen clearly inFigs. 6, 7, and 8, where the bearing capacity as a function ofthe eccentricity ratio is depicted. For all three friction anglesapplied in this project, the bearing capacity drops markedlyfor an eccentricity ratio in the interval 0.25–0.35, and thegreater the surcharge the greater the drop. Also shown inFigs. 6, 7, and 8 is the bearing capacity according to Euro-code 7 (CEN 2004), the bearing capacity equation being ap-plicable up to an eccentricity ratio of 0.33. It can be seenfrom the graphs that especially for small friction angles andhigh values of the surcharge, this may — in comparisonwith the lower-bound values — lead to underdesign. In gen-eral, for smaller values of the eccentricity ratio, Eurocode 7underestimates the bearing capacity except in, for practicaldesign, the very rare case where there is no surcharge at all.This underestimation of the bearing capacity is due to thefact that the bearing capacity according to Eurocode 7 — asis common practice — is found by superposition of the con-tribution from the selfweight of the soil and the surcharge.These two contributions are found individually from calcula-tions based on two different failure mechanisms, and the re-sult is therefore inherently incorrect and furthermore on thesafe side. This explains why the bearing capacity valuesfound in this study — although being lower-bound values —in general are greater than the Eurocode 7 values. This hasalso been reported by other authors, e.g., Michalowski(1997) and Zhu et al. (2003).As mentioned in the introduction of this paper, other re-

searchers have put forward proposals with regard to the de-

sign for great eccentricities. Figures 9 and 10 show thebearing capacities for a friction angle 4 = 30° using themethods suggested by Prakash and Saran (1971) and Pur-kayastha and Char (1977), respectivley. It can be seen, aswith Eurocode 7, that especially for great eccentricitiesand higher values of the surcharge, there are large discrep-ancies between their bearing capacities and the equivalentvalues found in the present study, the latter being thesmaller.For values other than B = 1.0 m and g = 20 kN/m3, the

bearing capacity of a strip foundation can be found by utiliz-ing the fact that the bearing capacity factors Nq and Ng take onthe same values for the same values of 4 and l (Michalowski1997). The surcharge qs can be expressed by the equation

½4� qs ¼ lgB

and this leads to the following expression for the bearing ca-pacity:

½5� qult ¼ lgBNq þ 0:5gB0Ng

¼ lgBNq þ 0:5gB 1� 2e

B

� �Ng ) qult

gB

¼ lNq þ 0:5 1� 2e

B

� �Ng

and accordingly the bearing capacity of a strip foundation,having a width B and a soil unit weight g, can be foundfrom the equation

½6� qult ¼ qult;0gB

ð20Þð1Þwhere qult,0 is the bearing capacity at the relevant eccentricityratio read from either Table 2 or the lower-bound values fromFigs. 7, 8 or 9.As an alternative to the classical bearing capacity equation,

a newer approach of expressing the ultimate load of a stripfooting has been introduced. The method was first describedby Butterfield and Ticof (1979), and later Gottardi and But-terfield (1993) carried out a large number of small-scale testswith surface foundations, which showed that combinations ofthe vertical force, horizontal force, and moment at failure lieon a unique surface — the failure envelope.For the special case where the horizontal force is equal to

zero, the failure envelope reduces to the V–M plane, whichcan be normalized resulting in a (V/Vmax)–(M/BVmax) planewhere Vmax is the ultimate vertical force at zero eccentricity,V is the actual vertical force acting at a distance e from thecentre of the footing, and B is the total width of the founda-tion. M is the moment at foundation level, and as this isequal to the actual vertical force V multiplied by the eccen-tricity e, with known values of Vmax, V, and B the eccentric-ity e causing the footing to fail can be found from thediagram. The interaction diagrams shown in Figs. 11 through15 have been made on the basis of the lower-bound values inTable 2.Gottardi and Butterfield (1993) found that the failure plane

for a surface foundation could be represented very well by asecond-order parabola with the equation

½7� M

BVmax

¼ 0:36V

Vmax

1� V

Vmax

� �



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The maximum value for M/BVmax is equal to 0.09 and occursat V/Vmax = 0.50. The graph representing this equation is shownin Fig. 11 together with graphs representing the lower-boundvalues for 4 = 30° and surcharge ratios = 0, 0.25, 0.50, and1.00. It can be seen that compared with Gottardi and Butterfield(1993), the lower-bound values are conservative and a possiblereason for this discrepancy may be that the results reported byGottardi and Butterfield are based on small-scale 1g tests, inwhich the friction angle is strongly influenced by the stresslevel resulting in relatively large bearing capacities at large ec-

centricities. Lower-bound calculations of a surface strip footingof the same size, width = 0.10, based on a stress-dependentfriction angle for a sand having the same relative density, i.e.,Id = 0.84, carried out by the authors Krabbenhoft et al. (2012),show results similar to the ones obtained by Gottardi and But-terfield. Also in Fig. 11 it can be seen that the maximum valueof M/BVmax increases with increasing surcharge ratio, the maxi-mum point moving to the right, indicating the peak momentbeing mobilized at a higher vertical load. This observation isconsistent with the findings by Gourvenec et al. (2008).

Table 2. Ultimate vertical loads (in kPa) over the effective width B′ = B – 2e.

l = qs/gB

e/B 4 (°) l = 0.00 l = 0.05 l = 0.10 l = 0.20 l = 0.25 l = 0.40 l = 0.50 l = 1.00 ∞a

0.00 25 63.5 84.7 101.2 129.7 144.6 180.1 206.9 317.6 105.630 142.7 178.3 207.2 256.5 285.4 344.7 394.5 595.8 181.735 327.0 388.3 440.8 531.9 591.5 693.5 793.1 1126.5 316.6

0.10 25 50.1 72.1 88.1 116.2 130.9 166.6 193.6 305.8 108.230 116.3 152.1 180.7 230.0 257.9 318.4 368.8 575.6 191.035 258.9 330.3 381.9 473.8 532.8 636.4 738.4 1079.7 346.9

0.20 25 37.7 57.4 72.3 96.7 110.0 142.9 167.2 272.9 103.630 86.9 120.6 146.7 192.9 217.4 276.3 321.5 517.9 186.535 194.1 261.7 310.3 396.1 447.3 550.2 642.2 976.4 342.5

0.25 25 32.0 49.4 62.1 84.6 95.2 125.9 145.9 243.1 94.230 72.2 104.8 129.2 172.5 193.0 252.3 290.7 476.0 179.035 166.5 229.3 276.8 361.1 400.5 513.9 586.6 940.0 334.9

0.30 25 25.0 41.0 51.3 68.0 75.9 96.5 110.2 170.6 47.530 56.0 85.8 106.1 142.6 161.4 209.6 244.8 402.3 154.735 128.4 187.8 229.8 302.5 345.3 440.8 517.0 828.1 316.4

0.35 25 18.7 33.2 42.2 55.8 61.4 75.1 82.4 108.1 16.4b

30 41.4 68.1 81.1 110.5 112.6 131.6 142.0 165.3 59.435 95.9 145.8 179.6 237.7 271.5 349.6 409.2 663.2 253.7

0.40 25 12.7 25.1 32.8 43.6 47.3 56.2 59.2 57.0 14.8b

30 28.1 50.6 64.7 71.4 73.2 74.3 74.0 69.9 17.9b

35 64.3 105.2 132.2 173.9 192.7 238.3 264.8 374.2 28.6b

0.45 25 6.2 16.0 21.2 26.3 25.7 16.7b 18.1b 23.7b 17.1b

30 13.1 32.0 42.6 57.0 34.4 31.5 28.2 20.1 17.5b

35 30.9 65.6 86.4 116.7 126.0 157.8 169.8 217.5 19.5b

ag = 0; q = 10 kPa.bOptimal solution was not reached.

Fig. 2. Failure mechanism for e = 0.00.



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It has been found that a fourth-order polynomial with theequation

½8� f ðxÞ ¼ a1x4 þ a2x

3 þ a3x2 þ a4x

where x = V/Vmax and f(x) = M/BVmax, provides a good fit tothe failure envelope for all the computed lower-bound values.The coefficients ai are given in Table 3 as functions of thefriction angle, 4, and the surcharge ratio, l, and it has been






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verified that for values of the friction angle other than theones applied in this study, linear interpolation can be used indetermining the coefficients ai.

ExampleA strip foundation, B = 3.0 m, is subjected to a vertical

force = 1000 kN/m acting at a distance e from the centre of the

Fig. 6. Friction angle angle 4 = 25°. Bearing capacity according to Eurocode 7 (CEN 2004) and lower-bound limit analysis.




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foundation. The foundation level is 1.0 m below ground level,the angle of friction is 30°, and the soil unit weight is 18 kN/m3. Determine the magnitude of the moment M that the founda-tion can sustain.

The surcharge ratio is calculated

l ¼ ð18 kN=m3Þð1 mÞð18 kN=m3Þð3 mÞ ¼ 0:33


Fig. 9. Friction angle 4 = 30°. Bearing capacity according to Prakash and Saran (1971) and lower-bound limit analysis.



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Fig. 10. Friction angle 4 = 30°. Bearing capacity according to Purkayastha and Char (1977) and lower-bound limit analysis.

Fig. 11. (V/Vmax)–(M/BVmax) plane for 4 = 30°. Dashed line refers to Gottardi and Butterfield (1993).



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Fig. 12. (V/Vmax)–(M/BVmax) plane for surcharge ratio = 0.

Fig. 13. (V/Vmax)–(M/BVmax) plane for surcharge ratio = 0.25.



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From Table 2 the following is found:

e ¼ 0 and l ¼ 0:25 ) qult;0 ¼ 285:4 kPa; e ¼ 0 and l ¼ 0:40 ) qult;0 ¼ 344:7 kPa

l ¼ 0:33 ) qult;0 ¼ 285:4þ 344:7� 285:4

0:40� 0:25ð0:33� 0:25Þ ¼ 317:3 kPa

Equation [6] gives

qult ¼ 317:3ð18Þð3Þð20Þð1Þ kPa ¼ 856:0 kPa

The vertical force Vmax is then found

Vmax ¼ ð856:0 kPaÞð3:0 mÞ ¼ 2568 kN=m

Figure 11 is used to find M/BVmax

V

Vmax

¼ 1000

2568¼ 0:389

) M

BVmax

¼ 0:092

) M ¼ 0:092ð30 mÞð2569 kN=mÞ ¼ 708:7 kN �m=m

The eccentricity e is calculated

e ¼ 708:7 kN �m=m

1000 kN=m¼ 0:709 m

Alternatively, eq. [8] can be used and the coefficients aifor l = 0.33 can be found from Table 3 by linear interpola-tion: a1 = –0.12, a2 = 0.26, a3 = –0.56, a4 = 0.416

M

ð3:0Þð2568Þ ¼ �ð0:12Þð0:389Þ4 þ ð0:26Þð0:389Þ3� ð0:56Þð0:389Þ2 þ ð0:416Þð0:389Þ

¼ 0:09 ) M ¼ 690:8 kN �m=m

The discrepancy between the two results for the moment isabout 2.5%, which is acceptable.The soil stress over the effective width is

B0 ¼ B� 2e ¼ 3� 2ð0:709Þ ¼ 1:58 m

s ¼ 1000 kN=m

1:58 m¼ 631:9 kPa

This stress may be checked with the bearing capacity,which can be obtained from Table 2 by linear interpolationusing the values indicated in Table 4. The results from theinterpolation are shown in bold font.

q1 ¼ q11 þ q12 � q11

0:40� 0:25ð0:33� 0:25Þ ¼ 248:8 kPa

q2 ¼ q21 þ q22 � q21

0:40� 0:25ð0:33� 0:25Þ ¼ 224:6 kPa

q0 ¼ q1 þ q2 � q1

0:25� 0:20ð0:226� 0:20Þ ¼ 236:2 kPa

Using eq. [6] gives the bearing capacity

qult ¼ 236:2ð18Þð3Þð20Þð1Þ kPa ¼ 637:7 kPa � 631:9 kPa

ConclusionLower-bound, finite-element-based calculations of an ec-

centrically, vertically loaded strip footing in cohesionless soilwith friction angles 25°, 30°, and 35° and with varying sur-charges have been carried out, and the calculations have re-

Table 3. Coefficients ai in the fourth-order polynomium.

4 (°) l a1 a2 a3 a425 0 –0.2115 0.5580 –0.7567 0.410225 0.25 –0.1069 0.2311 –0.5234 0.399125 0.50 –0.042 76 0.038 79 –0.3702 0.374225 1.00 –0.016 22 –0.090 36 –0.2453 0.351930 0 –0.2115 0.5580 –0.7567 0.410230 0.25 –0.1400 0.3173 –0.5967 0.419430 0.50 –0.084 33 0.1573 –0.4804 0.407530 1.00 –0.0562 0.03518 –0.3739 0.394935 0 –0.2115 0.5580 –0.7567 0.410235 0.25 –0.1604 0.3815 –0.6550 0.434035 0.50 –0.1344 0.2622 –0.5500 0.422935 1.00 –0.084 25 0.1500 –0.4982 0.4325

Table 4. Interpolation values.

l = 0.25 l = 0.33 l = 0.40e/B = 0.20 q11 = 217.4 q1 = 248.8 q12 = 276.3e/B = 0.226 — q0 = 236.2 —e/B = 0.25 q21 = 193.0 q2 = 224.6 q22 = 252.3



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sulted in design tables and charts applicable to practical de-sign. The results have also shown that for smaller eccentric-ities, design according to Eurocode 7 (CEN 2004) and othersimilar methods provides bearing capacities, which ingeneral — although being slightly smaller — are in goodagreement with the lower-bound values. For greater eccen-tricities — e/B > 0.25–0.30 depending on the friction angleand surcharge ratio — however, the lower-bound values aresmaller than the Eurocode 7 values, and considerably smallerthan the bearing capacities suggested by both Prakash andSaran (1971) and Purkayastha and Char (1977). For the caseof no surcharge, the results are in good agreement with thoseobtained by the effective width method.

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