Corrier d Savoir N 8, Mars 4, pp.-5 UNDRAINED BEARING CAPACITY OF EMBEDDED STRIP FOOTINGS UNDER VERTICAL AND ORIZONTAL LOADS D. BENMEDDOUR,. YAIA-CERIF, M. MELLAS Department of Civil and ydralic Engineering, University of Biskra, BP 45 Biskra 7, Algeria ABSTRACT This paper is concerned with the ndrained bearing capacity of embedded strip footing nder inclined loading (i.e. combined vertical and horizontal). A series of nmerical comptations sing the finite-difference code Fast Lagrangian Analysis of Contina (FLAC) was carried ot to evalate the failre envelopes in vertical force horizontal force (V-) plane, sing both probe and swipe analyses. The adopted approach involves a nmerical soltion of the eqations governing elasto-plastic soils. The soil is modeled by an elasto-plastic model with a Tresca criterion. The reslts are presented in terms of the failre envelope in vertical and horizontal loading plane. KEYWORDS: bearing capacity, inclined loads, failre envelope, nmerical modeling, embedded fondation. INTRODUCTION The bearing capacity of fondations has been extensively stdied by several methods; these methods may be classified into the following for categories: () the limit eqilibrim method (e.g., [-3]); () the method of characteristics, (e.g., [4-5]), (3) the limit analysis method, which incldes pper bond and lower bond theorems (e.g., [6-8]), and (4) nmerical methods that are based on either the finite-element or the finite-difference approaches [9-]. In practice, the bearing capacity of a strip footing is generally evalated sing the sperposition eqation proposed by Terzaghi []; this eqation is valid for a sitation where the shallow strip footing is sbjected to centered vertical loads, which involves a symmetric failre mechanism. Exact soltion for the ndrained vertical bearing capacity for a srface strip footing on idealized plastic material is well established. Expressed in terms of bearing capacity factor N c =q /c, where q is the limiting vertical stress and c the representative ndrained shear strength, N c for strip footing is eqal to 5.4 (Prandtl []). A closed-form exact soltion defining ltimate limit states nder combined V loading for a srface fondation on a Tresca material was obtained by Green []. owever, no exact soltion is available for the vertical bearing capacity of an embedded strip. Undrained vertical bearing capacity of shallowly embedded fondations has been addressed extensively throgh empirical, analytical and nmerical stdies for a range of fondation/soil interface condition (e.g., Skempton, [3]; Meyerhof, [4]; ansen, [5]; olsby and Martin, [6]; Salgado et al., [7]; Edwards et al., [8]). owever, very little work has addressed inclined bearing capacity of embedded strip footing and no exact soltion has been identified. For a strip srface footing nder inclined loading, the ndrained bearing capacity is calclated as follows: q c N i () c c Where c is the ndrained shear strength, N c is the bearing capacity factor and i c is the load inclination factor. For vertical loading the inclination factor has a vale of and the soltion is identical to Prandtl s soltion. In the literatre, there are expressions for the ndrained inclination factor, Table smmarises the expressions proposed by different athors to evalate the inclination factor for ndrained bearing capacity Capacity nder the interaction of vertical and horizontal loads is conveniently represented by a failre envelope V, load space. Recently, Gorvenec [9]; Gorvenec and Barnett []; Bransby and Randolph [], investigated the failre srfaces in (, V, and M) load space throgh analytical and nmerical stdies of embedded strip footing. Their reslts show that the size and shape of failre envelopes defining the ndrained capacity of shallow fondations nder general loading are dependent on embedment ratio. Université Mohamed Khider Biskra, Algérie, 4
D. BENMEDDOUR & al In this paper, a series of nmerical comptations sing the finite-difference code FLAC [] are carried ot to determine the shape of the failre envelopes in vertical force horizontal force (V-) plane of an emended strip footing on an ndrained soil. The nmerical reslts are compared with the available reslts in the literatre Table: Expressions of inclination factor. Athor Meyerhof [] ansen [4] Vesić [3] Green [4].5 N c i c 9.5.5 cos N c The evalation of the ndrained bearing capacity for the embedded strip rigid footing nder vertical and horizontal loads is based on sbdividing the soil into a nmber of zones, and specifying eqal vertical or horizontal velocities to the zone representing the footing. To simlate the rigid footing, the vertical and horizontal displacements of nodes which discretize the strip footing are constrained in the vertical and horizontal directions. It is worthwhile noting that refinement of the mesh and the choice of a small velocity does prodce slightly better reslts, and the importance of the mesh size in bearing capacity comptations was demonstrated earlier by Frydman and Brd [9]. A series of nmerical comptations have been carried ot to test the inflence of the mesh size and the magnitde of the velocity applied at the footing nodes. Figre shows a typical finite-difference mesh sed in the analysis of strip footing with embedment ratios D/B=.5. In all cases, the mesh in the footing neighborhood is refined to captre significant displacement gradients. The bondary EFG of the rigid footing is connected to the soil via interface elements defined by Colomb shearstrength criterion. The soil was modelled as a Tresca material (c = kpa, υ=.49, E =4 MPa and γ=5 kn/m 3 ). It shold be noted that the vales of the elastic parameters had a very small effect on the bearing capacity (Mabroki et al. []). NUMERICAL MODELING PROCEDURE In this paper, the finite-difference code FLAC was sed to reach the ndrained bearing capacity for embedded strip footings nder vertical and horizontal loads. The embedment ratio D/B of (srface), 5,.5 and were considered, where D is the depth of embedment and B is the footing width. In the crrent modeling stdy the width B of the footing is m. Becase of the absence of loading symmetry, the entire soil domain of width 4B and depth B is considered. The bondary conditions are shown in Figre. The base of the model is constrained in all directions. The right and the left vertical sides are constrained in the horizontal direction only. h R : Footing load α: load inclination D R α V B L Rogh base Figre : Problem geometry and bondary conditions 4m E F 8m Figre : Finite-difference mesh for the case of D/B=.5. Both swipe and probe analyses were carried ot to identify the (-V) failre envelope (where and V are respectively the horizontal and vertical ltimate footing loads). Swipe tests, introdced by Tan [3], are convenient, as a complete failre envelope in a two-dimensional loading plane can be determined in a single test. The swipe tests involve first bringing the fondation to vertical bearing failre, and sbseqently applying horizontal velocity while not allowing the footing to move vertically. In the probe analyses, after applying a vertical niform stresses (smaller than q ) at the base of the footing; damping of the system is introdced by rnning several cycles ntil a steady state of static eqilibrim is developed in the soil. Then a controlled horizontal velocity is applied to the nodes sitated at the footing. Displacement is increased ntil failre is reached 3 G
Undrained bearing capacity of embedded strip footings nder vertical and horizontal loads 3 NUMERICAL RESULTS 3. Inclination factor i c The load inclination factor defined by normalization of V lt with respect to the limit load for the corresponding vertically loaded footing case V lt,α=. Figre 3 presents the inclination factors fond in present stdy for srface footing by probe analyses and those proposed by Meyerhof [], ansen [5], Vesić [3] and Green []. It shows that i c decreases as the load inclination α increase. The vales of i c obtained by analytical expressions of ansen [5], Vesić [3] and Green [] diverges from the finite difference predictions with increasing load inclination α, where the vale of i c proposed by Green [] are in excellent agreement with ansen s expression. The nmerical prediction obtained sing the finite difference code FLAC is in good agreement with the inclination factor proposed by Meyerhof []. Figre 3 presents the inclination factors fond by the present stdy sing probe analyses for different vales of D/B ratio. The figre shows that i c decreases as the embedment ratio D/B increase. difference analyses for srface footing (D/B=) and ansen s, Vesic s and Green s eqations for different load inclinations α. The failre loads define a failre srface in the V plane. The stdy indicates that there is a critical angle of inclination, measred from the vertical direction, above which the ltimate horizontal resistance of the fondation dictates the failre of the fondation. When the inclination angle is more than the critical vale, the vertical force does not have any inflence on the horizontal capacity of the fondation; the critical angle is predicted to be 7. The non-dimensional failre envelope predicted by the present nmerical analyses is compared in Fig. 4, with those of ansen [5], Vesić [3] and Green []. It is noted that for vertical loads less than half the ltimate vertical load V lt the footing fails when the shear strength along the soil footing interface is flly mobilized, giving sliding failre at the ltimate horizontal failre load of lt. The Vesic s soltion nderestimates the normalized loads at low vales and overestimates them at higher horizontal loads.. D=, D/B= D=.5, D/B=5 4 35 3 Present stdy ansen (97) Vesic (975).8 D=, D/B=.5 D=, D/B= 5 ic.6.4 (KN/m) 5 5..4.6.8 tan(α) 5 5 5 V (KN/m).8 Present stdy Meyerhof (963) ansen (97) Vesic (975) Green (954).8 Present stdy ansen (97) Vesic (975) Green (954).6.6 ic.4 /lt.4.4.6.8 tan (α).4.6.8 V/Vlt Figre 3: Inclination factor as a fnction of the load inclination α from probe analyses Comparison with existing expressions Inclination factor as a fnction of D/B Figre 4: Comparisons of (V-) failre envelopes for srface footing (D/B=): comparison with ansen s, Vesic s and Green s expression. Normalized failre envelopes 3. Failre envelope The dots in Fig. 4 represent pairs of failre horizontal () and vertical (V) loads obtained from the finite Figre 5 represents the ltimate limit states normalized by the ltimate limit loads, V/V lt and / lt, indicating the shape and relative size of the failre envelopes from probe and swipe analyses for range of D/B ratio of (srface), 3
D. BENMEDDOUR & al 5,.5 and. The size of the normalized envelope redces with increasing embedment ratio. From the figre probe analyses reslts are in good agreement with the swipe analyses. These reslts prove that the shape of the failre envelopes are similar and not niqe dependent on embedment ratio D/B. orizontal load governs failre of the fondation for vales of V less than abot.5v/v lt. A pre sliding mechanism is observed in this region, with = lt. /lt.8.6.4 D/B=, 5,.5 and 3.3 Soil deformation mechanisms Figre 6 shows the contors of maximm shear strain for different load inclination. The plots clearly show the trianglar elastic wedge nderneath the footing base and demonstrate that there are different mechanisms for different load inclination. Under pre vertical (α=) loading failre path is symmetrical and similar to the mechanism proposed by Terzaghi []. An elastic wedge zone is located immediately below the bottom of the footing pshed the soil in tow symmetrical zone. owever, more α>, more an asymmetrical doble-wedge mechanism prevails, with the footing moving with the soil in the larger wedge zone. The size of the shear zone decreases with increasing vales of the load inclination (α). Form figres 6, it is noted that the mechanism of deformation for the embedded fondation is larger than that for the srface fondation. It is worthwhile noting that the vale of the maximm magnitde of the displacement vectors varies with the embedment for larger embedment the maximm displacement is higher..4.6.8 V/Vlt Figre 5: Comparison between the present V failre envelope from probe and swipe analyses (α= ) (α= ) (α=5.69 ) (α=3.63 ) (α=7. ) (α=4.99 ) Figre 6: Contors of maximm shear strain for different load inclination D/B=, D/B=5. 4
Undrained bearing capacity of embedded strip footings nder vertical and horizontal loads 4 CONCLUSIONS The ndrained bearing capacity nder inclined loading of embedded strip footing has been investigated. The stdy proves that the embedment affect the vales of inclination factors i c. The calclations of combined vertical and horizontal loading are smmarized in the form of failre envelopes for different vales of D/B. Under inclined loading, the shape of the failre envelope is slightly dependent of embedment ratio. The reslts clarify the redcing of the size of the failre envelope with the increasing of embedment. The contors of maximm shear strain for different load inclination and for different vales of D/B proved that the size of the shear zone increases with increasing vale of the depth. REFERENCES [] Terzaghi, K. Theoretical soil mechanics. New York: Wiley, 943. [] Meyerhof, G.G. Some recent research on the bearing capacity of fondations. Can Geotech J 963;, 6 6. [3] Vesić A.S. Bearing capacity of shallow fondations. In: Winterkorn F, Fang Y, editors. Fondation engineering handbook. Van Nostrand Reinhold, 975. [4] ansen J. B. A general formla for bearing capacity. Dan Geotech Inst 96;, 38 46. [5] Sokolovskii, V.V. Statics of soil media (translated from the 94 Rssian edition). London: Btterworths, 96. [6] Shield, R.T. Stress and velocity fields in soil mechanics. Jornal of Mathematical Physics 954; 33(), 44 56. [7] Chen, W.-F. Limit analysis and soil plasticity. Elsevier, Amsterdam, 975. [8] Michalowski, R.L., and Shi, L. Bearing capacity of footings over two-layer fondation soils. Jornal of Geotechnical Engineering 995; (5), 4 48. [9] Frydman, S. & Brd,.J. Nmerical stdies of bearing capacity factor N γ. J. Geotech. Geoenviron. Engng ASCE 997; 3 (), 9. [] [Mabroki, A., Benmeddor, D., Frank, R., Mellas, M. Nmerical stdy of the bearing capacity for two interfering strip footings on sands. Compters and Geotechnics ; 37, (4), 43 439. [] Prandtl L. Uber die arte Plastischer Korper. Nachr. Ges. Wiss. Goettingen Math. Phys 9; Kl.,74 85. [] Green, A. P. The plastic yielding of metal jnctions de to combined shear and pressre. J Mech. Phys. Solids 954; (3), 97. [3] [Skempton, A. W. The bearing capacity of clays. Proc. Bilding Research Cong. London, 95;, 8 89. [4] Meyerhof, G.G. The bearing capacity of fondations nder eccentric and inclined loads. Proc 3rd Int. Conf. Soil Mech. Fond. Engng, Zrich, 953;, 44 445. [5] ansen J. B. A revised and extended formla for bearing capacity. Danish Geotech Inst Bll. 97; 8, 5. [6] olsby, G. T., and Martin, C. M. (3). Undrained bearing capacity factors for conical footings on clay. Géotechniqe 3; 53 (5), 53 5 [7] Salgado, R., Lyamin, A.V., Sloan, S.W., Y,.S. Two- and three-dimensional bearing capacity of fondations in clay. Géotechniqe 4; 54(5), 97 36. [8] Edwards, D.., Zdravkovic, L., Potts, D.M. Depth factors for ndrained bearing capacity. Géotechniqe 5; 55 (), 755 758. [9] Gorvenec, S. Effect of embedment on the ndrained capacity of shallow fondations nder general loading. Géotechniqe 8; 58(3), 77 85. [] Gorvenec, S.M. Barnett, S. Undrained failre envelope for skirted fondations nder general loading. Géotechniqe ; 6 (3), 63 7. [] Bransby, M.F., Randolph, M.F. Combined loading of skirted fondations. Géotechniqe 998 48 (5), 637 655. [] FLAC Fast Lagrangian Analysis of Contina, version 5.. ITASCA Conslting Grop, Inc., Minneapolis; 5. [3] Tan, F.S.C. Centrifge and theoretical modelling of conical footings on sand. PhD thesis, University of Cambridge, 99. [4] Bransby, M.F., Randolph, M. F. (999). The effect of embedment depth on the ndrained response of skirted fondations to combined loading. Soils Fond 999; 39(4), 9 33. 5