The Bearing Capacity of Foundations under Eccentric and Inclined Loads

Transcription

1 Session 4/24 The earing Capacity of Foundations under Eccentric and Inclined Loads Capacité ptante des sols de fondation sous charges excentrées et obliques by G. G. M e y e r h o f, Ph.D., M.Sc. (Eng.), F.G.S., A.M.I.C.E., A.M.I. Struct. E., uilding Research Station, Garston, Watfd, Herts., England Summary The auth s recently published they o f the bearing capacity of foundations under a central vertical load is extended to eccentric and inclined loads. First, an analysis is given f eccentric vertical loads on a hizontal foundation and is compared with the results of labaty tests on model footings on clay and sand. In the second section the they is extended to central inclined loads on hizontal and inclined foundations and compared with the results o f some model tests on clay and sand. Finally, it is shown how these methods o f analysis can be combined f foundation loads which are both eccentric and inclined and some test results are presented. Sommaire La théie antérieure de la fce ptante des fondations sous charge centrale et verticale publiée récemment par l auteur est étendue aux charges excentrées et obliques. Premièrement, une analyse est donnée pour des charges excentrées verticales sur fondations hizontales et elle est comparée avec les résultats d essais en labatoire sur fondations modèles d argile et de sable. Dans la deuxième section la théie est étendue aux charges centrales obliques et elle est comparée avec des résultats obtenus avec modèles d argile et de sable. Enfin il est démontré comment ces méthodes analytiques peuvent être combinées pour des charges qui sont à la fois excentrées et obliques et les résultats de quelques d essais sont présentés à l appui. Introduction Foundations are frequently subjected to eccentric and inclined loads due to bending moments and hizontal thrusts acting in conjunction with the vertical loading. The bearing capacity they recently published by the auth (Meyerhof, 1951) can readily be extended to cover such loading conditions, and the present paper gives an outline of the methods together with the results of some tests with model footings on clay and sand. Thus f a shallow hizontal strip foundation of width and depth D carrying a vertical load Q with an eccentricity e on the base (Fig. 1), it may be assumed that the load acts centrally on a foundation of effective contact width ' = 2<?. (1) t earing Capacity of Foundation with Eccentric Load They. W h e n a foundation carries an eccentric load, it tilts towards the side of the eccentricity, and the contact pressure below the base is generally taken to decrease linearly towards the heel from a m a x i m u m at the toe. At the ultimate bearing capacity of the foundation the distribution of contact pressure is not even approximately linear, and a very simple solution of the problem is obtained by assuming that the contact pressure distribution is identical to that indicated previously (Meyerhof, 1951), f a centrally loaded foundation but of reduced width. Fig. 1 Plastic Z ones N ear R ough Strip Fou ndation with Eccentric Load Z ones plastiques près d une sem elle à surface rugeuse sous charge excentrée 440

2 If the remaining width -' is igned, which is somewhat conservative, the cresponding zones of plastic equilibrium in the material on the side of the eccentricity are the same as f a similar centrally loaded foundation. (The shear zones are shown in Fig. 1.) O n this basis f a material of density y, unit cohesion c and angle of internal friction <p the bearing capacity can be represented by Q = q = q' (2a) (2b) where <7 = cn ca ' N y (3) and Ncq and Nyq are the resultant bearing capacity facts f a central load (Meyerhof, 1951) and depend mainly on <p and the depth ratio DI1of the foundation. The above expressions give only the base resistance to which must be added any skin friction (Ca + Ps cos <5, see Fig. 1) on the shaft to obtain the total bearing capacity of the foundation. The suggested procedure can be extended to a rectangular foundation of length L and width, carrying a load Q with eccentricities ex and ey on the maj axes, and to other areas as shown in Fig. 2 by finding the minimum effective contact area A' (with straight boundary across the base) such that its centroid coincides with that of the load. Then Q=XqA' (4 ) L.... l i SINGLE ecc tm T R lc ITY RECTAN GLE (SQUARE S im i u A «) OOULE e c c e n t r ic it y -18 * Fig. 3 E C C E N T R I C I T Y e / ( a ) L O O S E A N D c o m p a c t P A C K IN G S e c c e n t r i c i t y e */e. c0 ) d e n s e p a c k i n g earing Capacity of Footings with Eccentric Vertical Load on Sand Capacité ptante des fondations sur sable sous charge verticale excentrée In der to check the they when the shearing strength of the soil is known independently, some tests were made at the uilding Research Station. Footings of 1 in. width and various shapes were loaded to failure under different eccentricities on the surface of soft remoulded London clay and medium H a m River sand in a loose and dense packing (posity of 45 and 37 per cent, respectively). The average shearing strength of the clay was c = 2 lbs./in2 and f the sand <p= 36 (loose) and <p= 48 (dense) from unconfined compression and direct shearing tests, respectively. The experimental procedure of the model tests was similar to that described previously (Meyerhof, 1948, 1951), and a typical footing after failure is illustrated by Fig. 4. The test results of the footings on clay (Fig. 5) show that the average bearing capacity (maximum load/footing area) decreases linearly, with increase in eccentricity, to zero f ej = 0.5; similarly f any given eccentricity <?v,the bearing capacity decreases with greater eccentricity e. These results Fig. 2 Effective C ontact A rea of Foundations with Eccentric Load Aire de contact effectif des fondations sous charge excentrée where A is the shape fact (Meyerhof, 1951) depending on the average length/width ratio L ' ' of the contact area, and q is given by equation (3). F foundations whose depth is greater than about their width appreciable lateral fces are induced on the shaft by tilting under the load. These fces modify the plastic zones and increase the bearing capacity; their effect can be estimated as f rigid cantilever sheet piles (Terzaghi, 1943). Experiments'. The only published tests results of eccentrically loaded foundations appear to be those from an extensive investigation in elgium (Ramelot and Vandeperre, 1950). Circular and square footings up to 16 in. wide were loaded at various depths in compact sand whose angle of internal friction at the particular packing was unftunately not determined. The experimental results f surface and shallow footings (Fig. 3) are consistent with the they by taking. cp= 44, which would be a reasonable angle. Shallow footings were only tested with relatively large eccentricities when the they is conservative because it neglects the resistance due to the lateral fces on the shaft. Fig. 4 Failure o f Strip Footing witli Ecccnlric Vertical Load on Sand Rupture de l em pattem ent sur sable sous charge verticale excentrée 441

3 Fig. 5 («O S T R I P F O O T IN O (M C IR C U L A R (N D S a U P.* & FOOTINGS earing C apacity of Footings with Eccentric Vertical Load on Clay Fce ptante des fondations sur argile sous charge verticale excentrée compare well with the estimates when an allowance is made f some increase in bearing capacity due to the penetration required f mobilization of the shearing strength as f centrally loaded footings (Meyerhof, 1951). The bearing capacity of circular and square footings is about 20 per cent greater than that of strips at the same eccentricity, as found (Meyerhof, 1951) f central loads. Fig. 5 also shows that the customary method of assessing the bearing capacity from the m a x i m u m toe pressure is rather conservative. F single eccentricities of the load the contact width length at failure was, within experimental limits, given by equation (1), while f double eccentricities the centroid of the contact area at failure coincided with the point of application of the load, as had been assumed in the they. The average bearing capacity of the footings on sand (Fig. 3) decreases approximately parabolically, with increase in eccentricity, to zero f ej = 0.5; f a given ex, the bearing capacity decreases approximately linearly with greater ey. These results are in fair agreement with the theetical estimates; f large eccentricities on dense sand the observed bearing capacity is somewhat greater than estimated due to the greater angle of internal friction with smaller pressure on the failure surface. The bearing capacity of circular and square footings is the same as that of strips f loose sand but is about 30 per cent less than that of strips on the surface of dense material, as found (Meyerhof, 1951) f similar central loads. The customary method of analysis is reasonable f small eccentricities but unsafe f greater eccentricities owing to the rapid decrease of bearing capacity with smaller effective contact width. The contact area at failure was similar to that of footings on clay, and f dense sand the failure surface width at ground level decreased practically linearly with greater eccentricity as expected. While the tests on clay and sand indicated that the middle third rule is rather arbitrary, they suppt the practice of designing shallow foundations with central loading if possible since the ption outside the effective contact area can be igned. bearing capacity is tilted and the adjacent zones are modified accdingly. T w o main cases m a y be considered, namely, foundations with a hizontal base and foundations with a base nmal to the load (i.e. base inclined at a to the hizontal). The cresponding zones of plastic equilibrium in the material are shown in Fig. 6 and solutions f the ultimate bearing capacity q are derived in the appendix (A. 1 and A. 2). The solution f a hizontal foundation (appendix A. 1) can be expressed in terms of the vertical component of the bearing capacity qv = q COS a cncq + y Nyq (5) where the bearing capacity facts Ncq and Nyq depend on <p, DI and a. These bearing capacity facts, inclusive of any skin friction, are given in Figs. 7 a and 8 a f a shallow strip foundation in purely cohesive (<p= o) and cohesionless (c = o) materials, respectively; they decrease rapidly with greater inclination a to zero f a surface footing if a = 90 on purely cohesive material and if a = <pon cohesionless soil, when failure occurs by sliding on the base. It should be noted that f foundations on clay the base adhesion c'a m a y vary between 0 and c depending on the degree of softening of the soil (Meyerhof, 1951), while f cohesionless soil the angle of base friction 5'as <p the cresponding limiting facts are given in Figs. 7 a and 8 a. h ~ 8 '"1 E # 2 F i //N V f / 's V 1 / L S ii a. 90 -? C I A & ' V ' (6) Hizontal base with large inclination o f load D 1 _ earing Capacity of Foundation with Inclined Load They: Under a central foundation load inclined at an angle a to the vertical, the central shear zone at the ultimate 442 Fig. 6 P lastic Z ones near R ough Strip Fou ndation with Inclined Load Z ones plastiques près d un em pattem ent à surface rugeuse sous charge oblique

4 The solution f an inclined foundation with a base nmal to the load (appendix A. 2) can be expressed in terms of the resultant bearing capacity q = cnc,j + y Ny,, (6) The bearing capacity facts, exclusive of any skin friction, are given in Figs. 7b and 8b f a shallow strip foundation in purely cohesive and cohesionless materials, respectively; they decrease rapidly with greater inclination a to the passive earth pressure coefficients of a smooth vertical wall f a = 90. It is of interest to note that f a given a an inclined foundation has a greater bearing capacity than a hizontal base, which suppts the practice of designing shallow foundations with a base nmal to the resultant load if possible. The bearing capacity of foundations of other shapes under inclined loading can at present only be based on empirical evinence to obtain shape facts A in conjunction with equations (5) and (6) on account of the variable boundary conditions of the problem. The theetical contact pressure distribution at failure is similar to that of a foundation with vertical load. Experiments: In view of limited previous experimental evidence the bearing capacity has been determined f different inclinations of a central load on hizontal footings as befe 7. FOUNDATION DEPTH/WIOTM vl Ct ITAl ) D / * cc CL 8 FO R IN T E R N e O T. DEPTHS r, u. u. S ' y A v 3 Z AL c e < i Cl. 4 u X > N D <3 v 7. a > Oi 4 LLI <D * > Ul < > FO R N O T M S E E FW.C a) E a - o o o' 4 0 * 6 0 * 8 0 * O * I. o ' 80* IN C L I N A T IO N O F LO A D 01 IN C L IN A T IO N O F F O U N D A T IO N < ( a i H O R IZ O N T A L FOUNDATION (M INCLIN ED FOUNDATION Fig. 9 Arrangem ent o f M odel Test on F ooting with Inclined Load Arrangem ent d essais sur fondation sous charge oblique Fig. 7 Fig. 8 earing C apacity Facts f Strip F oundation w ith Inclined Load in Purely Cohesive M aterial Facteurs de la capacité ptante pour em pattem ent sous charge oblique en matière purement cohérente 1«) HORIZONTAL FOUNDATION Cb) INCLINED FOUNDATION earing C apacity Facts f Strip Fou ndation with Inclined Load in C ohesionless Material Facteurs de la capacité ptante pour em pattem ent en so l pulvérulent sous charge oblique with a rough base on the same clay and sand (but in a compact packing with posity of 38 per cent and <p= 45 ). In the tests on clay the inclined load was increased to failure; in the tests on sand a vertical load was applied and kept constant while the hizontal load applied by a second proving ring was increased to failure (Fig. 9). In both cases the footing remained sensibly hizontal throughout the test. The test results of the strip footings on clay (Fig. 10) are in reasonable agreement with the estimates. The bearing capacity of square footings was about 20 per cent greater than that of strips at small inclinations, as found previously (Meyerhof 1951) f vertical loads, the difference becoming small f an inclination exceeding about 25 when failure occurred by sliding as would be expected theetically. The observed bearing capacity of the strip footings on sand (Fig. 11) confmed with the theetical estimates and approached zero f an inclination equal to the angle of internal friction g> = 45, as would be expected. The bearing capacity of square footings was about 30 per cent less than that of strips f a vertical load, as found previously (Meyerhof; 1951) f surface loads on compact sand, the difference decreasing to zero beyond an inclination of about 15. The present analysis was also checked by the observation that the failure surface width at ground level decreased steadily with greater inclination of the load and approached zero f a =

5 - ^ Fig. 10 Fig. 11 *s a ÀE [ S T % V V a X U P E X P E lm EN T A L R ESU LTS: S T R I P a / - 6 ) * S Q U A R E O T H E O R E T IC A L R E S U L T S : S T R I P C EN T R I C x / C ITY 10 20* 30* 4 0 SO" IN C LIN A T IO N OF LOAO OL CO 140 : X E C C E * T R IC IT Y 'K ' s t r J N. E X P E R i ME NTAL R E SULTS: S T R IP ( L / = 6 ) S Q U A R E T H E O R E T IC A L R E S U L T S : S T R I P ( 4.= 4 5 ) S lo S 0 IN C LIN A T IO N OF LOAD ol earing C apacity o f F ootings with Inclined Load on Clay Capacité ptante des fondations sur argile sous charge oblique earing capacity o f footings with inclined load on sand Capacité ptante des fondations sur sable sous charge oblique earing Capacity of Foundation with Eccentric Inclined Load They. W h e n a foundation carries an eccentric inclined load an estimate of the bearing capacity can be obtained by combining the above methods of analyses. Thus f a shallow strip foundation with a fward eccentricity of loading (a is positive, i.e. eccentricity in direction of hizontal component of load) an effective contact width ' (equation 1) is used in equations (5) (6) and the total bearing capacity is given by equation (2). Similarly, f a double eccentricity on a rectangular other area the effective contact area and shape fact are used as in equation (4). If the eccentricity is backward (a is negative, i.e. eccentricity in opposite direction to hizontal component of load), failure of the soil occurs either on the side of the eccentricity (small eccentricity, method as above but using negative a in analysis) on the opposite side (large X a eccentricity, method as above with positive a); the bearing capacity is given by the lower estimate. Experiments'. Hizontal model footings on clay and sand as in section 2 were loaded to failure with a single fward eccentricity of ej = 0.25 and different inclinations of the load; a typical footing after failure is illustrated by Fig. 12. The test results are given in Figs. 10 and 11 f clay and sand, respectively. The bearing capacity was about one-half of that of cresponding centrally loaded footings in accdance with the they, which was suppted by the observed contact area and mechanism of failure. Preliminary experiments with a backward eccentricity of loading were also found to be in reasonable agreement with the estimates. Conclusion The previous bearing capacity they of foundations under a central vertical load has been extended to eccentric and inclined loads. The they, which indicates that the bearing capacity decreases rapidly with greater eccentricity and inclination of the load, is suppted by the results of loading test with model footings on clay and sand. Acknowledgment The auth is indebted to his colleagues, particularly Mr. L. F. Cooling M.Sc., f helpful criticism and Mr.. J. Catterall.Eng., f assistance in carrying out most of the model tests. The wk was carried out as part of the research programme of the uilding Research oard of the Department of Scientific and Industrial Research and the paper is published by permission of the Direct of uilding Research. Appendix earing Capacity of Hizontal Strip Foundation with Inclined Load The region above the failure surface of a shallow rough strip foundation with load inclined at a to vertical is assumed to be divided into a central elastic zone AC, a radial shear zone A CD and a mixed shear zone ADEF (Fig. 6 a). The stresses in these zones can be found as shown {Meyerhof, 1951) f a vertical load, by replacing the resultant of the fces on the shaft AF and the weight of the adjacent soil wedge AEF by the equivalent stresses p0and s0, nmal and tangential, respectively, to the plane AE inclined at ft to the hizontal. O n this basis the vertical component of the bearing capacity can, in the first instance, be represented by q C O S a =-= cnc + p0n + y Ny (7) == Qv + Fig Failure o f Strip Footing with Eccentric Inclined Load on Clay Em pattem ent sur argile sous charge excentrée et oblique w h e r e q[, = cnc + p0n, (8) qv -=v- Ny (9) T and Nc, N and Ny are the general bearing capacity facts. Determination of Nc and Nq In zone AC with angle»/'at A, the shearing strength Sp under the nmal pressure pp on AC is Sp = c + p'p tan q>. Hence from M o h r s diagram

6 Qi and / Qv = c + pfi tan ip COS cp + pp tan cp cos cp [sin (2y> <p) + sin cp] + p'p (10) cos (2y> cp) cot a (11) from which y>can be determined from any given a, cp, c and pj, (obtained from equations 12 and 13). In zones ACD and ADE with angle 0 = fi >/ v and angle??, respectively, at A, it was shown {Meyerhof, 1951) that Pp t(c + Pi tan cp) e20 tan <p c] cot and Pi = c -(- Pi tan cp [sin (2i; + <?>) sin cp + p0 cos Ip where >}can be determined from the given ratio sjp0. Substituting equations (12) and (13) into (10) <7 = c + Po cot <p q'v = cnc + p0nq [1 + sin <psin {2y>?>)] 1 sin cpsin (2»; + <p) e 2 6 ta n <p ] 1 -)- sin <psin (2y> (p)» 2 0 t a n <p 1 sin cpsin (2»? + cp) (12) (13) + (14) from equation (8) where Nc and Nq have the values given in the square brackets above. The hizontal component q'h of the bearing capacity cannot exceed the shearing resistance on the base, i.e. q'h = q' sin a = q'v tan a < c ' + ^ t a n < 5 ' (15) where c'a = unit base adhesion and S' = angle of base friction. F greater inclinations a when q'h governs, equation (14) must therefe be replaced by tan a tan <5' snr y> '2P[j y^lcosiy cp) - cos{y> <p)> sin yi cos {y> cp) cos cp (16) obtained from (15). Determination of Ny: The minimum passive resistance Pp acting at cpto the nmal on AC in the zone ACDE can be found either by a numerical step-by-step computation {Caquot and Kerisel, 1949) by a semi-graphical procedure {Meyerhof, 1951) based on the logithmic spiral method. Then it can be shown that y <7, = y y r = Nv (17) from equation (9) where Ny has the value given in the square bryckets above. The above solution holds only f a ^ (5' (see equation 15). Determination of Resultant earing Capacity. The vertical component of the resultant bearing capacity is qv = cncq + y Nyq (18) where Ncq (depending on Nc and Nq) and Nyq (depending on Ny and Nq) are the resultant bearing capacity facts, and is computed from the above solutions by determining the foundation depth parameters (/?, p0 and i0) f various depths D as shown {Meyerhof, 1951) f a vertical load. F large inclinations a when qh governs, the hizontal component of the passive earth pressure on the front of the foundation is added to the shearing resistance on the base given by equation (15); and if in addition the foundation has a rough shaft, the foundation is part of the central zone ACF (Fig. 6b). It has therefe been found convenient to include the skin friction vertical component of the passive earth pressure on the shaft in the bearing capacity facts (Figs. 7a and 8a). earing Capacity of Inclined Strip Foundation with ase Nmal to Load F a shallow rough strip foundation of width and depth D of the upper edge of the base inclined at an angle a to the hizontal (Fig. 6c) the zones are similar to those of a hizontal foundation with y>= 45 + <p/2 and with 0 = fi a rj (p/2. Using the same approach as above, the bearing capacity facts Nc and N in the relation q = cnc + Po Nq + y Ny (19) are obtained by substituting these values of y>and equation (14). Similarly it is found that Ny = 4Pp sin j45 ^ tan I45, y2 2 2 j COS a into (20) where Pp is the minimum passive resistance obtained as indicated earlier. The resultant bearing capacity q = cncq + y Nyq (21) is determined from these solutions as befe, and the bearing capacity facts are given in Figs. 7 b and 8 b. References Caquot, A. and K erisel, J. (1949): Traité de M écanique des Sols. G authier-v illars, Paris, p. 85. M eyerhof, G. G. (1948): A n Investigation o f the earing C apacity o f Shallow Footings on D ry Sand. Proc. Second Int. C onf. Soil M ech., vol. 1, p M eyerhof, G. G. (1951): The Ultim ate earing Capacity o f Foundations. G éotechnique, vol. 2, p R am elot, C. and Vandeperre, L. (1950): Travaux de la C om m ission d Etude des F ondations de Pylônes. Com pt. Rend. R ech., I.R.S.I.A., russels, N o. 2. Terzaghi, K. (1943): Theetical Soil M echanics. J. W iley, N ew Yk, p