3B/16. L a force portante des fondations en coins

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1 3B16 T h e U ltim a te B e a r in g C a p a c ity o f W e d g e -sh a p e d F o u n d a t io n s L a force portante des fondations en coins by P ro fe sso r G. G. M e y e r h o f, D. S c., P h. D., F. A.S.C.E., M.E.I.C., A.M.I.C.E., H ead, D e p a rtm e n t o f C ivil E ngineering, N ova S cotia T echnical C ollege, H alifax, N.S., C an ada Sum m ary The previous theory of the bearing capacity of foundations is extended to wedge-shaped bases and cones. The analysis is compared with the results of tests on cones and model piles of different roughnesses and with various shapes of tips in clays and sands. Som m aire La théorie antérieure de la force portante des fondations est étendue aux bases en coins et en cônes. L analyse est comparée avec les résultats d'essais sur cônes et modèles réduits de pieux de rugosités différentes et avec des pointes de formes diverses dans les argiles et les sables. Introduction Piles frequently have pointed rather than fiat tips, and cone penetrom eters are used in the field and laboratory. The bearing capacity theory previously published by the A u thor (1951, 1953, 1955) can readily be extended to cover such loading conditions. T he present paper gives an outline of the m ethods and the results o f som e tests on cones and m odel piles in clays and sands. B earin g C ap a city of W edges W hen a foundation w ith a w edge-shaped base carries a central load, the zones o f plastic flow in the soil a t failure are sim ilar to those o f an inclined strip foundation for w hich a solution o f the ultim ate bearing capacity was derived previously (M eyerhof, 1953). Thus, for a perfectly sm ooth wedge w ith a sem i-angle a (Fig. 1) the region above the failure surface on each side o f the foundation centre line is assum ed to be divided into a plane shear zone A CD, a radial shear zone A D E and a m ixed shear zone A E F G (shallow wedge) o r a plane shear zone A E F (deep wedge). A s the roughness of the wedge increases, the angle ^ at A in zone A CD decreases as under an inclined load on a horizontal base (M eyerhof, 1953). F o r a perfectly rough wedge (Fig. 1) a central elastic zone A CD form s a false base on a blunt wedge when the bearing capacity is identical to th a t o f a horizontal base (M eyerhof, 1955), w hile for a sharp wedge this elastic zone coalesces w ith the wedge. T he stresses in the zones o f plastic equilibrium can be found as show n for a horizontal foundation (M e y e rh o f, 1951, 1953) b y replacing the w eight o f the soil wedge A FG by the equivalent stresses p n an d s, norm al and tangential, respectively, to the plane A F inclined a t an angle 3 to the horizontal. T he bearing capacity can be represented by (T erzagh i, 1943) where c = cohesion, y = unit weight o f soil, B = w idth o f foundation, and 7VC, N g and N y = bearing capacity factors depending on 3, angle o f internal friction 0 and depthwidth ratio D B of foundation. For shallow foundations (D B ^ I) the stress p 0 = yd, w here D = base depth of wedge, while for deep foundations (D jb > 4 to 10, depending on 0 ) Po = K tfd (2) w here K b = earth pressure coefficient on shaft near base, which is ab o u t 0-5 fo r sands and 1-0 for clays (M e y e rh o f 1951, 1959). SMOOTH BASE (a ) SHALLOW BLUNT WEDGE OR CONE cn r_ yb P,N, + N y (1) Fig. 1 Cb-) DEEP SHARP WEDGE OR CONE Plastic Zones Near Wedge-Shaped Foundations. Zones plastiques près des fondations en coins. 105

2 <* oc ( a ) S H A L L O W D E P T H Fig. 2 Bearing C ap acity Factors N c and N cr. Facteurs de force portante N e et N cr. ( b ) G R E A T D E P T H T he bearing capacity factors are given in Figs. 2 to 4 for the lim iting conditions o f perfectly sm ooth and perfectly rough wedges a t shallow and great depths. T he factors for sm ooth wedges decrease rapidly w ith sm aller sem i-angles a, but for a < 30, approxim ately, the factor N y increases again. F o r rough wedges the factors are sensibly unaffected by the wedge angle (false base) except for ab o u t a < 30 when the factors increase rapid ly w ith sm aller angles. T he factors for sm ooth wedges are m uch sm aller than those for rough wedges. B earing capacity factors for interm ediate degrees o f roughness can be found by linear interpolation between the above lim its w ith good ap proxim ation, and such factors decrease w ith sm aller a to a m inim um and then increase again. T he above expressions give only the base o r wedge resistance to which m ust be added any skin friction (c(l 4- p s sin 8, see Fig. 1) on the shaft to obtain the total bearing capacity of the foundation. 106 B earin g C ap a city o f Cones A t the ultim ate bearing capacity o f a cone plastic flow of the soil induces circum ferential stresses, which raise the bearing capacity above th a t for a corresponding wedge. T he previous solution for the bearing capacity o f circular foundations in purely cohesive soils (M e y e rh o f, 1951) has been extended in the A ppendix to derive corresponding bearing capacity factors N cr for perfectly sm ooth and perfectly rough cones, w hich are show n in Fig. 2. T h e factors for rough cones vary w ith a in a sim ilar w ay to those of wedges an d the shape factor (ratio o f conew edge bearing capacity) is sensibly independent o f a. T he factors N cr fo r sm ooth cones do n ot vary appreciably w ith a ; they are less than those fo r rough cones and o f the sam e order as fo r rough blunt wedges (a > 30 ), Bearing capacity factors for cones of interm ediate roughnesses can be interpolated linearly. For cohesive soils w ith internal friction the bearing capacity

3 oc. ot ( a ) S H A L L O W D E P T H Fig. 3 C b ) Bearing Capacity Factors N Qand N qr. Facteurs de force portante N q et N Qr. G R E A T D E P T H of cones can a t present only be obtained from em pirical shape factors in conjunction w ith eq. ( 1) to give the cone resistance = <=N y B P»N + N y r (3) On the assum ption th a t the shape factors are the sam e as observed fo r circular foundations w ith horizontal bases (M e y e rh o f, 1951, 1955), the bearing capacity factors for perfectly rough cones are given in Figs. 2 to 4. W hile the factors N cr and N Qr for cones are greater than those for wedges, as w ould be expected, the sem i-em pirical factors N yr are sm aller, although an approxim ate theory for circular footings (B e re z a n tz e v, 1952) gives the opposite result. This difference appears to be due to the effect o f the interm ediate principal stress, w hich raises the actual bearing capacity of wedges relative to that o f cones in frictional soils. Thus, for circular surface footings on sands the em pirical shape factors are less than unity com pared w ith theoretical values exceeding 2 ; this w ould correspond to an increase in 0 u n d er strip foundations o f som e 14 per cent (30 < 0 < 45 ), w hich is in reasonable agreem ent w ith the am ount o f ab o u t 10 per cent found by com paring som e plane strain and triaxial com pression test results (Bishop, 1957). Experim ents with Cones and Piles Som e loading tests w ere m ade, first a t the Building R esearch S tation and m ore recently a t the N ova S cotia Technical College, using either brass (sem i-rough) o r sanded (rough) cones and m odel piles o f and 1 in. dia. w ith tips o f various angles, w hich w ere pushed in to soft rem oulded clays (c = 2 to 3 lb.in2) and m edium sands o f various densities ( 0 = 35 to 45 ). T he experim ental procedure was sim ilar to th a t described previously (M ey erh o f, 1948, 1951). 107

4 V> l f FOR LEGEf-10 SEE FIG.2 t 4 5 FACT ORS FOR SHALLOW DEPT H k A y the tip is less im portant. T he interpretation o f laborato ry cone tests on clays is, how ever, difficult due to the unknow n am o u n t o f adhesion and lip o f the m a te ria l; thus ignoring consolidation and tim e effects, the resistance o f a 60 cone m ay be only one-half o f that o f a perfectly rough cone w hich w ould be preferable in practice.! 100 > 80 j GO ) c w V V 5 y 1 4F> - - N;40 x ^ L y Conclusions The previous bearing capacity theory o f foundations w ith horizontal bases has been extended to w edge-shaped bases and cones. T he theory, which indicates th a t the p o in t resistence o f piles w ith sm ooth tips decreases and w ith rough tips increases as the cone angle decreases, is supported by the results o f loading tests o n cones and m odel piles in clays and sands. ^ü " ^ " Acknowledgem ent T he early laborato ry investigations w ere carried o u t at the B uilding Research Station o f the D epartm ent o f Scientific and Industrial R esearch and the results are published by permission of the D irector of Building Research Fig. 4 Bearing Capacity Factors NY and N Y Facteurs de force portante NY et NYr. c*. 8 0 T he test results for clays (Fig. 5a) show th a t the cone resistance and point resistance o f brass piles agree well w ith theoretical estim ates based on perfectly rough tip s ; ancillary pure torsion tests on the cones gave an adhesion o f ab o u t 0 8 c, w hich is likely to be increased by vertical load. T he theory for perfectly sm ooth cones can be com pared w ith the results o f shallow indentation tests using lubricated steel cones in copper and aluminium ( D u g d a l e, 1954); the experim ental cone resistance is som ew hat less than predicted unless an allow ance is m ade for the raised lip around the indentation (Fig. 5a). Sim ilar indentation tests w ith 60 cones in cohesive-frictional soils ( E v a n s, 1950) also support the theoretical relationships and indicate a skin friction of about 20 lo 80 per cent of the shearing strength (Fig. 5b). E x ploratory m odel tests w ith rough piles in com pact sand indicated th a t the p oint resistance increases little w ith sm aller cone angles (H ab ib, 1953); this is supported by the present test results, which show th a t th e observed p o in t resistance of rough piles is som ew hat greater than estim ated (Fig. 6 ). The m easured point resistance o f brass piles in sand is in fair agreem ent w ith estim ates based on a skin friction o f about 0 5 0, which com pares well w ith the results o f direct shearing tests under the sam e conditions. A lthough large-scale tests w ould be useful as a check, the proposed m ethods o f analysis are probably sufficiently accurate fo r practical purposes. F o r steel (sem i-rough) piles and penetrom eters the point resistance decreases, w hile for concrete (rough) piles the p oint resistance increases, as the cone angle o f the tip decreases (sharper points). Since the ultim ate bearing capacity o f piles in cohesionless soils is largely due to p o in t resistance, the shape o f the tip m ay have a considerable influence on the bearing capacity and penetration resistance in such soils and should be taken into account in estimates. In cohesive soils the bearing capacity of piles is m ainly due to skin friction and the shape of IG A * K S PILES IN CLAYS (PRESENT TESTS) * LUBRICATED S TEEL CONES IN METALS CDU6DALE 1954) o T H E ORY scpe RF. R OUGH CONE B A S E DEPTHWIDTH DB tl T H E 3RY (unc ORRE N (PER F. SM iotu, - 1 *> CORR ECTEÍ FOR H P 8 D SEMI-ANGLE OF TIP (.Q) CONE RESISTANCE AND POINT RESISTANCE OF PILES IN CLAS ; ^ : 60 S T E E L CONES (EVANS 1950) 40 d s 30 z 20 T H E O R Y o (PfcR F.RO CON E > V io - "V T H E DRY H 8 (PER :. S K OOTH CON :) S ANGLE OF SHEARING RESISTANCE <p (b) CONE RESISTANCE IN COHESIVE SOILS Fig. 5 Cone and Point Resistance of Piles in Cohesive Soils. Résistance de cône et de pointe des pieux dans les sols cohérents.

5 GOO E P E R IM E N T A L R E S U L T S S A N D E D C O N E S * B R A S S C O N E S T H E O R E T IC A L R E S U L T S P E R F E C T L Y R O U S H C6 = < ) S E M I- R O U S H C O N E Cb = <j>2.) RELATIVE D EN SITY DEfy S E? = 4 5 ) -i } ' COM PAC! ( * = 41 ) t>0 v, LOC S E ( -Gii 35 ). Fig SEMI-ANGLE OF TIP «Point Resistance of Piles in Sands. Résistance de pointe des pieux dans les sables. A P P E N D I Bearing Capacity of Cones in Purely Cohesive M aterial O n the assum ption th a t the plastic zones on radial planes o f cones are identical to those on transverse sections o f wedges (Fig. I) and th a t th e circum ferential stresses are equal to the m inor principal stresses, it w as show n (M e y e rh o f, 1951) th a t a t failure the vertical contact pressure on the base qx a t any radius r = x from the foundation axis w ith cylindrical coordinates (r, z) is 9* = 9 + c ^ lo g c J - I y j... (4) = q + A q ----(4a) w here q = bearing capacity o f sim ilar wedge (eq. 1), A q = contact pressure due to circum ferential stresses a t failure, x and x ' = radial coordinates o f C ' at beginning and F ' a t end, respectively, o f the slip line (parallel to failure surface CD EF) governing the contact pressure qx. The bearing capacity factor N cr in eq. (3) is then given by 8 r 012 N = N c + - p. A q x d x (5) J 0 w hich integration m ust be carried o ut num erically w ith A q given by the last term of eq. (4). T h e results o f this analysis show th a t the bearing capacity increases alm ost linearly with depth (o r (5), and the factors N cr a re given in Fig. 2 for the lim its o f surface cones ({5 = 0) and deep cones ( 3 = 90 ). References [1] B e r e z a n t z e v, V. G. (1952). A x ia l Sym m etrical Problem o f the Lim it Equilibrium Theory of Earthy Medium, Moscow. [2] B i s h o p, A. W. (1957). Discussion on Soil Properties and their Measurement, Proc. Third Int. Con. Soil Meclt., vol. 3, p [3] D u g d a l e, D. S. (1954). Cone Identation Experiments, JL Meclt. Phys. Solids, vol. 2, p [4] E v a n s, I. (1950). The Measurement o f the Surface Bearing Capacity of Soils in the Study of Earth-Crossing Machinery, Geotechnique, vol. 2, p. 46. [5] H a b ib, P. (1953). Essais de charge portante de pieux en modèle réduit, Ann. Inst. Tech. Bat. Trav. Pub!., Paris, vol. 6, p [6] M e y e r h o f, G. G. (1948). A n Investigation o f the Bearing C ap acity o f Sh allow Footings on D ry Sand, Proc. Second Int. Con. Soil Mech., vol. 1, p (1951). The U ltim ate Bearing C ap acity o f Foundations, Geotechnique, vol. 2, p (1953). The Bearing C ap acity o f Foundations U n d er Eccentric and Inclined Loads, Proc. Third Int. Con. Soil Mech., vol. 1, p (1955). Influence o f Roughness o f Base and Ground-W ater C onditions on the Ultim ate Bearing C ap a city o f Fo u n d a tions, Geotechnique, vol. 5, p (1959). Com paction o f Sands and Bearing C ap a city o f Piles, JL Soil Mech. and Found. Div., A.S.C.E., vol. 85. N o. S M 6, p [7] T e r z a g h i, K. (1943). Theoretical Soil Mechanics, W iley, N ew Y o rk. 109