The ultimate bearing capacity of wedge-shaped foundations - La force portante des fondations en coins

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The ultimate bearing capacity of wedge-shaped foundations - La force

portante des fondations en coins

The Associate Committee on Soil and Snow

Mechanics is one of about thirty special committees which

assist the National Research Council in its work.

Formed

in 1945 to deal with an urgent wartime problem involving

soil and snow, the Committee is now perfonning its intended

ta sk of co-ordinating Canadian research studies concerned

with the physical and mechanical properties of the terrain

of the Dominion.

It

does this through subcommittees on

Snow and Ice, Soil Mechanics, Muskeg and Pennafrost.

The Committee, which consists of about twenty-five Canadians

appointed as individuals and not as representatives. each for

a 3-year term, has funds available to it for making research

grants for work in its fields of interest.

Inquiries

wi

11 be

welcomed and should be addressed to: The Secretary, Associate

Committee on Soil and Snow Mechanics, c/o Division of

Building Research, National Research Council, Ottawa, Ontario.

This publication is one of a series being produced by the

Associate Committee on Soil and Snow Mechanics of the National Research

Council.

It

may therefore be reproduced, without amendment, provided

that the Division

i

st o ld in advance and that full and due acknowledgment

of this publication is always made.

No abridgment of this report may

be published without the written authority of the Secretary of the ACSSM.

NATIONAL RESEARCH COUNCIL

CANADA

ASSOCIA TE COMMITTEE ON SOIL AND SNOW MECHANICS

CANADIAN PAPERS PRESENTED AT THE FIFTH

INTERNA TIONAL CONFERENCE ON SOIL MECHANICS

AND FOUNDATION ENGINEERING,

PARIS,

JULY 1961

TECHNICAL MEMORANDUM NO.72

OTTAWA

JANUARY 1962

PREFACE

The Fifth International Conference on Soil Mechanics

and Foundation Engineering was held in Paris, France, from 17 to

22 July 1961.

The first such conference was held in 1936 as a part

of the tercentenary celebrations of Harvard University, Cambridge,

Mass.

The incidence of war necessitated the gap of twelve years

between the first two meetings.

The second conference was held in

Rotterdam in 1948, and the third was held in Zurich in 1953.

The

fourth was held in London in 1957.

Seven Canadians were present at the Harvard rn e e

ti.ng,

This number has increased over the years and over 25 were present

at the conference in Paris.

The Associate Committee on Soil and

Snow Mechanics of the National Research Council is pleased to publish

the reprints of the eleven Canadian papers which were included in the

official proceedings.

The International Society of Soil Mechanics and Foundation

Engineering is composed of national sections.

The executive body for

the Canadian Section is the Associate Committee on Soil and Snow

Mechanics of the National Research Council.

The principal function of

the Canadian Section is to assist in the further development and

appli-cation of soil mechanics throughout Canada.

Enquiries with regard to

its work will be welcome; they may be addressed to the Secretary,

Associate Committee on Soil and Snow Mechanics, National Research

Council, Ottawa 2, Canada.

Robert F. Legget,

Chairman.

Ottawa

3B/16

The Ultimate Bearing Capacity of Wedge-shaped Foundations

La force portante des fondations en coins

by Professor G. G. MEYERHOF, D. Sc., Ph. D., F. A.S.C.E., M.E.I.e., A.M.I.e.E., Head, Department of Civil Engineering, Nova Scotia Technical College, Halifax, N.S., Canada

Summary

The previous theory of the bearing capacity of foundations is extended to wedge-shaped bases and cones. The analysis is com-pared with the results of tests on cones and model piles of different roughnesses and with various shapes of tips in clays and sands.

Sommaire

La theorie anterieure de la force portante des fondations est etendue aux bases en coins et en cones. L'analyse est comparee avec les resultats d'essais sur cones et modeles reduits de pieux de rugosites differentes et avec des pointes de formes diverses dans les argiles et les sables.

Introduction I '--8 (2) F D 90°-

't'

D SMOOTH BASE

(o.) SHALLOW BLUNT WEDGE OR CONE

tions (D/B

<

1) the stress Po = yD, whereD = base depth

of wedge, while for deep foundations (D/ B:> 4 to 10, depend-ing on $)

where Kb c= earth pressure coefficient on shaft near base,

which is about 0'5 for sands and 1·0 for clays (MEYERHOF 1951, 1959).

(I)

Piles frequently have pointed rather than flat tips, and cone penetrometers are used in the field and laboratory. The bearing capacity theory previously published by the Author (1951, 1953, 1955) can readily be extended to cover such loading conditions. The present paper gives an outline of the methods and the results of some tests on cones and model piles in clays and sands.

Bearing Capacity of Wedges

When a foundation with a wedge-shaped base carries a central load, the zones of plastic flow in the soil at failure are similar to those of an inclined strip foundation for which a solution of the ultimate bearing capacity was derived previously (MEYERHOF, 1953). Thus, for a perfectly smooth wedge with a semi-angle\I.(Fig. 1)the region above the failure surface on each side of the foundation centre line is assumed to be divided into a plane shear zone ACD, a radial shear zone ADE and a mixed shear zone AEFG (shallow wedge) or a plane shear zone AEF (deep wedge). As the roughness of the wedge increases, the angle '':' at A in zone ACD decreases as under an inclined load on a 'horizontal base (MEYERHOF, 1953). For a perfectly rough wedge '(Fig. 1) a central elastic zone ACD forms a false base on a blunt wedge when the bearing capacity is identical to that of a horizontal base (MEYERHOf, 1955), while for a sharp wedge this elastic zone coalesces with the wedge.

The stresses in the zones of plastic equilibrium can be found as shown for a horizontal foundation (MEYERHOF, 1951, 1953) by replacing the weight of the soil wedge AFG by the equivalent stresses Po and so, normal and tangential, respectively, to the plane A F inclined at an angle ｾ to the horizontal. The bearing capacity can be represented by (TERZAGHI, 1943)

where c = cohesion, y = unit weight of soil,

B = width of foundation, and N_{c ,} N_Q and Ny = bearing capacity factors depending on ｾＬ angle of internal friction $ and depth/width ratioD/ B of foundation. For shallow

founda-SMOOTH BASE ROUGH BASE

(b) DEEP SHARP WEDGE OR CONE

Fig. 1 Plastic Zones Near Wedge-Shaped Foundations Zones plastiques pres des fondations en coins.

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---.--ANGLE OF

INTERNAL FRltTION

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5EM' - ANGLE

rA

(b) GREA"T DEPTH

Fig. 2 Bearing Capacity Factors Nc and Ncr.

Facteurs de force portanteN';etNcr.

The bearing capacity factors are given in Figs. 2 to 4 for the limiting conditions of perfectly smooth and perfectly rough wedges at shallow and great depths. The factors for smooth wedges decrease rapidly with smaller semi-angles

o;

but for o:

<

30°, approximately, the factor Ny increases again. For rough wedges the factors are sensibly unaffected by the wedge angle (false base) except for about o:

<

30° when the factors increase rapidly with smaller angles. The factors for smooth wedges are much smaller than those for rough wedges. Bearing capacity factors for intermediate degrees of roughness can be found by linear interpolation between the above limits with good approximation, and such factors decrease with smaller oc to a minimum and then increase again.

The above expressions give only the base or wedge resis-tance to which must be added any skin friction (c,

+

P_ssin

a,

see Fig. 1) on the shaft to obtain the total bearing capacity of the foundation.

Bearing Capacity of Cones

At the ultimate bearing capacity of a cone plastic flow of the soil induces circumferential stresses, which raise the bearing capacity above that for a corresponding wedge. The previous solution for the bearing capacity of circular found-ations in purely cohesive soils (MEYERHOF, 1951) has been extended in the Appendix to derive corresponding bearing capacity factors Ncr for perfectly smooth and perfectly rough cones, which are shown in Fig. 2. The factors for rough cones vary with o: in a similar way to those of wedges and the shape factor (ratio of cone/wedge bearing capacity) is sensibly independent of o: The factors Ncr for smooth cones do not vary appreciably with « : they are less than those for rough cones and of the same order as for rough blunt wedges(«

>

30°). Bearing capacity factors for cones of intermediate rough-nesses can be interpolated linearly.

For cohesive soils with internal friction the bearing capacity

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Fig. 3 Bearing Capacity Factors Nq and Nq r •

Facteurs de force portante Nq et Nor'

of cones can at present only be obtained from empirical shape factors in conjunction with eq. (1) to give the cone resistance

.... (3)

On the assumption that the shape factors are the same as observed for circular foundations with horizontal bases (MEYERHOF, 1951, 1955), the bearing capacity factors for perfectly rough cones are given in Figs. 2 to 4. While the factors Ncr and _{N qr} for cones are greater than those for

wedges, as would be expected, the semi-empirical factors

NY r are smaller, although an approximate theory for circular

footings (BEREZANTZEV, 1952) gives the opposite result. This difference appears to be due to the effect of the inter-mediate principal stress, which raises the actual bearing capacity of wedges relative to that of cones in frictional

soils. Thus, for circular surface footings on sands the empiri-cal shape factors are less than unity compared with theoretiempiri-cal values exceeding 2; this would correspond to an increase in $ under strip foundations of some 14 per cent (30°

<

$

<

45°), which is in reasonable agreement with the amount of about 10 per cent found by comparing some plane strain and triaxial compression test results (BISHOP, 1957).

Experiments with Cones and Piles

Some loading tests were made, first at the Building Research Station and more recently at the Nova Scotia Technical College, using either brass (semi-rough) or sanded (rough) cones and model piles of land 1 in. dia. with tips of various angles, which were pushed into soft remoulded clays

(c = 2 to 3 Ib./in2_{) and medium sands of various densities}

($ = 35° to 45°). The experimental procedure was similar to that described previously (MEYERHOF, 1948, 1951).

1000 800 "00

Conclusions

The previous bearing capacity theory of foundations with horizontal bases has been extended to wedge-shaped bases and cones. The theory, which indicates that the point resis-tence of piles with smooth tips decreases and with rough tips increases as the cone angle decreases, is supported by the results of loading tests on cones and model piles in clays and sands.

Acknowledgement

The early laboratory investigations were carried out at the Building Research Station of the Department of Scientific and Industrial Research and the results are published by permission of the Director of Building Research.

the tip is less important. The interpretation of laboratory cone tests on clays is, however, difficult due to the unknown amount of adhesion and lip of the material; thus ignoring consolidation and time effects, the resistance of a 60° cone may be only one-half of that of a perfectly rough cone which would be preferable in practice.

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ANGLE OF SHEARING RESISTANCE <1'

(b) CONE RE51STANCE IN COHESI\lE SOILS Fig. 5 Cone and Point Resistance of Piles in Cohesive Soils.

Resistance de cone et de pointe des pieux dans les sols coherents.

200

40° GO° 800

SEMI-ANGLE OF TIP o,

(0.) CONE RESISTANCE AND POINT RE51STANCE

OF PILES IN CLfl..'(S

STEELCONES (EVANS 1950y

.

r:J...= 300

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--

I--BRASS CONES AND PILES

..

IN CLAYS (PRESENT TESTS) ",

\'

LUBRICATED STEEL CONES

I

I\.

IN MItTA.LSlOUGDALE 1954)0

ｾＢ

_I'--..

BASE.DEPTH/WIDTHDIB

\

THECRy__

If"

-'"

>4-ｾ

RF. " OUG CONE

ｾ

0 THEpRY ｾ ｾ

ｬＮｾｴ

ｏｾｬ ;rIEJl

-<, (PEfF, SMpOTH

--ｾ ""- _/---

--

,...-0,

ＮＮＮ｡ｾ coRRECTE ""- FOR LIP 8 IG 14 12 10 w u 2 4: I-.n

i7l

ｾ

z

8

1- 0

zi/i

- uJ

6

ｾＺｉ

Q::O OW uJ

4

z

o

u 20 SEMI- ANGLE cJ..

Fig. 4 Bearing Capacity Factors N,and N_{Yr .}

Facteurs de force portanteN,etN_{Yr .}

The test results for clays (Fig. 5a) show that the cone resistance and point resistance of brass piles agree well with theoretical estimates based on perfectly rough tips; ancillary pure torsion tests on the cones gave an adhesion of about 0'8 c, which is likely to be increased by vertical load, The theory for perfectly smooth cones can be compared with the results of shallow indentation tests using lubricated steel cones in copper and aluminium (DUGDALE, 1954); the exper-imental cone resistance is somewhat less than predicted unless an allowance is made for the raised lip around the indentation (Fig. 5a). Similar indentation tests with 60° cones in cohesive-frictional soils (EVANS, 1950) also support the theoretical relationships and indicate a skin friction of about 20 to 80 per cent of the shearing strength (Fig. 5b).

Exploratory model tests with rough piles in compact sand indicated that the point resistance increases little with smaller cone angles (HABIB, 1953); this is supported by the present

test results, which show that the observed point resistance of rough piles is somewhat greater than estimated (Fig. 6). The measured point resistance of brass piles in sand is in fair agreement with estimates based on a skin friction of about

0'5

CP,

which compares well with the results of direct shearing tests under the same conditions.

Although large-scale tests would be useful as a check, the proposed methods of analysis are probably sufficiently accurate for practical purposes. For steel (semi-rough) piles and penetrometers the point resistance decreases, while for concrete (rough) piles the point resistance increases, as the cone angle of the tip decreases (sharper points). Since the ultimate bearing capacity of piles in cohesionless soils is largely due to point resistance, the shape of the tip may 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 capa-city of piles is mainly due to skin friction and the shape of

E'lC.PERIMEN1AL ｒｅｓｕｌｔｓｾＭ SANDED C.ONE5 X BRASS CONES • THEORETICAL REｓｕｌｔＵｾﾭ PERfECTL'( ROUGH

<6

=

<P)

SEMI-ROUGH CONE

(0

=

!/l/2.)

2000

UJ:I:

vI-ｚｾ

«0

"'-w I/lI/'I ｉＯｬｾ wCO

ex: )(.

\ ) -Zt: -1/'1

Oz

O-UJ

o

1000

800

<;;00

400

300

200

100

80

bO

\

It _RELATIVE DENSITY;-ｾ

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ｏｅｾ

sE

q,

］ｾ 5·) ｾ /"

;

Ｌｾ

-

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r":

.- COMPAC

<;=

41°)

•

; <,

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•

K

l( LO( SE

<1'=

35j

·

/ '

·

<;

..-1---" ｾＮ 20° 40° <;;0° 80°

SEMI-ANGLE OF TIP

oc..

Fig. 6 Point Resistance of Piles in Sands.

Resistance de pointe des pieux dans les sables.

APPENDIX References

(4)

Bearing Capacity of Cones in Purely Cohesive Material

On the assumption that the plastic zones on radial planes of cones are identical to those on transverse sections of wedges (Fig. I)and that the circumferential stresses are equal to the minor principal stresses, it was shown (MEYERHOF, 1951) that at failure the vertical contact pressure on the base

qx at any radius r = x from the foundation axis with cylin-drical coordinates (r, z) is

" F ' d

q

+

C(lOge: -

j

C' / )

q

+

Llq .... (4a)

whereq = bearing capacity of similar wedge (eq. 1), Llq =

contact pressure due to circumferential stresses at failure,

x and x' = radial coordinates of C' at beginning and F'

at end, respectively, of the slip line (parallel to failure surface

CDEF) governing the contact pressureqx'

The bearing capacity factor Ncr in eq. (3) is then given by

8 ,BI2

u.;

= Ne T CB2.

I

/1qxdx (5)

•./ 0

which integration must be carried out numerically with Llq

given by the last term of eq. (4). The results of this analysis show that the bearing capacity increases almost linearly with depth (orｾＩＬ and the factors N_erare given in Fig. 2 for the limits of surface conesＨｾ = 0) and deep conesＨｾ = 90°).

[1] BEREZANTZEV, V. G. (1952). Axial Symmetrical Problem of the Limit Equilibrium TheoryofEarthy Medium, Moscow.

[2] BISHOP, A. W. (1957), Discussion on Soil Properties and their Measurement, Proc. Third Int. Con! Soil Mech.,

vol. 3, p. 103.

[3] DUGDALE, D. S. (1954). Cone Identation Experiments,

fl. Mech. Phys. Solids, vol.2, p.265.

[4] EVANS, I. (1950). The Measurement of the Surface Bearing Capacity of Soils in the StudyofEarth-Crossing Machin-ery, Geotechnique, vol.2, p. 46.

[5] HABIB, P. (1953). Essais de charge portante de pieux en modele reduit, Ann. Inst. Tech. Bat. Trav, Publ., Paris, vol. 6, p. 361.

[6] MEYERHOF, G. G. (194g). An Investigation of the Bearing Capacity of Shallow Footings en lory Sand, Proc. Second

Int. Con! Soil Mech., vol, 1,p. 237.

(1951). The Ultimate Bearing Capacty of Foundations,

Geotech nique, vol, 2,p. DI.

(1953). The BearingCapacity of Fcundations Under Eccen-tric and Inclined Load', Proc. Third Int. Con! Soil Mech., vol. 1, p. 440.

(1955). Influence of Roughness of Base and Ground-Water Conditions on the Ultimate Bearing Capacity of Founda-tions, Geotech nique, vol, 5,p.227.

(1959). Compaction of Sands and Bearing Capacity of Piles,

fl. Soil Mech. and Found.ot«,A.S.C.E.,vol, 85, No.SM 6,

p. 2291-1.

[7] TERZAGHI, K. (1943). Theoretical Soil Mechanics, Wiley,

New York.