Session 4/24
The Bearing Capacity of Foundations under Eccentric and Inclined Loads Capacité portante des sols de fondation sous charges excentrées et obliques by G. G. M e y e r h o f , Ph.D., M.Sc. M.Sc. (Eng (Eng.) .), , F.G.S., A.M.I.C.E., A.M.I.C.E ., A.M. A .M.I. I. Stru Struct ct. . E., E., Building Resear Res earch ch Station, Station, Garston, Garsto n, Watford, Watfo rd, Herts Herts., ., Engl En glan and d
Summary
Sommaire
The author’s recently published theory of the bearing capacity of foundations under a central vertical load is extended to eccentric and inclined loads. First, an analysis is is given for eccentric vertical loads on a horizontal foundation and is compared with the results of laboratory tests on model footings on clay and sand. In the second section the theory is extended to central inclined loads on horizontal and inclined foundations and compared with the results of some model tests on clay clay and sand. Finally, it is is shown how these methods of analysis can be combined for foundation loads which are both eccentric and inclined and some test results are presented.
La théorie antérieure de la force portante des fondations sous charge centrale et verticale publiée récemment par l’auteur est éten due aux charges excentrées et obliques. Premièrement, une analyse est donnée pour des charges excentrées verticales sur fondations horizontales et elle est comparée avec les résultats d’essais en labora toire sur fondations modèles d’argile et de sable. Dans la deuxième section la théorie 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 peu vent 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 inclin clined ed loads due to bending m o m e n t s and an d horizontal horizontal thru thrust sts s acting in conjunction conjunc tion with w ith the vert vertic ical al loading loading. . T h e bearing capacity theory recently published by the author ( Meyerhof,\ 1951) can readil readily y be exten e xtended ded to cover such loading conditi conditions, ons, an d the prese present nt paper give gives s an outl outlin ine e of the the meth me thod ods s togethe together r with the res resul ults ts of som so m e tests with mod m odel el footings footings on clay and an d sand.
Thus for a shallow horizontal strip foundation of width B and depth depth D carrying carrying a vert vertic ical al load Q with an eccentricity o n the base (Fig (Fig. . 1), it m a y be ass a ssum umed ed that that the load acts acts e on centra centrally lly on a foun foundat datio ion n of effe effect ctiv ive e contact widt w idth h B' = B — 2
(1) 1)
t
Bearing Capacity Capacity of Fou ndat ion with Ecce Eccent ntri ric c Lo ad foun dation on carr carrie ies s an eccentri eccentric c load, load, it Theory. W h e n a foundati tilts towa to ward rds s the side of the ec eccen centri tricit city, y, and an d the contact contact pressure belo be low w the the base is generally taken to decrease linear linearly ly towards the hee heel l from fr om a ma m a x i m u m at the the toe. At the the ulti ultima mate te bearing bearing capacity of the foundati found ation on the distribu distribution tion of contact pressure is not even e ven approxim approximate ately ly line linear, and an d a very simple solution of the problem is obtained by assuming that the contact pressure distribution is identical to that indicated previously (Meyerhof, 1951), 1951), for a centr centrall ally y loaded loade d founda fou ndatio tion n but of reduced reduc ed width. width. 44 0
Fig. Fig. 1 Plastic Plastic Zones Near R ough Stri Stripp Foundation with Eccentric Eccentric Load Zones plastiques près d’une semelle à surface rugeuse sous charge excentrée
If the remaining width B-B' is ignored, which is somewhat B-B' is conserv conservativ ative, e, the correspo correspondi nding ng zones zones of plast plastic ic equilibrium in the material on the side side of the eccen eccentri tricit city y are the sa sa me as for for a simil similar ar cent centra rall lly y loaded loade d foundation. (The (Th e shear zones zone s are sh o w n in Fig. Fig. 1.) O n this basis basis for for a material of density density y, y, p the bearing unit cohesion c and an d angle angle of internal internal frict friction ion < capacity can can be represented by
Q = qB
(2a)
or
qB' qB'
=
(2b)
where <7 = c N ca
B' B'
(3 )
N y „
Nyq are the resultant and Ncq and Nyq resultant bearing bear ing capacity capacity factors factors for a central central load l oad (Meyerhof, 1951 1951) ) and depend mainly on < p and an d the depth rati ratio o DIB DIB1 1of the foundation. T h e abov ab ove e expressions give give only the base r resi esista stance nce to which whic h must mu st be add a dded ed any an y skin fri frict ction ion (Ca cos <5, see Fig. 1) on (Ca + Ps cos the shaft shaft to obtain the total total bearing bearing capacity of the foundation. foundation. The suggested procedure can be extended to a rectangular B, carrying a load Q with foundation founda tion of length length L and width B, eccentricities ex and ey on o n the majo ma jor r axes, axes, and to other other area areas s as sho sh o w n in Fig Fig. . 2 by b y findin finding g the the mi m i n i m u m effective contact contact area A' (with str strai aight ght boun bo unda dary ry across the base) such that its centroid coincides with that of the load. load. T hen Q=XqA'
(4 )
- 18|*
L ....li
SINGLE ecCtMTRlCITY
OOUBLE e c c e n t r i c i t y
R E C T A N G L E ( S Q U A R E S im i u A « )
Fig. Fig. 2
ECCEN TRICITY
(a )
Fig. Fig. 3
L O O SE SE
AND
c o m p a c t
e„/B
PACKINGS
e c c e n t r ic i t y c0)
d e n s e
e*/e.
p a c k i n g
Bearing Capacity of Footings with Eccentric Eccentric Vertical Load o n Sand Capacité portante des fondations sur sable sous charge verticale excentrée
In order to check the theory wh w h e n the shearing strength strength of the soil is k n o w n independently, so m e tes tests were wer e m a d e at the the Building Resear Res earch ch Stati Station on. . Footings Footings of 1 in. width wid th and an d various shape shapes s wer were e load loaded ed to fail failur ure e under und er diffe differe rent nt eccentrieccentricities on the surface of soft remoulded London clay and me di u m H a m River River sand sand in in a loos loose e and dense packing packing (por (poroosity of 45 an a nd 37 per cent, cent, res respe pect ctiv ivel ely) y). . T h e average average shearing shearing strength of the clay clay was c = 2 lbs./in ./in2 2and for the sand sa nd < 6° p= 36° p= 48° (dense) (loose) and < (dense) from fr om uncon u nconfine fined d com compre pressi ssion on and direct direct shearing shearing tes tests, respec respectiv tively. ely. T h e experi experimen menta tal l procedure of the model tests was similar to that described previously (Meyerhof, 1948, 1951), 1951), and an d a typical typical footing after after failure is illustrate illustrated d by b y Fig. 4. T h e test r res esul ults ts of the footings o on n clay (Fig (Fig. . 5) sh sh o w that the average bearing capaci capacity ty (m ( m a x i m u m load/f load/footi ooting ng are area) a) decrea decreases ses linearl linearly, y, with with increase increase in eccentrici eccentricity, ty, to zero for an y given given eccentricity eccentricity
Effectiv Effectivee Contact Area of Foundations with Eccentric Load Aire de contact effectif des fondations sous charge excentrée
wher wh ere e A is the shape shap e factor factor (Meyerhof, 1951) dependi depe nding ng on the the average length/width ratio L '¡B' of of the contact ar area ea, , and and q is given by equat equ ation ion (3). For foundations whose depth is greater than about their width wid th appreciable late latera ral force forces s are induce ind uced d on the shaft shaft by tilting unde un der r the load. load. Thes Th ese e forces modi mo dify fy the pla plast stic ic zones zones and increase the bearing beari ng capacit capacity; y; their their effe effect ct can ca n be be estimated as for rigid cantilever sheet sheet piles (Terzaghi, 1943). only published published tests resu result lts s of eccentrieccentri Ex Experiments'. ts'. T h e only call cally y loaded foundations found ations appea a ppear r to be those from fr om an exten extensi sive ve investigation in Belgium (Ramelot and Vandeperre, 1950). Circular and square square footings footings up to 16 in in. wide wide wer w ere e loaded loade d at various depths in compact sand whose angle of internal friction at the particular packing was unfortunately not determined. T h e experimental experi mental resu result lts s for surface surface and shallow footings (Fig. (Fig. 3) are consistent with the theory theory by taking. taking. cp = 44°, which whi ch woul wo ul d be a reasonable angle angle. . Shallow Sha llow footin footings gs were we re only only tested tested with wit h relat relative ively ly large ecce eccentr ntrici icitie ties s wh e n the theory theory is conservative be becau cause se it neglects neglects the resistan resistance ce due du e to the lateral lateral forces on on the shaft. shaft.
Fig. 4
Failure of Strip Footin g witli witli Ecccnlric Vertical Load on Sand Rupture de l’empattement sur sable sous charge verticale ex centrée 441
bearing bearing capacity is tilted and an d the adjacent zone z ones s are modifi modified ed accor accordi ding ngly ly. . T w o main ma in cases cases m a y be con consid sidere ered, d, namely, namely, foundations with a horizontal base and foundations with a base no n ormal ma l to the load load (i.e. base inclined at a to the horizontal). T h e correspo correspondi nding ng zones zo nes of pla plast stic ic equilibrium in the material material are sho sh o w n in Fig. Fig. 6 and an d solutio solutions ns for for the ultimate ultimate bearing bearing capacity q are derived derived in in the appen appendix dix (A. 1 and A. 2). T h e soluti solution on for a horizont horizontal al foundation foundat ion (appendix A. 1) can be expressed in terms of the vertical component of the bearing capacity
qv = q COS a B cNcq + y — Nyq
( «O «O
Fig. 5
STR IP
F OOTINO
(M CIRC UL AR
( \N \N D S a U P . * & F OO OO TI TI NG NG S
Bearing Capa city of Footings with Eccentric Vertical Load on Clay Force portante des fondations sur argile sous charge verticale excentrée
comp co mpar are e well well with the esti estima mate tes s wh e n an allowance is m a d e for som so m e increase increase in bearing capacit capacity y due due to the penetrati penetration on required for mobilization of the shearing strength as for central centrally ly loaded loa ded footings (Meyerhof, 1951 1951). ). T h e bearing bearing capa ca pacity of circu circular lar and square square footings footings is abou ab out t 20 20 per per cent greater than than that of stri strips ps at the sa sam e ecce eccent ntri rici city ty, , as fou f ound nd (Meyerhof, 1951) for for central central loads. loads. Fig. Fig. 5 also show ho ws that the cust cu stom omar ary y me th od of asse assess ssin ing g the the bearing bearing capac capacity ity from fro m the m a x i m u m toe pressure pressure is rather conservative. For single single ecce eccentr ntrici icitie ties s of the load the contact width or length at failure was, within experimental limits, given by equation (1), while for double eccen eccentri tricit cities ies the centroid of the contac con tact t area area at failure failure coin coincided with the point of applicati application on of the load, load, as had h ad been bee n assu as sume med d in the the theory theory. . T h e averag average e bearing capacity capacity of the the footing footings s on o n san s and d (Fig. (Fig. 3) 3) decreases approximately parabolically, with increase in eccentric centricity ity, , to zero zero for eJB 0.5; for a given ex, the bearing eJB = 0.5; capacity decreases approximately linearly with greater ey. Thes Th ese e resu result lts s are in fai fair agre agreem emen ent t with w ith the theoretic theoretical al est estimates; mates; for large large ecce eccent ntri rici citi ties es on dens dense e sand san d the observe obser ved d bearing capac capacity ity is s om ew ha t greater greater than estimated due du e to the greater angle angle of of interna internal l fric fricti tion on with smaller pressure pressure on the fai failu lure re surfac surface. e. T h e bearing bearing capacity of circula circular r and square square footings is is the sa s ame as that of stri strips ps for for loose sand sa nd but but is is abou ab out t 30 per p er cent les less than than that that of stri strips ps o n the surface surface of dense material material, , as found fou nd (Meyerhof, 1951) for similar central load loads. s. T h e custo cu stomar mary y m met et ho d of anal analys ysis is is reasonable for for small small eccentr eccentrici icitie ties s but but unsa unsafe fe for greater ecce eccentr ntrici icitie ties s ow o w in g to the rapid decrease of bearing bearing capacity with w ith smaller effe effect ctiv ive e contact width. T h e contact area at f fai ailu lure re was simila similar r to that that of footings on clay, and for dense sand the failure surface width at ground level decreased practically linearly with greater greater ecce eccentr ntrici icity ty as expected. Whil hi le the test tests on clay and an d sand sa nd indicated indicated that the “midd mi ddle le third rule”is rather arbitrary arbitrary, , they support the practice of designing shallow foundations with central central loading loading if possible since since the portion portion outside the effe effect ctiv ive e contact area area can can be ignored.
(5)
wher wh ere e the bearing capacity capacity fact factors ors Ncq and Nyq Nyq depend on
E //N \\ V f /
's V 1\ /
4 5 ° -# 2 F \
h ~ 8 '" 1 i LSii
a.
A & 'V ' B
D 1_
90° 0°? C (6) Horizontal base with large inclination of load
Bearing Bearing Capacity of Found atio n with Incli Incline ned d L oa d central al foundation foundati on load incl inclin ined ed at an Theory: U n d e r a centr angle a to the vert vertic ical al, , the central shear zone zo ne at the ultimate ultimate 442
Fig. 6
Plastic Zones near near Rough Strip Strip Founda tion with Inclined Inclined Load Zones plastiques près d’un empattement à surface rugeuse sous charge oblique
T h e solut solution ion for for an incl inclin ined ed foundation with a base norma nor mal l to the the load (appendi (ap pendix x A. 2) can be expressed in terms of the resultant resultant bearing capacity
B q = cNc,j + y — Ny Ny,,
(6)
T h e bearing bearing capacity fact factor ors, s, exclusive exclusive of any a ny skin friction, are given given in Figs Figs. . 7b 7 b and 8 b for a shallow shal low stri strip p foundatio foundation n in purely cohesive cohesive an a nd cohesionless materials materials, , respectively; respectively; they decrease rapidly with wi th greater inclina inclination tion a to the passive earth pressure coeff coeffici icien ents ts of a smoot h vert vertic ical al wall for a = 90°. Itis of intere interest st to note note that for a given given a an incli incline ned d foundation founda tion has a greater greater bearing capacity than a horizontal base, base, which whi ch supports the practic practice e of designing designing shallow foundation founda tions s with a base no n ormal to the resultant resultant load load if possib possible. le. T h e bearing capacity capacity of foundations of other other shapes under und er inclined loading can at present only be based on empirical evinence to obtain shape sh ape factor factors s A in conjunction with equations equatio ns (5) and an d (6) on account accou nt of the varia variabl ble e boun bo unda dary ry condition conditions s of the proble problem. m. T h e theoreti theoretical cal contact contact pressure dist distribut ribution ion at failure is similar to that of a foundation with vertical load. o f limited limited previous experimental Exp Exper erim imeents: ts: In view of evidence evide nce the bearing capacity has been b een determin dete rmined ed for dif diffe fere rent nt inclinat inclinations ions of a central central load o n horizontal footings as before before FOUNDATION DEPTH/WIOTM
7. \
cc 8
D/B *
\ \
i 4 u
Ct ITAl )
r, S'
u.
X
---
___
>N
<3 7. Oi 4
D \
a -o
<
C . l .
\ \ \
B * E
>
AL c ■e
Z
FOR NOTM SEE FW.Ca)
CL
\ BA v 3
y
—
FOR INTERNeOT. DEPTHS
\
\
u.
—
vl
v >
a
Ul <£>
LLI
o
¿o '
40*
INC LINA TION
( a i HOR IIZON ZON TAL
Fig. 7
60*
80*
OF LOAD
01
F OUNDATI OUND ATION ON
20°
INCLINA TION
40 *
I. o'
OF
(M INCLINED
80*
F O U N D AT AT IO IO N <
F OUNDA OUN DATI T ION ON
Bearing Capacity Factors for Strip Strip Foundation with Inclined Inclined Load in Purely Cohesive Material Facteurs de la capacité portante pour empattement sous charge oblique en matière purement cohérente
1«) HORIZ ONTA L FOUNDATION
Fig. 8
O
Cb) INCLINED
FOUNDATION
Bearing Capacity Factors for Strip Strip Founda tion with Inclined Inclined Load in Cohesionless Material Facteurs de la capacité portante pour empattement en sol pul vérulent sous charge oblique
Fig. 9
Arrangement o f Model Test on F ooting with Incline Inclinedd Load Arrangement d’essais sur fondation sous charge oblique
with a rough rou gh base base on the the same sam e clay clay and an d sand san d (but (but in in a comp co mpact act p = 45°). In the packing packin g with porosi porosity ty of 38 per cent cent and < the tests on clay the inclined inclined load load wa w as increase increased d to fail failur ure; e; in the te tests on sand sand a vert vertic ical al load w as applied an a nd kept k ept constant while the horizontal load applied by a second proving ring was increased increased to fail failure ure (Fig. (Fig. 9). In bot both h cases the footing remai rem aine ned d sensi sensibly bly horizontal horizontal through thro ughou out t the test. T h e test result results s of the strip footings on o n clay (Fig. (Fig. 10) are in reasonable agree agr eeme ment nt with the esti estimat mates. es. T h e bearing capacity of square footings footings was abo a bout ut 20 per cent greate greater r than that of str strip ips s at small inclinat inclination ions, s, as fou fo und previousl previously y ( Mey Meyeer hof\ 1951) for ver verti tica cal l loads, loads, the the differen difference ce beco be com ming small small for an inclin inclinati ation on exceeding exceedi ng abou ab out t 25° w h e n fail failur ure e occurred by slidin sliding g as wo would be expected expected theo theore reti tica call lly. y. T h e obser observed ved bearing bearing capacity capacity of the the strip footings footings on sand san d (Fig (Fig. . 11) 11) confo con form rmed ed with the theo theoret retica ical l estimates estimates and a p proa pr oach ched ed zero for an inclination inclination equal equal to the angle of inter internal nal friction g> = 45°, as wou wo uld be expected expected. . T h e bearing beari ng capacity of square footings was about 30 per cent less than that of stri strips ps for a vertica vertical l load, as fou fo und previously previously ( Me 1951) ) Meyerho rhof ; 1951 for surface surface loads on comp co mp ac t sand, the diffe differe renc nce e decreasing decreasing to zero beyo be yond nd an incli inclinat nation ion of abou ab out t 15°. T h e present anaan alysis was also chec ch ecke ked d by b y the observati observation on that the fai failu lure re surface width at ground level decreased steadily with greater incl inclina inatio tion n of the load and a nd appr approa oach ched ed zero for for a = 45° 5°. 443
E X P E Ri ME NTAL R E SULTS: STRIP (L/B= 6) SQUARE EXPEBlMENTAL RESULTS: S T R IP IP a / B - 6 ) SQUARE
* O
\ *s \ ÀE ^ % V \ a
- C E N T R I CITY
[
ST UP
\
\
V
THEORETICAL RESULTS: S T R I P ( 4 . = 4 5 ° )-----------
CO 140
THEORETICAL RESULTS: STRIP ---------------
X a
:\ \ X \ EC C E* T R I C I T Y
Cx/B
K° 'B
s t r
° a X
'
Conclusion
S JN .
10°
20*
INCL INA TIO N
30*
40°
OF LOAO OL
SO"
lo°
20®
eccentricity, method as above with positive a); the bearing capacity is given by the lower lowe r estimate. estimate. mo del l footing footings s on clay clay and an d sand Ex Experiments'. Horizontal mode as in section section 2 were we re loaded loaded to fail failur ure e with a single single forw forwar ard d eccentricity of eJB = 0.25 and differe different nt inclinations inclinations of the load; load; a typical footin footing g after after failure failure is illustra illustrated ted by Fig. 12. T h e test resu result lts s are given in in Figs Figs. . 10 an a nd 11 for clay and sand, resp respec ecti tive vely ly. . T h e bearing capacity capacity was wa s abou ab out t onehalf of that that of correspondin corresp onding g cent centra rall lly y loaded footings footings in in accordan accor dance ce with with the the theo theory ry, , whic wh ich h was w as supported by the observed contact area and an d mec m ec ha ni sm of failure. Preliminary Preliminary experiments with a back ba ckwa ward rd ecce eccent ntri rici city ty of loading loadin g were wer e also also foun fo und d to be in reasonable agre ag reem emen ent t with with the esti estima mate tes. s.
50°
40 °
S0°
INCLINATI ON OF LOAD oL
Fig. Fig. 10 10
Bearing Bearing Capacit Capacityy of Footings with with Incli Inclined ned Load on Clay Clay Capacité portante des fond ations sur argile argile sous charge oblique
Fig. 11
Bearing Bearing capaci capacity ty of footings footings with with incline inclinedd load on sand Capacité portante des fondations sur sable sous charge oblique
T h e previous previous bearing capa capacity city theory theory of foundations under und er a central vertical load has been extended to eccentric and inclined inclined loads. loads. T h e theory, whic which h indicates indicates that the bearing capacity decreases rapidly with w ith greater ecce eccentr ntrici icity ty an a n d inclination nation of of the load, load, is is suppo sup porte rted d by the resu result lts s of loading loading test with mod m odel el footin footings gs on clay and an d sand. sand. Acknowledgment
Bearing Capacity of o f Fou nda tio n with Eccentr Eccentric ic Incl Inclin ined ed Load founda tion carr carrie ies s an eccentri eccentric c incli incline ned d Theory. W h e n a foundation load an estimate estimate of the bearing bearing capaci capacity ty can be obtained by by combinin comb ining g the above abo ve met m ethod hods s of ana analy lyse ses. s. Thus Th us for for a shall shallow ow strip foundation with a forward eccentricity of loading (a is positi positive, ve, i.e. eccentric eccentricity ity in direction of horizont horizontal al co co m p on e n t B' (equati of load) an an effect effectiv ive e contact wid width th B' (equation on 1) is used used in equation equations s (5) (5) or (6) and the tota total l bearing bearing capacity is is given by by equati equation on (2). Simila Similarly rly, , for a doub double le eccentr eccentrici icity ty o n a rectan rectangugular or other area the effective contact area and shape factor are used use d as in equation equat ion (4). If the eccen eccentri tricit city y is bac ba ckward opposi site te direction direction to horiz horizonta ontal l (a isnegative, i.e. eccentricity in oppo compon ent of load load), ), fail failur ure e of the the soil occurs either either on the side of the ecce eccentr ntrici icity ty (small ecc eccen entr tric icit ity, y, me m et ho d as abov ab ove e but using negative a in analys analysis) is) or on on the opposi opposite te side (larg (large e
The a aut utho hor r is indebted indebted to his his colleagues colleagues, , particu particularl larly y Mr. Mr .
L. L. F. F. Co Cooling M.Sc., for helpful criticism and Mr. B. B. J. J. Ca Cat B.Eng., for assis assistan tance ce in in carrying carrying out mo m ost of of the mo de l terall B.Eng., tests. T h e w o r k w was as carried carried out as part part of the the research propr og r a m m e of the the Build Building ing Research Board Boar d of the the Depart Dep artmen ment t of Scientific ific and Indust Industria rial l Resea Re search rch and an d the pape pa per r is published by permission of the Director of Building Building Research.
Appendix Bearing Capacity of Horizontal Str Strip ip Foun dat ion with Inclined Load The region above the failure surface of a shallow rough stri strip p found foundati ation on with wi th load load inclined at a to verti vertica cal l is is ass as sume um ed to be divided into a central ela elastic stic zone zo ne AB radial shear ABC, a radial zone ACD and a mixed shear zone ADE (Fig. 6 a). T h e ADEF (Fig. stre stress sses es in these these zones can c an be fou f ound nd as s h o w n {Meyerhof, 1951) for a vertic vertical al load, by b y replacing replacing the resultant of the forces forces on the shaft AF a nd the weight of the adjacent so soil AF wedge AEF AEF by the equivalent stresses p0 and s0, normal and ft to tangential tangential, , respect respectivel ively, y, to the plan plane e AE inclined at ft to the horizo horizonta ntal. l. O n this bas basis is the ver verti tica cal l com pone nt of the bearing capacity capacity can, in the first instance, instance, be represe represented nted by — qCOS
a
cNc + == cNc
B p0N p0N„ + y — Ny Ny
(7)
or
==
Qv +
where
q[, = cNc + p0N p0N ,, B qv -= v- Ny T
and Nc,
Fig. 12
444
Failure of Strip Footing with Eccentric Inclined Load on Clay Empattement sur argile sous charge excentrée et oblique
(8)
(9)
bearing capacity fact factor ors. s. N and Ny Ny are the general bearing with ang angle »/'at De Deter termina inatio tion of of Ncan Ncand Nq\ In zone AB ABC with the shearing strength under unde r the t he norm no rmal al pressure pres sure A, A, Sp pp pp on He nce e from M o h r ’ s diagram AC AC is is Sp = c + p'p tan q>. Henc
c + pf pfi tan ip [sin (2y> — < sin cp] + p'p p) + sin COS cp
Qi
(10)
+ pp pp tan
Qv =
cp
cos cp
cos (2 y>—
cp) cot a
(11)
from which y>can be determined from fro m any given a, cp, c and fr om equations 12 and an d 13). pj pj, (obtained from In zones AC with ang angle le 0 = 180° 1 fi ACD and ADE ADE with fi — >/ — v and ang angle le ??, respectivel respectively, y, at A, it was show sh ow n {Meyerhof, 1951) A, it that
Pp — e20 tan < p— c] cot t( t(c + Pi tan cp) e2
( 12)
and
c ( Pi tan cp [sin [sin (2i; + ) — sin cp\ + p0 cos p I
Pi =
where > det ermined from fr om the given given rati ratio o } can be determined Substituting Substituting equations equati ons (12) (12) and an d (13) (13) into (10 (10) ) <7„ = c +
cot < p
[1 + sin < p sin {2y> — ?>)] p) 1 — sin sin cpsin (2»; + <
1 ) sin < p sin (2y> — (p) Po 1 — sin sin cp sin sin (2»? + cp)
» 2 0 t a n
e 2 6 ta
(13)
sjp0 jp 0.
n
___ ___
]
+ (14)
or
q'v = cNc cNc + p0Nq p0Nq from fr om equation (8 (8) where whe re Nc and Nq have the values giv given en in Nq have the square brackets above. T he hori horizo zont ntal al comp co mpon onen ent t q'h of the bearing capacity cannot can not exceed the shearing resistance on the base, i.e.
q'h = q' sin a = q'v tan a
compone nt of the resu resulta ltant nt bearing capacity capacity is
B qv = cNcq + y — Nyq
a n d
/
De Deter termina ination of Re Resulta ltant Bearing Ca Capacity ity. T h e vertic rtical al
tan a — tan tan <5'
where Nc Ncq (depending on Nc and Nq) and Nyq Nyq (depending on bearing capacity capacity factors factors, , an and is Ny Ny and Nq) are the resultant bearing comp co mput uted ed from fro m the the above solu soluti tion ons s by determining determining the the foundation depth parameters (/?, p0 and i0) for various depths D as shown {Meyerhof, 1951) for a vertic vertical al load. For large large inclinations a when qh governs, governs, the the horizon horizontal tal co m po nent ne nt of the passive earth pressure on the front of the foundation is adde ad ded d to the shearing shearing resi resist stan ance ce on the base given by equati e quation on (15) (15); ; and an d if in addition the foundat foundation ion has a rou rough gh shaft, the foundat foun dation ion is part of the central central zone zo ne AB 6b). ABCF (Fig. 6b). It has therefore been found convenient to include the skin fric fricti tion on or vert vertic ical al co mponent of o f the passive earth pressure on the shaft shaft in in the bearing capacity capacity factors factors (Fi (Figs gs. . 7a 7 a and 8a). 8a).
Bearing Capacity Capaci ty of Inclin Inclined ed Str Strip ip Foun Fo unda dati ti on with Base Nor mal to Load For Fo r a shallow roug ro ugh h str strip foundation of width B and depth upp er edge edg e of the base inc inclin lined ed at at an angle a to the D of the upper horizontal (Fig. (Fig. 6c) the zones zones are similar similar to to those of a horihorizontal zontal foundation with w ith y>= 45° +
B q = cNc + Po Nq + y — Ny Ny
(19)
are obtained by substituting these values of equa equati tion on (14). (14). Similarly it is foun fo und d that
y>and
into
(15) ssin in j^4 45° + — 4Pp sin
unit base adhesion c'a = unit angle of base base friction. S' = angle
For greater inclinations a when must mu st theref therefore ore be replaced replaced by
(18)
Ny = governs, equation equation (14) (14) q'h governs,
(16)
obtained obtained fr om (15). passiv ive e resi resist stan ance ce Pp De Deter termina inatio tion of Ny: Th e m i n i m u m pass acting at cp to the the normal on AC in the zone zon e ACD can be AC in ACDE can found either by a numerical stepbystep computation {Ca1949) or by a semigraph semigraphical ical procedure proced ure quot and Kerisel, 1949) {Meyerhof, 1951) based on the logorithmic spiral method. T he n it it can be sh ow n that that snr y> sin yi yi cos {y>—cp) „ yB '2P[j ________ - cos{y>— s{y>—
— (17) cos cp y^l cos iy — cp)
<7, = y
or \ r = y—B N v
from fr om equation equation (9 (9) where wher e Ny the square Ny has the value given in the bryckets above. T h e above ab ove soluti solution on holds holds only for for a ^ (5' (see equa equati tion on 15).
yB2
---- tan I45° , 2
\
2
j COS a
(20)
where Pp is the mi m i n i m u m passive resis resista tanc nce e obtained as indicated earlier. T h e result resultant ant bearing capacity capacity
B q = cNc cNcq + y — Nyq
(21)
is determin dete rmined ed fr f rom these these solution solutions s as befor before, e, and an d the bearing capacity factors are given in Figs. Figs. 7 b and 8 b.
References Caquot, A.
and K er is el , J. (1949): Traité de Mécanique des Sols. Gauthier-Villars, Paris, p. 85. M ey er h of , G. G. (1948): An Investigation Investigation of the Bearing Capacity o f Shallow Footing s on Dry Sand. Sand. Proc. Second Int. Conf. Soil Mech., vol. 1, p. 237. M ey er h of , G. G. (1951): The U ltimate Bearing Bearing Capacity of Founda tions. Géotechnique, vol. 2, p. 301. R am el ot , C. and Vandeperre, L. (1950): Travaux de la Comm ission d’Etude des Fond ations de Pylônes. Com pt. Rend. Rech., I.R I.R .S.I.A ., Brussels, No. 2. Terzaghi, K. (1943): Theoretical Soil Mechanics. J. Wiley, New York, p. 355.
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