Session 4/24
Meyerhof, G. G. (1953). “The Bearing Capacity of Foundations Under Eccentric and Inclined Loads,” Proceedings, 3rd International
Conference on Soil Mechanics and Foundation Engineering, Vol. 1, 440–445.
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. ( Eng. ) , F. G. S. , A. M. I . C. E. , A. M. I . St r uct . E. , Bui l di ng Resear ch St at i on, Gar st on,
Wat f or d, Her t s. , Engl and
Summar y
Sommai r e
The author’s recently published theory o f the bearing capacity of
foundations under a central vertical load is extended to eccentric and
inclined loads. First, an analysis 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 o f som e
model tests on clay and sand. Finally, it is shown how these methods
o f analysis can be combined for foundation loads which are both
eccentric and inclined and som e test results are presented.
La théorie antérieure de la force portante 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
horizontales et elle est comparée avec les résultats d’essais en laboratoire sur fondations modèles d’argile et de sable. D ans 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 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.
I nt r oduct i on
Foundat i ons ar e f r equent l y subj ect ed t o eccent r i c and i n
cl i ned l oads due t o bendi ng moment s and hor i zont al t hr ust s
act i ng i n conj unct i on wi t h t he vert i cal l oadi ng. The bear i ng
capaci t y t heor y r ecent l y publ i shed by t he aut hor ( Meyerhof,\
1951) can r eadi l y be ext ended t o cover such l oadi ng condi t i ons,
and t he pr esent paper gi ves an out l i ne of t he met hods t oget her
wi t h t he r esul t s of some t est s wi t h model f oot i ngs on cl ay and
sand.
Thus f or a shal l ow hor i zont al st ri p f oundat i on of wi dt h B
and dept h D car r yi ng a vert i cal l oad Q wi t h an eccent r i ci t y
e on t he base (Fi g. 1), i t may be assumed t hat t he l oad act s
cent r al l y on a f oundat i on of ef f ect i ve cont act wi dt h
B' =
B — 2<? .
(1)
t
Bear i ng Capaci t y of Foundat i on wi t h Eccent r i c Loa d
Theory. Wh e n a f oundat i on carr i es an eccent r i c l oad, it
ti l ts t owar ds t he si de of t he eccent ri ci t y, and t he cont act
pr essur e bel ow t he base i s gener al l y t aken t o decr ease l i near l y
t owar ds t he heel f r om a ma x i mu m at t he t oe. At t he ul t i mat e
bear i ng capaci t y of t he f oundat i on t he di st r i but i on of cont act
pr essur e i s not even appr oxi mat el y l i near, and a ver y si mpl e
sol ut i on of t he pr obl em i s obt ai ned by assumi ng t hat t he
cont act pr essur e di st r i but i on i s i dent i cal t o t hat i ndi cat ed
pr evi ousl y (Meyerhof, 1951) , f or a cent r al l y l oaded f oundat i on
but of r educed wi dt h.
440
Fig. 1
P lastic Z o nes N ea r R ou gh Strip F ou n d a tio n w ith E ccen tric L oad
Z o n es p lastiq ues près d ’un e sem elle à su rface rugeuse sou s charge
ex cen trée
I f t he r emai ni ng wi dt h B-B' i s i gnor ed, whi ch i s somewhat
conser vat i ve, t he cor r espondi ng zones of pl ast i c equi l i br i um
i n t he mat er i al on t he si de of t he eccent r i ci t y ar e t he same as
f or a si mi l ar cent r al l y l oaded f oundat i on. ( The shear zones
ar e shown i n Fi g. 1.) On t hi s basi s f or a mat er i al of densi t y y,
p t he bear i ng
uni t cohesi on c and angl e of i nt er nal f ri ct i on <
capaci t y can be r epr esent ed by
Q = qB
or
=
qB'
wher e
<7
=
( 2a)
c N ca
( 2b)
B'
(3 )
N y„
and Ncq and Nyq ar e t he r esul t ant bear i ng capaci t y f act ors
p
f or a cent r al l oad (Meyerhof, 1951) and depend mai nl y on <
and t he dept h r at i o DIB1of t he f oundat i on.
The above expr essi ons gi ve onl y t he base resi st ance t o whi ch
must be added any ski n f ri ct i on (Ca + Ps cos <5, see Fi g. 1) on
t he shaf t t o obt ai n t he t ot al bear i ng capaci t y of t he f oundat i on.
The suggest ed pr ocedur e can be ext ended t o a r ect angul ar
f oundat i on of l engt h L and wi dt h B, car r yi ng a l oad Q wi t h
eccent r i ci t i es ex and ey on t he maj or axes, and t o ot her ar eas
as shown i n Fi g. 2 by f i ndi ng t he mi ni mu m ef f ect i ve cont act
ar ea A' ( wi t h st r ai ght boundar y acr oss t he base) such t hat
i ts cent r oi d coi nci des wi t h t hat of t he l oad. Then
Q=XqA'
(4 )
-
L
18|*
. . . . l i
SINGLE ecC tM T R lC IT Y
OOUBLE
RECTA N G LE (SQ UA RE S im i
Fig. 2
e c c e n t r ic it y
u A « )
E C C E N T R IC IT Y
(a )
Fig. 3
LO O S E
AN D
c o m p a c t
e „ /B
P A C K IN G S
e c c en t r ic it y
c0 )
d e n s e
e */ e .
p a c k in g
Bearing C apacity o f F ootings w ith Eccentric V ertical L oad on Sand
C ap acité p o rta n te des fond atio ns su r sable sous charge verticale
excentrée
I n or der t o check t he t heor y when t he shear i ng st r engt h of
t he soi l i s k nown i ndependent l y, some t est s wer e made at t he
Bui l di ng Resear ch St at i on. Foot i ngs of 1 i n. wi dt h and
var i ous shapes wer e l oaded t o f ai l ure under di f f er ent eccent r i
ci t i es on t he sur f ace of sof t r emoul ded London cl ay and
medi um Ha m Ri ver sand i n a l oose and dense packi ng ( por o
si t y of 45 and 37 per cent , respect i vel y) . The aver age shear i ng
p= 36°
st r engt h of t he cl ay was c = 2 l bs. / i n2and f or t he sand <
p = 48° ( dense) f r om unconf i ned compr essi on
( l oose) and <
and di r ect shear i ng t est s, respect i vel y. The exper i ment al
pr ocedur e of t he model t est s was si mi l ar t o t hat descr i bed
pr evi ousl y (Meyerhof, 1948, 1951) , and a t ypi cal f oot i ng af t er
f ai l ure i s i l l ust r at ed by Fi g. 4.
The t est resul t s of t he f oot i ngs on cl ay ( Fi g. 5) s how t hat
t he aver age bear i ng capaci t y ( max i mum l oad/ f oot i ng ar ea)
decr eases l i nearl y, wi t h i ncr ease i n eccent ri ci t y, t o zer o f or
eJB = 0. 5; si mi l ar l y f or any gi ven eccent r i ci t y <?v, t he bear i ng
capaci t y decr eases wi t h gr eat er eccent ri ci t y e . These r esul t s
Effective C on ta ct A re a o f F o u nd atio ns w ith Eccentric L oad
A ire de co n ta ct effectif des fond ation s sous charge excentrée
wher e A i s t he shape f act or (Meyerhof, 1951) dependi ng on t he
aver age l engt h/ wi dt h rat i o L '¡B' of t he cont act ar ea, and q
i s gi ven by equat i on (3).
For f oundat i ons whose dept h i s gr eat er t han about t hei r
wi dt h appr eci abl e l at eral f or ces ar e i nduced on t he shaf t by
t i l t i ng under t he l oad. These f or ces modi f y t he pl ast i c zones
and i ncr ease t he bear i ng capaci t y; t hei r ef f ect can be est i mat ed
as f or r i gi d cant i l ever sheet pi l es (Terzaghi, 1943) .
Experiments'. The onl y publ i shed t est s resul t s of eccent r i
cal l y l oaded f oundat i ons appear t o be t hose f r om an ext ensi ve
i nvest i gat i on i n Bel gi um (Ramelot and Vandeperre, 1950) .
Ci r cul ar and squar e f oot i ngs up t o 16 i n. wi de wer e l oaded at
var i ous dept hs i n compact sand whose angl e of i nt er nal
f r i ct i on at t he par t i cul ar packi ng was unf or t unat el y not de
t er mi ned. The exper i ment al r esul t s f or sur f ace and shal l ow
f oot i ngs ( Fi g. 3) ar e consi st ent wi t h t he t heor y by t aki ng.
cp = 44° , whi ch woul d be a r easonabl e angl e. Shal l ow f oot i ngs
wer e onl y t est ed wi t h rel at i vel y l ar ge eccent ri ci t i es when t he
t heor y i s conser vat i ve because i t negl ect s t he r esi st ance due t o
t he l at er al f or ces on t he shaf t .
Fig. 4
Failure o f Strip F oo tin g witli E cccnlric Vertical L oad on Sand
R up ture de l ’em pattem en t sur sable sou s charge v erticale excen trée
441
bear i ng capaci t y i s t i l t ed and t he adj acent zones ar e modi f i ed
accor di ngl y. Two mai n cases ma y be consi der ed, namel y,
f oundat i ons wi t h a hor i zont al base and f oundat i ons wi t h a
base nor mal t o t he l oad (i.e. base i ncl i ned at a t o t he hor i zont al ) .
The cor r espondi ng zones of pl ast i c equi l i br i umi n t he mat er i al
ar e shown i n Fi g. 6 and sol ut i ons f or t he ul t i mat e bear i ng
capaci t y q ar e der i ved i n t he appendi x ( A. 1 and A. 2).
The sol ut i on f or a hor i zont al f oundat i on ( appendi x A. 1)
can be expr essed i n t er ms of t he vert i cal c omponent of t he
bear i ng capaci t y
qv = q CO S a
B
cNcq + y — Nyq
(« O
Fig. 5
S T R IP
F O O T IN O
(M
C IR C U L A R
(\ N D
S a U P .* &
FO O T I N G S
B earing C a pacity o f Fo otin gs w ith E ccentric V ertical Load on Clay
F orce p o rta n te des fonda tion s sur argile sous charge verticale
excentrée
compar e wel l wi t h t he est i mat es when an al l owance i s made
f or s ome i ncr ease i n bear i ng capaci t y due t o t he penet r at i on
r equi r ed f or mobi l i zat i on of t he shear i ng st r engt h as f or
cent r al l y l oaded f oot i ngs (Meyerhof, 1951) . The bear i ng capa
ci t y of ci r cul ar and squar e f oot i ngs i s about 20 per cent gr eat er
t han t hat of st r i ps at t he same eccent r i ci t y, as f ound (Meyerhof,
1951) f or cent r al l oads. Fi g. 5 al so shows t hat t he cust omar y
met hod of assessi ng t he bear i ng capaci t y f r om t he ma x i mu m
t oe pr essur e i s r at her conser vat i ve. For si ngl e eccent ri ci t i es of
t he l oad t he cont act wi dt h or l engt h at f ai l ur e was, wi t hi n
exper i ment al l i mit s, gi ven by equat i on (1), whi l e f or doubl e
eccent ri ci t i es t he cent r oi d of t he cont act ar ea at f ai l ur e coi n
ci ded wi t h t he poi nt of appl i cat i on of t he l oad, as had been
assumed i n t he t heor y.
The aver age bear i ng capaci t y of t he f oot i ngs on sand ( Fi g. 3)
decr eases appr oxi mat el y par abol i cal l y, wi t h i ncr ease i n ec
cent ri ci t y, t o zer o f or eJB = 0. 5; f or a gi ven ex, t he bear i ng
capaci t y decr eases appr oxi mat el y l i near l y wi t h gr eat er ey.
These r esul t s ar e i n f ai r agr eement wi t h t he t heor et i cal est i
mat es; f or l ar ge eccent ri ci t i es on dense sand t he obser ved
bear i ng capaci t y i s somewhat gr eat er t han est i mat ed due t o
t he gr eat er angl e of i nt er nal f ri ct i on wi t h smal l er pr essur e on
t he f ai l ure surf ace. The bear i ng capaci t y of ci r cul ar and squar e
f oot i ngs i s t he same as t hat of st ri ps f or l oose sand but is
about 30 per cent l ess t han t hat of st ri ps on t he sur f ace of
dense mat er i al , as f ound (Meyerhof, 1951) f or si mi l ar cent r al
l oads. The cust omar y met hod of anal ysi s i s r easonabl e f or
smal l eccent ri ci t i es but unsaf e f or gr eat er eccent ri ci t i es owi ng
t o t he r api d decr ease of bear i ng capaci t y wi t h smal l er ef f ect i ve
cont act wi dt h. The cont act ar ea at f ai l ur e was si mi l ar t o t hat
of f oot i ngs on cl ay, and f or dense sand t he f ai l ur e sur f ace
wi dt h at gr ound l evel decr eased pract i cal l y l i near l y wi t h
gr eat er eccent r i ci t y as expect ed. Whi l e t he t est s on cl ay and
sand i ndi cat ed t hat t he “ mi ddl e t hi r d r ul e” i s r at her ar bi t r ar y,
t hey suppor t t he pr act i ce of desi gni ng shal l ow f oundat i ons
wi t h cent r al l oadi ng i f possi bl e si nce t he por t i on out si de t he
ef f ect i ve cont act ar ea can be i gnor ed.
(5)
wher e t he bear i ng capaci t y f act or s Ncq and Nyq depend on <
p,
DIB and a.
These bear i ng capaci t y f act ors, i ncl usi ve of any ski n f ri ct i on,
ar e gi ven i n Fi gs. 7 a and 8 a f or a shal l ow st r i p f oundat i on
i n pur el y cohesi ve (<p = o) and cohesi onl ess (c = o) mat er i al s,
r espect i vel y; t hey decr ease r api dl y wi t h gr eat er i ncl i nat i on a t o
zer o f or a sur f ace f oot i ng i f a = 90° on pur el y cohesi ve mat er i al
pon cohesi onl ess soi l , when f ai l ure occur s by sl i di ng
and i f a = <
on t he base. I t shoul d be not ed t hat f or f oundat i ons on cl ay t he
base adhesi on c'a ma y var y bet ween 0 and c dependi ng on t he
degr ee of sof t eni ng of t he soi l (Meyerhof, 1951) , whi l e f or
p\ t he cor
cohesi onl ess soi l t he angl e of base f ri ct i on 5' as <
r espondi ng l i mi t i ng f act or s ar e gi ven i n Fi gs. 7 a and 8 a.
I
E
4 5° -# 2 F
//N \\ V f /
's V 1\
/
\
h ~ 8 '" 1
i
L S ii
a.
A & 'V ' B
D
1_
90° -? C
(6) H o riz on tal base with large in clin a tion o f load
Bear i ng Capaci t y of Foundat i on wi t h I ncl i ned Load
Theory: Under a cent r al f oundat i on l oad i ncl i ned at an
angl e a t o t he vert i cal , t he cent r al shear zone at t he ul t i mat e
442
Fig. 6
P lastic Z o n es near R o u gh S trip F ou n d atio n w ith In clined L oad
Z o nes p lastiq u es près d ’un em p attem ent à su rface rugeu se so u s
ch a rge ob liq u e
The sol ut i on f or an i ncl i ned f oundat i on wi t h a base nor mal
t o t he l oad ( appendi x A. 2) can be expr essed i n t er ms of t he
r esul t ant bear i ng capaci t y
B
q = cNc,j + y — Ny,,
(6)
The bear i ng capaci t y f act ors, excl usi ve of any ski n f ri ct i on,
ar e gi ven i n Fi gs. 7b and 8b f or a shal l ow st ri p f oundat i on i n
pur el y cohesi ve and cohesi onl ess mat er i al s, r espect i vel y; t hey
decr ease r api dl y wi t h gr eat er i ncl i nat i on a t o t he passi ve ear t h
pr essur e coef f i ci ent s of a smoot h vert i cal wal l f or a = 90° .
I t i s of i nt er est t o not e t hat f or a gi ven a an i ncl i ned f oundat i on
has a gr eat er bear i ng capaci t y t han a hor i zont al base, whi ch
suppor t s t he pr act i ce of desi gni ng shal l ow f oundat i ons wi t h a
base nor mal t o t he r esul t ant l oad i f possi bl e.
The bear i ng capaci t y of f oundat i ons of ot her shapes under
i ncl i ned l oadi ng can at pr esent onl y be based on empi r i cal
evi nence t o obt ai n shape f act or s A i n conj unct i on wi t h equat i ons
(5) and (6) on account of t he var i abl e boundar y condi t i ons of
t he pr obl em. The t heor et i cal cont act pr essur e di st r i but i on at
f ai l ure i s si mi l ar t o t hat of a f oundat i on wi t h ver t i cal l oad.
Experiments: I n vi ew of l i mi t ed pr evi ous exper i ment al
evi dence t he bear i ng capaci t y has been det er mi ned f or di f f erent
i ncl i nat i ons of a cent r al l oad on hor i zont al f oot i ngs as bef or e
FO U N D A T IO N
7.
cc
\
\
8
\
\
u.
FO R
I N T E R N e O T . D EP T H S
—
—
Z
IT A l
r,
S'
)
B *
>
A L
i
FO R
N O T M
SEE
F W .C a )
E
■
c
■ e
a - o
<
Cl .
\
4
u
Ct
CL
u.
BA
v3
y
*
\
\
vl
D EPT H / W I O T M
D / B
___
- - -
\
\
X
> N
<3
7.
Oi
4
D
\
v
>
a
Ul
<£>
LLI
<D
¿o'
o
4 0 *
IN C L IN A T IO N
( a i H O R IZ O N T A L
Fig. 7
LO A D
8 0 *
01
F O U N D A T IO N
O
2 0 °
4 0 *
IN C L IN A T IO N
(M
IN C L IN ED
I. o '
O F
80*
F O U N D A T IO N
<
FO U N D A T IO N
B earing C ap acity F ac tors for Strip F ou n d a tion w ith Inclined L oad
in P urely C o h esive M aterial
Facteurs de la c ap acité p ortan te p ou r em pattem ent s ou s charge
o bliq u e en m atière p urem ent coh ére nte
1 «) H O RIZ O N TA L
Fig. 8
6 0 *
O F
F O U N D A T IO N
Cb )
IN C L IN ED
FO U N D A T IO N
B earing C apa city F actors for Strip F ou n d atio n w ith Inclined
L o ad in C o h e sion le ss M aterial
Facteurs de la ca pa cité p ortan te pour em pattem ent en s o l p u lvérulent sou s charg e o bliq ue
Fig. 9
A rran gem ent o f M odel T est on F oo tin g with In clined Load
A rrangem ent d ’essais sur fo n dation sou s ch arge ob liq u e
wi t h a r ough base on t he same cl ay and sand ( but i n a compact
p = 45° ). I n t he
packi ng wi t h por osi t y of 38 per cent and <
t est s on cl ay t he i ncl i ned l oad was i ncr eased t o f ai l ure; i n
t he t est s on sand a vert i cal l oad was appl i ed and kept const ant
whi l e t he hor i zont al l oad appl i ed by a second pr ovi ng r i ng
was i ncr eased t o f ai l ure ( Fi g. 9). I n bot h cases t he f oot i ng
r emai ned sensi bl y hor i zont al t hr oughout t he test.
The t est resul t s of t he st ri p f oot i ngs on cl ay ( Fi g. 10) ar e
i n r easonabl e agr eement wi t h t he est i mat es. The bear i ng
capaci t y of squar e f oot i ngs was about 20 per cent gr eat er t han
t hat of st r i ps at smal l i ncl i nat i ons, as f ound pr evi ousl y ( Meyerhof\ 1951) f or ver t i cal l oads, t he di f f er ence becomi ng smal l f or
an i ncl i nat i on exceedi ng about 25° when f ai l ure occur r ed by
sl i di ng as woul d be expect ed t heoret i cal l y.
The obser ved bear i ng capaci t y of t he st ri p f oot i ngs on sand
(Fi g. 11) conf or med wi t h t he t heor et i cal est i mat es and ap
pr oached zer o f or an i ncl i nat i on equal t o t he angl e of i nt er nal
f ri ct i on g> = 45° , as woul d be expect ed. The bear i ng capaci t y
of squar e f oot i ngs was about 30 per cent l ess t han t hat of
st ri ps f or a vert i cal l oad, as f ound pr evi ousl y (Meyerhof; 1951)
f or sur f ace l oads on compact sand, t he di f f er ence decr easi ng
t o zer o beyond an i ncl i nat i on of about 15° . The pr esent ana
l ysi s was al so checked by t he obser vat i on t hat t he f ai l ure sur
f ace wi dt h at gr ound l evel decr eased st eadi l y wi t h gr eat er
i ncl i nat i on of t he l oad and appr oached zer o f or a = 45° .
443
E X P E R i M E N T A L R E SU LTS:
S T R IP
(L / B = 6 )
X
S Q U A R E
E X P E B lM EN T A L R E SU LT S :
S T R IP
a / B - 6 )
S Q U A R E
\
R ES U L T S :
---------------
*s\
ÀE
- ^
% V
\
a
- C EN T R I C IT Y
[
ST
U P
\
\
V
T H E O R E T IC A L
R ES U LT S:
S T R I P ( 4 .= 4 5 ° ) - - -- -- - -- --
CO 140
O
T H E O R E T IC A L
S T R IP
*
a
:\
\
C x / B
X
\
E C C E * T R IC IT Y
K°
' B
s t r
°
a
X
'
Concl usi on
S
J N
10°
20 *
IN C L IN A T IO N
3 0 *
O F
4 0 °
LO A O
OL
SO "
lo °
.
20®
IN C L IN A T IO N
eccent r i ci t y, met hod as above wi t h posi t i ve a); t he bear i ng
capaci t y i s gi ven by t he l ower est i mat e.
Experiments'. Hor i zont al model f oot i ngs on cl ay and sand
as i n sect i on 2 wer e l oaded t o f ai l ur e wi t h a si ngl e f or war d
eccent r i ci t y of eJB = 0. 25 and di f f er ent i ncl i nat i ons of t he
l oad; a t ypi cal f oot i ng af t er f ai l ure i s i l l ust rat ed by Fi g. 12.
The t est resul t s ar e gi ven i n Fi gs. 10 and 11 f or cl ay and sand,
r espect i vel y. The bear i ng capaci t y was about one- hal f of t hat
of cor r espondi ng cent r al l y l oaded f oot i ngs i n accor dance wi t h
t he t heor y, whi ch was suppor t ed by t he obser ved cont act ar ea
and mechani sm of f ai l ure. Pr el i mi nar y exper i ment s wi t h a
backwar d eccent r i ci t y of l oadi ng wer e al so f ound t o be i n
r easonabl e agr eement wi t h t he est i mat es.
5 0 °
OF
4 0 °
LO A D
S0 °
oL
Fig. 10
Bearing C ap acity o f F oo tin g s w ith In clined L o a d o n Clay
C ap acité p orta n te des fon d atio n s sur argile so u s charge ob liq u e
Fig. 11
B earing capa city o f footin gs w ith in clin ed loa d o n sand
C ap acité po rta n te des fo n d ation s sur sab le sou s charge ob liqu e
The pr evi ous bear i ng capaci t y t heor y of f oundat i ons under
a cent r al ver t i cal l oad has been ext ended t o eccent r i c and
i ncl i ned l oads. The t heor y, whi ch i ndi cat es t hat t he bear i ng
capaci t y decr eases r api dl y wi t h gr eat er eccent r i ci t y and i ncl i
nat i on of t he l oad, i s suppor t ed by t he r esul t s of l oadi ng t est
wi t h model f oot i ngs on cl ay and sand.
Ac k nowl edgment
Bear i ng Capaci t y of Foundat i on wi t h Eccent r i c I ncl i ned
L oad
Theory. Wh e n a f oundat i on car r i es an eccent r i c i ncl i ned
l oad an est i mat e of t he bear i ng capaci t y can be obt ai ned by
combi ni ng t he above met hods of anal yses. Thus f or a shal l ow
st ri p f oundat i on wi t h a f or war d eccent r i ci t y of l oadi ng (a i s
posi t i ve, i .e. eccent ri ci t y i n di r ect i on of hor i zont al component
of l oad) an ef f ect i ve cont act wi dt h B' ( equat i on 1) i s used i n
equat i ons (5) or (6) and t he t ot al bear i ng capaci t y i s gi ven by
equat i on (2). Si mi l ar l y, f or a doubl e eccent r i ci t y on a r ect angu
l ar or ot her ar ea t he ef f ect i ve cont act ar ea and shape f act or
ar e used as i n equat i on (4). I f t he eccent ri ci t y i s backwar d
(a i s negat i ve, i .e. eccent r i ci t y i n opposi t e di r ect i on t o hor i zont al
component of l oad) , f ai l ur e of t he soi l occur s ei t her on t he
si de of t he eccent r i ci t y ( smal l eccent r i ci t y, met hod as above
but usi ng negat i ve a i n anal ysi s) or on t he opposi t e si de ( l arge
The aut hor i s i ndebt ed t o hi s col l eagues, par t i cul ar l y Mr .
L. F. Cooling M. Sc. , f or hel pf ul cri t i ci sm and Mr . B. J. Catterall B. Eng. , f or assi st ance i n car r yi ng out most of t he model
t ests. The wor k was car r i ed out as par t of t he r esear ch pr o
gr amme of t he Bui l di ng Resear ch Boar d of t he Depar t ment of
Sci ent i f i c and I ndust r i al Resear ch and t he paper i s publ i shed
by per mi ssi on of t he Di r ect or of Bui l di ng Resear ch.
Appendi x
Bear i ng Capaci t y of Hor i zont al St r i p Foundat i on wi t h
I ncl i ned Load
The r egi on above t he f ai l ure sur f ace of a shal l ow r ough
st ri p f oundat i on wi t h l oad i ncl i ned at a t o ver t i cal i s assumed
t o be di vi ded i nt o a cent r al el ast i c zone ABC, a r adi al shear
zone ACD and a mi xed shear zone ADEF ( Fi g. 6 a). The
st r esses i n t hese zones can be f ound as s hown {Meyerhof,
1951) f or a vert i cal l oad, by r epl aci ng t he r esul t ant of t he
f or ces on t he shaf t AF and t he wei ght of t he adj acent soi l
wedge AEF by t he equi val ent st r esses p0 and s0, nor mal and
t angent i al , respect i vel y, t o t he pl ane AE i ncl i ned at ft t o t he
hor i zont al . On t hi s basi s t he ver t i cal component of t he bear
i ng capaci t y can, i n t he fi rst i nst ance, be r epr esent ed by
— qC O S
a
B
=--= cNc + p0N„ + y — Ny
or
(7)
== Qv +
w h ere
q[,
cNc + p0N,,
B
qv -= v- ■Ny
=
T
444
F ailure o f S trip F ootin g w ith E ccentric Inclined Load on Clay
E m pattem e nt su r argile sous ch arge excentrée e t oblique
(9)
Nc, N and Ny ar e t he gener al bear i ng capaci t y f act ors.
Determination of Nc and Nq\ I n zone ABC wi t h angl e »/' at
A, t he shear i ng st r engt h Sp under t he nor mal pr essur e pp on
AC is Sp = c + p'p t an q>. Hence f r om Mo h r ’s di agr am
and
Fig. 12
( 8)
c + pfi t an ip
[ si n (2y> — <
p)
COS cp
Qi
and
+
/
Qv =
+ si n cp] +
p'p
( 10)
The vert i cal
B
qv = cNcq + y — Nyq
pp t an cp
cos ( 2y>— cp) cot a
cos cp
y>can be det er mi ned f r om any
(11)
f r om whi ch
gi ven a, cp, c and
pj, ( obt ai ned f r om equat i ons 12 and 13).
/— v
I n zones ACD and ADE wi t h angl e 0 = 180° -1- fi — >
and angl e ??, respect i vel y, at A, i t was s hown {Meyerhof, 1951)
t hat
Pp — t (c + Pi t an cp) e20tan<p— c] cot
( 12)
and
c -(- Pi t an cp
[ si n (2i ; + <?>) — si n cp\ + p0
cos Ip
}can be det er mi ned f r om t he gi ven r at i o sjp0.
wher e >
Pi
Determination of Resultant Bearing Capacity.
component of t he r esul t ant bear i ng capaci t y is
( 13)
=
( 18)
wher e Ncq ( dependi ng on Nc and Nq) and Nyq ( dependi ng on
Ny and Nq) ar e t he r esul t ant bear i ng capaci t y f act ors, and i s
comput ed f r om t he above sol ut i ons by det er mi ni ng t he f oun
dat i on dept h par amet er s (/?, p0 and i 0) f or var i ous dept hs D
as s hown {Meyerhof, 1951) f or a ver t i cal l oad. For l ar ge
i ncl i nat i ons a when qh gover ns, t he hor i zont al c omponent of
t he passi ve ear t h pr essur e on t he f r ont of t he f oundat i on i s
added t o t he shear i ng r esi st ance on t he base gi ven by equat i on
( 15); and i f i n addi t i on t he f oundat i on has a r ough shaf t ,
t he f oundat i on i s par t of t he cent r al zone ABCF (Fi g. 6b) .
It has t her ef or e been f ound conveni ent t o i ncl ude t he ski n
f ri ct i on or vert i cal component of t he passi ve ear t h pr essur e
on t he shaf t i n t he bear i ng capaci t y f act or s ( Fi gs. 7a and 8a) .
Subst i t ut i ng equat i ons (12) and ( 13) i nt o ( 10)
<7„ = c
+
Po
cot
<p
[1 + si n <
psi n {2y> — ?>)]
1 — si n cpsi n (2»; +
psi n (2y> —
1 -)- si n <
1 — si n cpsi n ( 2»? +
(p)
cp)
» 2 0
<p)
t a n
e 26
t a n
<p
___
]
<p
+
( 14)
or
q'v = cNc + p0Nq
Bear i ng Capaci t y of I ncl i ned St r i p Foundat i on wi t h Base
No r mal t o Load
For a shal l ow r ough st ri p f oundat i on of wi dt h B and dept h
t he upper edge of t he base i ncl i ned at an angl e a t o t he
hor i zont al ( Fi g. 6c) t he zones ar e si mi l ar t o t hose of a hor i
zont al f oundat i on wi t h y>= 45° + <p/2 and wi t h 0 = 135° +
fi — a — rj — (p/2. Usi ng t he same appr oach as above, t he
bear i ng capaci t y f act or s Nc and N i n t he r el at i on
D of
f r om equat i on (8) wher e Nc and Nq have t he val ues gi ven i n
t he squar e br acket s above.
The hor i zont al component q'h of t he bear i ng capaci t y cannot
exceed t he shear i ng r esi st ance on t he base, i .e.
B
q = cNc + Po Nq + y — Ny
q'h = q' si n a = q'vt an a
ar e obt ai ned by subst i t ut i ng t hese val ues of
equat i on (14). Si mi l ar l y i t i s f ound t hat
<c ' +^ t a n<5 '
( 15)
c'a = uni t base adhesi on
S' = angl e of base f ri ct i on.
gr eat er i ncl i nat i ons a when q'h
wher e
and
For
must t her ef or e be r epl aced by
Ny
gover ns, equat i on (14)
( 16)
t an a — t an <5'
obt ai ned f r om (15).
Determination of Ny: The mi n i mu m passi ve r esi st ance Pp
act i ng at cp t o t he nor mal on AC i n t he zone ACDE can be
f ound ei t her by a numer i cal st ep- by- st ep comput at i on {Caquot and Kerisel, 1949) or by a semi - gr aphi cal pr ocedur e
{Meyerhof, 1951) based on t he l ogor i t hmi c spi r al met hod.
Then it can be shown t hat
„ yB
<7, =
y
snr y>
'2P[j ________
y ^ l c os i y — cp)
-cos{y>—<p)> —
si n yicos {y>—cp)
cos
cp
( 17)
or
y B \r
=—
Nv
f r om equat i on (9) wher e Ny has t he val ue gi ven i n t he squar e
br ycket s above.
The above sol ut i on hol ds onl y f or a ^ (5' (see equat i on 15).
4Pp si n j^445°
=
( 19)
y> and
i nt o
+ —
yB2
- - - - t an I 45°
\
2
,
2
j COS a
(20)
wher e Pp i s t he mi ni mum passi ve r esi st ance obt ai ned as
i ndi cat ed earl i er.
The r esul t ant bear i ng capaci t y
B
q = cNcq + y — Nyq
(21 )
is det er mi ned f r om t hese sol ut i ons as bef or e, and t he bear i ng
capaci t y f act or s ar e gi ven i n Fi gs. 7 b and 8 b.
Ref er ences
C a quot, A . and K erise l, J. (1 949): T raité de M écaniq ue des S o ls.
G au th ier-V illars, Paris, p. 85.
M eyerh of, G. G. (19 48): A n Investigation o f the Bearin g C ap acity o f
S h a llow F o o tin gs on D ry Sand. Proc. Sec on d Int. C onf. S o il M ech.,
vol. 1, p. 237.
M eyerh of, G. G. (1951): T h e U ltim ate Bearing C ap acity o f F ou n d a tion s.
G éo tech n iq u e, v ol. 2, p. 301.
R am e lo t, C. and V andeperre, L . (1950): T ra vau x de la C om m ission
d ’Etude des F on d ation s d e P y lôn es. C om p t. R en d . R ech ., I.R .S .I.A .,
B ru ssels, N o . 2.
Terzagh i, K. (194 3): T h eor etical S oil M echan ics. J. W iley, N ew Y ork ,
p. 355.
445