ML20100D038
| ML20100D038 | |
| Person / Time | |
|---|---|
| Site: | Vogtle  |
| Issue date: | 11/10/1962 |
| From: | Kapila I INDIA |
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| References | |
| NUDOCS 8504010058 | |
| Download: ML20100D038 (12) | |
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4 - EARTHQUAEE RESISTANT DESIGN OF RETAINING WALLS
, I. P. Kapila * !
SYNOPSIS "this paper relates the determination of active and passive pressures exerted by cohe- ! sionless.meterial under earthquake conditions. The introduction of earthquake inertia forces to Coulomb's sliding wedge yields the Mononobe Okabe formula. Modifg:stions of known gra. l phical methods to include the effect ofearthquake forces are presented in this paper. INTRODUCTION The pressure, both active and passive exerted by cohesionless material against a retain-ing structure in the absence of earthquake conditions may be determined by the well known
) Coulomb's Wedge Theory. The earthquake problem introduces an inertia force corresponding I to the acceleration imparted to the mass oi the sliding wedge. The solution of the resulting force diagram yields the pressure exerted against the the structure;which s can be expressed by the Mononobe.Okabe Formula (I,2). Several graphical methods.are available for pressure
, deter ninstion from Coulomb's Wedge Theory. M.odiScation of two.such, methods,are presea-
- f. ted in this paper to. include the effect of eathquake force.
0.. t: ACTIVE PRESSURE - ' ' h 4 Coulomb's Formula : In Fig I(a). AB is the fill-side .fsce of the ' retaining structure, . ,' inclined at an angle a: to the vertical. The surface of the Sil'AD'is , inclined to the horizon. , g* tal at an angle 3. If the retaining structure is removed, the materialpf the 611will settle anally .) : l gc at the angle of repose which is close to the angle ofinternal friction, represented by the plane BD. ?,, According to the Coulomb's, Wedge Theory, however, when the retaining, structure moves for- , p
- : i l k,,.plane ward only slightly, the surface of and rupture willbenot formtoatcorrespond plane ,BD immed '
l of rupture will lie somewhere between AB BD. It may assumed i ! I[,Uo the plane BC in Fig. I {a). ..The active pressure.'apost the 'trinining structu'riifchnise6y .] Ythe weight of the wedge ABC (assuming earthquake force to be. absent) and it is . desired to . 5, "Eletermine the location of the plane BC whkh irill cau'se tbc maximum reaction against the . face AB. ,
- Dire. tor. Bhakra B4aa Dan Desiga Direeserste, New Delhi.
Y
? I mypymewww= ~
-_ , , - - = _ . . . - . _ . , . - - - . - . - - , . . - . _ - . _ . - . - , . , , . - ,,.,-m., ,..-,_,,,,--.--.,.,.m- -
_--_w,_._--,-,-,, , ,_.., . . ..,...n.. - , . ,
1
. .. j
. 3
. Second Earthquake Symposium : University of Roorkee 7 98 ; ;
. \
to the normal, where 3 The reaction against the face AB will be inclined at an angle si j Os is the angic of wall friction between the cohesionless material I structure. The reaction against the plane BC will be inclined at an ang -
.....'.: ..4 i
as, sere es es 4.M I I 5 . i 4 -s . _, , , 1.
.7
., A 4 t F
" 3 ' 'e ,
* * . L- . g , ,
_ s
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.a d::t -
/ .
, - e, .) 'y- ....
q d 4
- A
, , s ts / s , s .
g a s
~'.. ' ..
Fig.1(a) ,,i s f1 . . . . '
..i. . f
- s. . . s . e.
....~.' ..o
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h E a - o ;' y,
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.... . . f<y- r ,. ,
t: ,
. a .:n 2 .
f . , . . s
. ;i, .: . v: i ? * . .< /
- < .i -wr. r . e .
Fig.1 (c) *
..,s. : i . . ... .. Fig.1 (h)
- t. : ., , i. . ,
'. l.a
' d the iveight of wedge is W, Pg a
l . ,,
,If,these.two reactions are designated Pg: . nd . . .R. . respectively
. . , . . .. . . . . . an
'l
.s FI 6:diy bei det'erinined from the triangle'of fo Id(.
D'ai)Iith .../.I[c)bl.' i," I't9 M' If $ib AF aid.CE in F sitich that escii'mskWarIs'nsie $d* LS7 . maximum active pressure would result when I * (1) __ . - - AABC = 6BCE .. . .~.." -
- . . g., g . . . . . -
(N and 2 -=
-',b y ={- g'd _
L'
I M i ', Kapita on Eanhquake '- Resisyn,at Design of RetainingWans s..
, s. - - ,. - .
. C
.W. .,
a
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~
S( !.. ' qs . y ggti d
) ~ . .
"'"l* *
- k l) h ;
5 I I' ' g (IfdvI ' ;, da# -
.g > L[ --- . gL mi ,
e'g .% *\ o ii 4 d,
\ :
l
*+. .\ . i ft 4
) $ /. g ,
g . Fig. 2 (c) Fig.2(b) D -
'l
$strisce of fiH h.e*N i i.
~ /. . . ..
C u . - y < . a l 9 " E
.- )-
A s e\ 9 a **
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er ..c:
;: s ... .
a L
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,, \
.G - \
r '
,l b .
... .v',.y.y 3 y .
t -
' . . I Ja .
-e hi ,..' , / /. ./. /* / / 8 e . \.9 ,
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). irtW' / t I' .
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m .54 '54 164U4 8r N *
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100 ' Second Earthquake Symposium : University of Roorkee the mazimum active pressure would be
- 17 x8 Sin (90 -(x('e +) (3) where y is the unit weight of the cohesionless material. This may also be expressed as P. "4 y.h8 Cos' (e-c)" " ,
,g :
' Tin (e +e ) Sinte --3)
Co6*x. Cos(ci--x) 1 r- co,(g , j ,cy c,,g;;,) / (43
~
which is the Coulomb's Formula for setive pressure.
,g j
- The Mononobe-Okabe Formu!a : For earthquake conditions, additionalinertia forces 1*
' * ' ' ce=W and c c,W would be IEiposed on the mass of the sliding wedge (Fig. 2) where
- l
~ -
cem and ac, are the horizontal and vertical seismic cocmcients respectively. The " l resultant force, W , is thus D i W, - -(Icae,) W tWe (5) where e == tan ' ** 1.t: x, (6) { According to the Mononobe's theory,the effect of the accelerations due to an earthquake is to modify the direction of the gravity force, which would be equivalent to a rotation of the vertical and horizontal planes of reference through an angle e in the same direction. This effect may be introduced by plane BD makihg an angle (e - e)with the horizontal as indicated in Fig. 2(a). The resulting force polygon would be as shown in Fig. 2(c) If AF and CE in Fig. 2 (a) are drawn such that each makes .an angle (90 - ei - x
-8) with BD,it can be shown that the maximum active pressure: would result when the l requirements of Eq. (I) and Eq. (2) as for the non-seismic condition are satisfled. The maximuss active pressure would be P
A , 1. I l** 1 7 xi. sin [ 90 - (m + ei + e ) J (7)
. Cos 9 This assy also beexpressed as
~
l
, P4 = } (! m ee.) yh' C3 , W whare Cg si the active pressure coefficient l
~
" Cos' (e - E - 8) -
CA ...* ~ Case Cos'c. Cos(ei+m+6) I + L.Cos(e +cc+9)Cos(3-sj j J J Sin (e +ci) m $* f e-3-e) i G l 1 This is the Mononobe.Oirabe Formula for active pressure,
,-..r--, ._, , - . .,,,._m.-_
- _ _ _ ~ _ _ _ _ . _ . ._. _ _ . . . . . _ _ _
t
. l
', , ,, , ,,,,, n.w n' 3 i "
101 Kapila on Earthquake Resistant Design of Retaining Walls g *~ ' rs ' g(..
- f . c GRAPHICAL METHOD, . .. .- * ~
Culm' ann's Method
" * "Culmann's graphical, methodifor 4he~ determinailon oT fetive pressure is prmore
. gen'eral application ihan sevral pthets described de- teal books 'en Sdil' Mechanic
.*r .
essentially comprises the ' construction of.the tnangle of fori:es I for each' of t The veijtor representing the - weight Of the tiedge ik' plot'ted on h: planes of rupt'u're. Fig. - (1). The vec' tor r'e,pr,esenting the aqtive pressure is then dra'wn s6 (90-Op!-d-)'with .the , plan.e ,BDi The triansle of: forces is thus that whir.h'is f p ~ T assumed plane of ruptur$. th.e plane BD,and the active prenurc'vectohirah
- The locus'of the, point of intersection of the active pressure ' vector with thE correspbud ~
e of rupture is designdt'eil as the "Culmann Line".Theehaaimuid distan& 6e' tween E r the tangeht to ths CulniaE'LInc drawn parallel to the plane BD, measure
- pressure vector'is thl'activ'e pressure to thk force se, ale adopted in th
. The modificati6ds,,,necessary in Culmann's graphical construction to inclu fe the e
- g. 7 of earthquake' forces uould be readily apparent " from thgrgceding,di c,ussion regarding A
Mononobe,-Okabeformula. These arett. ' ' Q. e
~
i)
- The plane BD is to be drawn,such that .it is inclined at an angle 10-6)'t s
horizontal. : - ii)*l The active pressure vector is to be drawn at )an
,.. withengld' the (90
*- pfani BD. }, ,, ,..p. .., .
I' It would be seen that the resulting triangle of forces conforms to that shown in " Fig. 2(c) This modification is illustrated in the example on Fig. p. ,, ,., t.
.w Melbye's Metho'd '
l r Another graphical tnethod ' based on : Coulomb's Wedge Theo
- proposed by A. Melbye(4).This,is directed.towards drawing the' plane of ruptu
( requirments of Eq. (1) are met, and then scaling,off the length 'a' to 'the lideir scal
- construction. In' Fig.1,' if'E, G .is drawri parallel to AD, then N mus't lie ori the plane of rupt fer Eq. I to be satisfied. Funhermore,,n can be shown that the locus of the point 'l the intersection of the diagonals GC and AE of the parallelogmm ACEG)'sillbe parallel to BD N innd will bisect AIR' The g;raphical constructi 9 n, therefore, merely tequir'es dra
~ "* *hich biiedts Ad I s'ad [s' parallel to BD, and then determinining by trial the plane of rupture
#* BC which yields Gl=l{ ,,The modi 6 cations necessary'tol Include t ,
. forces would be, ~
N *(IE i T'oe pla'ne BD is to.$drMen so as to rnake an' angle ","" (c'-e) b t l
~ -
n (ii) ' AF,is to be'7raN to make on angle (90-et-ac-e)Wth"BD.
- It'would be'seen that the requirepents of,Eq.(I) remain unchanged under'the'e
~ eogditions. The value br'x' as'd'etermined by this construction eiin be substituted The application of this method is iDustrated in Fig. 4.
- ts yeild maximum active pressure.
l , . l l . j r l
T , %'*$ Y >-'
~
.Yn,...-
-- :- vil'
- (NRLN;f$WE& l '.w
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*s 4
Second Earthqu' aire Symposiumi University of Roorkee
.f6 102 %
fr .
'i; DATA: Maximum active pressure vector = 930 kips h = I00 fr.
(1 -0.05_) . 930 kips cc = 35' P. = cos 9* p p = 30' i
= 894.5 kips Y es = 15' s
. . 3 = 15' 1 cem = 0.1$ (acceleration away from retaining structure)
- -- - - - : oc, = 0.05 (acceleration downwstds)
..v g . 9= -
m ..
.y,= ' 132 lbs,/cft.
la. < ..
- .h. - . was a w
.. . t
- .:2- ,.!!
- i. :
e sd ;.- - au e m, a s
*///.' ',,, w ac / n ,
* }.
r
/
' ' ~~
/ ' -
0 .p C<' N s g,/ , ,, , ,,7
'mia a e / /. /
/ // /; ~~~M- j
,. ,.s, m.~ oc e. . ..a a ,
/~~~
~~~~ ~~ , . . -g .
* / *e M '; . ,/,9;'., ~~~ ,. ;
. / .
> i .,,si,,,.
, n.o..a . e,
~ ad ..- f ..._...,. ,
eds ts.it dw .. >
, . adt % sL% := . 4,
,-Q OC.EDURE -
7 3-ai d addi ' Draw-BD to make an angle (c--e) wi;h the horizontst. k I~sNp'la'nes'of rupture Ba, Bb etc., and compute weight of3we i d..', WW. s'nN plot on any convenient an.force(:u .. scale M~. 4 Ba',Bb'on ** 4,- BD. .
,cc. >.e) 3 31: y . . , , . . .. ...
onesqui .A.',.
'sie at sa,, ang}e. -(90*a.e. ~. 3. n. ~ s .
w e-
.v;.e:sise
- ~ ,ive pr,ess.u.r.e
.a a- . vectors.-n from
.. ta.hunes oT r,upture. . u .,,.m .
,to .o ..s ,
- y ses stersect correspond.mg as ,
ubb.w, theJocus of tbe intersection of assumed planes .of --: ins activg;g,ssure vector (The, culaman. I.ine), . . 1:A. and de,termi i 1 ,
- st2. cu edr*rst.or,.pa,1,atie,, to. BE.
i $ t'i . :: *.. : - "" Fig.3-ModiSed Culmann's Construction for ActivePressure. . .. O.
.4 l A
~
~s s
\
i
- .; . - =. -
m-- - - - - - v
,1 103 Kafila 'on Earthquake Resistant Desihn of Retaining WaEs w
-' DATA: Cos. (si+ ac+s) i:O ' h- - 100 ft. Pe = { (12ce, W ces e i
. oc . 33 Q.w ..
s = 30* = } (1-0.05).132. (165.5): .,.- Cos. 59' I '
,, , 33
== 895.6 kips
. . 3 = 15'
,j Acceleration away from retaining structu.c = 0.15 j l jg%
- 3. . ata =
= Acceleration downwards = 0.05
.! 1 ac. .
~'
0
- 9'
- . . - L y = 132 lbs./cfr.
-) -
=
y 1 ~ ~ .
- g. ..
.y
- ; s.A f, o
,,.u....,
e c
- 1 .
c" *e
) e ,
f 1 . .#
*=*-e m- 9. .. ,
i I ~ . . . , . . , , .,,, I . e PROCEDURE
- 1. Draw BD to make an angle (0-9) with the horizontal. ,,
- 2.- Bisect AB, AH = BH.
- 3. Draw HJ parallel to BD. ..
^
'4 " Draw AF to niak'e an angle (90*-ei-T-9) with BD. . , r.
& . .. .. .e
- 5. Determine phoe of rupture BC by trial such that..GI = IC. .
3h 6. Draw CE parallel to AF and measure length to linear scale pf ponstruction. p *
.. .. .;.... :. , . i le. .. . .i ::5 ' ~::Mi g.. Fig:
.=:.... .... . ;
.'4'-Modi . ..
6ed . Melbye's Construction fotf,5}iveptaig,,,, f
. .=
*;.
- Besad on papet ahlaing Aetive Thurst on the beek of a Well Realaing Cohseles le Meterlat** by A Melbye. Civil Engineerlag & Publie Works Revie w Yel. 50, No. See, Jely IM
' ~ ' - - - - - , . ~ , ,., _ _ _ _
***'Wv"m mswm,,,,we ,,
, 'I rg r-
- -+-- -
. ~
N j l4 i lj
,o i
3 .3-] ' [i -
,104 Second. Earthquake Symposins : University of R5erkee .
- .4F k 4 6
PASSIVE PRESSURE
* - O ' f y
- . .....z.. ^
- ,: Coulomb's Formylae:-The determination of the passive pressure follows ,the same ,g] a basic considerations as for actikpressure. with the. main point of difference beiig that the plane of Ll' ' '
rupture ~be such as will result in the minimum pressure 'against a the ret' ining st'ructure. The h J. plane BD shown in Fig. I and'2 to be above the horizontal plane for the active pressure con- 3
- ditios would be below the horizontal plane for passive pressure and horizontal earthquake
.t
. = -
,,,. l
~
' inertia forces would have to be assumed acting away from the structtir'e and' noi 't'oM the g l
- r. ,* . sirbCfure. =
V . , , i i . 'I P, = Co's 6 I "I* "'" E* " ' '*" " *** ')
~
c . g .,
***'**C****"'- g ,.. p ,
o3ye
,s.
;~
x ..
,. r 2
0 & .sr . - 1 e y
~
j
\'\ \ ~ =.-
% %s
%R
\ f e i a,: ";",,,
a .> l j
!:" V \ \ \ \ '
\\ \ -
-; a s.\ .*
r.- . h'
'h. [ .\
l --- t,o s
- A
- 81 j2 N -
'. =
.'s.k , , ,
.,.y, -. , . . ,
, j -3 f
- PROCEDURE * . .s _-
.s.
1 Draw DBD. to make angle (0 -9) with horizontal. , ,,.,.{ f.h h k-- 2 Assume planes, of rupture.Ba, Bb, etc. and compute weighi of wedge abs,
, .'4 ?
.T,'
.,f
. -s1 ABb, etc. and plot Ba' 3' Draw pass!'v'c'obiirs'from preh'sM, Bb' toat"s anyn En'gic' convenient (902e+.2force r+9) 3 j scale ]$
~
a'/b' ' etc. M! .y..
.shteNect coki,,N3idg' asis'aidNnes of reptire?: isunt.r; O wed .S ,
, yl 1_ ] ~'
YC 4' Draw the locus of, ~the intersection of the assumed ru '
'l s and the. !d .
correipdading'p' asst've presst$re vedors (The t)e*~ 3l7 Li$ptu
.'Culmann" =
* )'and determine
- .---minimum passive pressure vector. -
-~-- l/j -
, ' '( Fig. 5-Modified Culmann's Construction for Passive Pressure.
- 4 a g ,'
$C J I
+ E , ;5
~
' w. - .ww#enrM*e J
' - " L .iu . .J l g g g
45: i.. i $: . 3 ~r
. .Kapila on Earthquake Resistant Design of Rem n ng ii Wads 105
[
. I.P. -
i Cos. (ei-E+9) (1 t%, )
- Ip aus { 7 1,. pg g
! T. . =. -
, c- ;
o $,- . i
, I a
- m, , *
- i i . e i m .. .
,. c.
~' s
' ~
?
s., *** * *' **' R C I A " O-g* -
. c.,*sg.T A r**8'8 s re " f e , it Of
- pp er owe w 8
, g y
, se, . . . .. . -
,. f .
I PROCEDURE i 1. Draw DBD' to make an angle (3 -9) with tbc horizontal.
- 2. Bisect AB,' AH=BH.
[";~ 3. Draw JHJ parallelto DBD'. ,
- 4. Draw - A F G to make angle (90*+ E i-ac+9) with DBD'
'.a..
a
- 5. Determine plane of ,ruptune BC by trial such that GI=IC. .-
- 6. Draw CE parallelto AG and measure length tolinear scale orco Fig. 6--Modified Melbye's Construction .. .for Passive Pressure. -'
.
- t i i *. .. .
( .;
.. . .t .
e '
- A The analytical solution for passivehh pressure due to cohesion not earthquake'. forces
') .
that the requirements of Eq. (l) continue to hold and that w - kg includ.e.d in. t the. analysis ten s o s '
- 4 ' ** ' :3.* M i
**.;.* **t-Pressure is ,,, v s .. . .T ..
. (10).
.nto"'M. -
I, = { 7 x* Cos (cc-e. )8 m
* 'e
* :*ll ***MI'* l 2 i u, ., a ; i 1 t. t e
- e .n " ' - i '36*
- l f Wan htaaningCohemian "
less Etaterk!
' * -1sw. . . :
*Bemed on paper, ** Deteet.aining Active Thrust on the back o a by A.Libye Civil Ensincerisig & Publie n'oebe hvi., Voi, H, No, %Jely I a I ^
~~;;ay.
m l l . 106' Second br'thquak'e Symposium : University of Roorkee 1 t-This niay also be expressed as } i U
! 7.hs cos:(a + c) ~'
h ~ 3 I8 -
? Cos' cr Cos ie - c ) 1- IL Co '"I(e 9'I.E"!U e) Cos(e-3) E. 3) L ii ,
. The Mononobe-Okabe Formula When earthquake inertis forces are included, the passise pressure is y
- I i
(12) u P, = t (le x,) e Cos. (cc - c t - 9) ... ... . ... Where 6 is as defined in Eq. (6). [ ' This may also be expressed as I
*. (13)
P, = i (!
- x ) 7.h8.C. ... ... ... .
'I where C,is the passive pressure coefficient
). 1 C,= - --
8 '
- ,s Co@. Cm'x. Cos (c -x-9) 1- -
C - 14)
- This is the Mononobe-Okabe Formuls for passive pressure.
GRAPHICAL METHODS The Culmann and Melbye graphical methods for passive pretsure determination foIIo the same basis described earlier, and the modifications considered necessary for active pressure would apply for including the effects of earthquake forces. The application of these two gra- I
's phical methods for earthquake resistant design is indicated in Figs. 5 and 6.
1 - REFEENCES; [ 1; Mononohe, N. and H. Matsuo. (1929). " On Determination of Earth' Pressure During , Earthquakes" Proceedings World Erig*meering Congress, Tokyo,1929. -
- 2. Oksmoto, S.,(1956) " Bearing Capacity of Sandy soils and Late'ral Earth Pressure
*during Earthquake". Prxeedings of the World Confeirence on l'arthq'uske Ensoeeri ~ .
ing. Berkeley. Calif. June 1956. A - a i fj_-?_"j$ff hh? = $Pm
- - ~ ~ - - - - - . _ . . _ _ _ _ _ _ _ _ _ _
- \
... .. l f* .' ~
g .
- be.
- 4 .',: 207 e .
h Kapita on Earthquake Rasistant Desist. of Retaining Wails
's t i f ,
Materials l i' i l
- . 3. Kapila, l.P.. and S.S. Maini, (1962). " Active Pressurt Exerted by
.~ !
under Eanhquske Conditions." Indian Concrete Journal, Vol. 36 7 .{,.
^
- f. f4. Capper. P.L and W.F. Cauie, (1962) "The Mecha
! "- F.N. Spon, t.ondon.
.jI . 1 Melbye, A (1961) " Determining the Maximura Active Review Thrust
? .
Cohesionless Material" Civil EngineerMg and Public Work:
} Retsining f; Vol. 56. No. 660. July 1961.
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~ . . .
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