DEPARTMENT OF CIVIL ENGINEERING AND ENGINEERING MECHANICS
In P ar tia l Fulfillm ent of the Requirements . For the D egree of
MASTER OF SCIENCE WITH A MAJOR IN CIVIL ENGINEERING
In the G raduate C ollege
THE UNIVERSITY OF ARIZONA
1 9 7 4
STATEMENT BY AUTHOR
This th e s i s h as been subm itted in partia l fu lfillm ent of re quirem ents for an advanced degree at The U niversity of Arizona and is d ep o s ited in the U niversity Library to be made av a ilab le to borrowers under ru les of the Library.
Brief quo ta tions from th is th e s is are a llow able w ithout sp e c ia l p e rm iss io n , provided th a t accu ra te acknowledgm ent of source is m a d e . R equests for perm iss ion for ex tended quotation from or reproduction of th is m anuscrip t in whole or in part may be granted by the head of the major departm ent or the Dean of the G raduate C ollege when in h is ju d g ment the proposed use of the m ateria l is in the in te re s ts of sch o la r sh ip . In a ll o ther in s ta n c e s , how ever, perm iss ion m ust be obta ined from the au thor.
SIGNED:
APPROVAL BY THESIS DIRECTOR
This th e s i s h a s been approved on the date shown below:
iLPH M . 'RICHARD P rofessor of C iv il Engineering
and Engineering M echan ics
ACKNOWLEDGMENTS
I w ish to e x p re ss my g ra titude to my re se a rc h d irec to r . Prof.
Ralph M. Richard, for h is encouragem ent, a s s i s t a n c e , and gu idance dur
ing th is r e se a rc h and throughout my g radua te s tu d ie s . In add it ion , I
ex p re ss my a p p rec ia t io n to Prof. Robert L. Sogge for h is support and
advice and p a r t ic u la r ly for h is g en e ro s i ty in providing the com puter p ro
grams u se d in th is s tu d y . Thanks are ex tended to Prof. H a s s a n A. Su ltan
for h is h e lp fu l su g g e s t io n s and for serv ing as an ad d it io n a l member of
the exam ining com m ittee .
S p ec ia l th an k s are e x p re sse d to my w ife , Sandra Rae, for her
p a t ie n c e and encouragem ent and for h e r a s s i s ta n c e during the p repara
t io n of th is m an u sc rip t .
i l l
TABLE OF CONTENTS
Page
LIST OF ILLUSTRATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . v
1 „ Region Inc luded in the A nalysis . . . . . . . . . . . . . . . . . . . 15
2 . F in i te -e le m e n t Id e a l iz a t io n for Grid 1 and Grid 2 . . . . . . . . 16
3 . P a r tia l F in i te -e le m e n t Id e a l iz a t io n for G rid 3 and Grid 4 ... .. 19
4 . ' Bending Moments for Loading C a s e s 1 to 4 . . . . . . . . . . . . 24
5 . Bending Moments for Loading C a s e s 5 to 8 . i . . . . . . . . . . 25
6. Normal Forces for Loading C a s e s 1 to 4 . . . . . . . . . . . . . . 26
7. Normal Forces for Loading C a s e s 5 to 8 . . . . . . . . . . . . . . 27
8. Moment Variation for Various Loading C onditions . . . . . . . . 29
9 . D iam etric Change for Various Loading C o n d it io n s . . . . . . . . 29
10. Bending Moments for Loading C a s e s 9 to 12 . . . . . . . . . . . 31
11. Bending Moments for Loading C a s e s 13 to 16 . . . . . . . . . . 32
12. Bending M oments for Loading C a s e s 17 to 20 . . . . . . . . . . 33
13. Bending Moments for Loading C a s e s 21 to 24 . . . . . . . . . . 34
14. Normal Forces for Loading C a s e s 9 to 12 . . . . . . . . . . . . . 35
15. Normal Forces for Loading C a s e s 13 to 16 . . . . . . . . . . . . 36
16. Normal Forces for Loading C a s e s 17 to 20 . . . . . . . . . . . . 37
17. Normal Forces for Loading C a s e s 21 to 24 . . . . . . . . . . . . 38
18. In te rp o la tio n of C o eff ic ien ts for S t if fn esses Not G ivenin In fluence C h a r t s . ................... 41
19. Some C a s e s for A nalysis by In fluence C harts ................ 43
20. Effect of In te rface Slip on Bending M o m e n t s ................. 47
21. Effect of In te rface Slip on Normal F o r c e s ................. 48
v
ABSTRACT
F in i te -e le m e n t a n a ly s is u s ing e la s t ic m ate ria l p ropert ies is
u se d to ev a lu a te the e f fec t of l ive loads on shallow r ig id condu its „ It
i s shown th a t the a ssu m p tio n th a t l ive lo ad s are uniformly d is tr ib u ted
oyer the su rface w ill u n d eres tim a te the moments and o v eres t im a te the
normal fo rces in the in c lu s io n for p a r t ia l loading cond it ions „
Extending the in fluence line c o n c e p t , a c o l le c t io n of in fluence
c h a r ts is deve loped for th e d a ta o b ta ined from the f in i te -e le m e n t a n a ly
s i s . U sing the c h a r t s , nondim ensional bending moment and normal force
in fluence c o e f f ic ie n ts c a n be determ ined for any su rface lo a d in g . The
in fluence ch a r ts may be app lied d irec tly to the b a lan ced d es ig n method
and .a re a v a lu ab le ad ju n c t to th is p rocedure .
N either of the assu m p tio n s u se d in e la s t i c s o lu t io n s , full
s l ip p ag e or no s l ip p a g e , is s e e n to h av e any p a r t ic u la r advantage in
es tim a tin g the bending moments and normal forces for shallow buried
rig id c o n d u i t s .
CHAPTER 1
INTRODUCTION
Today the trend is toward g rea te r concern for the e s th e t ic and
environm enta l qua lity of our u t i l i ty and t ra n sp o r ta t io n s y s t e m s . The
an tiq u a ted com bined sew er sy s tem s of many large c i t i e s are being s y s
te m a tic a l ly r e p l a c e d . The buried s truc tu re is rece iv ing more a t ten t io n
a s an a l te rn a t iv e to the som etim es u n s ig h tly appearan ce of overhead
and g ro u n d -lev e l u t i l i ty i n s t a l l a t i o n s .
The s o i l - c u lv e r t system is one of the more im portant of th e se
underground i n s t a l l a t i o n s . Large sums of money are being in v e s te d in
cu lv e r t in s ta l la t io n s each y e a r . Due to the m agnitude of th e se expen
d itu res any u n n e c e s sa ry c o n se rv a t ism in des ign ing a s o i l - c u lv e r t s y s
tem should be s e r io u s ly q u e s t io n e d .
Current d e s ig n p ra c t ic e does not provide for a r e a l i s t i c a s s e s s
ment of the sa fe ty and re l ia b i l i ty o f buried cu lv e r ts under an assum ed
loading c o n d it io n . The m ost commonly u sed d e s ig n te c h n iq u e s are la rg e
ly em p ir ica l . The s e le c t io n of co n d u its is b a se d on y ea rs of experience
and the o b se rv a tio n of the perform ance of in - s e rv ic e c u l v e r t s . A g rea t
d e a l of re se a rc h h a s b een done in the p a s t s ev e ra l y ea rs ; how ever,>-
l i t t le of th is know ledge h a s found i ts way into eng ineering p r a c t i c e .
This is due to the d e f in it io n of-cu lvert supporting s treng th and to the
fa c t th a t too l i t t le is known of the e x a c t nature of the load ing and the
r e su l ta n t s t r e s s e s d eve loped in the sy s te m .
2
The r e s u l t s of th is w ork , u sed in co n ju n c tio n with the d e s ig n
p rocedures d e sc r ib e d in a la te r c h a p te r , provide an im portant s tep toward
m aking an a c c u ra te ev a lu a t io n of cu lv e r t sa fe ty p o s s ib le .
The em pirica l d e s ig n p ro ced u res u sed today hav e been made
n e c e s s a r y by th e com plex ity of the s o i l - c u lv e r t sy s te m . Soil p roperties
are non lin ea r and s t r e s s d ep en d en t. They are a ffec ted by the tim e-
dependen t p re se n c e of pore w ate r , and many are s e n s i t iv e .to d is tu rb an ce
by o u ts id e fo rc e s . The system is c h a ra c te r iz e d by a high degree of in
de te rm in an cy . In ad d it io n , boundary cond it ions in troduced when lim iting
th e e x te n t of the system to be an a ly zed a s w ell as in te rface p roperties
b e tw een the so i l and the cu lv e r t are h igh ly com plex .
The follow ing assu m p tio n s are made:
1 . Only sha llow buried condu its are co n s id e re d . For pu rposes of
th is s tudy a sha llow co n d u it sh a l l be one buried a t a depth of
one d iam e te r .
2 . In the developm ent of the in f lu en ce ch a r ts in C hap te r 4, no s lip
i s a llow ed a t the s o i l - c u lv e r t in te r fa c e . P a r tia l and full s lip a t
th e in te rface are t re a ted s e p a ra te ly .
3 . A nalys is i s lim ited to r ig id s tru c tu res ; th u s , buckling is not a
c o n s id e ra t io n .
4 . The so il medium, is a s s ig n e d l in e a r e la s t ic hom ogeneous prop
e r t i e s , and the p re se n c e of w ate r is e x c lu d e d .
5 . The a c tu a l th re e -d im e n s io n a l system is t re a te d as a tw o-
d im ensiona l p lan e s tra in problem . The p lane c o n s id e red is
ta k e n p e rp en d icu la r to the long itud ina l ax is of the condu it .
6 . W eigh t of the so il medium is n e g le c te d .
The follow ing s ign conven tion is adhered to throughout th is
th e s i s . P os it iv e bending moments produce te n s io n on the ou te r su rface
o f the condu it w a l l . T en s ile normal fo rces in the condu it w a ll are p o s i
t iv e . The X -coord inate d ire c t io n is p o s i t iv e to the r ig h t , the Y-
co o rd in a te d ire c t io n i s p o s i t iv e downward, and the o r ig in is tak en a t the
ce n te r of the co n d u it . The angle 9 u sed to lo ca te po in ts along the c i r
cum ference of the condu it is m easured in the co u n te rc lo ck w ise d ire c t io n
from the p o s i t iv e X a x is .
CHAPTER 2
LITERATURE SURVEY
The shallow buried condu it is of in te re s t a s a d e s ig n problem
in i t s own r ight and as an in te rm ed ia te c a s e which o ccu rs during the
ea r ly c o n s tru c t io n p h a s e s of d eep er c o n d u i t s . In both c a s e s of shallow
co n d u its > the live lo ad s c o n s t i tu te the m ost s ig n if ican t loads app lied to
the s y s te m . In i t ia l s t r e s s e s due to th e so il f il l are u s u a lly su ff ic ien tly
sm all to be n e g le c te d . Of th e re se a rc h th a t h a s been under taken , su r
p r is in g ly l i t t le h a s b een devo ted to th is s p e c ia l c a s e .
An e x te n s iv e s ta te of the art e v a lu a t io n of p ipe cu lver t t e c h
nology h a s b e e n g iven by K r iz e k e ta l . (1971). The report o r ig ina ted as
an effort to develop new d e s ig n p r o c e d u re s . H ow ever, there was not
su ff ic ie n t inform ation a v a i la b le to develop a com ple te ly new d es ig n p ro
c e d u re . Severa l recom m endations were m a d e .
The M ars to n -S p an g le r theory for determ ining lo ad s due to the
so il prism above the condu it h a s b een p re se n te d by Spangler and Handy
(1973). According to th is theory , the loads developed on buried condu its
are dependen t on the re la t iv e se tt le m e n t of the so il prism above the c o n
duit with re s p e c t to the surrounding s o i l , c h a ra c te r is t ic s of the i n s t a l l a
t io n , w eigh t and h e ig h t of the so il prism above the c o n d u it , and the
fr ic t ion deve loped b e tw een the so il prism above the co n d u it and the s u r
rounding m edium . The e ffec t of l ive lo ad s is b a se d on the B oussinesq
so lu t io n . They have d is c u s s e d the e f fe c ts of arching and bedding along
4
with various m ethods of determ ining the a v a i la b le supporting s trength o f
c o n d u i t s . They have a lso d e sc r ib e d the " im perfect d itch" te c h n iq u e ,
which is u sed to reduce the v e r t ic a l load th a t m ust be supported by the
c o n d u it . In th is method", a lay e r of lo o se or soft m ate r ia l is in s ta l le d
above the c o n d u it . This in su re s g rea te r re la t iv e se tt lem en t of the so il
prism d irec tly over the co n d u it , the reby genera ting upward f ic tion fo rces
on the s id e s of the prism and reducing the load on the c o n d u i t .
E la s t ic formulas for the s t r e s s e s developed in condu its and the
surrounding medium under the in fluence of a uniformly d is tr ib u ted su rface
load were p roposed by Burns and Richard (1964). The theory is deve loped
for c a s e s of fu ll s lip and no s lip a t the s o i l - s t ru c tu re in t e r f a c e . D e f le c
t io n re la t io n s are a lso g iv en . Very rap id a t ten u a tio n of s t r e s s e s in the
so il medium w as r e p o r te d . T h u s , the th e o r ie s p re se n te d are equally
v a l id for shallow and deep s o i l - c u lv e r t s y s te m s .
D ar and Bates (1974) u sed c l a s s i c a l e la s t i c theory to develop a
s e t of equa tions for s t r e s s e s and deform ation which app ly w ith in the c u l
vert w a lls as w ell as in the surrounding medium. A uniformly d is tr ib u ted
su rface load is a s s u m e d . These r e se a rc h e rs have done an e x ten s iv e a n
a ly s is of cu lv e r t behav io r as a ffec ted by the s t i f fn e ss p ro p ertie s of both
the in c lu s io n and the surrounding so il medium. Except in the v ic in ity
im m ediately a d ja c e n t to th e s o i l - s t ru c tu re in te r fa c e , e x c e l le n t agreem ent
with the r e su l ts of Burns and Richard (1964) were r e p o r te d .
- Abel, M ark, and Richards (1973) reported the re s u l t s of a f in i te -
e lem en t a n a ly s is of the so i l - c u lv e r t system for e l l ip t ic a l p i p e s . Their
work inc luded experim en ta l s tu d ie s us ing p h o to e la s t ic te c h n iq u e s . They
d i s c u s s e d the im portance of s lip and s truc tu ra l f le x ib i l i ty in inducing
so il a rch ing in l igh t of the e l l ip t ic a l geom etry; how ever, only the fu l l -
s lip and n o - s l ip c a s e s w ere c o n s id e re d . The use of so ft bedding m ateri
a l w as found to provide no a d v a n ta g e .
An in c rem en ta l c o n s tru c t io n a n a ly s is of the deep buried s o i l -
c u lv e r t sy s tem h a s b een performed by L e r t lak san a (19 73). N onlinear
s o i l p ro p ert ie s and the p re se n c e of pore w ate r in the so il f i l l were c o n
s id e red „ B ased on the r e s u l t s of h is s tu d y , L e rtlaksana conc luded th a t
th e M ars to n -S p an g 1 er procedure for c a lc u la t in g v e r t ic a l p re s su re s on
r ig id c u lv e r ts y ie ld s c o n se rv a tiv e r e s u l t s . The Iowa d e f le c t io n formula
for computing crow n d e f le c t io n s in f le x ib le condu its w as a lso found to be
ex trem ely c o n se rv a t iv e „
P a r tia l d is tr ib u ted loading con d it io n s for sha llow buried rig id
co n d u its w ere in v e s t ig a te d by Anand (1974). He dem onstra ted th a t the
moments p roduced by a p a r t ia l load in th e v ic in i ty of the condu it are un
d e res t im a ted by the e l a s t i c i ty so lu t io n . The normal fo rces s im ilarly p ro
duced are o v e re s t im a te d . This is a m atter for some c o n ce rn , s ince the
bending moment is the more im portant c o n s id e ra t io n in the d e s ig n of
r ig id c u l v e r t s . F in a l ly , he conc luded th a t the th e o re t ic a l e la s t ic i ty s o lu
t io n is not a p p lic a b le to the sha llow s o i l - c u lv e r t s y s te m . H ow ever, th is
co n c lu s io n i s in co rrec t , a s w ill be show n in C hap ter 4 .
Current D es ig n P rac tice
Except for very large condu its , d e s ig n p ra c t ic e to da te has b een
b a s e d on the M ars to n -S p an g le r th eo r ie s developed during the 1920 's and
1 9 3 0 's . This theory c o n s i s t s of the de term ina tion of to ta l v e r t ic a l loads
assu m ed to a c t on the condu it and a p p lic a t io n of load fac to rs th a t re la te
required lo ad -ca rry in g a b i l i ty to th a t a v a ilab le in m anufactured conduit
s e c t io n s . An appropria te factor of sa fe ty is in c lu d e d .
The to ta l v e r t ic a l earth load Wc ac ting on the condu it is g iven
by
Wc = CiYBj2 (1)
in w h ich 'V is the unit w eight of the fill m a te r ia l , Bj is the horizon ta l
d im ension e ith e r of the d itch at the top of the pipe or the o u ts id e d iam
e te r of the p ipe depending on the method of in s ta l la t io n , and is the
load c o e f f ic ie n t . C i is determ ined from an appropriate ch a r t and is a
function of the geom etry of the s o i l - c u lv e r t sy s tem , the method of in
s ta l la t io n , and the m ate ria l p roperties of both the so il f ill and the
c u lv e r t .
The live load tran sfe rred to the cu lver t Wt is determ ined from
Wt = ( l /£ ) I c C t P (2)
in which ^ is the length of the conduit s e c t io n on which the load is com
p u ted , tak en as the length of a s ing le s e c t io n of p ipe or an e ffec tive
length in the c a s e of very long s e c t io n s , a s with c a s t - i r o n pipe; Ic is
an im pact factor; P is a co n cen tra ted load applied at the f ill surface;
and Cj- is the load c o e f f ic ie n t , b a sed on the B oussinesq th e o ry . The
w eigh t of w ater and the pipe may a lso be added .
The v e r t ic a l p re ssu re d is tr ib u tio n on any ho rizo n ta l p lane below
a co n cen tra ted load is b e l l s h a p e d . As the depth below the surface in
c r e a s e s , the d is tr ib u t io n becom es more uniform. Thus, the assum ption
of a uniform d is tr ib u tio n of live load p re ssu re applied to the conduit is
r e a so n a b le for deep i n s t a l l a t i o n s . H ow ever, for shallow c o n d u its , th is
assum ption ca n y ie ld r e su l ts which are su b s ta n t ia l ly d iffe ren t than the
8
true co n d it io n , p a r t ic u la r ly i f the condu it is large enough th a t s ig n if ic a n t
v a r ia t io n in the p re s su re d is tr ib u t io n w ill o ccu r over the w idth of the
s t ru c tu re .
C onduit s treng th is commonly sp e c if ie d in term s of the D - l o a d .
This is the th re e -e d g e s treng th e x p re s s e d in terms of th e load supported
pe r un it foot of d iam e te r . Using th e D - lo a d a llow s sp e c i f ic a t io n s to be
w ritten in a form th a t is indep en d en t of condu it d ia m e te r . The th re e -e d g e
s treng th is m odified through the u se of load f a c to r s , Lf, tak ing into a c
coun t la te ra l earth p re s s u re s th a t develop in - s i tu and bedding con d it io n s .
The sa fe supporting s treng th of the r ig id conduit i s th en the D -lo ad m ul
t ip l ie d by the d iam eter and a load fac to r d iv ided by an appropria te fac to r
of s a fe ty . A s e c t io n is s e le c te d th a t h a s a safe supporting strength
g re a te r th a n the to ta l ap p lied v e r t ic a l lo a d .
W hile the procedure is c o n v e n ie n t , i t does not provide a d ire c t
re la t io n sh ip be tw een the requ ired s treng th and the sa fe supporting
s t r e n g th . D es ig n by th is method therefo re tends to be very c o n s e rv a t iv e .
P roposed D es ig n M ethods
S evera l r e se a rc h e r s have p roposed a l te rn a t iv e p rocedures for
the d e s ig n of buried c o n d u i ts . Some are simply m od ifica tions of the
M ars to n -S p an g le r m ethods, w hile o thers adopt a fresh approach to the
p ro b le m . Two of th e se a l te rn a t iv e s are p re se n te d in the following d i s
c u s s io n .
Ring C om press ion M ethod
The ring co m p ress io n method p roposed by W atk in s (1966) is
b a se d on the a s s e r t io n th a t the d e s ig n of c irc u la r condu its is con tro lled
by condu it d e fo rm a tio n s . These deform ations fall into two c a te g o r ie s ,
w all c rippling and ring d e f lec t io n , with ad d it iona l co n s id e ra t io n g iven
to hand ling and p lacem en t.
Crippling is con tro lled by lim iting the maximum com press ive
s t r e s s in the conduit w all T/Ac to a va lu e th a t is l e s s th an the ring com
p re s s iv e s trength S by some factor of s a f e t y . T is the co m p ress iv e force
in the condu it w a l l , and Ac is the c r o s s - s e c t io n a l a rea of the w all per
unit of le n g th . For a r ig id so il (friction angle = 90), the ring com p res
s ive strength is eq u iv a len t to Syp, the y ie ld strength of the conduit ma
t e r ia l . For the c a s e of h y d ro s ta t ic so il 0 = 0 ) , the ring com press ive
s trength is e x p re s se d as
S = (l/E) (D /k )2 (3)
where E is the modulus of e la s t i c i ty of the conduit m a te r ia l , D is the d i
am eter, and k is the rad ius of gyra tion of the conduit w all per unit l e n g th .
It is apparen t th a t th is equa tion is of the same form as the Euler column
formula, e x cep t th a t the condu it d iam eter is su b s t i tu te d for the column
le n g th . W atk ins (1966) provided a p lo t of the ring co m p ress io n strength
as a function of the condu it p roperties and the so il f r ic tion a n g le . The
maximum co m p ress iv e s t r e s s in the condu it w all , T/Aq , is determ ined
from a th ru s t fac to r, which is a function of the so il and ring s t i f fn e s s e s
and K, the ra tio of ho r izo n ta l to v e r t ic a l p ressu re in la te ra l ly confined
so il due to a v e r t ic a l so il p re ssu re pv . For a c o n se rv a t iv e d e s ig n , take
K = 1 and the re la t io n sh ip s T = 3 /4 pvD for f lex ib le co n d u its or T = pvD
for rigid co n d u its can be u s e d . For more accu ra te d e s ig n s , v a lu es of the
th ru s t fac tor can be ob ta ined from a p lot provided by W a tk in s .
10
A quick c o n se rv a t iv e e s tim a te of ring d e f lec t io n for f lex ib le
co n d u its is ob ta ined by the c o n se rv a tiv e assum ption th a t the ring d e f le c
tio n fac to r is g iven by A y / D = € # where e is the v e r t ic a l so il s tra in from
a conso lid o m ete r t e s t , and Ay is the v e r t ic a l change in the conduit d iam
e te r . This is the r e s u l t of tak ing K = 0, th a t i s , no la te ra l p ressu re a c ts
on the co n d u it . G rea te r accu racy may be ob ta ined by using the p lot pro
v ided by W a tk in s . Ring d e f lec t io n s for nonflexib le co n d u its are e s t i
mated by applying m odification fac to rs to the ring d e f lec t io n factors for
f lex ib le c o n d u i ts .
B alanced D es ig n M ethod
The b a lan ced d e s ig n method proposed by Allgood and T akahash i
(1972) is b a se d on ring co m p ress io n and e la s t ic th e o ry . The procedure
perm its the c o n s id e ra t io n of a ll s t i f fn e s s e s and g e o m e tr ie s , arching ,
and b ack p ack in g . C o n s id e ra tio n is g iven to a l l modes of fa i lu re , and
p ro v is io n is made for the com putation of fac to rs of sa fe ty for each m ode.
The u se of the D -lo ad s p e c if ic a t io n for the s e le c t io n of the proper c o n
duit is e lim ina ted and the em phas is p la c e d on the more fam iliar approach
of s e le c t in g a s e c t io n b ase d on the requ ired e x te n s io n a l and bending
s t i f f n e s s e s . Using th is method, the d e s ig n e r can a t ta in a d es ired
"b a lan ce" be tw een the fac to rs of s a fe ty of the various fa ilure m odes.
The following is a b rie f summary of the s te p s in th is des ign
p rocedure .
1. Estim ate the arching and c a lc u la te the normal th ru s t in the c o n
duit w all by the modified ring com press ion theory as
N = pv (1 - A) (D/2) (4)
where A is the a rc h in g , pv is the v e r t ic a l s t r e s s a t the crown
e le v a t io n , and D is the mean cy lin d er d iam eter .
2 . C a lc u la te the s t i f fn e s s , El, required by the handling c r i te r ion
^ 0 .0433 (5)
where E is the modulus of e la s t i c i ty of the condu it m ateria l
and I is the moment of in e r t ia per unit l e n g th .
3 . D eterm ine the moment using the th e o re t ic a l e l a s t i c so lu tion as
M = Cmpv (l - A) D2 (6)
where C m is a co e f f ic ie n t dependen t on the so il and conduit
p r o p e r t i e s .
4 . Using e la s t i c fo rm u las , com pute the s t i f fn e ss El required to
r e s i s t the normal force N and bending moment M . Choose the
con tro lling s t i f fn e ss v a lu e .
5. Compute the arching and compare it to the a ssu m ed v a lu e . I t
e ra te over s te p s 1-5 as n e c e s s a r y .
6 . Determ ine conform ance with d e s ig n c r i te r ia and the factors of
sa fe ty for the various p o s s ib le failure m odes. These include
d e f le c t io n , w all c ru sh in g , seam s treng th , and b u c k l in g .
The arching term is inc luded in Eqs. 4 and 6 to co rrec t some of the w eak
n e s s e s of the e la s t i c so lu tion , which does not take into accoun t such
fac to rs as the nonhom ogeneity of the so il medium, non linearity of the
s t r e s s - s t r a i n c h a r a c te r i s t i c s , and the time e f fe c ts . In ad d it io n , the
boundary cond it ions in troduce c o n s tra in ts th a t do not e x i s t in the ac tu a l
in s ta l la t io n . Allgood (19 72) p re sen ted a d e ta iled developm ent of arching
re la ted to buried c y l in d e r s .
The ch a r ts developed in su b seq u en t ch ap te rs are d irec tly a p
p l ic a b le to th is d es ig n method in the de term ination of the normal force
and bending moment.
CHAPTER 3
RESEARCH TOOLS
The com puter program and f in i te -e le m e n t grids u se d in th is
s tudy are the product of a g rea t d ea l of prelim inary in v e s t ig a t io n . Pro
gram SSI w as p repared by Sogge ( n . d . ) . No attem pt i s made here to p re
se n t an ex h au s t iv e a n a ly s is of the f in i te -e le m e n t method or the
developm ent of the p rog ram . H ow ever, a summary of the program docu
m enta tion is p re se n te d . The f in i te -e le m e n t grids are the end re su l t of
se v e ra l r e f in e m e n ts . The prime o b jec t iv e in th e i r developm ent was to
o b ta in a h igh degree of accu racy in the v ic in i ty of the co n d u it .
Program SSI
Program SSI, "S o il-S tru c tu re I n te r a c t io n ," com putes the d i s
p la c e m e n ts , s t r e s s e s , and moments in a f in i te -e le m e n t system com pris
ing so il and s tru c tu ra l e le m e n ts .
Program F ea tu res
The program s y n th e s iz e s and so lv e s the eq u a tio n s for e q u il ib r i
um, force deform ation, and geom etric com patib ili ty for the so i l - s t ru c tu re
continuum id e a l iz e d a s an a s sem b lag e of d isc re te e lem en ts . The m atrix
s t i f fn e ss method is u s e d . This procedure h a s been p re se n te d by Z ien-
k iew icz (1971) and by o ther w o rk e rs . In order to m inim ize s to rage r e
qu irem en ts , a banding tech n iq u e is u se d to store the system equa tions in
12
13
the s t i f fn e ss a r r a y . The eq u a tio n s are th en so lved us ing a banded sym
m etrica l equa tion so lv e r .
For s o i l - s t ru c tu re sy s te m s , two degree o f freedom nodes are
predom inant in the f in i te -e le m e n t id e a l iz a t io n . H ow ever, the system is
u su a l ly an a ly zed assum ing th ree d eg rees of freedom a t each node. Since
th is would be w a s te fu l of com puter s to r a g e , a m odifica tion th a t m akes
u se of a v e c to r of the cum ulative system d eg rees of freedom a t each node
is in c lu d e d . Using th is p ro c e d u re , nu ll d eg rees of freedom are e lim i
n a ted and s to rage is r e se rv e d for th o se d eg rees of freedom th a t are
p r e s e n t .
C o n s tan t s tra in t r ia n g le s (TRIMS) are u se d to model the so il
medium. These e lem ents are programed to include non lin ea r strength
p r o p e r t ie s , as d e sc r ib e d by D uncan and Chang (1970). Elements are
a v a i la b le to s im ula te the s o i l - s t ru c tu re in te rface c h a r a c te r is t ic s . These
e lem en ts are programed to include non lin ea r behav io r , as d esc rib ed by
Clough and D uncan (1971). L inea riza tio n of both the tr ian g u la r and in
te r fa c e e lem ents is accom plished by a s s ig n in g a fa ilu re ra tio va lue
Rf = 0 and in se r t in g a l in e a r va lue of the e la s t i c m o d u lu s . P o is so n 's
ra tio p m ust a lso be s p e c if ie d .
The s truc tu re is id e a l iz e d a s a se r ie s of in te rc o n n ec ted beam
e lem ents w hose m ateria l behavior is approxim ated by a l in ea r s t r e s s -
s tra in r e la t io n . Boundary d isp lace m e n ts of ad jacen t t r ian g u la r and beam
e lem en ts are not c o m p a t ib le , s in ce th e ir d isp lace m e n t p a t te rn s are c h a r
a c te r iz e d by f i r s t - and th ird -o rd e r d isp lace m e n t fu n c t io n s , r e sp e c t iv e ly .
In the lim it, convergence re s u l t s as the m esh s ize i s made f ine r. . For
c o a rse m e s h e s , the so il is s tif fe r th an a c tu a l , re su lt in g in low er mo
m ents in the s t ru c tu re .
Program M odif ica tion
In order to fit the very large system of equa tions re su lt in g from
the f in i te -e le m e n t m esh es u se d in to the av a i lab le com puter c en tra l mem
ory , s e v e ra l m odifica tions of Program SSI were n e c e s s a ry .
.1 . Common arrays, are modified to co n ta in the d a ta .
2 . F ile len g th s are sp e c if ie d to override th o se a s su m ed by the
CDC com puter sy s te m .
3 . Element d a ta are s to red ex te rn a l ly on a tap e f i l e . The order of
en ter ing e lem en t d a ta is not im portan t, s in ce th e order in w hich
the system eq u a tio n s are g en e ra ted is im m aterial as long as
th ey are co rrec t ly s to red .
4 . The portion of the m ain program th a t com putes the v ec to r of
cum ula tive system d eg rees of freedom i s rem oved. These r e
s u l ts are o b ta ined beforehand and are subm itted as d a ta .
5. BAR and STRBAR subrou tines are rem oved. In ad d it io n , the
INFACE and STRESI sub rou tines are rem oved, e x c e p t for th o se
c a s e s w here they are ac tiv e in the a n a ly s is .
6 . Statements, to perform n o n d im ensiona liza tion of th e conduit
moments and normal fo rces are added in subrou tine STRESB.
W ith th e s e m o d if ic a t io n s , more c e n tra l memory is r e se rv e d for the s to r
age of the s t i f fn e ss e q u a t io n s , w hile the in teg rity of the program as a
too l for the a n a ly s is o f non linear so il behav io r i s p re se rv e d .
F in ite -e lem en t M odels
Four f in i te -e le m e n t grids are u sed in the a n a ly s is of the s o i l -
condu it sy s te m . The com plete reg ion is shown in F ig . 1. In a ll g r id s ,
beam e lem en ts are u sed to model the condu it and c o n s ta n t - s t r a in t r ia n
gu lar e lem en ts are u sed to model the surrounding so il medium.
The region inc luded for a n a ly s is is sym m etrical about a v e r t ic a l
line through the c e n te r of the condu it (the dashed portion is excluded
h e r e ) . Taking advan tage of th is symmetry, only the r ight h a lf of the r e
gion is m odeled for Grid 1 and Grid 2. These are shown in F ig . 2 . As a
re s u l t of th is sym m etrical c o n s tru c t io n , only loading con d it io n s th a t are
a lso sym m etrical about th is v e r t ic a l ax is ca n be a n a ly z e d . In both of
th e se g r id s , the s truc tu re is co n s tru c te d of 24 beam e lem en ts and the
surrounding so il medium is made up of 4 74 tr iangu la r e le m e n ts . These
e lem en ts are in te rco n n ec ted at 2 71 node p o in ts . In add it io n to th is .
8d 2d
Figure 1. Region Included in the A nalysis
Figure 2. F in ite -e lem en t Id ea l iza tio n for Grid 1 and Grid 2
17
Grid 2 in c lu d e s 24 in te rface e lem ents b e tw een the so il and the s t ru c tu re .
S ince th e s e e lem en ts are a s s ig n e d a zero th ic k n e s s , the ap pearance of
both g rids is id e n t ic a l . Grid. 2 h a s a to ta l of 522 e lem en ts and 296 co n
nec ting node p o i n t s . U tiliz ing Grid 2 , the e ffec t of s lip a t the s o i l -
s truc tu re in te rface c a n be in v e s t ig a te d .
A m ain po in t of in te re s t in th is s tudy is to in v e s t ig a te the e f fec t
of nonsym m etrica l load ing c o n d i t io n s . Therefore , i t is n e c e s s a ry to d e
velop an ad d it io n a l model th a t in c lu d es the cu lver t and medium on both
s id e s of th e c e n t ra l ax is (the d ash ed portion of the reg io n is om itted
h e r e ) . S ince th e p rev ious m odels requ ire very nearly a l l of the com
p u te r c e n tra l memory a v a i l a b le , Grid 3 m ust n e c e s s a r i ly be made co n
s id e rab ly c o a r s e r . The m esh is kep t r e la t iv e ly fine im m ediate ly a d ja c e n t
to the condu it in order to p rese rve a good degree of a c c u racy in the c o n
duit s t r e s s e s .
It i s a lso n e c e s sa ry to determ ine the e ffec t of the boundary
co n d it io n s on the s t r e s s e s induced in the condu it by lo ad s p laced near
po in ts of support. Grid 4 is co n s tru c te d for th is p u r p o s e . The entire
reg io n , inc lud ing an e x te n s io n eq u iv a len t in width to two conduit diam
e te rs (dashed portion of the reg ion in F ig . 1) is inc luded in th is m o d e l .
A com parison of the condu it moments produced by lo ad s a t various d i s
ta n c e s from the la te ra l support po in ts u s in g Grid 3 and Grid 4 is m a d e .
The r e s u l t s show th a t the e ffec t of the boundary co n d it io n s is in s ig n if i
c a n t for loads ap p lied no neare r th an two d iam eters from the supports .
Grid 4 c a n therefore be u se d for po in t lo ad s p laced up to four d iam eters
to the r igh t of a l ine through the c e n te r of the co n d u it .
18
F ig . 3 is a p a r t ia l v iew of the f in i te -e le m e n t id e a l iz a t io n for
Grid 3 and Grid 4 . The le f t -h a n d portion of both is a mirror image of
th a t show n. The e x te n s io n of the reg ion for Grid 4 is not show n. In bo th ,
the condu it c o n s is t s of 18 beam e le m e n ts , the so il medium is made up of
tr ian g u la r e lem en ts and no in te rface is p rov ided . Grid 3 h a s 266 nodes
and 496 e lem en ts ; Grid 4 h a s 269 nodes and 502 e le m e n ts .
Following the p rep ara tio n of node po in t and c o n n e c tiv i ty d a ta ,
the maximum node po in t s ep a ra t io n w as m inimized us ing an in te ra tiv e
schem e s im ila r to th a t p roposed by Grooms (1972).
V erification of M odels
V erification of the com puter program and the f in i te -e le m e n t
grids d e sc r ib e d in the preced ing s e c t io n is accom plished by performing
e la s t i c a n a ly s is of problem s with known c l a s s i c a l s o lu t io n s . The com
pu ted re s u l t s are th en com pared with the known s o lu t io n s .
Three s e p a ra te problem s w ere u se d in the te s t in g of Grids 1 and
2 . F i r s t , the so il e lem en ts are a s s ig n e d zero m ateria l s t i f fn e ss p roper
t i e s leav ing only the s t ru c tu re . An a n a ly s is of the co n d u it as a c i rc u la r
ring su b jec te d to a co n cen tra ted ax ia l load is perfo rm ed . The r e su l ts are
in e x c e l le n t ag reem ent with the e l a s t i c so lu tion as p re se n te d by Roark
(1965). Secondly , the beam e lem ents rep resen ting the condu it are a s
s igned zero s t i f fn e ss p ro p ertie s leav ing only the s o i l . This is an a lyzed
a s a s e m i- in f in i te p la te with a h o le . The s t r e s s d is tr ib u t io n through the
medium is th en com pared with the known so lu tion g iven by Timoshenko
and G oodier (1970). Again there is e x c e l le n t .a g re e m e n t . F in a lly , the
beam e lem en ts are a s s ig n e d very large s t i f fn e ss p ro p e r t ie s . This is
Figure 3 . Partia l F in ite -e lem en t Id ea liza tio n for Grid 3 and Grid 4
20
an a ly zed and com pared to the known so lu tio n for a s e m i- in f in i te e la s t i c
p la te with a r ig id cy lin d r ic a l in c lu s io n as p roposed by G oodier (1933).
■Again, the medium s t r e s s e s are in e x c e l le n t agreem ent with the known
so lu t io n , ex c e p t in the lay e r im m ediately a d ja c e n t (0 .20 diam eters) to
th e in c lu s io n . This is a t tr ibu ted to an in ab il i ty to m ain ta in su ff ic ien t
s ig n if ic a n t f igures in the com puter sy s tem to model com plete r ig id i ty .
S in c e .i t w as n e c e s s a ry to c o n s tru c t G rids 3 and 4 with l e s s
d ef in it ion in the so il medium away from the co n d u it , v e r if ic a t io n of
th e s e g rids by the lim iting c a s e approach u se d above w ill in d ica te l e s s
a c c u racy in the s t r e s s e s ob ta ined for the medium. A nalys is of the s tru c
ture a s a c i rc u la r ring under an a x ia l load ag a in is in e x c e l le n t ag ree
ment with the known so lu t io n . S ince th is study is concerned with mo
m ents and forces in the co n d u it , l e s s accu racy o f the s t r e s s e s in the
so il medium away from the condu it is of no c o n s e q u e n c e . In order to
verify the accu racy of th e se grids in ob ta in ing the d e s ire d moments and
normal fo r c e s , an a n a ly s is of the s o i l - c u lv e r t system is performed and
com pared to the known so lu tio n p roposed by Burns and Richard (1964).
E xce llen t agreem ent is o b ta in ed .
Anand (19 74), u s ing s im ilar a n a l y s e s , co n c lu d e s th a t the Burns
and Richard so lu tion is not va lid for the c a s e of a sha llow buried con
d u i t . H ow ever, a n a ly s e s performed us ing Grid 1 in d ic a te th a t the f in i te -
e lem ent model u sed by Anand w as overrig id . In ad d it io n , he did not co n
s id e r the full d is tr ib u ted load assu m ed in the e la s t i c so lu t io n . This
work is p re se n te d in d e ta i l in C hap te r 4 .
CHAPTER 4
DISCUSSION OF RESULTS
The a ssu m p tio n of a uniformly d is tr ib u ted su rface load over an
in fin ite e x ten t is a common fac to r in the th e o re t ic a l e l a s t i c so lu tions
th a t have b een p roposed for the s o i l - c u lv e r t s y s te m . H ow ever, as w as
s ta te d e a r l ie r , a n a ly s e s b a se d on th is assum ption may se r io u s ly under
e s t im a te the s t r e s s e s induced in shallow buried c o n d u i ts . In order to in
v e s t ig a te the e ffec t of various p a r t ia l loading con d it io n s on a shallow
co n d u it , se v e ra l c a s e s are an a ly zed us ing the f in i te -e le m e n t grids p re
v io u s ly d e s c r ib e d .
In a l l c a s e s , the condu it and so il m ateria l are a ssum ed to b e
have e l a s t i c a l ly . The so il is a s s ig n e d an e la s t ic modulus v a lu e , Eg =
200 k s f = 9 . 58x10® N/m2 and a Pois s o n 's ra t io , ^as = 0 . 4 . These
v a lu e s are re p re se n ta t iv e of den se g ranu lar s o i l s . The co n d u it is a s
sumed to be made of co n cre te with an e l a s t i c modulus v a lu e , E = 2 .5
x 10® p s i = 17 .25 x 10® kN/m2 .
The loading c a s e s c o n s id e red in the a n a ly s is , a long with the
f in i te -e le m e n t grid u se d , are sum m arized in Table 1. The conduit s t i f f
n e s s , e x p re s se d as the ra tio of the in te rio r d iam eter, d , to w all th ic k
n e s s , t , i s in d ica ted for each c a s e .
The in fluence of uniformly d is tr ib u te d loads of in te n s i ty , p ,
c en te re d about a v e r t ic a l l ine through .the condu it are c o n s id e red in the
f ir s t e ig h t c a s e s . The loads are of varying leng th , b . C a s e s 9 to 24
21
Table 1. Load and C onduit S tiffness for Analyzed C a s e s
22
Load
3ase Grid D e sc r ip t io n L ocation In te n s i ty d/i
1 1 D is tr ib u ted b = d P 402 1 D is tr ib u ted b = 2d P 403 1 D is tr ib u ted b = 3d P 404 D is tr ib u ted b = 4d P 405 1 D is tr ib u ted b = 5d P 40
6 D is tr ib u ted b = 6d P 407 1 D is tr ib u ted b = 7d P 408 1 D is tr ib u ted b = 8d P 409 3 C oncen tra ted C en te r P = pd 40
10 3 C o n cen tra ted 0 . 250d P = pd 40
11 3 C o n cen tra ted O.SOOd P = pd 4012 3 C o n ce n tra ted 0 . 875d P = pd 4013 3 C o n cen tra ted 1 . 250d P = pd 4014 4 C oncen tra ted 1 . 750d P = pd 4015 4 C o n cen tra ted 2 . 625d P = pd 40
16 4 C o n cen tra ted 4 . OOOd P = pd 4017 3 C o n cen tra ted C en te r P - pd 2018 3 C o n cen tra ted 0 . 250d P = pd 2019 3 C o n cen tra ted O.SOOd P = Pd 2020 3 C o n cen tra ted 0 . 875d P = pd 20
21 3 C o n cen tra ted 1 . 250d P = Pd 2022 4 C oncen tra ted 1 . 750d P = pd 2023 4 C oncen tra ted 2 . 625d P = pd 2024 4 C o n cen tra ted 4 . OOOd P — pd 2025 2 D is tr ib u ted b — 8d P 40
23
c o n s id e r a co n c e n tra te d load of in te n s i ty , P = pd, lo ca ted with r e sp e c t
to the c e n te r of the co n d u it , as show n. Two condu it s t i f fn e s s e s are c o n
s id e re d . For C a s e s 1 -1 6 , d / t = 40 , w hile for C a s e s 17 -24 , d / t = 20. In
the l a s t c a s e , the e f fec t of s lip a t the in te rface is a n a ly z e d .
P artia l D is tr ibu ted Loads
The bending moments induced in the conduit for the various d i s
tr ibu ted loading c a s e s are p re se n te d in nondim ensional form in F ig s . 4
and 5. S im ilarly , the normal fo rces are g iven in F ig s . 6 and 7.
For p u rp o ses of com parison , the e la s t i c formulas p roposed by
Burns and Richard (1964) for no in te rface s lip are u sed to c a lc u la te the
bending m om ents, M, and normal fo rc e s , _N, in the c o n d u i t . For the m a
te r ia l and geom etric p roperties s ta te d the equa tions are g iven in nondi
m ensional form as follows
M /p d 2 = 0 .0001 + 0 .0059 co s 28 (7)
N /pd = 0 .4961 + 0 .0992 c o s 29 (8)
where 9 is m easured around the c ircum ference of the c o n d u it . If the a p
propria te v a lu es are in se r te d for 9 and the re su lt in g moments and normal
force are p lo t te d , the th e o re t ic a l e la s t i c so lu tion ag rees very c lo se ly
with the r e su l ts of loading c a s e 8. F i g s . 4 and 5 show th a t the bending
moments in c re a se rap id ly for the f irs t two c a s e s and th en appear to lev e li
off. H ow ever, as the length b of the load in c re a se s beyond four d iam
e te r s , there is a reduc tion in the m agnitude of the moments and the th e o
re t ic a l so lu tion is ap p roached . F i g s . 6 and 7 show a g radual in c re a se in
the normal fo rces through the f irs t five loading c a s e s . Beyond th is p o in t ,
the change a t the crown and b ase is more g ra d u a l . There is v ir tua lly no
d / t = 40 1.0x10
0.0
- - - - V
Case 2 y
\ Cose \ Case\4 ^
OX)
0 .5x10
1.0x10
Figure 4 . Bending Moments for Loading C a s e s 1 to 4
25
d / t = 40
0.0
Case 5
-U
Case 7 \ \Case 8 (elastic so
0.0
0 .5x10
Figure 5. Bending Moments for Loading C a s e s 5 to 8
Figure 6. Normal Forces for Loading C a se s 1 to 4
27
- 0.6
d / t = 40
0.2
0.0
Case 5
0.0
- 0.2
CaseCase 8 (feiqstio' solution)*
- 0 . 4
- 0.6
Figure 7. Normal Forces for Loading C a s e s 5 to 8
28
change a t the s p r in g l in e . The normal fo rces a lso approach the th eo re t
ic a l so lu t io n .
F ig . 8 shows the v a r ia t io n of the bending moments a t the
crow n, sp r in g l in e , and b a se of the c o n d u it . The bending moment a t the
crown is s e e n to reach a maximum s lig h tly before th o se a t the sp ring line
and b a s e . The maximum v a lu e s are about tw ice a s la rge as th o se p re
d ic ted by the th e o re t ic a l so lu t io n . Some feeling for the movement tak ing
p la c e c a n be ob ta ined from Fig . 9, w hich show s the change in d iam eter
a long the h o r iz o n ta l ( sp r in g lin e ) . The d iam etric shorten ing along the v e r
t ic a l follow s the same path but is 3 .5 to 4 .0 p e rcen t g re a te r in m agn itude .
The above o b se rv a t io n s c a n be in te rp re ted by cons ide ring the
re la t iv e s t i f fn e s s of the so il-s tru c tu re system under lo a d . For loads over
a short length (Case 1), the large so il m ass ad ja c e n t to the conduit is
e ffec t iv e in r e s is t in g d isp la c e m e n ts and thereby a s s i s t s in carrying the
lo ad . Thus, com pared to c a s e s with lo ad s of in te rm ed ia te ex ten t , the
bending moments in the condu it are low . Loads of in term ed ia te length
(C ases 2-5) becom e the dom inant fac to r cau s in g g rea te r movement in th e i
so il a d ja c e n t to the prism above the co n d u it . This motion reduces the
arching e ffec t in the so il and induces the larger moments o b se rv ed . The
loads rem ain dom inant as the full load cond it ion is app roached . How
ever , here the loads provide a confin ing su rface o v e rp re s su re , thus in
c re a s in g the re la t iv e s t i f fn e ss of the so il and reducing the m om ents.
The normal fo rces prove to be re la t iv e ly in se n s i t iv e to the arching e f
f e c ts and are more d ire c t ly dependen t on the load in te n s i ty .
As d i s c u s s e d in C hap te r 3, the p re se n c e of the supports along
the s id e s of the reg ion a f fe c ts the a c t io n of loads p la c e d nea rb y . T h u s ,
29
0.0150
V ln me0.0100
0.0050
0.0
d / t = 40
-0 .0 0 5 0Base
0.0100
Crown
-0 .0 1 5 08d2d
load length, b
Figure 8. Moment Variation for Various Loading Conditions
0.0100
Ad 0.0050 (ft)
0.0
d / t = 40
4 d 6 d 8 d2d0load length, b
Figure 9. Diametr ic Change for Various Loading Condit ions
30
the fo rces and moments reported for C a s e s 6 and 7 are s lig h tly g rea te r
th an a c tu a l . In C a s e 8, the full load e ffec t is ach iev ed for a load leng th ,
b , som ew hat shorter th a n a c tu a l . The r e su l ts for th e se c a s e s are q u a l i
ta t iv e ly c o r re c t , a s w ill becom e more apparen t in the d is c u s s io n th a t
fo llo w s .
C o n cen tra ted Loads and Influence C harts
The p rev ious d i s c u s s io n i s lim ited to the e ffec t of sym m etrical
p a r t ia l d is tr ib u ted loads on buried c o n d u i t s . In order to com plete the
s tudy of the e ffec t of p a r t ia l loading co n d i t io n s , unsym m etrica lly lo c a te d
co n c e n tra te d su rface lo ad s are c o n s id e red in the following d is c u s s io n .
F ig s . 10 through 13 show the bending moments induced in the
condu it for the v a r io u s co n cen tra ted loading c a s e s s tu d ie d . These r e
s u l ts show th a t the moments produced in the s tif fe r condu it (d /t = 2 0 )
are co n s id e ra b ly g re a te r in m agnitude th an th o se in the more f lex ib le
c o n d u it . The more f lex ib le s tru c tu re , b e c a u se i t d e f le c ts to a g rea te r
e x ten t under load , encourages the developm ent of arching in the s o i l .
The so i l c a r r ie s the load around the co n d u it , th u s , low er moments are
deve loped in the s tru c tu re . The maximum moments d eve loped at the
crown for co n c e n tra te d loads lo ca ted up to one d iam eter from the c e n te r
are a s much as two and o n e -h a l f t im es g rea te r than p red ic ted by the
e l a s t i c so lu tio n s th a t assu m e a full uniform load on the su rfa c e .
The normal fo rces developed by co n cen tra ted lo ad s are shown in
F ig s . 14 through 17. A com parison of th e s e p lo ts show s th a t the induced
normal fo rces are s l ig h t ly h igher for the rigid conduit but become nearly
equa l for both s t i f fn e s s v a lu e s as the load moves away from the ce n te r
31
d / t = 40 - 2
- 2
C ase / 9
Figure 10. Bending Moments for Loading C a s e s 9 to 12
32
d / t = 40
e 15
Case it
Figure 11. Bending Moments for Loading C a s e s 13 to 16
33
= 20—2 J h
—CL£L
XX
\ x j \
Figure 12. Bending Moments for Loading C a s e s 17 to 20
34
d /t = 20
- 2 . 0 x l Q
-Case 2,1 v
Case y23 Case 24
Z Z
Figure 13. Bending Moments for Loading C a s e s 21 to 24
35
d / t = 40
Case 19 Case j\0
Case 11 Case
Figure 14. Normal Forces for Loading C a s e s 9 to 12
36
d / t = 40
^ Case ) Case/14
A—
Figure 15. Normal Forces for Loading C a s e s 13 to 16
37
- 0 . 3
d /t = 20
I X
Case 19 Case 2C
Figure 16. Normal Forces for Loading C a s e s 17 to 20
38
d /t = 20
X\
Case 21
Case ]
Case 23 ^ Case 2;
Figure 17. Normal Forces for Loading C a s e s 21 to 24
39
of the co n d u it . Some te n s io n is developed near the crown for loads lo
c a te d near the c e n te r of the co n d u it . This due to the assum ption of fix ity
be tw een the so il and the s t ru c tu re . The forces developed at the crown
and invert are c o n s id e ra b ly lower th an th o se p red ic ted by an a n a ly s is
b a se d on the full su rface load a ssu m p tio n .
H aving ob ta ined th e se d a ta , in fluence c h a r ts are made up by
p lo tting the bending moment and normal force in fluence of each load
a g a in s t the lo ca t io n of the load with r e s p e c t to a v e r t ic a l line drawn
through the c e n te r of the co n d u it . The c u r v e s , or in fluence l in e s , are
p lo tted at in te rv a ls of 15 d eg rees around the c ircum ference of the c o n
d u it .
The in fluence c h a r ts are nondim ensional p lo ts of Cm = M /pd^
and C n = _N/pd on the v e r t ic a l ax is a g a in s t the load lo c a t io n along the
h o r iz o n ta l . Cm and C n are the bending moment and normal force c o e f f i
c i e n t s , r e s p e c t iv e ly . The bending moment ch a r ts are p re se n te d in Ap
pendix A along with a l l the da ta u sed to co n s tru c t the c h a r t s . S im ilarly ,
the normal force c h a r ts are p re sen ted in Appendix B along with the a p
p lic a b le d a ta .
It is observed th a t the in fluence cu rves are not zero at 4 d . This
in d ic a te s th a t loads lo ca ted more than four d iam eters from the cen te r
have an e f fec t on the bending moments and normal fo rces in the co n d u it .
If ex tended beyond four d iam e te rs , the cu rves will approach zero at a
poin t where the load is su ff ic ien tly far away from the condu it th a t it h a s
no in fluence on the bending moment and normal fo rce . H ow ever, in su f
f ic ie n t da ta were o b ta ined to determ ine the lo ca tio n of th a t po in t .
40
The in fluence ch a r ts may be app lied to any loading co n d it io n .
The moment, in g e n e ra l , is g iven by
M = C mp d 2 . (9)
H ow ever, for a co n c e n tra te d load , P = pd , so the eq u a tio n becom es
M = C mPd. (10)
For d is tr ib u ted lo a d s , the moments are ob ta ined by in teg ra ting the c h a r ts
over the loaded leng th , b . S im pson 's ru le for num erical in teg ra tio n c a n
be co n v en ien tly a p p l ie d . T h u s ,
M = Cmp d 2 (11)
where
Cm Cm • (12)
Cm and C m are the bending moment c o e f f ic ie n ts . S im ilarly , the normal
fo rce , in g e n e ra l , is g iven by
N = C npd. (13)
For a c o n cen tra ted load , P = pd , and th is reduces to
N = C nP. (14)
For d is tr ib u ted lo a d s , in teg ra tio n is performed as befo re , and
N = C npd (15)
where
C n = / C n . (16)
C n and C n are the normal force c o e f f i c i e n t s .
For r ig id co n d u its with s t i f fn e s s e s o ther than th o se g iven by the
c h a r t s , an in te rp o la tio n may be ca rr ied ou t, as shown in F ig . 18. The
c a s e shown is for the spring line moment due to a co n c e n tra te d load
above the c e n te r of the co n d u it . Using both char t s t i f fn e s s v a lu e s , the
bending moment or normal force c o e f f ic ie n ts are determ ined for the
41
0.0400
0.0300
0.0200
0.0100
0.04.01.0 3.00.0 2.0
In (d/t)
Figure 18. In te rpo la tion of C o e ff ic ien ts for S t i f fn e sse s Not G iven in Influence C har ts
42
d e s ire d lo a d in g . The com puted in fluence c o e f f ic ie n ts are p lo tted a g a in s t
In ( d / t ) . A s tra ig h t l ine is p a s s e d through the po in ts and a c lo se approx
im ation to the c o e f f ic ie n t for the d e s ire d s t i f fn e ss is o b ta in e d . D a ta for
d / t of 11 .5 and 30 are o b ta ined as a ch eck in F ig . 18. This procedure is
ve r if ied for a range of s t i f fn e s s e s from d / t = 11 .5 to d / t = 40 .
Thus, for r ig id condu it s t i f f n e s s e s , the bending moment and
normal force d iagram s c a n be c o n s tru c te d by computing the appropriate
c o e f f ic ie n ts from the c h a r ts a t 1 5 -deg ree i n t e r v a l s . The range of s t i f f
n e s s e s for which the v a l id i ty of in te rp o la t io n is e s ta b l i s h e d in c lu d es
m ost com m ercia lly av a i la b le concre te s e c t io n s conforming to ASTM
(C 7 6 -7 2 ) .
For loads to the r igh t of the c e n te r , the cu rv es for the d es ired
ang les are u s e d . For lo ad s to the le f t , the cu rves corresponding to the
supplem ent of th is angle are u s e d . The ch a r ts are p re se n te d in a format
th a t in c lu d es the cu rv es for supplem entary an g les on the same c h a r t .
Some exam ples w ill serve to c la r ify the ap p l ic a t io n of thex- -
c h a r t s . Four ty p es of problem s are show n in F ig . 19. Assume in each
th a t the moment a t 9 = 60° from the h o r iz o n ta l is d e s i r e d . . A po in t load
is co n s id e re d in F ig . 19a. The moment is determ ined by going to the
appropria te c h a r t , F ig . A -3 (Appendix A) for an angle of 9 = 600 and o b
ta in in g the va lue of Cm for the appropria te s t i f fn e s s . In th is c a s e , if
d / t = 20 and the load is a t 1 . 25d, Cm ?= - 0 .0 1 5 0 . Eq. 10 is then u sed to
com pute the moment. For the sym m etrical d is tr ib u ted load in F ig . 19b,
the co e f f ic ie n t Cm is o b ta ined from Eq. 12, where the in teg ra tio n is p e r
formed on the curve for 9 = 60° , correspond ing to the load on the r ig h t ,
and the curve for 9 = 120°, co rrespond ing to the load on the le f t . Due
to symmetry, both are in teg ra te d over a length of 0 to l .O d . The moment
1.250'l.Odl.Od
1 0.75d r 1.25dr
Figure 19. Some C a s e s for A nalysis by Influence Charts
(a), point load; (b), sym m etrically d is tr ibu ted load; (c), unsymmetrically d is t r ib uted load; and (d), randomly d is tr ibu ted load . ax
co
44
a t 9 = 60° from the h o r izo n ta l is th en com puted us ing Eq. 11. For unsym -
m etr ica l d is tr ib u ted lo ad s a s in F ig . 1 9 c , the same procedure is follow ed
e x c e p t th a t in teg ra tio n is performed only over the length a f fec ted by the
l o a d . For the c a s e show n, the curve for 6 = 60° is in teg ra te d from 0 to
1 .2 5 d and the curve for 9 = 120° is in teg ra te d from 0 to 0 , 7 5 d . It should
be c le a r th a t loads skew ed to one s ide only m ust be c o n s id e re d . Loads
skew ed to the o p p o s ite c a n be accom m odated by changing the poin t of
v iew . Random d is tr ib u ted lo a d s , such as th o se shown in F ig . 19d c a n
be an a ly zed by co n s id e r in g the load to be made up of a s e r ie s of sho rte r
uniform l o a d s .
The normal fo rces can be com puted in a s im ila r m anner by us ing
the ch a r ts in Appendix B .
The accu racy o b ta ined in com puting the c o e f f ic ie n ts for d is t r ib
u ted lo ad s us ing a num erical in teg ra tio n procedure w ill depend on the
care with which po in ts are s e le c te d and the s iz e of the in te rv a l be tw een
p o in ts . M oments for s e v e ra l of the d is tr ib u ted loading con d it io n s a n a
lyzed in th is study (C ases 1-8) are com puted with the c h a r t s . D iffer
e n c e s of l e s s th a n 1 p e rc e n t are o b ta in ed us ing S im pson1 s rule with an
in te rv a l of 0 . Id .
The v a r ia t io n in bending moments due to the in fluence of p a r t ia l
d is tr ib u ted loads of p ro g re s s iv e ly in c re a s in g la te ra l e x te n t may be further
v isu a l iz e d by exam ining the in fluence c h a r t s . Reference is made to the
moment in fluence cu rves for d / t = 40 and 9 = 0° and 180° g iven in F ig .
A -7 (Appendix A). It is o b se rv ed th a t the a re a under the curve is p o s i
t iv e up to 1 .9 d for 9 = 0° and up to 1 .3 for 9 = 180°; both are nega tive
beyond th a t p o in t . Thus, the moment at the spring line w ill in c re a se with
45
an in c re a se in the la te ra l ex ten t of the load until the a rea becom es n e g a
t iv e . Beyond th a t p o in t , the moment w ill d e c r e a s e . This co rresponds to
F ig . 8 where the moment a t the spring line beg ins to d e c re a se betw een
b = 3d and 4 d .
The ch a r ts may be d irec tly app lied in the b a lan ced d es ig n
method p re se n te d in C h ap te r 2. T h u s . E q . 4 is eq u iv a len t to
N = C n (l - A)P (17)
for c o n cen tra ted l o a d s , and
N = C n (l - A)pd (18)
for d is tr ib u ted l o a d s . E q . 6 is eq u iv a len t to
M = C m (l - A)Pd (19)
for c o n cen tra ted l o a d s , and
M = C m (l - A)pd2 (20)
for d is tr ib u ted l o a d s . In th e se e q u a t io n s , the term (1 - A) is used to
e s tim a te the e ffec t of a rch ing , A , due to backpack ing
In te rface Slip
The e f fec t of in te rface s lip is determ ined by including a layer
of in te rface e lem ents be tw een the so il and the condu it , as d esc rib ed in
C hap te r 3 . Relating s t r e s s to d isp la c e m e n t, th e se e lem en ts are a s s ig n e d
a sh ea r s t i f fn e ss va lue of Ej = 170 kef = 2 .67 x 107 N /m 3 . This mod
u lus va lue is r e p re se n ta t iv e of the fr ic tion developed be tw een a rough
su rface and a g ranu lar b a c k f i l l .
The sh ea r strength of the in te r fa c e , r f , is p roportional to the
normal force acting on the in te r fa c e , dfn , and may be e x p re s se d as a
function of the ang le of w all f r ic tion , 8 , or rf = <5"n t a n S . For a rough
s u r f a c e , the la rg e s t angle of fric tion th a t ca n be developed is eq u iv a len t
. 46
to the so il ang le of fr ic t ion In th is a n a ly s is , the la rg e s t ang le d e
ve loped i s l e s s th an 10°, thus in d ica tin g th a t the l in e a r portion of the
s t r e s s - s t r a i n curve h a s not b een ex ceed ed and the assu m p tio n of e la s t i c
behav io r for the in te rface is v a l id .
F ig . 20 show s the e ffec t of full s l ip , as c a lc u la te d by the e l a s
t ic so lu tio n p roposed by Burns and Richard (1964) and p a r t ia l s lip (Case
25) com pared to the n o - s l ip cond it io n (Case 8). The moments for full
s l ip are g re a te r in m agnitude than for no s l ip by approx im ate ly 10 p e r
c e n t . For the a c tu a l c a s e as id e a l iz e d by the f in i te -e le m e n t a n a ly s is ,
th e re is only a s l ig h t in c re a se in the moments as com pared to the no
s lip c a s e .
F ig . 21 s im ila r ly shows the e f fe c t of full and p a r t ia l s lip on the
normal force in the co n d u it . For bo th , the normal force is in c re a se d at
the crown and the b a s e and d e c re a se d a t the sp ring line as com pared to
the n o - s l ip co n d it io n . For full s l ip , th e change is about 20 p e rc e n t ,
w h ile for p a r t ia l s lip the change is only 10 p e r c e n t .
Slip r e l e a s e s the s truc tu re and a llow s the so il to se t t le around
the co n d u it . Thus, the condu it is forced to carry more load in b en d in g ,
w hile the normal force d is t r ib u te s more ev en ly . N either the fu ll-s l ippage
or no-s lippage a ssu m p tio n is s e e n to offer any p a r t ic u la r advan tage in
e s tim a tin g the a c tu a l co n d it io n . The fu l l - s l ip con d it io n g iv es a c o n s e r
v a tiv e e s tim a te of bending m om ents. The n o -s l ip c o n d it io n g ives a c o n
se rv a t iv e e s tim a te of the maximum normal fo rces .
0.0075
d /t = 400.0050
0.0025
0,0
0.0025
-0 .0 0 5 0
-0 .0 0 7
Case 18, No Slippage Casq 25 \
-0 .0 0 7 5
-0 .0 0 5 0
-0 .0 0 2 5
0.0
0.0025
0.0050
Figure 20. Effect of In te rface Slip on Bending Moments
48
-0.60
d / t = 40
- 0 .4 0
-0 .3 0
- 0.20
- 0.10
0.0
Case 8 / No Slippage I Case \\ Full Slippage \
0.0
0.10
- 0.20
- 0 .3 0
-0 .4 0
Figure 21. Effect of In te rface Slip on Normal Forces
CHAPTER 5
CONCLUSIONS
Based on the r e s u l t s of th is s tu d y , the fo llow ing c o n c lu s io n s
are drawn:
1. M ethods cu rren tly u se d in the d e s ig n of buried condu its tend to
be qu ite c o n s e rv a t iv e .
2 . Solu tions b a se d on e la s t i c theory are ap p licab le to shallow
buried r ig id condu its for uniformly d is tr ib u ted loads over an
in fin ite e x te n t .
3 . E la s t ic so lu tio n b a se d on the assu m p tio n of a uniformly d i s
tr ib u ted load u n deres tim ate the bending moments and o v e re s t i
mate the normal fo rces produced by p a r t ia l su rface loads for
sha llow buried c o n d u i ts .
4 . The in f lu en ce line co n cep t c a n be ex tended to buried c o n d u i ts .
In fluence c h a r ts are deve loped which c a n be u t i l iz e d to d e te r
mine the bending moment and normal force a t any point along
the c ircum ference of the sha llow conduit for any d es ired lo a d
ing c o n d it io n . The procedure is d irec tly a p p lic a b le to the b a l
anced d e s ig n m ethod.
5. N either the fu ll-S lip or n o - s l ip a ssum ption h a s any p a r t icu la r
advan tage in approximating the true c a s e where some slip o c
curs a t the s o i l - s t ru c tu re in te r f a c e .
Recomm endations for Further Research
R esearch p u b lish ed s in ce the m id -1 9 6 0 's h a s provided c o n s id
erab le in s ig h t into the behav io r of both shallow and deep buried c o n d u i t s .
H ow ever, sev e ra l a s p e c ts of the s o i l - c o n d u i t system rem ain to be
s tu d ie d . N oncircu lar geom etr ies and m ultip le condu it in s ta l la t io n s have
rec e iv e d only lim ited a t te n t io n . In te rm ed ia te s t i f f n e s s e s , such as for
p la s t i c m a te r ia ls , need to be s tu d ied so th a t d es ig n s c a n be. ca rr ied o u t .
Experim ental d a ta from f u l l - s c a le or model t e s t s are bad ly needed in or
der th a t the re la t iv e m erits of d e s ig n p ra c t ic e s c a n be e v a lu a te d .
APPENDIX A
CHARTS AND DATA FOR MOMENT INFLUENCE COEFFICIENTS
The c o l le c t io n of cu rves and ta b le s p re sen ted in th is appendix
is an e x te n s io n of the in fluence line co n c e p t app lied to the so i l - c o n d u i t
s y s te m . Each curve re p re se n ts the e f fec t of a unit su rface load on the
bending moment a t a sp ec if ied poin t along the c ircum ference of the
buried co n d u it . C o l le c t iv e ly , the cu rves are referred to as the moment
in fluence c h a r t s . They c a n be u sed to determ ine the in f luence of any
su rface load , co n cen tra ted or d is t r ib u te d , sym m etrically or unsym m et-
r ic a l ly p l a c e d , on the bending moment in the s t ru c tu re .
It should be noted th a t th e se c h a r ts are ob ta ined using e la s t ic
m ateria l p roperties as fo llow s: for the cu lvert,E = 2 .5 x 10^ p s i =
17. 25 x 1C)6 k N / m 2 and for the so il medium, Es = 200 k s f = 9 . 58 x 10^
N/m2 and p s = 0 .4 .
51
52
0.0100
A----0.0
^ 0 = 270
- 0.0100
m
- 0.0200
O d / t = 20
A d / t = 40
—0.0300
-0 .0 4 0 0
Distance from Center, Diameters
Figure A - l . Moment In fluence Curves for 9 = 90° and 270°
53
0.0100
0.00500 = 105
—A-----— -A—. _ _
0.00 = 75
-0 .0 0 5 0
0.0100'
- 0.0200
- 0.0250
° d /t =-0 .0 3 0 0
A d /t =
-0 .0 3 5 0
Distance from Center, Diameters
Figure A-2. Moment Influence Curves for 9 = 75° and 105°
0.0150
0.0100
0 = 1200.0050
0.0
0 = 60
-0 .0 0 5 0
- 0.0100
-0 .0 1 5 0O d /t = 20
A d /t = 40- 0.0200
-0 .0 2 5 0
-0 .0 3 0 0
Distance from Center, Diameters
Figure A - 3 . Moment Inf luence Curves for 0 = 60° and 120°
55
0.0200
0.0100
Q = 135
A --A0.0050
0.0
Cm
-0 .0 0 5 0
- 0.0100
O d/ t =
-0 .0 1 5 0
A d /t =
- 0.0200
Distance from Center, Diameters
Figure A-4 . Moment Influence Curves for 9 = 45° and 135°
56
0.0250
O d /t = 20
d /t = 400.0200
0.0150
0.0100
Cm
0.0050
150
0.0
0 = 30
-0 .0 0 5 0
Distance from Center, Diameters
Figure A -5. Moment Influence Curves for 9 = 30° and 150°
57
0.0250
O d/ t = 20
A d /t = 40
0.0200
0.0150
0.0050m
0= 1650.0
A— — A —
-0 .0 0 5 0
Distance from Center, Diameters
Figure A-6. Moment Influence Curves for 9 = 15° and 165°
0.0300
0.0250
O d/ t = 20
A d /t = ̂00.0200
0.0150
m
0.0100
0.0050
0.0
0= 180
-0 .0 0 5 0
Distance from Center, Diameters
Figure A - 7. Moment Influence Curves for 9 = 0 ° and 180°
59
0.0250
O d / t =
A d /t =
0.0200
0.0150
0.0100
•m
0.0050
0.0
- 0.0050
Distance from Center, Diameters
Figure A-8. Moment Inf luence Curves for 9 = 195° and 34 5°
60
0.0200
A d /t = 40
0.01000 = 330 ^
A'
0.0" * &-J---- a -------- A-
<?= 210
- 0.0100
Distance from Center, Diameters
Figure A-9 . Moment Inf luence Curves for 0 = 210° and 330°
0.0120
0.0080
0 = 3150.0040
- -A---------A——A
0.0Cm
"A- --0.0040
0 = 225/O d /t = 20
-0.0080A d/1 = ̂0
- 0.0120 ~TDistance from Center, Diameters
Figure A - 10. Moment Influence Curves for 0 = 225° and 315°
61
0.0100
0.0050
0. 0J
Cm -0.0050
- 0.0100
-0.0150
- 0.0200
Figure A - l l . Moment Inf luence Curves for 0 = 240° and 300°
0.00500 = 285 v
0.0
< 0 = 255
-0.0050
- 0.0100
O d / t =
-0.0150A d / t =
- 0.0200
-0.0250
Distance from Center, Diameters
Figure A-12. Moment Inf luence Curves for 9 = 255° and 285°
Table A - l . Bending Moments for Various Loading C o n d i i t io n s , d / t = 40
Ac C r o s s - s e c t i o n a l a rea of condui t wal l per unit length
Bj H or izon ta l d im ension of ditch or condui t
b Length of d is t r ibu ted surface load
Ci Load co e f f ic ien t for so i l load
Cm/ Cm Bending moment coe f f ic ien ts
Cn» Cn Normal force co e f f ic ien ts
c t Load co e f f ic ien t for l ive load
D M ean d iam eter of condui t
d In te rna l d iameter of pipe
E Modulus of e l a s t i c i t y of condu i t mater ia l
El Modulus of e l a s t i c i t y of in te r face
Es Modulus of e l a s t i c i t y of so il medium
I Moment of ine r t ia of unit longi tud ina l s e c t io n of condui t wall
Ic Impact fac tor
•K Ratio of hor izon ta l to v e r t ic a l p re s su re in a l a te ra l ly confined so i l
k Radius of gyra t ion of condu i t wal l per unit length
Lf Load fac tor
t Length of s e c t io n of condui t on which load is computed
M Bending moment in condui t wal l per unit length• 79
Symbol Explanation
N Normal force in condui t wal l per unit length
P Concen tra ted surface load
pv Ver tical so i l p re ssu re
p D is t r ibu ted load in ten s i ty
Rf Failure ra t io
5 Ring com press ive s trength
Syp Yield s trength of condui t mater ia l
T Normal th rus t in condui t wal l
t Condui t w a l l th i c k n e s s
W c Vertical load appl ied to the condu i t due to so i l f i l l
Wt Vertical load app l ied to pipe due to live load
Ad Diametr ic change
Ay Vertica l change in d iameter
6 Angle of wal l f r ic t ion
0 Angle m easured from hor izon ta l in coun te rc lockw ise d i rec t ion
$ Soil f r ic t ion angle
7 Unit weight of so i l medium
€ Vertical Soil s t r a in
Ps Pols s o n 1 s rat io for so i l
<5n Normal force ac t ing on the in te r face
T f Shear s trength of the in terface
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