U. S. Department of Agriculture
Soil Conservation Service
,
Engineering Division
Technical Release No.
Design Unit
December, 1969
SINGLE CELL RECTANGULAR CONDUITS
CRITERIA AND PROCEDUF3S FOR S T R U C m DESIGN
42
PREFACE
This t e c h n i c a l r e l e a s e p r e s e n t s t h e c r i t e r i a and procedures e s t a b l i s h t
f o r t h e s t r u c t u r a l design of s i n g l e c e l l r e c t a n g u l a r conduit c r o s s sect i o n s . The c r i t e r i a f o r t h e s e designs has developed over a p e r i o d of
y e a r s and has been d i s c u s s e d a t v a r i o u s Design Engineers' meetings.
A p r e l i m i n a r y s e t of designs obtained as computer output, was p r e s e n t e d
and discussed a t t h e Ehgineering and Watershed Planning Unit-Washington
S t a f f Design C o n s u l t a t i o n i n Columbus, Ohio d u r i n g July 14-18, 1969.
Subsequently, a d r a f t o f t h e s u b j e c t t e c h n i c a l r e l e a s e dated August 14,
1969, was s e n t t o t h e Engineering and Watershed Planning Unit Design
Engineers f o r t h e i r review and comment.
This t e c h n i c a l r e l e a s e was prepared by M r . Edwin S. A l l i n g of t h e Design
Unit,.Design Branch a t & a t t s v i l l e , Maryland. He a l s o wrote t h e computer
program.
TECHNICAL RELEASE
NUMBEB 42
SINGLE CELL RECTANGULAR CONDUITS
CRITERIA AND PROCEDURES FOR STRUCTURAL DESIGN
Contents
Computer Designs
S e c t i o n Designed
b a d s S p e c i f i e d by User
Design Mode
Loads on Rectangular Conduits
External Loads
Conduits on e a r t h foundations
Conduits on rock foundations
I n t e r n a l Water Loads
Basic S e t s of Loads
Methods of Analysis f o r Indeterminate Moments
General Considerations
Dissymmetry of Sidewalls
Consideration t h a t tb 2 (tt + 1)
Consideration t h a t t s b > tst
Slope D e f l e c t i o n Equation
Slope D e f l e c t i o n Method
Design C r i t e r i a
Slab Thicknesses Required by Shear
Location of C r i t i c a l Section f o r Shear
Design of Top Slab
Design of Sidewall
Conduits founded on e a r t h , without i n t e r n a l water l o a d s
Conduits founded on rock, without i n t e r n a l water loads
Convergence procedure - conduits on rock
Limit of s h e a r c r i t e r i a
Conduits w i t h i n t e r n a l water loads
Summary shear design of s i d e w a l l t h i c k n e s s
Design of Bottom Slab
Summary of Shear Design
Page
Analyses of Corner Moments
Unit Load Analyses
Evaluation of Ekternal Load Corner Moments
Analyses f o r I n t e r n a l Water Loads
Sense of Corner Moments
S t e e l Required by Combined Bending Moment and Direct Force
Treatment of Bending Moment and D i r e c t Force
Usual case, d i r e c t f o r c e at l a r g e e c c e n t r i c i t y
Unusual case, d i r e c t compressive f o r c e a t small
eccentricity
Unusual case, d i r e c t t e n s i l e f o r c e a t small e c c e n t r i c i t y
Load Combinat ions P r o d u ~
i n g Maximum Required Areas
Maximum moment p l u s a s s o c i a t e d d i r e c t f o r c e
Maximum d i r e c t f o r c e p l u s a s s o c i a t e d moment
Procedure a t a S e c t i o n
Location 4 - negative bending moment w i t h l o a d i n g Bl-IC#3
Location 6 - n e g a t i v e bending moment w i t h loading Bl-IC#3
Location 1 - p o s i t i v e bending moment w i t h l o a d i n g B2-LC#l
Location 7 - p o s i t i v e bending moment w i t h l o a d i n g B3-LC#2
maximum d i r e c t compression
Bottom s l a b , s e c t i o n a t midspan
Bottom s l a b , s e c t i o n a t midspan - m a x i m u m d i r e c t t e n s i o n
-
31
31
31
32
33
34
34
36
37
37
38
39
40
41
42
Anchorage of P o s i t i v e S t e e l
P o s i t i v e S t e e l a t Face of Support
P o s i t i v e S t e e l a t Corner Diagonals
Top corner diagonal
Bottom corner diagonal
Spacing Required by F l e x u r a l Bond
Relation t o Determine Required Spacing
Load Combinat ions Producing Minimum Required Spacing
Procedure a t a S e c t i o n
Location 1 - w i t h l o a d i n g El-LC#1
Location 1 - w i t h loadings B2-IC#l and B 3 - ~ # 1
Location 3 - w i t h l o a d i n g B~-Lc#O
Location 4 - w i t h loadings B1-LC#1 and Bl-LC#2
Summary of Design
51
Appendix A
53
Appendix B
55
iii
Figures
Figure
Figure
1
2
Figure 3
Figure 4
Figure 5
Figure 6
Figure 7
Figure 8
Figure 9
Figure 1 0
Figure 11
Figure 1 2
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
13
14
15
16
17
18
19
20
21
22
23
24
Figure 25
Figure 26
Figure 27
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
28
29
30
31
32
33
34
35
Conduit c r o s s s e c t i o n and s t e e l l a y o u t .
Load combinations producing maximum v a l u e s a t
c r i t i c a l locations.
Treatment of i n t e r n a l water l o a d s .
Assumed v a r i a t i o n i n moment of i n e r t i a .
Conduit spans and member thicknesses.
Unequal s l a b t h i c k n e s s e s .
Sidewall b a t t e r .
12
Development of Slope Deflection Equation.
13
Designations f o r analyses.
Location of c r i t i c a l shear.
17
Loads f o r shear t h i c k n e s s e s design of t o p s l a b .
Load d i s t r i b u t i o n s and shear diagrams f o r same
t o t a l l a t e r a l load, WS.
Sidewall shear, conduits on rock foundation.
Partial s i d e w a l l shear diagram.
F o s s i b l e s i d e w a l l s h e a r diagrams w i t h Vex i ~ c l u d e d .
Loading f o r sidewall thickness, c o n t r o l from top.
Loading f o r s i d e w a l l thickness, c o n t r o l from bottom.
Loads f o r s h e a r t h i c k n e s s of bottom s l a b .
Loadings f o r u n i t l o a d analyses.
Fixed end moments f o r h y d r o s t a t i c loading.
Sense o f moments from Slope Deflection s o l u t i o n .
Range of v a l u e s of moment and d i r e c t f o r c e .
Moments i n t o p s l a b w i t h loading ~ 1 - E # 3 .
Components of d i r e c t f o r c e i n t o p and bottom s l a b s
w i t h l o a d i n g I31-IC#3.
Moment and d i r e c t f o r c e i n sidewall w i t h loading
B1-IC#3.
Moments i n t o p s l a b w i t h loading ~ 3 - ~ # l .
Components of d i r e c t f o r c e i n t o p and bottom s l a b s
w i t h loading B2-LC#l
Moments i n s i d e w a l l w i t h loading ~ 3 - E # 2 .
Moments i n bottom s l a b w i t h loading Bl-E#3.
Existence of t e n s i o n i n bottom s t e e l .
Top c o r n e r geometry.
R e s u l t a n t s on t o p corner diagonal.
Points of i n f l e c t i o n i n t o p s l a b w i t h l o a d i n g ~ 1 - ~ # 1 .
Points of i n f l e c t i o n w i t h loadings ~ 2 - E # 1and ~ 3 - E # 1 .
Summary flow c h a r t of design process.
Tables
Table 1
Table 2
Table 3
Table 4
Table 5
Table 6
Loadings f o r midspan moments.
Loadings f o r moments a t face of supports.
Additional loadings f o r f a c e moments.
Loadings f o r maximum d i r e c t f o r c e s .
Loadings f o r t e n s i o n i n bottom corner i n s i d e s t e e l .
Loadings f o r f l e x u r a l bond.
34
35
36
37
45
49
NOMENCLATlTRE
Not all nomenclature i s l i s t e d . Hopefully, t h e meaning of any u n l i s t e d
nomenclature may be a s c e r t a i n e d from t h a t given.
equivalent a r e a of r e i n f o r c i n g s t e e l ; t o t a l required s t e e l p e r
f o o t of width
gross a r e a of column
a r e a of r e i n f o r c i n g s t e e l
r a t i o used t o o b t a i n p r o p e r t i e s of non-prismatic members
i d e n t i f i c a t i o n f o r f i r s t b a s i c s e t of loads, t h o s e w i t h t h e
conduit empty
width of r e i n f o r c e d concrete member; r a t i o used t o o b t a i n
p r o p e r t i e s of unsymmetrical non-prismatic .members
o u t s i d e width of conduit
carry-over f a c t o r ; a parameter i n t h e equation f o r equivalent
a x i a l load; a c o e f f i c i e n t i n t h e expression f o r allowable
f l e x u r a l bond s t r e s s
carry-over f a c t o r f o r bottom s l a b
carry-over f a c t o r from end J t o end K
l o a d c o e f f i c i e n t f o r p o s i t i v e p r o j e c t i n g conduits
carry-over f a c t o r f o r s i d e w a l l
carry-over f a c t o r f o r t o p s l a b
nominal diameter of r e i n f o r c i n g bar; a parameter i n t h e equation
f o r equivalent a x i a l load
e f f e c t i v e depth o f r e i n f o r c e d concrete member
d
-
t/2
e f f e c t i v e depth f o r balanced working s t r e s s e s
e f f e c t i v e depth of sidewall a t bottom f a c e
e f f e c t i v e depth of sidewall a t t o p f a c e o r a t dst from t o p f a c e
u n i t dead weight c a r r i e d by bottom s l a b
u n i t dead weight of t o p s l a b
modulus of e l a s t i c i t y
e c c e n t r i c i t y of d i r e c t f o r c e measured from c e n t e r of s e c t i o n
c o e f f i c i e n t i n t h e cubic equation used t o l o c a t e t h e n e u t r a l
axis
compressive s t r e s s i n concrete
compressive s t r e n g t h of concrete
s t r e s s , i n reinforcing s t e e l
a paramet,-.r i n t h e equation f o r equivalent a x i a l load
v e r t i c a l distqmce from t o p of conduit t o t o p of embankment
\
- v e r t i c a l d i s t a n c e from p h r e a t i c s u r f a c e t o t o p of embankment
E
-
v e r t i c a l d i s t a n c e from t o p of conduit t o p h r e a t i c s u r f a c e
E
c l e a r h e i g h t of conduit
=
moment of i n e r t i a
i n t e r n a l water p r e s s u r e head measured from t h e bottom of t h e
top slab
r r a t i o used i n r e i n f o r c e d concrete r e l a t i o n s
s
c o e f f i c i e n t of l a t e r a l p r e s s u r e s a t r e s t
= ratio
used i n r e i n f o r c e d concrete r e l a t i o n s ; s t i f f n e s s c o e f f i c i e n t
E
span l e n g t h
5
c l e a r span
= load
-
E
-
combination number one
bottom s l a b span
span between p o i n t s of i n f l e c t i o n
= sidewall
E
-
span
t o p s l a b span
moment
design moment a t A
r a moment i n
ES-28
design moment a t t o p corner diagonal
r e x t e r n a l load corner moment a t B f o r L C # ~
= design
= design
moment a t f a c e of t h e t o p support of t h e s i d e w a l l
moment at f a c e of t h e support of t h e t o p s l a b
corner moment at B f o r u n i t load on bottom s l a b
= corner
moment a t B f o r u n i t l o a d on s i d e w a l l s
r corner moment a t B f o r u n i t l o a d on t o p s l a b
=
~g~~
-
corner moment at B f o r p r e s s u r e head l o a d i n g
r corner moment a t B f o r h y d r o s t a t i c s i d e w a l l l o a d i n g
moment a t J i n span JK
= fixed
M&
E
M
,:
end moment a t B f o r h y d r o s t a t i c s i d e w a l l loading
f i x e d end moment a t J i n span JK
f i x e d end moment f o r span L '
Ms
= equivalent
mF
= fixed
N
I
Nb
= direct
moment, moment about a x i s a t t h e t e n s i o n s t e e l
end moment c o e f f i c i e n t
d i r e c t f o r c e on a s e c t i o n
f o r c e i n bottom s l a b
NBk
Nc
Ns
Nt
n
P
p
pb
pbi
pg
phi
---
vii
d i r e c t f o r c e on t o p corner diagonal.
d i r e c t compressive f o r c e
d i r e c t force i n sidewall
z d i r e c t force i n top slab; d i r e c t t e n s i l e force
modular r a t i o ; an i n t e g e r ; number of b a r s p e r f o o t of width
a x i a l column l o a d
I
u n i t load
=
unit l o a d on bottom s l a b
u n i t load on bottom s l a b f o r LC&
gross s t e e l r a t i o
h o r i z o n t a l u n i t l o a d corresponding t o L C # ~
phd z u n i t load f o r p r e s s u r e head l o a d i n g
phy
ps
psf
=
maximum unit load f o r hydrostatic sidewall loading
u n i t l o a d on s i d e w a l l
pounds p e r square f o o t
psi z u n i t l o a d on s i d e w a l l f o r
LC#I
pt
z u n i t l o a d on t o p s l a b ; s t e e l r a t i o f o r temperature and shrinkage
R
-reaction
S
Sb
SJK
S,
St
s
t
tb
---E
stiffness
s t i f f n e s s of bottom s l a b
s t i f f h e s s a t end J i n span JK
s t i f f n e s s of s i d e w a l l
s t i f f n e s s of t o p s l a b
z spacing of r e i n f o r c i n g s t e e l
= thickness
t h i c k n e s s of bottom s l a b
average t h i c k n e s s of s i d e w a l l
t,
tsb = t h i c k n e s s of s i d e w a l l a t t h e bottom
tst = t h i c k n e s s of s i d e w a l l a t t h e t o p
tt
u
V
Vb
Vgt
Vex
Vf
Vp
= thickness
of t o p s l a b
r allowable flexural. bond s t r e s s i n concrete
--
-shear
-
shear a t bottom f a c e of s i d e w a l l
s h e a r a t face of support of t o p s l a b
e x t r a shear i n s i d e w a l l due t o l o a d i n g on t o p s l a b
= shear
a t t o p f a c e of s i d e w a l l
s h e a r at s e c t i o n of maximum p o s i t i v e moment; shear a t p o i n t of
i n f l e c t ion
viii
--
d i s t a n c e from c e n t e r of t o p support of sidewall t o s e c t i o n of
maximum po s it i v e moment
E
e c c e n t r i c i t y of d i r e c t f o r c e measured from t h e t e n s i o n s t e e l
E
buoyant u n i t weight of embankment
E
-
moist u n i t weight of embankment
=
s a t u r a t e d u n i t weight of embankment
shear at t o p f a c e of sidewall
r allowable shear s t r e s s i n concrete
-- t o t d
load on sidewall
c l e a r width of conduit
u n i t weight of water
=- r o t a t i o n a t support J
p r o j e c t i o n r a t i o f o r p o s i t i v e p r o j e c t i n g conduits
TECHNICAL RFILEASE
NllMBER 42
SINGLF: CELL RECTANGULAR CONDUITS
CRITERIA AND PROCEDURES FOR STRUCTURAL DESIGN
Computer Designs
The S o i l Conservation Service annually designs a number of cast-in-place
r e c t a n g u l a r conduits f o r use i n p r i n c i p a l and emergency spillways passi n g through e a r t h embankments. Thorough design of t h e s e r e c t a n g u l a r
conduit c r o s s s e c t i o n s by manual methods i s a time consuming process.
Because of t h e s t a t i c a l indeterminacy, t h e i n t e r a c t i o n of member t h i c k nesses w i t h moments makes d i r e c t design d i f f i c u l t . Af'ter an adequate s e t
of member t h i c k n e s s e s i s obtained, s t e e l requirements must be completely
determined.
A computer program w r i t t e n i n FORTRAN f o r IBM 360 equipment was developed
t o perform t h i s design t a s k . The program executes t h e complete s t r u c t u r a l
design of s i n g l e c e l l r e c t a n g u l a r conduit c r o s s s e c t i o n s given t h e c l e a r
height and width of t h e conduit, two load combinations, and t h e design
mode. The program was used i n t h e p r e p a r a t i o n o f Technical Release No.43
"Single C e l l Rectangular Conduits - Catalog of Standard Designs."
This t e c h n i c a l r e l e a s e documents t h e c r i t e r i a and procedures used i n t h e
computer program. The t e c h n i c a l r e l e a s e may be u s e f u l as a r e f e r e n c e and
a s a t r a i n i n g t o o l f o r s i m i l a r s t r u c t u r a l problems.
Section Designed
Figure 1 d e f i n e s t h e c r o s s s e c t i o n a l shape of t h e conduit and shows t h e
assumed s t e e l l a y o u t . No attempt i s made h e r e i n t o completely i d e n t i f y
t h e computer output l i s t i n g nor a l l of t h e a s s o c i a t e d nomenclature.
This i s done i n Technical Release No. 43..
The computer program determines t h e required t h i c k n e s s e s of t h e t o p and
bottom s l a b s and t h e thicknesses at t h e t o p and bottom of t h e sidewalls.
Then t h e computer o b t a i n s t h e minimum acceptable s t e e l a r e a s and maximum acceptable s t e e l spacings a t each of t h e f o u r t e e n l o c a t i o n s shown
i n Figure 1. I n t h e case of p o s i t i v e c e n t e r s t e e l , t h e spacings a c t u a l l y
computed a r e t h o s e r e q u i r e d a t t h e r e s p e c t i v e p o i n t s of i n f l e c t i o n . The
computer a l s o determines whether o r not any of t h e p o s i t i v e s t e e l r e q u i r e s
d e f i n i t e anchorage a t t h e corners of t h e conduit.
Loads S p e c i f i e d by User
The design of r e c t a n g u l a r conduit s e c t i o n s by t h e program i s independent
of t h e methods by which t h e user determines h i s e x t e r n a l l o a d s . The
user s p e c i f i e s , o r s e l e c t s , u n i t pressures i n two combinations of e x t e r n a l
l o a d s . These l o a d combinations a r e defined a s :
LC#l i s t h e l o a d combination having t h e maximum p o s s i b i e v e r t i c a l
u n i t l o a d combined w i t h t h e minimum h o r i z o n t a l u n i t load c o n s i s t e n t
w i t h t h a t v e r t i c a l u n i t load.
lX#2 i s t h e l o a d combination having t h e maximum p o s s i b l e
h o r i z o n t a l u n i t l o a d combined w i t h t h e minimum v e r t i c a l
u n i t load c o n s i s t e n t w i t h t h a t h o r i z o n t d u n i t load.
The computer design i s adequate f o r t h e s e two load combinations a s w e l l
as a number of o t h e r s constructed from them. See page 5 f o r a more
g e n e r a l discussion of l o a d s and load combinations f o r r e c t a n g u l a r conduit s.
H
TSTOP
ANCHOR*
-
Figure 1. Conduit c r o s s s e c t i o n and s t e e l l a y o u t .
Design Mode
The program designs conduit s e c t i o n s i n accordance w i t h t h e design mode.
A design mode c h a r a c t e r i z e s t h e conditions f o r which t h e conduit i s
designed. Four modes a r e e s t a b l i s h e d :
--
00
e a r t h foundation, no i n t e r n a l water l o a d
e a r t h foundation, w i t h i n t e r n a l water l o a d r 01
rock foundation, no i n t e r n a l water l o a d
10
rock foundation, w i t h i n t e r n a l water l o a d E 11
The type of foundation assumed in t h e design governs t h e number of
e x t e r n a l l o a d combinations t h a t a r e considered. I n t e r n a l water l o a d s
a r e included i n t h e design of p r e s s u r e conduits. Note t h a t t h e s e
conduits flow f u l l only i n t e r m i t t a n t l y . This r e p r e s e n t s a more severe
case t h a n e i t h e r a conduit t h a t never contains i n t e r n a l water o r a
conduit t h a t always contains i n t e r n a l water a t d e s i g n pressure.
Loads on Rectangular Conduits
Loads on conduits a r e of two types - e x t e r n a l loads and i n t e r n a l l o a d s .
For t h e purposes of t h i s t e c h n i c a l r e l e a s e ; e x t e r n a l l o a d s a r e due t o
embankment mat e r i d and contained water p l u s any surcharge water above
t h e embankment s u r f a c e and a l s o include t h e dead weight e f f e c t s of t h e
conduit, i n t e r n a l l o a d s a r e due t o water w i t h i n t h e conduit. The cond u i t must b e designed t o s a t i s f a c t o r i l y r e s i s t a number of p o s s i b l e
loading conditions which may occur over t h e l i f e of t h e s t r u c t u r e .
Ekternal Loads
Loads a r e assumed uniformly d i s t r i b u t e d on t h e t o p s l a b , sidewalls,
and bottom s l a b except f o r t h e bottom s l a b of conduits founded on
rock. Appropriate r e c o g n i t i o n i s made of t h e f a c t t h a t a c t u a l s i d e w a l l l o a d s a r e t r a p e z o i d a l and may vary from n e a r l y t r i a n g u l a r t o
n e a r l y uniform d i s t r i b u t i o n s .
Conduits on e a r t h foundations. - Load combinations producing t h e maximum
requirements f o r c o n c r e t e thicknesses, s t e e l a r e a s , and bond should be
used i n design. Q u a l i t a t i v e influence l i n e s m y b e employed t o h e l p
a s c e r t a i n t h e s e c r i t i c d l o a d combinations. The o r d i n a t e s of an i n fluence l i n e give t h e v a l u e of some f u n c t i o n a t a s p e c i f i c l o c a t i o n
caused by a u n i t l o a d anywhere on t h e s t r u c t u r e . Thus an i n f l u e n c e
l i n e may ( a ) suggest t h e o r e t i c a l loading p a t t e r n s , ( b ) v e r i f y t h e i n c l u s i o n of v a r i o u s l o a d combination;, and ( c ) i n d i c a t e t h e load combinat i o n producing t h e maximum value o f a f u n c t i o n .
The t h r e e influence l i n e s f o r MA, MB, and MC i n Appendix A ( s e e Figure 9
f o r l o c a t i o n of s e c t i o n s A, B, and C ) show t h a t t h r e e l o a d combinations
must be considered i n t h e design of r e c t a n g u l a r conduits on e a r t h .
These t h r e e load combinations a r e designated as LC#1, E # 2 , and E#3.
They a r e shown i n Figure 2. LC#l and IC#2 have been defined on pages 1
and 2. If pvl, ph,, pv2, and ph2 a r e t h e v e r t i c a l and h o r i z o n t a l ext e r n a l u n i t l o a d s e x c l u s i v e of conduit dead weight e f f e c t s , t h e n by
those definitions
L C # ~ has simultaneously high v e r t i c a l and l a t e r a l p r e s s u r e s . This load
combination i s not u s u a l l y of concern i n t h e design of c i r c u l a r o r o t h e r
curved conduits but it should be considered f o r r e c t a n g u l a r conduits
s i n c e it causes l a r g e negative corner moments. For t h i s work, IC#3 i s
taken as
Appendix B i n d i c a t e s one s i t u a t i o n i n which f o u r l o a d combinations m a y
e x i s t a t v a r i o u s times. In t h i s s t t u a t i o n , E#l would correspond t o
developed condition - moist, while L C # ~ would correspond t o i n i t i a l
condition - s a t u r a t e d .
Conduits on rock foundations. - Many suggestions a r e made concerning t h e
d i s t r i b u t i o n of l o a d s on t h e bottom s l a b s of conduits founded on rock.
One of t h e common proposals i s t h a t t h e load should be assumed t o vary
l i n e a r l y from zero at midspan t o a maximum value a t t h e supports. The
l i m i t of t h i s approach i s t o assume t h e bottom l o a d i s concentrated a t
t h e s i d e w a l l s and t h e bottom s l a b i t s e l f i s not loaded. Although t h i s
assumption may sound severe, it a c t u a l l y i s not much more so than t h e
assumption of l i n e a r i t y s i n c e moments i n a span a r e l a r g e l y produced by
t h a t p a r t of t h e l o a d neax t h e c e n t r a l p o r t i o n of t h e span. Thus t h r e e
a d d i t i o n a l load combinat ions e x i s t , t h e y a r e designated L C # ~ , L C # ~ ,
and LC#% and a r e shown i n Figure 2. The l a t t e r t h r e e load combinat i o n s a r e t h e same a s t h e former t h r e e except t h a t t h e y have no v e r t i c a l
p r e s s u r e on t h e bottom s l a b .
Figure 2.
Load combinations producing maximum values a t c r i t i c a l l o c a t i o n s .
When conduits a r e founded on rock, t h e conduits must be able t o s a f e l y
r e s i s t a l l s i x load combinations s i n c e t h e a c t u a l contact and i n t e r a c t i o n
w i t h t h e rock i s unknown. When conduits have l a r g e c l e a r spans and a r e
founded on rock, it may be more economical t o use a f i x e d ended frame
r a t h e r than t h e closed r e c t a n g u l a r shape.
I n t e r n a l Water b a d s
I n t e r n a l water l o a d s should be considered i n t h e design of r e c t a n g u l a r
p r e s s u r e conduits. Although i n t e r n a l water l o a d s w i l l seldom d r a s t i c a l l y
a f f e c t t h e design, t h e y w i l l always i n p r e s s u r e conduits cause e i t h e r
an i n c r e a s e i n concrete t h i c k n e s s , o r an i n c r e a s e i n t h e t h e o r e t i c a l
amount of t e n s i o n s t e e l r e q u i r e d a t some s e c t i o n , o r both. I n t e r n a l
water l o a d may be handled conveniently by t r e a t i n g it as two d i s t i n c t
loadings
( 1 ) Water t o t o p of conduit, f i l l i n g t h e conduit, b u t not under
pressure, and
( 2 ) Loading due t o p r e s s u r e head, \, above t h e bottom of t h e t o p
slab.
The f i r s t l o a d i n g amounts t o h y d r o s t a t i c l o a d i n g on t h e s i d e w a l l s only
s i n c e l o a d and r e a c t i o n on t h e bottom s l a b cancel each o t h e r .
Figure 3.
Sidewall
m d r o s t a t i c Loading
E'ressure Head
Loading
Phy = 7whc
phd = 7whw
Treatment of i n t e r n a l water l o a d s .
The second l o a d i n g c r e a t e s a uniform d i s t r i b u t i o n all around. The magn i t u d e of % may be r e l a t e d i n some way t o t h e h e i g h t of embanlanent
over t h e conduit, it i s taken t h e r e f o r e as a f u n c t i o n of pv2, o r
where pv2 i s i n psf and h, i s i n f t .
Sketches of t h e d e f l e c t e d shape of t h e conduit caused by t h e h y d r o s t a t i c
l o a d i n g show t h a t moments a t t h e midspans a r e p o s i t i v e ( t e n s i o n on t h e
i n s i d e of t h e conduit) f o r t h e t o p and bottom s l a b s and a r e n e g a t i v e
( t e n s i o n on o u t s i d e of t h e c o n d u i t ) f o r t h e s i d e w a l l s . However, deflectec
shapes caused by t h e p r e s s u r e head loading show t h a t midspan moments
in any span may t a k e on e i t h e r s i g n depending on t h e proportions of t h e
conduit. It i s apparent t h a t moments due t o i n t e r n a l water loads a r e
a d d i t i v e t o some of t h e moments of i n t e r e s t due t o t h e e x t e r n a l l o a d s .
Basic S e t s of Loads
Hence, when a conduit i s t o c a r r y i n t e r n a l water, t h e r e a r e t h r e e b a s i c
s e t s of l o a d s which should be considered t o determine t h e c r i t i c a l combin a t i o n of l o a d s f o r each f u n c t i o n i n v e s t i g a t e d . These b a s i c s e t s a r e
identified as
( ~ 1 External
)
l o a d s only, conduit empty
( ~ 2 )Ekternal l o a d s p l u s i n t e r n a l water l o a d s when conduit i s flowing
f u l l as an open channel, and
( ~ 3 Ekternal
)
l o a d s p l u s i n t e r n a l water l o a d s when conduit i s flowing
f u l l under p r e s s u r e head, hw.
Therefore, t h e problem i s t o determine which e x t e r n a l load combfnation
should be combined w i t h which i n t e r n a l load, i f any, t o produce t h e maximum e f f e c t f o r each f u n c t i o n . The i n t e r n a l l o a d i n g i s e i t h e r t h e s i d e w a l l
h y d r o s t a t i c loading o r t h e s i d e w a l l h y d r o s t a t i c l o a d i n g p l u s the pressure
head loading.
LC#@, shown i n Figure 2, i s included t o t a k e c a r e of t h o s e c a s e s f o r
which it i s d e s i r a b l e t o consider minimum e x t e r n a l l o a d s i n combination
w i t h i n t e r n a l water loads. For t h i s work, LC#@ is taken a s
Method of Analysis f o r Indeterminate Moments
Genc?ral Considerations
-
Slol.)e Deflection i s s e l e c t e d a s t h e method of a n a l y s i s f o r use i n t h i s
worl.r because t h e method y i e l d s e x p l i c i t s o l u t i o n s . Although Moment
Distx i b u t i o n i n some i n s t a n c e s y i e l d s exact s o l u t i o n s , i n general it
is Et method of successive approximations which i s summed a f t e r some
f i n ji t e number of cycles of d i s t r i b u t i o n a r e performed. Hence, where a
larg:e number of s o l u t i o n s a r e required, an e x p l i c i t method of a n a l y s i s
has an advantage over a method r e q u i r i n g i t e r a t i v e procedures.
I n Etccordance w i t h u s u a l theory, t h e a n a l y s i s assumes s t r a i g h t - l i n e s t r e s s
d i s ti r i b u t i o n on a c r o s s s e c t i o n . This assumption i s l e s s w e l l s a t i s f i e d
a s t;he thickness-to-span r a t i o i n c r e a s e s . For t h i c k members, t h e e f f e c t
of :;hearing s t r a i n s i s important.
Meml:Iers a r e assumed non-prismatic having a c o n s t a n t moment of i n e r t i a
wit1i i n t h e c l e a r span and having moments of i n e r t i a which approach i n f i n jity o u t s i d e t h e c l e a r span. This assumption, when used w i t h t h e s i d e w a l ls, introduces approximations discussed beiow. Figure 4 shows t h i s
v a r jl a t i o n i n moments of i n e r t i a .
Figure
4.
Assumed v a r i a t i o n i n moment of i n e r t i a .
is symmetry of Sidewalls
See Figure 5 which i d e n t i f i e s t h e nomenclature concerning spans and
member thicknesses. The conduit l a c k s symmetry of shape about any h o r i z o n t a l a x i s because o f two i n e q u a l i t i e s . These a r e
Both i n e q u a l i t i e s cause t h e sidewalls t o l a c k symmetry and t h e r e f o r e
cause t h e sidewall s t i f f n e s s e s , c a r r y over f a c t o r s and f i x e d end moments
t o be d i f f e r e n t a t t h e t o p and bottom of t h e sidewalls. F i n a l moments
and shears a r e a f f e c t e d t o some e x t e n t . Two questions a r i s e ; f i r s t ,
how s e r i o u s i s t h e e f f e c t and second, how should t h e s i d e w a l l be t r e a t e d ?
Figure 5.
Conduit spans and member t h i c k n e s s e s .
Some i n s i g h t i n t o t h e e f f e c t of t h i s l a c k of symmetry of shape on t h e
f i n a l moments may be obtained by considering t h e e f f e c t of varying t h e
s t i f f n e s s e s of e i t h e r t h e t o p s of t h e s i d e w a l l s o r t h e bottoms of t h e
sidewalls. Note t h a t both i n e q u a l i t i e s produce t h e same s o r t of r e sult s i n c e both r e p r e s e n t , a s compared t o e q u a l i t y s i t u a t i o n s , a dec r e a s e i n s t i f f n e s s of t h e t o p s and an increase i n s t i f f n e s s of t h e
bottoms of t h e s i d e w a l l s ,
Consideration t h a t tb 2 ( t t + l ) . - The bottom s l a b thickness w i l l exceed t h e t o p s l a b t h i c k n e s s because t h e bottom s l a b c a r r i e s a s l i g h t l y
l a r g e r dead l o a d t h a n t h e t o p s l a b and t h e bottom s l a b has 1 inch more
concrete cover over t h e o u t s i d e s t e e l t h a n does t h e t o p s l a b . The
importance of t h i s i n e q u a l i t y may be eva l u a t e d q u a l i t a t i v e l y by observing
Figures 29-31 on pages 152-154 i n
"continuous Frames of Reinforced concrete"
by Cross and Morgan where aL
=
I
-(t ),
2 t
1
bL = -(tb),and L = hc + $(tt+tb).These
2
c h a r t s can be used t o o b t a i n s t i f f n e s s e s
and c a r r y over f a c t o r s f o r various a and b
values. Values o f f i x e d end moments can
be computed by s t a t i c s knowing
gf= &( L ' ) ~
where Lf
=
L
- aL - bL.
Note t h a t tb w i l l not d i f f e r from tt by
Unequal s l a b more than a few inches a t most. It seems
thicknesses.
c l e a r t h a t t h e e f f e c t of t h i s i n e q u a l i t y can
not be g r e a t w i t h i n t h e e x i s t i n g l i m i t s of r e l a t i v e values of
and t t .
Thus it i s concluded, it i s not necessary t o make t h e refinement t h a t
tb # t t . To o b t a i n v a l u e s f o r design use
Figure 6.
ti
Consideration t h a t tSb> t,t. - The t h i c k n e s s o f t h e sidewall i n c r e a s e s
i n a downward d i r e c t i o n because of t h e n e c e s s i t y of providing b a t t e r on
t h e o u t s i d e s u r f a c e s of t h e w a l l s . The
importance of t h i s i n e q u a l i t y may be e v a l k- %t
I
uated q u a l i t a t i v e l y by observihg F i g m e 27
I
on page 150 of Cross and Morgan o r by obs e r v i n g Charts 2 and 3 on pages 258 and
259 i n " S t a t i c a l l y Indetermina ;e S t r u c t u r e s f '
by Anderson. The source of t h e s e c h a r t s i s
L
noted a s t h e Portland Cement Association,
t h e y can be found i n s e v e r a l t e x t s . The
parameters of i n t e r e s t can be obtained
I
from t h e r i g h t hand s i d e s of t h e p a i r s o f
Graphs 1, 2, and 3. Note t h a t t s b and tst
a r e not g r e a t l y d i f f e r e n t within t h e c l e a r
tsb
~i~~~~
7. sidewall batter. height of t h e s i d e w a l l s i n c e t h e b a t t e r i s
about 318 i n c h p e r f o o t . Thus it i s concluded, it i s not necessary t o make t h e refinement t h a t t s b f tst. To
o b t a i n v a l u e s f o r design assume %he t h i c k n e s s of t h e sidewall t s constant
and use
1
ts = -(t t+tsb)
2 s
4
- k-
Slope Deflection Equation
The Slope Deflection Equation used i n t h i s a n a l y s i s i s derived f o r t h e
assumption t h a t members a r e symmetrical about t h e i r c e n t e r l i n e s and a r e
subjected t o applied l o a d s and end r o t a t i o n s only. That is, members do
n o t undergo r e l a t i v e t r a n s l a t i o n of t h e i r ends. The usual Slope Deflect i o n s i g n convention i s followed; clockwise r o t a t i o n s a r e p o s i t i v e ,
clockwise j o i n t moments a r e p o s i t i v e .
Figure 8.
Development of Slope Deflect ion Equation.
however, due t o symmetry
SJK = SKJ = S and CJX = CKJ = C
then
MJK
=
Ma -
s(@J+ ceK)
where
i s t h e moment at J i n span JK
Ma
i s t h e moment a t K i n span JK
M~~
is t h e f i x e d end moment a t J i n span J K
M&
i s t h e f i x e d end moment a t K i n span J K
Sa
i s t h e s t i f f n e s s a t end J i n span J K
SKJ i s t h e s t i f f n e s s a t end K i n span JK
CJK i s t h e c a r r y over f a c t o r from end J t o end K
CKJ i s t h e c a r r y over f a c t o r from end K t o end J
OJ i s t h e end r o t a t i o n at support J
OK i s t h e end r o t a t i o n a t support K
Slope Deflection Method
Both t h e loading- on,. and t h e shape of, t h e s e s i n a e c e l l r e c t a n g u l a r
conduits a r e symmetrical about t h e v e r t i c a l c e n t e r l i n e of t h e s t r u c t u r e .
Hence no j o i n t can t r a n s l a t e . There are two unknown displacements, say
Figure 9. Designations f o r analyses.
t h e two r o t a t i o n s
eB and
OD s i n c e by symmetry
eI
=
-BBand OF
=
-OD.
Two j o i n t ( s t a t i c a l moment ) equations a r e required t o determine t h e s e
r o t a t i o n s . Say
EMB
=
MBI
EMD = MDB
+ Mm
=
0
+ MDF
=
0
By t h e Slope Deflection Equations:
F
MBI = MBI - st (1F
MBD = MBD
ct )eB
- sS(eB-I-cSeD)
and
These expressions f o r moment may be s u b s t i t u t e d i n t h e j o i n t equations
from which eB and eD may be determined. The values of eB and OD can
then be i n s e r t e d i n t h e Slope Deflection Equations y i e l d i n g f o r t h e
The various fixed end moments, s t i f f n e s s e s , and c a r r y over f a c t o r s i n volved may be computed from t h e r e l a t i o n s
F
Fixed end moment = M~ = mipiL;
i
Stiffness
=
si
=
ki E I ~ / L CC
~ kiti3/li
Carry over f a c t o r = Ci
The c o e f f i c i e n t s mi, ki, and Ci may be obtained from Table 2-5, page 2-9
of Technical Release No. 30, o r from t h e expressions
where ( a ) i s defined i n Figure
4
.
I n t h e a n a l y s i s , take f o r t o p s l a b , sidewall, and bottom s l a b r e s p e c t i v e l y
1
tt = tt
Lt =
+ tst
"tLt = 5 t s t
The r e s u l t a n t expressions f o r MBI and MDF may b e checked a g a i n s t known
s o l u t i o n s f o r s p e c i a l cases, f o r i n s t a n c e MK i n Tables 2-4 and 2-6 of
i n ES-28.
Technical Release No. 30, a l s o qband
Design C r i t e r i a
Materials
Class 4000 concrete and intermediate grade s t e e l a r e assumed.
Working S t r e s s Design
Design of s e c t i o n s is i n accordance w i t h working s t r e s s methods.
allowable s t r e s s e s i n p s i a r e
Extreme f i b e r s t r e s s i n f l e x u r e
Shear, ~ / b da t ( d ) from face of support
Flexural Bond
tension t o p bars
*
other tension bars
Steel
i n tension
i n compression, a x i a l l y loaded columns
Minimum Slab Thicknesses
Top s l a b and sidewalls
Bottom s l a b
The
f c = 1600
v =
70
u
=
u
=
3 . 4 m / ~
4 . 8 m / ~
f s = 20,000
f , = 16,000
1 0 inches
11 inches
Sidewall B a t t e r
Approximately 3/8 in. per foot, using whole inches f o r sidewall thicknesses
at t o p and bottom of t h e sidewall.
Temperature and Shrinkage S t e e l
The minimum s t e e l r a t i o s a r e
f o r outside faces
f o r inside faces
Slabs more than 32 inches t h i c k a r e taken a s 32 inches.
Web Reinforcement
The n e c e s s i t y of providing some type of s t i r r u p o r t i e i n t h e s l a b s
because of -bending a c t i o n i s avoided by
( 1 ) l i m i t i n g t h e shear s t r e s s , d s a measure o f diagonal tension,
s o t h a t web s t e e l i s not required, and
(,2) providing s u f f i c i e n t ef f e c t i v e depth of s e c t ions so t h a t compres
s i o n s t e e l i s not required f o r bending.
Cover f o r Reinforcement
S t e e l cover i s everywhere 2 inches except f o r o u t s i d e s t e e l i n t h e
bottom s l a b where cover i s 3 inches.
Spacing of Reinforcement
The maximum permissible spacing of any reinforcement i s 18 inches.
*In some cases shear may be c r i t i c a l a t t h e f a c e of t h e support, s e e
page 17.
-
Slab Thicknesses Required by Shear
Consideration of v a r i o u s influence l i n e s f o r s h e a r a t t h e f a c e s of
supports i n d i c a t e t h a t w i t h t h e exception of s h e a r i n t h e sidewalls
of conduits on rock, shear t h i c k n e s s design o f a p a r t i c u l a r s l a b may
be performed assuming t h e s l a b i s a simple span. This i s obviously
t r u e of t o p and bottom s l a b s due t o symmetry of l o a d and shape of t h e
s t r u c t u r e about t h e v e r t i c a l c e n t e r l i n e of t h e s t r u c t u r e . It i s an
approximation when applied t o sidewall design s i n c e t h e corner moments
a t t o p and bottom a r e i n general unequal. The moments a r e unequal because n e i t h e r t h e l o a d s nor t h e shape of t h e s t r u c t u r e i s symmetrical
aboat any h o r i z o n t a l l i n e .
Location of C r i t i c a l Sections f o r Shear
In ordinary beams and s l a b s t h e shear w i t h i n d i s t a n c e ( d ) from t h e f a c e
of a support i s l e s s c r i t i c a l than t h a t a t t h e d i s t a n c e ( d ) from t h e
support. The A C I Code t h e r e f o r e s t i p u l a t e s c r i t i c a l shear a s t h a t l o cated a d i s t a n c e ( d ) from t h e support. However, w i t h r e a c t i o n condit i o n s as i l l u s t r a t e d i n s k e t c h ( b ) of Figure 10, diagonal t e n s i o n
cracking can t a k e p l a c e a t t h e f a c e of t h e support. Computation of
c r i t i c a l shear a t d i s t a n c e ( d ) does not apply i n such cases, c r i t i c a l
shear i s l o c a t e d a t t h e f a c e of t h e support.
( a ) Shear c r i t i c a l a t
d from f a c e
( b ) Shear c r i t i c a l a t
face
Figure 10. Location of c r i t i c a l s h e a r .
When i n t e r n a l water l o a d s a r e included i n t h e design of r e c t a n g u l a r cond u i t s , r e q u i r e d shear t h i c k n e s s is sometimes c o n t r o l l e d by shear a t t h e
f a c e of t h e support.
Design of Top Slab
The t o p s l a b of a p r e s s u r e conduit i s s u b j e c t t o two p o s s i b l e c o n t r o l l i n g
loads a s shown i n Figure 11.
Figure 11. Loads f o r shear
t h i c k n e s s design of t o p s l a b .
Shear i n t h e t o p s l a b depends on t h e dead weight of t h e s l a b . The dead
weight depends on t h e t h i c k n e s s which i s i n i t i a l l y unknown. Hence a
convergence procedure is d e s i r a b l e . The l a r g e r of t h e two computed
thicknesses c o n t r o l s .
tt
= d
+
2.5
dwt = 150tt/12
from t h e f i r s t loading, s h e a r c r i t i c a l a t d i s t a n c e ( d )
1
v
$4.1
+
dvt )we -
=
(pvl + ~t )d/12
bd
from t h e second loading, s h e a r c r i t i c a l a t f a c e
If (phd
- Pv, - Qt) < 0,
t h e second shear does not e x i s t .
relations
pv, = v e r t i c a l u n i t l o a d of E#1, i n p s f
pv2 = v e r t i c a l u n i t l o a d of E#2, i n psf
= 62.4 &, i n p s f
Phd = 7w
dwt = dead weight of t o p slab, i n psf
in f t
Wc
= c l e a r width of conduit,
b
= 1 2 inches
d
= e f f e c t i v e depth, i n inches
tt
= thickness of t o p s l a b , i n inches
v
= allowable shear s t r e s s =
70 p s i
In t h e s e
Design of Sidewall
The s i d e w a l l design f o r required shear t h i c k n e s s is more complex t h a n
t h a t of t h e t o p s l a b . Several problems need consideration, among t h e s e
are
(1)
(2)
(3)
(4)
s i d e w a l l loads a r e t r a p e z o i d a l ,
s i d e w a l l t h i c k n e s s increases w i t h depth,
c r i t i c a l s i d e w a l l shear f o r conduits on rock i s usually a
f u n c t i o n of t h e maximum e x t e r n a l l o a d s on t h e t o p s l a b , and
i n t e r n a l water l o a d s f o r p r e s s u r e conduits may produce
c r i t i c a l sidewall s h e a r a t e i t h e r t h e f a c e of t h e t o p
support o r t h e f a c e of t h e bottom support.
Conduits founded on e a r t h , without i n t e r n a l water loads. - This case i s
t h e simplest t o design, As previously noted, t h e a c t u a l e x t e r n a l l a t e r a l
l o a d s a r e t r a p e z o i d a l ranging from n e a r l y t r i a n g u l a r f o r conduits near
t h e s u r f a c e t o n e a r l y uniform f o r conduits a t g r e a t depth. What i s needed
i s some way of conservatively approximating t h e shear diagram over those
p o r t i o n s of it t h a t may be c r i t i c a l .
0.WWS
7
a
t-
Figure 1 2 . Load d i s t r i b u t i o n s and shear diagrams
f o r same t o t a l l a t e r a l load, Ws.
Figure 12 shows t h a t t h e assumption of a uniform l o a d gives conservative
shear values between O.OL and O.3L from t h e t o p and between 0.125L and
0.5L from t h e bottom. Usually conservative s h e a r values between 0.3L
and 0 . 6 ~from t h e t o p may be obtained by using t h e i d e a l i z e d curve i n d i cated. Recalling t h a t shear s t r e s s i s c r i t i c a l a t ( d ) from t h e f a c e of
a support, t h e n shear between O.OL and 0.125L from t h e bottom w i l l never
be c r i t i c a l f o r all but perhaps those conduits having very l a r g e c l e a r
heights.and small l o a d s . Further, s i n c e s i d e w a l l thicknesses over t h e
lower h a l f of t h e sidewall a r e g r e a t e r than over t h e upper h a l f , s i d e w a l l
thickness w i l l not be c o n t r o l l e d by shear between 0.125L and 0 . j from
~
t h e bottom.
The following procedure f o r determining t h e s i d e w a l l t h i c k n e s s f o r t h e
conduits t h e r e f o r e a p p l i e s . Compute t h e required e f f e c t i v e depth on
t h e assumption of a uniformly loaded simple span loaded w i t h p
If
t h e c r i t i c a l s e c t i o n a t ( d ) from t h e f a c e of t h e t o p support, 42
1s more
t h a n 0.3L from t h e top, recompute t h e r e q u i r e d e f f e c t i v e depth on t h e
b a s i s of t h e i d e a l i z e d curve. With ( d ) known compute t h e s i d e w a l l
t h i c k n e s s at t h e top, tSt, and t h e s i d e w a l l t h i c k n e s s a t t h e bottom,
tsb, t a k i n g i n t o account t h e required b a t t e r . Equations f o r t h e s e
computations a r e given below i n combinat i o n w i t h requirements f o r cond u i t s on rock.
-
The influence
Conduits founded on rock, without i n t e r n a l water l o a d s .
l i n e f o r shear VB i n t h e sidewall, given i n Appendix A, shows t h a t
l o a d s on t h e t o p s l a b produce t h e same kind of shear a t t h e t o p of t h e
s i d e w a l l a s does t h e s i d e w a l l loading, however l o a d s on t h e bottom s l a b
produce t h e opposite kind o f shear. When conduits a r e on rock, l o a d s
on t h e t o p s l a b a r e p o s s i b l e when t h e bottom s l a b i s not loaded. Thus
LC* produces t h e maximum s h e = - a t t h e t o p of t h e s i d e w a l l . This conc l u s i o n i s v e r i f i e d by Figure13which shows t h e d e f l e c t e d shape and
sense of end moments on t h e s i d e w a l l due t o l o a d i n g on t h e t o p s l a b .
These end moments produce an e x t r a shear which adds t o t h a t due t o t h e
s i d e w a l l loads.
load on t o p s l a b
1
sidewall
load
Vex
Vex
Figure 13. Sidewall shear, conduits on rock foundation.
This e x t r a shear, Vex, can be added t o t h e s h e a r diagram f o r s i d e w a l l
l o a d s and t h e procedures given above f o r conduits on e a r t h foundations
may be followed. Figure14shows a p o r t i o n of t h i s combined shear d i a gram r o t a t e d 90 degrees.
Figure
14. P a r t i a l sidewall shear diagram.
-
Assuming t h e c r i t i c a l s e c t i o n f o r shear i s not more t h a n 0.3L from t h e
t o p ( l e f t end), t h e n
1
d 112
2 Ph2
+ Vex - Ph2 s t
v =
bdst
However, i f t h e c r i t i c a l s e c t i o n is more than 0.3L from t h e top, t h a t
is, i f
then g e t t h e new shear at t h e f a c e of t h e supportjwhich i s
7
1
d i v i d e t h i s by (-h + veX/ph2 + 0 . 1 ~ )t o o b t a i n t h e new e f f e c t i v e unit
2 c
load. The required e f f e c t i v e depth becomes
+ Vex
Vf
dst =
840 + (,
Vf
+
Vex
1/12
where Vex = e x t r a s h e a r from t o p s l a b load, i n l b s
Vf
= shear a t f a c e from i d e a l i z e d curve, i n l b s
+ tt/12,
L
= he
hc
=
c l e a r h e i g h t of conduit, i n f t
dst
=
e f f e c t i v e depth of sidewall a t dst from top, i n inches
in f t
These expressions may be used f o r conduits on e a r t h , i n which case Vex
i s equal t o zero, o r f o r conduits on rock.
Convergence procedure - conduits on rock. - When conduits a r e on rock,
i s i n i t i a l l y unknown s i n c e it depends on t h e
t h e c o r r e c t vdlue f o r ,V
sidewall end moments duzAto loading on t h e t o p s l a b . These end moments
a r e obtained from indeterminate analyses which a r e functions of t h e
thicknesses and span l e n g t h s of all members, q u a n t i t i e s which a r e i n
t h e process of being determined. Thus a convergence approach t o design
i s required here. The initial value of Vex i s s e t a t zero. Then t h e
remaining shear design i s completed. Indeterminate analyses a r e p e r formed and t h e end moments due t o t o p loads obtained. If M B and
~ ~ Mkt,
s e e page 27, a r e t h e moments i n f't -lbs a t B and D due t o a uniform l o a d
of u n i t y on t h e t o p s l a b , then Vex i s given by
This furnishes a new value f o r Vex which may be used i n t h e expressions
f o r e f f e c t i v e depth, d s t . Then t h e remaining she= design i s completed
and t h e cycle i s repeated a s many times a s is necessary. The process
is ended when t h e required sidewall thickness is unchanged from one
c y c l e t o t h e next.
Limit of s h e w c r i t e r i a . - A f u r t h e r complication may e x i s t when t h e
sidewall shear includes Vex > 0 . According t o t h e ACI Code, c r i t i c a l
Figure 15. R s s i b l e sidewall shear diagrams with Vex included.
s e c t i o n s f o r shear a r e l o c a t e d ( d ) from t h e f a c e of t h e support. As
i n d i c a t e d i n F i g u r e 1 5 i t i s p o s s i b l e f o r t h e computed value of dst t o
l o c a t e t h e c r i t i c a l s e c t i o n o u t s i d e of t h e span, he. If t h i s occurs,
t h e shear c r i t e r i a i s held i n v a l i d and t h e design i s terminated.
Conduits with i n t e r n a l water loads. - The sidewall
conduits may be determined by i n t e r n a l water l o a d s
mum e x t e r n a l load combinations. Further it i s not
whether shear at t h e t o p o r shear a t t h e bottom of
control.
thickness of p r e s s u r e
r a t h e r than by maxiknown beforehand
t h e sidewall w i l l
A t t h e top, maximum shear due t o i n t e r n a l water load occurs, i f it e x i s t s ,
when t h e e x t e r n a l loading on t h e sidewall i s a minimum. This can b e
taken as IC#l. A t t h e bottom, maximum shear due t o i n t e r n a l water l o a d
occurs, if it e x i s t s , when t h e e x t e r n a l loading on t h e sidewall i s a
minimum, t h e externa3 loading on t h e t o p s l a b i s a maximum, and t h e ext e r n a l loading on t h e bottom s l a b i s a minimum. This i s LC# f o r cond u i t s on rock and can be taken a s L C #f~o r conduits on e a r t h . Note
t h a t Vex i n t h i s case adds t o t h e shear a t t h e bottom.
For these computations,
since phi is of secondary
importance here, it is
treated strictly as a
uniform load with no
adjustment made for a
trapezoidal distribution.
Figure 16. Loading for sidewall thickness,
control from top.
and the shear stress at the top face is
givingdst =
IT,-840.
where
Vt
= shear at top face, in lbs
dst = effective depth of sidewall at top face, in inches
If Vt < 0 the desired shear does not exist.
Vex
The loading diagram when
thickness is controlled
from the bottom is similar
to that when control is
from the top except that
Vex due to top slab loads
must be added as shown in
Figure 17.
-
z
'
e
x
Vb
~ h y
~ h d
Figure 17. Loading for sidewall thickness,
control from bottom.
and t h e shear s t r e s s a t t h e bottom f a c e i s
(vb + veX)/bdsb
v
=
Vb
= shear a t bottom face, i n l b s
where
dsb
If Vb
=
e f f e c t i v e depth of sidewall a t bottom face, i n inches
< 0 the
d e s i r e d shear can not c o n t r o l .
S ~ m m a q , shear design of sidewall thickness. - In t h e general case t h e
thickness of t h e sidewall may be governed a t t h e t o p by e x t e r n a l loads,
at t h e t o p with i n t e r n a l water loads, o r at t h e bottom with i n t e r n a l
water loads. In t h e f i r s t and last instances e x t r a shear must b e
added f o r conduits on rock. The a c t u a l thickness required at t h e t o p
of t h e sidewall, t t, and t h e corresponding a c t u a l thickness required
a t t h e bottom of tRe sidewall, tsb, i s s e l e c t e d from t h e most c r i t i c a l
s e t of thicknesses determined from t h e two computed values of dst and
t h e one value of dsbo
Design of Bottom Slab
The bottom s l a b of a pressure conduit i s s u b j e c t t o two p o s s i b l e cont r o l l i n g loads a s shown i n Figure 18
Figure 18. Loads f o r shear thickness of bottom s l a b .
The bottom s l a b may c a r e t h e dead weight of t h e t o p s l a b and t h e s i d e w a l l s . Since tb i s as y e t unknown, t a k e th = tt + 1, SO
For t h e f i r s t loading, shear c r i t i c a l a t d i s t a n c e ( d )
L
bb = d
+ 3.5
For t h e second loading, if t h e conduit i s on e a r t h pup = pv2
t h e conduit is on rock pup = 0, t h e r e f o r e
tb = d
+ dwb,
if
+ 2.5
and, with shear c r i t i c a l at f a c e
.
where
dwb = dead weight on bottom slab, i n psf
tb = thickness of bottom s l a b , i n inches
d
I f (phd
= e f f e c t i v e depth, i n inches
- Pup) < 0,
t h e desired shear does not e x i s t .
The l a r g e r of t h e two computed thicknesses controls.
Summary of Sheas Design
The s e t of thicknesses tt, tst, tsb, and tb J u s t obtained represents
t h e minimum possible s l a b thicknesses consistent w i t h t h e selected
c r i t e r i a f o r shear as a measure of diagonal tension. This s e t i s t h e
f i r s t trial s e t of thicknesses f o r which indeterminate analyses can be
made and subsequent determinations of required s t e e l areas can be ~ b tained. It may be found necessary t o increase one o r more of t h e s e
thicknesses i n subsequent stages of t h e design.
AnaZyses of Corner Moments
With a l l conduit dimensions known, indeterminate analyses f o r moments
can be performed. Before t h i s i s done, it i s d e s i r a b l e t h a t all u n i t
loads on top, sides, and bottom s l a b s be evaluated f o r t h e seven load
c o ~ b i n a . ~ i _ opreviously
ns
established. these are
f o r which,
Unit Load Analyses
Corner moments f o r seven e x t e r n a l load combinations and two i n t e r n a l
water loadings a r e required. Rather than perform t h e s e nine analyses,
t h r e e u n i t l o a d analyses, with t h e magnitudes of t h e u n i t load taken
a s unity, may be used t o o b t a i n corn&r moments f o r a l l but one of t h e
i n t e r n a l - w a t e r loadings. The analyses needed a r e shown i n Figure 19
Figure 19. Loadings f o r u n i t load analyses.
The corner moments at B and D f o r each of t h e t h r e e u n i t loadings may
be obtained from t h e general s o l u t i o n s f o r MBI and Mw previously d e rived. These moments i n f t - l b s p e r l b of loading, a r e designated
M B ~ ~MDut,
,
MBus, Mks,
M B ~ ~and
, MDub r e s p e c t i v e l y . immediately
a f t e r MBut m d MDut a r e obtained, a revised estimate of Vex may be
computed when necessary a s discussed on page
21.
Evaluation of External Load Corner Moments
w be obtained as t h e sums
Corner moments f o r a given load combination m
of t h e respective unit load moments times t h e corresponding a c t u a l ext e r n a l loads. Thus, adopting t h e nomenclature MBi 7 MBIi and MDi =
where MB and MD t a k e on t h e same Slope Deflection slgns a s MBI and MDF
t h e moments i n f t - l b s a r e
Referring t o t h e discussions on pages 1 0 and 11, note t h a t t h e assumptions
r e l a t i n g t o sidewalls
a s w e l l a s t h e assumption
(3) sidewall loading i s uniform i n s t e a d of trapezoidal
may a l l cause t h e corner moments a t B t o be computed too high and t h e
corner moments a t D t o be computed too low. That is, t h e e r r o r s may be
a d d i t i v e and on the s a f e o r unsafe s i d e depending on t h e moment o r o t h e r
function under consideration. Therefore t o p a r t i a l l y account f o r t h e
e f f e c t s of t h e t h r e e assumptions, a second s e t of corner moments i s
due t o s i d e loads a r e a r b i t r a r i l y adjusted
This second s e t of e x t e r n a l load corner moments i s used only i n those
instances when it i s conservative t o t a k e a lower moment at B o r a
higher moment a t D.
Analyses f o r I n t e r n a l Water Loads
Moments due t o pressure head loading may be computed using t h e u n i t
load analyses
The minus signs a r e used s i n c e phd i s an outward a c t i n g load.
Moments due t o h y d r o s t a t i c sidewall loading may be computed from t h e
generd. solutions f o r MBI and MDF a f t e r t h e f i x e d end moments a r e obt ained
.
Figure 20. Fixed end moments f o r h y d r o s t a t i c loading.
From t h e f i g u r e , t h e f i x e d end moments i n f t - l b s a r e
where hc is i n f't and tt and tb a r e i n inches.
Sense o f Corner Moments
Confusion sometimes e x i s t s as t o t h e sense of various moments. The
Slope Deflection s i g n convention and t h e bending moment s i g n convent i o n a r e independent of one another. The Slope Deflection convention
i s concerned w i t h t h e sense of moments a c t i n g on j o i n t s ( o r on t h e ends
of members ). The bending moment convention i s concerned w i t h whether
a moment causes t e n s i o n on t h e i n s i d e o r o u t s i d e (sometimes t o p o r
bottom) of a member. Figure 2 1 shows t h e senses of various moments
when t h e s o l u t i o n s f o r MBi and M D i a r e p o s i t i v e i n accordance w i t h
t h e adopted Slope Deflection convention.
&
senses of moments a t B
when MBi i s p o s i t i v e
I
u
senses of moments a t D
when MDi i s p o s i t i v e
Figure 21. Sense of moments from Slope Deflection s o l u t i o n .
S t e e l Required by Combined Bending Moment and D i r e c t Force
A f t e r s l a b t h i c k n e s s e s a r e known and corner moments f o r e x t e r n a l and
i n t e r n a l l o a d s have been determined, t h e next s t e p i s t o compute s t e e l
a r e a s r e q u i r e d a t t h e f o u r t e e n l o c a t i o n s given i n Figure 1
In order
t o do t h i s , t h e r e must be a procedure by which r e q u i r e d a r e a s m y b e
obtained f o r any a c c e p t a b l e combination of moment and d i r e c t f o r c e . The
load combination o r combinations t h a t may produce maximum r e q u i r e d a r e a
f o r each design mode must a l s o be recognized.
.
Treatment of Bending Moment and D i r e c t Force
The procedure f o r determining r e q u i r e d a r e a s , t o be general, must handle
all c a s e s of M I 0 and N e i t h e r compression, tension, o r zero. This i s
i n d i c a t e d s c h e m a t i c d y i n Figure 22 which shows t h r e e s t a t i c a l l y
equivalent f o r c e systems.
Figure 22. Range of values of moment and d i r e c t f o r c e .
-
For t h e m a j o r i t y of
Usual case, d i r e c t f o r c e a t l a r g e e c c e n t r i c i t y .
combinations of moment and d i r e c t force, t h e procedure s p e c i f i e d i n
National Ehgineering Handbook Section 6, s u b s e c t i o n 4.2.2(c ) a p p l i e s
and is followed. Thus, t a k i n g compressive d i r e c t f o r c e and clockwise
22
as p o s i t i v e
moment in Figure
M,
=
M
+ Nd1'/12
and
from which k may be determined from t h e cubic equation
- Aka +
3
where
k2
+ kF
= F
then
and
where M
M,
= moment about mid-depth of s e c t i o n i n f t - l b s
= moment about t e n s i o n s t e e l , i n f t - l b s
N = d i r e c t force, i n l b s
dl' = d
t/2, i n inches
d = e f f e c t i v e depth, i n inches
fs = steel stress, i n psi
-
A = e q u l v a e n t area, i n sq. inches p e r f t o f width
A, = area, i n sq. inches p e r f t of width
A s s t a t e d i n t h e c r i t e r i a on page 16, s o l u t i o n s r e q u i r i n g t h e use of
compression s t e e l i n bending a r e not acceptable s i n c e t h i s i n t u r n
r e q u i r e s t h e use of s t i r r u p s o r t i e s around t h e compression s t e e l .
Compression s t e e l i s n o t r e q u i r e d i f t h e actual. e f f e c t i v e depth a t a
s e c t i o n is not l e s s t h a n t h e e f f e c t i v e depth required f o r balanced working s t r e s s e s . Thus, i f
<dbal
where
and
k = kbal =
j = jb,,
=
0.3902
0.8699
t h e e f f e c t i v e depth i s i n s u f f i c i e n t . If t h i s condition has not already
occurred more than an a x b r i t r a r i l y s e l e c t e d number of times ( c u r r e n t l y 9)
t h e thickness of t h e p a r t i c u l a r s e c t i o n and s l a b i s incremented so t h a t
New u n i t loads incorporating corrected dead load e f f e c t s a r e
d 2 dbal.
established, new indeterminate analyses a r e obtained, and new designs
a r e attempted.
Unusual case, d i r e c t compressive f o r c e at small e c c e n t r i c i t y . - In a
few cases, where t h e d i r e c t f o r c e i s compression and t h e moment i s
r e l a t i v e l y small, t h e normal t e n s ion c o n t r o l theory f o r obtaining required
a r e a does not apply. This occurs when
These cases a r e considered i n t h e compression c o n t r o l range i n accordance
w i t h A C I 318-63 s e c t i o n 1407(b). Equation (14-9) may be used t o d e r i v e
an equation f o r an equivalent axial. load from which t h e required s t e e l
area may be obtained. The equation f o r equivalent a x i a l load may be
written
See M I Publicat ion SP-3; from Table 26, f o r f s = 16000 p s i and
small pg values, t a k e CD = 4.0, from Table 23 f o r f c t = 4000 p s i ,
f s = 16000 p s i and sma.l.1 pg values, take G = 0.64.
For design,
P i s taken a s t h e l a r g e r of
From ACI Code s e c t i o n s 1402 and 1403
P
=
0.85~g(0.25f,'
+
f s pg)
o r t h e t o t a l required s t e e l a r e a i s
1
A =
id-
- 12000t)
Assuming t h e opposite s i d e of t h e s e c t i o n has a t l e a s t p t = 0.001,
i.e., 0.001bt = 0.012t sq. in. of s t e e l , t h e s t e e l a r e a required on
t h e s i d e under consideration i s
i f t exceeds 32 inches.
In t h e s e r e l a t i o n s
e = 1 2 M / ~ = e c c e n t r i c i t y of d i r e c t compressive force, i n inches
P = equivalent a x i a l load, i n l b s
f C t= 4000 p s i
A = gross a r e a of column, i n sq. inches
pg = gross s t e e l r a t i o
g
Unusual case, d i r e c t t e n s i l e f o r c e a t small e c c e n t r i c i t y . - It i s
p o s s i b l e f o r t h e d i r e c t f o r c e i n pressure conduits t o be tension. When
t h e d i r e c t f o r c e is t e n s i o n and t h e bending moment is r e l a t i v e l y small,
t h e normal t e n s i o n c o n t r o l theory does not hold. This occurs when
% S O
These cases a r e considered i n t h e d i r e c t t e n s i o n range.
quired s t e e l a r e a i s
The t o t a l r e -
Again, assuming t h e opposite s i d e of t h e s e c t i o n has at l e a s t pt = 0.001,
i.e., 0.001bt = 0.012t sq. inches of s t e e l , t h e s t e e l a r e a requlred on
t h e s i d e under consideration i s
A
=---
s
20000 0.384
i f t exceeds 32 inches.
Load Combinat ions Producing Maximum.Required Areas
Selection of t h e load combination o r combinations t h a t may produce maximum required s t e e l a r e a a t a given l o c a t i o n may be accomplished by experience, i n t u i t i o n , a n a l y s i s , o r combinations of t h e s e . It can not
always be ascertained beforehand whether a p a r t i c u l a r f u n c t i o n e x i s t s .
However, i f t h e f u n c t i o n e x i s t s , t h e load combination o r combinations
t h a t w i l l produce t h e f u n c t i o n can be determined.
Maximum moment plus a s s o c i a t e d d i r e c t f o r c e . - Table 1 l i s t s t h e b a s i c
s e t o r s e t s of loads and t h e e x t e r n a l l o a d combination f o r each design
mode which may give maximum required s t e e l a r e a s a t t h e midspans of
t h e t o p s l a b , sidewall, and bottom s l a b . These r e s u l t s were determined
by considering influence l i n e s of t h e types shown f o r MA and MC i n
Appendix A and by considering d e f l e c t e d shapes due t o i n t e r n a l wa?ier
loads.
.Table
Location
(see
Figure 1)
1
1
.
Loadings f o r midspan moments
Earth Foundation
No
Internal
Water
Rl-Lc#1
I
With
Water
Rock Foundation
No
Internal
Water
With
internal^
-- B2Water
-LC#1
or
B1-Lc#l
--
--
-A
~3 -E#l
Table 2 l i s t s t h e b a s i c s e t of l o a d s and t h e e x t e r n a l load combinat i o n s f o r each design mode which m y give maximum required s t e e l a r e a s
a t t h e f a c e s of supports. These r e s u l t s were determined by considering
influence l i n e s of t h e type shown f o r MB i n Appendix A and by considering deflected shapes due t o i n t e r n a l water loads. Alternate load combinations a r e given i n some cases. The influence l i n e s f o r moment
c o r r e c t l y i n d i c a t e t h e load combination producing maximum moment.
However, t h e a l t e r n a t e load combination, although producing a smaller
moment, m a y r e q u i r e a g r e a t e r s t e e l a r e a because of t h e smaller d i r e c t
f o r c e involved.
Table 2.
I
Location
(see
Figure 1)
Loadings f o r moments a t f a c e of supports.
Earth FC n d a t ion
From
Influence
Line
No
With
I n t e r n a l Internal
Water
Water
3
--
Rock Fou .dation
Alternate
With
Influence
With
1
Alternate
With
B~-LC#O
The c o ~ c l u s i o n sreached i n Table 2 neglect t h e f a c t t h a t t h e maximum
moment a t a f a c e of a support may be caused by a load combinat ion o t h e r
than t h e one producing t h e maximG corner moment. Thus Table 2 i s not
s u f f i c i e n t by i t s e l f t o always determine maximum required s t e e l a r e a s
a t f a c e s of supports. Table 3 supplements t h e previous t a b l e . It l i s t s
a d d i t i o n a l b a s i c s e t s of loads and e x t e r n a l load combinations which may
give maximum a r e a s a t f a c e s of supports. These a d d i t i o n a l r e s u l t s were
determined by considering influence l i n e s of t h e type shown i n Appendix A
f o r M a t f a c e of support i n t h e t o p slab.
Table 3
Locat ion
(see
Figure 1)
.
Additional loadings f o r f a c e moments
Earth Foundat ion
Rock Foundat ion
No
Internal
Water
With
Internd
Water
No
Internal
Water
W%th
Internal
Water
n-LC#l
B2 -IC#l
or
B3-LC#l
Bl-IC#1
B~-I;c#~
or
B3-IC#l
Maximum d i r e c t f o r c e plus a s s o c i a t e d moment. - Occasionally t h e maximum
required s t e e l a r e a i s governed by t h e maximum d i r e c t f o r c e p l u s a s s o c i a t e d moment r a t h e r t h a n by maximum moment p l u s a s s o c i a t e d d i r e c t f o r c e .
Maximum compressive d i r e c t f o r c e s occur with t h e conduit empty. Maximum
t e n s i l e d i r e c t f o r c e s i n t h e t o p s l a b and sidewalls occur, i f they occur,
with t h e conduit flowing f u l l under pressure. The bottom slab, i f t h e
conduit i s on rock, may c a r r y d i r e c t t e n s i o n when t h e conduit i s empty.
Table 4 l i s t s t h e b a s i c s e t of loads and e x t e r n a l load combinations producing t h e s e maximum d i r e c t f o r c e s .
-
/
Table
I
4
Member
lfop Slab
.
I
I
Loadings f o r maximum d i r e c t f o r c e s
Compression
I
Tens ion
r
On e a r t h I31-W#2
On rock EU-LC#~
Sidewalls
~1- ~ # 1
Bottom Slab
B~-I;c#~
B~-LC#
~3-LC#
On e a r t h B~-Lc#O
On rock B ~ - L c # ~
On rock 31-LC#%
Procedure a t a Section
With t h e corner moments known f o r all e x t e r n d and i n t e r n a l loads, t h e
design moment and d i r e c t f o r c e a t a given s e c t i o n may be computed by
s t a t i c s f o r each of t h e b a s i c s e t s of l o a d s and e x t e r n a l load combinat i o n s t h a t must be considered. This i s i l l u s t r a t e d below f o r s i x
l o c a t i o n s and loadings. With t h e moment and d i r e c t f o r c e a t a s e c t i o n
known, t h e required s t e e l a r e a of i n t e r e s t can be determined as des c r i b e d beginning on page 31.
Location 4 - negative bending moment w i t h loading ~ 1 - ~ # 3- .Care must
be exercised t o ensure t h a t s t a t i c a l moment, bending moment, and Slope
Deflection moment s i g n s a r e not confused and t o ensure t h a t u n i t s a r e
accounted f o r c o r r e c t l y . To o b t a i n t h e negative bending moment a t t h e
f a c e of t h e support of t h e t o p d a b f o r t h e indicated loading, observe
Figure 23 and note t h e Slope Deflection s i g n of ?4~3i s positive. Then
from s t a t i c s and symmetry, l e t t i n g MBt be t h e d e s i r e d moment
Figure 23. Moments i n t o p s l a b with loading B ~ - L C # ~ .
I f , as w r i t t e n , M
< 0 t h e desiked moment does n o t e x i s t . The
a s s o c i a t e d d i r e c t q o r c e i n t h e t o p s l a b may be obtained as t h e
sum of t h r e e components, s e e Figure 24
Figure 24. Components of d i r e c t f o r c e i n t o p and bottom s l a b s
with loading Bl-m#3.
Thus, noting t h e component due t o corner moments i s w r i t t e n f o r p o s i t i v e Slope Deflection s i g n s
N t = ps,(~s/2 +
tt/N
+ (%ij
+,
M
)/LS
where moments a r e in f t - l b s and
Ps3 = sidewall u n i t load, i n psf
Nt = d i r e c t f o r c e i n t o p s l a b , i n l b s
Lt
=
t o p s l a b span, i n f t
Ls
=
sidewall span, i n f t
wc
=
c l e a r width of conduit, i n f t
tt
= inches
With Mgt and Nt known, t h e required s t e e l a t l o c a t i o n 4 (negative s t e e l
a t f a c e o f support of t o p s l a b ) can be determined f o r t h e i n d i c a t e d
loading.
-#
Due
l o c a t i o n 6 - negative bending moment w i t h loading ~ 1 - L C
~ t.o
t h e l a c k of sidewall symmetry, t h e negative bending moment a t t h e
f a c e of t h e t o p support of t h e sidewall f o r t h e i n d i c a t e d loading
i s b e s t obtained by f i r s t computing t h e s i d e w a l l r e a c t i o n a t t h e
t o p and thkn using t h e f r e e body diagram shown i n Figure 25,
Thus from s t a t i c s , l e t t i n g MBs be t h e d e s i r e d moment
R = P ~ , L ~ +/ ~
(MW + MD, )/LS
39
and
N~ = p t s ( ~ t + tsJ12)/2
I f , a s w r i t t e n , MBs < 0 t h e d e s i r e d moment does not e x i s t . With MBs and
Ns known, t h e r e q u i r e d s t e e l a t l o c a t i o n 6 (negative s t e e l a t f a c e of
t o p s u p p r t of sidewall) can be determined f o r t h e i n d i c a t e d loading.
Location 1 - p o s i t i v e bending moment w i t h loading ~ 2 - ~ # 1- . This case
includes i n t e r n a l h y d r o s t a t i c sidewall loading due t o t h e conduit flowing f u l l a s an open channel.
I
..
Ptl
Figure 26. Moments i n t o p s l a b with loading B2-i%#l.
by Slope DeflectThe moment a t B due t h e h y d r o s t a t i c loading i s M
Thus
fl-om
s
a
i
c
s , l e t t i n g MA be
ion s i g n s , it i s a negative moment.
t h e d e s i r e d moment
R;
If, a s w r i t t e n , MA < 0 t h e d e s i r e d moment does not e x i s t . The a s s o c i a t e d
d i r e c t f'orce may be obtained a s t h e sum of four components, see Figure
27. The d i r e c t f o r c e i n t h e t o p s l a b decreases as t h e exzernal sidewall
load approaches a t r i a n g u l a r d i s t r i b u t i o n . Required s t e e l s e a i n c r e a s e s
a s t h e d i r e c t f o r c e decreases. The f i r s t component of t h e d i r e c t f o r c e
i s t h e r e f o r e adjusted t o p a r t i a l l y account f o r a t r a p e z o i d a l d i s t r i b u t i o r
Hence
Figure 27. Components of d i r e c t f o r c e i n t o p and bottom s l a b s
with loading B2-E#1
With MA and Nt known, t h e required s t e e l at l o c a t i o n 1 ( p o s i t i v e s t e e l
i n c e n t e r of t o p s l a b ) can be determined f o r t h e i n d i c a t e d loading.
-
p o s i t i v e bending moment w i t h loading ~ 3 - L C # -~ .This case
Location 7
includes i n t e r n a l water loads due t o t h e conduit flowing f u l l as a
p r e s s u r e conduit. It w i l l produce maximum required a r e a f o r conduits
on e a r t h foundations only when t h e proportions of t h e conduit a r e
such t h a t t h e moment a t t h e c e n t e r of t h e sidewall due t o i n t e r n a l
p r e s s u r e head i s p o s i t i v e and l a r g e r i n magnitude t h a n t h e moment due
t o i n t e r n a l h y d r o s t a t i c sidewall loading. Normally loading Bl- L C # ~
w i l l govern t h i s l o c a t i o n . With e i t h e r loading, t h e s e c t i o n of m a x i mum p o s i t i v e moment i s unknown due t o t h e l a c k of symmetry. This
section is l o c a t e d and t h e moment evaluated. The s t e e l required a t
t h i s s e c t i o n i s determined and recorded f o r l o c a t i o n 7.
--- -
From Figure 28 noting t h e corner moments a r e w r i t t e n f o r p o s i t i v e
Slope Deflection s i g n s and thus any negative values a r e automatically
correct
1
R = (pS2 - P ~ ~ ) L (~~ / h ~hc)
y (hc/3 + t b / 2 4 ) / ~ s
The s e c t i o n of maximum p o s i t i v e moment i s determined by s e t t i n g Vp = 0 and
solving f o r xp. With l i t t l e e r r o r , t h e l a s t term (xp - tt/24)/hc
may be
taken a s 112, then
Figure 28. Moments i n s i d e w a l l with loading ~ 3 - ~ # 2
F i n a l l y , l e t t i n g MC be t h e d e s i r e d moment
I n t h e s e expressions moments a r e i n f t - l b s , p r e s s u r e s a r e i n p s f ,
t h i c k n e s s e s a r e i n inches, and d i s t a n c e s i n c l u d i n g xp a r e i n f e e t .
If, as w r i t t e n , MC < 0 t h e d e s i r e d moment does not e x i s t . The a s s o c i a t e d d f r e c t force is
With MC and Ns known, t h e r e q u i r e d s t e e l a t l o c a t i o n 7 ( ~ o s i t i v es t e e l
nominally i n c e n t e r of s i d e w a l l ) can be determined f o r t h e i r d i c a t e d
loading. The t h i c k n e s s a t t h e s e c t i o n of maximum p o s i t i v e rr jment i s
Bottom s l a b , s e c t i o n a t midspan - maximum d i r e c t c o m p r e s s i o ~ . - The
maximum d i r e c t compression on t h e bottom s l a b i s produced by l o a d i n g
Bl-IC#3 and may be obtained a s t h e sum of t h r e e components, s e e
Figure 24 used above. The d i r e c t f o r c e i n t h e bottom slab i n c r e a s e s
a s t h e e x t e r n a l s i d e w a l l l o a d approaches a t r i a n g u l a r d i s t r i b u t i o n .
The f i r s t component of t h e d i r e c t f o r c e i s t h e r e f o r e a d j u s t e d t o p a r t i a l l y account f o r a t r a p e z o i d a l load d i s t r i b u t i o n .
If, as w r i t t e n , Nb 5 0 compressive d i r e c t f o r c e does not e x i s t i n t h e
bottom s l a b .
Figure 29.
Moments i n bottom s l a b w i t h loading ~ 1 - L C # ~ .
From s t a t i c s and symmetry, l e t t i n g MR be t h e a s s o c i a t e d moment and n o t ing by Slope Deflection s i g n s , t h e corner moment is negative.
1
ME
=
pb3h2
-
(- M ~ 3 )
The s i g n of ME i s unknown. I f ME i s p o s i t i v e as w r i t t e n , t h e r e q u i r e d
s t e e l computed from Nb and ME a p p l i e s t o l o c a t i o n 13. I f ME i s negative,
t h e required s t e e l 3 p p l i e s t o l o c a t i o n 14.
Bottom s l a b , s e c t i o n a t midspan - maximum d i r e c t t e n s i o n . - The maximum
d i r e c t t e n s i o n i n t h e bottom s l a b i s sometimes produced by loading B3-LC&).
If, a s w r i t t e n , Nb 2 0 t e n s i l e d i r e c t f o r c e does not e x i s t i n t h e bottom
s l a b . From s t a t i c s and symmetry, l e t t i n g ME be t h e a s s o c i a t e d moment
I f ME i s p o s i t i v e a s w r i t t e n , t h e r e q u i r e d s t e e l computed from Nb and
ME a p p l i e s t o l o c a t i o n 13. If ME i s negative, t h e r e q u i r e d a r e a a p p l i e s
t o l o c a t i o n 14.
Anchorage of Pos it i v e S t e e l
Safe p r a c t i c e r e q u i r e s t h a t t h e t e n s i o n i n any b a r at any s e c t i o n be
adequately developed on each s i d e of t h a t s e c t i o n . Thus t h e i n s i d e
( p o s i t i v e ) s t e e l at t h e corners of t h e conduit must be provided s u f f i c i e i
anchorage whenever it is e s t a b l i s h e d t h a t t e n s i o n e x i s t s i n t h e b a r unde:
some combination of loads. The anchorage may be provided by standard
hooks o r by embedment l e n g t h i f t h e r e i s enough distance.
P o s i t i v e S t e e l a t Face of Support
It i s not necessary t h a t s e p a r a t e analyses be performed t o e s t a b l i s h
whether t h e
steel
l o c a t i o n s 3, 5,
and 11 i s ever i n tensic
The determinations a r e made and recorded a t t h e time t h e required a r e a a1
t h e s e l o c a t i o n s i s computed. Whenever t h e t e n s i l e a r e a required a t one (
t h e s e l o c a t i o n s i s g r e a t e r than zero, then anchorage i n t o t h e support i s
required.
at
5,
P o s i t i v e S t e e l at Corner Diagonals
Tension may occur i n t h e i n s i d e s t e e l at t h e corner diagonals even thougl
it i s p o s s i b l e t e n s i o n never occurs i n t h e corresponding s t e e l at t h e
support face. Hence t h e e x i s t e n c e of t e n s i o n i n t h e i n s i d e s t e e l a t t h e
diagonals i s i n v e s t i g a t e d .
Referring t o Figure30 t e n s i o n w i l l e x i s t i n t h e bottom s t e e l
( a ) through ( e ) whenever:
f o r ( a ) , N i s t e n s i o n and M
f o r ( d ) , N i s zero and M
>
nT" skcLches
Nd1'/12
>0
f o r ( e ) , N i s compression and M
>~
( t / 2- d/3)/12
r e c a l l t h a t compressive d i r e c t f o r c e i s p o s i t i v e and t e n s i l e d i r e c t force
i s negative.
(c)
(dl
(e)
stence of t,ension i n bottom s t e e l .
The expression M > ~ ( t / 2- d/3)12 comes from Ms
= M + ~ d " / 1 2and d" = d - t / 2
Ms
> ~ ( 2 d / 3 ) / 1 2s i n c e
Top corner diagonal. - Tension i n t h e i n s i d e s t e e l a t t h e t o p corner
can occur only when i n t e r n a l water l o a d i n g i s included i n t h e design.
This i s loading B~-Lc#O. I n t h e a n a l y s i s of t h e corner diagonal, t h e
assumption i s made t h a t t h e r e s u l t a n t t e n s i l e f o r c e i n t h e i n s i d e s t e e l ,
i f t e n s i o n e x i s t s , i s l o c a t e d approximately a t t h e p o i n t on t h e diagonal
where t h e normal from t h e p o i n t of i n t e r s e c t i o n of t h e i n s i d e s t e e l s
p i e r c e s t h e diagonal. This p o i n t is conservatively taken as 3.535 inches
from t h e i n s i d e c o r n e r of t h e conduit.
Figure
31.'Top
corner geometry.
L e t t i n g MB be t h e moment on t h e corner diagonal and changing s i g n s s o
t h a t clockwise moments on t h e diagonal a r e p o s i t i v e
,
MB =
-
(MBO + MBhy
+
t h e components of t h e d i r e c t f o r c e on t h e diagonal a r e
Figure 32. R e s u l t a n t s on t o p corner diagonal.
The r e s u l t a n t d i r e c t f o r c e on t h e diagonal i s
NB
=
~
t
(
~
~~ / ~ ~ ( ~ ~ ) ~
~
/
t
~
)
With t h e r e s u l t a n t moment and d i r e c t f o r c e on t h e diagonal known, t e s t s
a r e performed t o determine i f conditions l i e w i t h i n t h e l i m i t s i n d i c a t e d
above, t h a t is, does t e n s i o n e x i s t on t h e i n s i d e s t e e l a t t h e diagonal.
Bottom corner diagonal. - Tension i n t h e i n s i d e s t e e l a t t h e bottom
corner can occur w i t h t h e loadings given i n Table 5.
Table 5
.
Loadings f o r t e n s i o n i n bottom corner i n s i d e s t e e l
E a r t h Foundation
No
Internal
Water
--
Rock Foundat i o n
With
Internal
Water
~3-=#0
No
kt e r n a l
Water
,
B;L - L C # ~
With
Internal
Water
~3-LC&
The a n a l y s i s f o r t h e bottom corner diagonal i s similar t o t h a t f o r t h e
t o p corner. For example, f o r loading ~ 3 - L C # l~e,t t i n g MD be t h e moment
on the corner diagonal
With MD and NDK known, t e s t s again d i s c e r n whether t h e i n s i d e s t e e l a t
t h e diagonal i s i n t e n s i o n .
Spacing Required by Flexural Bond
Flexural bond s t r e s s e s must b e held w i t h i n t o l e r a b l e values whenever a
b a r i s i n tension. It i s t h e r e f o r e necessary t o determine t h e maximum
sheas t h a t can e x i s t at any s e c t i o n under consideration when t h e s t e e l
under i n v e s t i g a t i o n at t h e s e c t i o n i s a c t i n g i n tension. This maximum
shear i s sometimes l e s s than t h e maximum shear t h a t can ever e x i s t a t
the section.
Relation t o Determine Required Spac i n
For s t e e l b a r s i z e s fi through #11 an: f o r f, ' = 4000 p s i , t h e allowable
f l e x u r a l bond s t r e s s is i n v e r s e l y proport ion& t o b a r diameter, D. Thus
t h e number of bars required per foot of width a t a s e c t i o n i s independent
of b a r s i z e . The number of bars required i s obtained by
e
Since t h e number of b a r s p e r f o o t can be determined, t h e allowable spacing of t h e b a r s i s obtained by
s = -1 -2
n
~
-7
n
-
f c~
d
V
f o r t e n s i o n t o p b a r s C = 3,4 and
s = 7,093d/~
f o r o t h e r tensioyt b a r s C = 4.8 and
s = 10,015d/~
where
s = c e n t e r t o c e n t e r spacing of bars, i n inches
d = e f f e c t i v e depth at t h e section, i n inches
V = shear a t t h e section, i n l b s .
Note t h a t it is t h e o r e t i c a l l y p o s s i b l e t o determine t h e minimum acceptabl
b a r s i z e a t a s e c t i o n when required s t e e l a r e a and spacing i s given. Thi
i s not done s i n c e it i s f e l t d e s i r a b l e t o allow t h e e x e r c i s e of judgement
i n t h e s e l e c t t o n of a c t u a l s i z e s and spacings of b a r s .
Load Comb i n a t ions Producing Minimum Required Spac i n &
The f l e x u r a l bond allowable s t e e l spacing at a p a r t i c u l a r l o c a t i o n i s con
puted only a f t e r it has been determined t h e t e n s i l e a r e a required in bend
ing at t h a t l o c a t i o n i s g r e a t e r than zero. Table 6 l i s t s t h e basic s e t
o r s e t s of loads and t h e e x t e r n a l load combbations t h a t may produce
m a x i m u m shear at t h e s e c t i o n under consideration when t h e i n d i c a t e d
s t e e l i s a c t i n g . I n some p l a c e s more than one l o a d i n g i s l i s t e d f o r
a p a r t i c u l a r s t e e l l o c a t i o n and design mode. I f i n a design, it is
determined t h a t t h e s t e e l i s i n t e n s i o n at t h a t l o c a t i o n t h e n t h e only
loadings i n v e s t i g a t e d t o determine t h e smallest allowable spacing a r e
t h e loadings which produce t e n s i o n i n t h e s t e e l f o r t h a t design.
Procedure a t a Section
The computation of bond spacing i s i l l u s t r a t e d below f o r t h r e e l o c a t i o n s in t h e t o p s l a b . Computations f o r l o c a t i o n s i n t h e s i d e w a l l s
and bottom s l a b a r e s i m i l a r except t h a t conditions i n t h e s i d e w a l l s
a r e complicated by t h e l a c k of symmetry. Note t h a t t h e spacings comput e d f o r t h e p o s i t i v e s t e e l a t l o c a t i o n s 1, 7, and 13 a r e r e d l y t h e
spacings required at t h e r e s p e c t i v e p o i n t s of i n f l e c t i o n as shown on
Figure 1.
-
-
Location 1 with l o a d i n g ~ l - U # l . I f t e n s i o n occurs i n t h e t o p s l a b
p o s i t i v e s t e e l f o r t h i s loading: t h e p o i n t s of i n f l e c t i o n a r e l o c a t e d ,
t h e shear at t h e p o i n t s of i n f l e c t i o n is obtained, and t h e required
bond spacing i s computed.
Figure 33.
m i n t s of i n f l e c t i o n i n t o p s l a b w i t h loading ~ 1 - L C # ~ .
From symmetry, assuming MA i s t h e moment a t t h e c e n t e r of t h e t o p s l a b
and
where dt i s t h e e f f e c t i v e depth of t h e t o p s l a b .
-
-
Location 1 w i t h loadings B2-U#1 and ~ 3 - L C # ~ .When i n t e r n a l water
l o a d i s included i n t h e design, it i s not known beforehand which of t h e
i n d i c a t e d loadings w i l l govern t h e spacing. Usually t h e moment a t t h e
c e n t e r of t h e t o p s l a b due t o t h e p r e s s u r e head loading i s negative,
when t h i s is so, l o a d i n g ~2-LC#lc o n t r o l s . However, i n some designs
Table
b c at ion
(see
Figure 1)
1
2
I
6
With
Internal
Water
~1- ~ # 1
B2-~#1
B3 -LC#l
--
Rock Foundat ion
No
Internal
Water
Bl-IC#1
With
Internal
Water
B2 -LC#l
~3- ~ # 1
--
--
--
B2 -LC#l
B3 -LC#l
~3-LC#
~1- ~ # 1
Bl-Ic#l
Bl-LC*
Bl-LC#1
Bl-m#2
~1-m#l
~1-LC#%'
B~-Lc#~
Bl -LC#
Bl-Ic#2
Bl-LC#2
I33 - ~ #
Bl-w#2
Bl-LC#@
5
7
I
No
Internal
Water
Bl-~#l
6
Loadings f o r f l e x u r a l bond
Earth Foundation
3
4
.
~3-m#O
~1-LC#
B2 -LC#1
B3 -E#1
~3-x#O
2
~ 3 - ~ # 2
BJ-LC#
Bl-IC#O
B~-Lc#~
Bl-LC#
131- ~ # 6
B~-LC#~
B~-LC@
B l -x#2
Bl-x#2
Bl-LC#
331- ~ # 5
B3 4 ~ # 5
~3 -LC&
=-mY
t h i s moment may be p o s i t i v e . For i l l u s t r a t i o n assume it. i s p o s i t i v e ,
t h e n l e t MA'' be t h e moment a t t h e c e n t e r of t h e t o p s l a b due t o l o a d i n g ~ 2 - L C #and
~ MA"' be t h e moment due t o loading B3-LC#l.
From Figure 34
Figure
34. Points of i n f l e c t i o n w i t h loddings B ~ - L C # ~and 133-~#1.
e i t h e r Vpf' o r Vpwl may c o n t r o l spacing.
$" =
,
3
(-)
Ptl
~
~
~
~
Lpt"
=
Thus
8~~" '
(
Ptl
- Phd
1 /2
1
The smaller spacing w i t h i t s d i s t a n c e from t h e c e n t e r of t h e span t o
t h e point of i n f l e c t i o n a r e taken a s t h e answer.
-
-
Location 3
with l o a d i n g ~3-LCfl.
I n some designs loadings ~ 2 - L C # ~
and/or B ~ - L c # ~w i l l cause t e n s i o n i n t h e s t e e l a t l o c a t i o n 3, i f t h i s
occ'ws, they w i l l r e q u i r e a smaller spacing t h a n loading ~3-LC#O.
The shear a t t h e f a c e of t h e support i n t h e t o p s l a b f o r loading
B3-LC#o i s
-
-
Location 4 with l o a d i n g s B l - K # l and Bl-LC#2.
I f both loadings cause
on1
t r o l s . However,
t e n s i o n i n t h e s t e e l a t l o c a t i o n 4, loading ~ 1 - ~c #
i n some designs only l o a d i n g Bl-LC#2 causes t e n s i o n i n t h i s s t e e l . In
e i t h e r event, t h e depth below t h e negative s t e e l i s checked t o determine
whether t h e s t e e l q m l i f i e s a s t e n s i o n t o p bars o r a s o t h e r t e n s i o n b a r s .
Summary of Design
Figure 33 p r e s e n t s a summary flow c h a r t showing t h e sequence of t h e
design process discussed on t h e preceding pages. The b a s i c l o g i c of
t h e computer program prepared and used t o design t h e Standard Single
C e l l Rectangular Conduits p a r a l l e l s t h i s summary flow c h a r t .
Begin design given
design mode
c l e a r spans
l o a d combinat ions
LC#l and L C # ~
Obtain s l a b thicknesses
f o r shears
Note :
Increment
thickness i f
any s e c t i o n
would r e q u i r e
compression
s t e e l i n bending
I
Obtain corner moments from
indeterminate analyses
Note :
Convergence (
sidewall thic
ness requirec
f o r rock
foundations
,
Obtain s t e e l areas a t c r i t i c a l l o c a t i o n s as required
by :
max. moment + d i r e c t f o r c e
max. d i r e c t compression f o r c e + moment
max. d i r e c t t e n s i o n f o r c e -t moment
Determine anchorage requirements
of p o s i t i v e s t e e l a t corners
Obtain s t e e l spacing a t c r i t i c a l l o c a t i o n s as required by f l e x u r a l bond
I
Design completed.
Compute Quantity i n cu. yds. per f t .
and w r i t e r e s u l t s
Figure 35. Summary flow c h a r t o f design process.
Appendix A
Some Q u a l i t a t i v e Influence Lines f o r Inward Acting Applied b a d s
The t h r e e influence l i n e s f o r moment, drawn below, suggest various theor e t i c a l loading p a t t e r n s and actual_ load combinations a s shown.
Theoretical
Loading P a t t e r n
Influence Line
for
positive
Mc
-
--
#
2
or
-
~ # 5
'-1111111 1 - 1
I
d
--.
-
~
A
(
-
fi-
for
negative
M~
C-
-
-i
- -
- t I i U + + i L
)
Load
Comb i n a t ion
Three a d d i t i o n a l influence l i n e s f o r d i f f e r e n t functions are given
below. Tlese, t o g e t h e r with t h e preceding influence l i n e s , may be
used as models t o o b t a i n o t h e r influence l i n e s as may be desired.
Influence Line
IL
for
VB
in
sidewall
-L
I
L
-"iJtop slab
Appendix B
I l l u s t r a t i o n of Existance of Load Combinat ions
'---------The
following shows one s i t u a t i o n
which g i v e s r i s e t o various load
combinations. Two conditions a r e
recognized.
(1) I n i t i a l ( c o n s t r u c t i o n ) condition
Shears along s i d e s of i n t e r i o r
prism have not y e t developed.
( 2 ) Developed (long term) condition
Shears have completely developed along s i d e s of i n t e r i o r
prism.
I n i t i a l Condition
-
moist
I n i t i a l Condition
-
saturated
Developed Condition
I
-
moist
3%
I
= C y b
pmc
-
from TR-5
( s e l e c t low K,)
Developed Condition
-
satura
?h
-
/--
.
( 3 IS~ an~approx
used f o r iLe s ign
( s e l e c t high
%)
.
..