n:( 1-n )3
sins (10)
l-nq( s - ~ ~ ) w i t h n = t / 2 R m and q = am/a,
For most pipe r e h a b i l i t a t i o n applications, the values of R ~ and R l are s u f f i c i e n t l y close to each other that the t e r m (R~/R,) 3 is b a s i c a l l y equal to unity, Similarly, the v a l u e of W can also be shown to be close to unity. Hence, the collapse p r e s s u r e enhancement factor can be a p p r o x i m a t e l y e x p r e s s e d as
H = ~ ( s i n ~ 3 si---~' (11)
It can be seen from the above e q u a t i o n that the e n h a n c e m e n t H is more or less independent of the encased pipe and the host p i p e g e o m e t r y and is simply a function of the gap size ratio m.
For a g i v e n v a l u e of m, Eqs (3), (4), (6), and (9) form a set of e q u a t i o n s for the d e t e r m i n a t i o n of the e n h a n c e m e n t factor H. The
p r o c e d u r e s to be followed are similar to those d i s c u s s e d e a r l i e r for the d e t e r m i n a t i o n of the final collapse pressure of the e n c a s e d pipe.
Symmetrical Collapse Mode
The equations for the symmetrical collapse m o d e can be d e r i v e d following the same procedures as those for the u n s y m m e t r i c a l collapse mode. The collapse of the encased pipe is still m o d e l e d as a p r o g r e s s i v e b u c k l i n g process (Figure 3). In this case, the e n c a s e d pipe first comes into contact w i t h the host pipe at two locations instead of one (Figure 3(a)). As the p r e s s u r e between the e n c a s e d pipe and the host pipe increases, the contact area or b u c k l e d c i r c u m f e r e n t i a l length increases in a symmetrical m a n n e r (Figure 3(b)). The final collapse occurs as shown in Figure 3(c). Using the same assumptions as in the case of u n s y m m e t r i c a l b u c k l i n g mode, the following g e o m e t r i c relations can be o b t a i n e d for the symmetrical b u c k l i n g mode.
2~R2-2R:(~~2 ~) ~2=p (12)
@sins=R1sin~ (13)
R e w r i t i n g Eqs (12) and (13) u s i n g the d e f i n i t i o n of the gap size ratio m, we have
J~ ~ • _
sin~ ~ 2 sins (14)
sin~ (15)
P=RIsins
It is important to p o i n t out that the only m a j o r d i f f e r e n c e b e t w e e n the
.ymm~rical and unIymmet~[eal buckling mod~a s in s sacond t~t-m of
Eqs (3) and (14). From this point onward, the b a l a n c e of the formulation for t h e symmetrical b u c k l i n g mode b e c o m e s identical to that of the u n s y m m e t r i c a l b u c k l i n g mode. The final e x p r e s s i o n of the e n h a n c e m e n t
LO AND ZHANG ON COLLAPSE RESISTANCE MODELING 103
factor H for the symmetric b u c k l i n g m o d e is the same as E q (9) except that u, 8, and m are n o w g o v e r n e d by E q (14) i n s t e a d of E q (3).
N u m e r i c a l S i m u l a t i o n s
U s i n g t h e e q u a t i o n s d e v e l o p e d for t h e u n s y m m e t r i c a l c o l l a p s e mode, F i g u r e 5 shows, as an i l l u s t r a t i v e example, t h e increase in t h e b u c k l i n g p r e s s u r e ~ as the b u c k l i n g p r o c e s s p r o g r e s s e s (or as t h e angle
decreases). C o l l a p s e of the e n c a s e d p i p e is a s s u m e d to have o c c u r e d w h e n no f u r t h e r increase in the b u c k l i n g p r e s s u r e is o b t a i n e d w i t h smaller v a l u e s of 8- The m a x i m u m b u c k l i n g p r e s s u r e is t h e n t a k e n as the collapse p r e s s u r e of the e n c a s e d pipe.
F i g u r e 6 shows t h e p r e d i c t i o n s by t h e two m o d e l s of t h e c o l l a p s e p r e s s u r e e n h a n c e m e n t factor as a f u n c t i o n of t h e r a t i o of t h e radial gap
size to t h e host p i p e inside radius. Here, the host p i p e is a s s u m e d to be s u f f i c i e n t l y r i g i d that no s i g n i f i c a n t change in t h e host p i p e inside d i a m e t e r has o c c u r r e d u n d e r the p r e s s u r e in the gap. The e n h a n c e m e n t factor in the figure is m o d i f i e d by t h e factor (RI/R~) 3 to remove t h e g e o m e t r i c d e p e n d e n c y of the results g i v e n in the figure. A v a l u e of equals to one is u s e d in c a l c u l a t i n g H.
It can be seen f r o m Figure 6 that t h e e n h a n c e m e n t factor decreases as t h e ratio of radial g a p size to host pipe inside radius increases.
F o r s m a l l e r g a p size ratios, s i g n i f i c a n t increase in c o l l a p s e resistance is indicated. However, it is important to point out that such a large i n c r e a s e in collapse r e s i s t a n c e can o n l y be r e a l i z e d if t h e crush s t r e n g t h of t h e e n c a s e d pipe is not exceeded. F o r g a p size ratios larger t h a n 5%, the e n h a n c e m e n t factor r e m a i n s m o r e or less c o n s t a n t and has a v a l u e b e t w e e n 4 and 5.
F i g u r e 7 shows t h e effect of t h e factor ~ on t h e e n h a n c e m e n t factor. For e n c a s e d p i p e s w i t h s t a n d a r d d i m e n s i o n ratios (SDR = 2 R m / t ) b e t w e e n 40 - i00, t h e effect on the v a l u e of t h e e n h a n c e m e n t factor by d r o p p i n g ~ is less t h a n 2.5 percent.
It is obvious t h a t w h e n t h e radial gap b e t w e e n t h e e n c a s e d p i p e and t h e host p i p e b e c o m e s s u f f i c i e n t l y large, t h e e n c a s e d p i p e w o u l d t e n d to b e h a v e m o r e like an u n s u p p o r t e d pipe. Hence, t h e p r e d i c t i o n s by t h e m o d e l s will not be v a l i d for v e r y large g a p size ratios.
F i g u r e 8 shows t h e r e l a t i o n s h i p b e t w e e n t h e e n h a n c e m e n t factor H and ~=. H e r e ~= is o n e - h a l f of t h e angle sustained b y t h e r e m a i n i n g u n b u c k l e d segment of t h e e n c a s e d p i p e just b e f o r e t h e final c o l l a p s e of t h e e n c a s e d pipe. It is interesting to note that for t h e same v a l u e of H, t h e v a l u e s of ~ for b o t h t h e symmetrical and u n s y m m e t r i c a l b u c k l i n g m o d e s are almost identical. However, as can be seen from Figure 6, this c o r r e s p o n d s to d i f f e r e n t radial g a p size ratios for t h e t w o c o l l a p s e modes.
C O R R E L A T I O N W I T H C O L L A P S E T E S T R E S U L T S
T h e p r e d i c t i o n s from the m o d e l s on t h e c o l l a p s e p r e s s u r e e n h a n c e m e n t factor are c o r r e l a t e d w i t h t h e test r e s u l t s g i v e n in [8].
D e t a i l s of the c o l l a p s e test p r o g r a m can be found in [8] and w i l l not be g i v e n here. However, for t h e c o m p l e t e n e s s of t h i s paper, t h e p e r t i n e n t test r e s u l t s are r e p e a t e d below.
T h e first set of test data was o b t a i n e d from e n c a s e d pipe samples m a d e w i t h t h r e e d i f f e r e n t epoxy r e s i n systems. T h e s e e n c a s e d pipes w e r e f a b r i c a t e d b y direct inversion into steel pipes of 12 inches inside
104 BURIED PLASTIC PIPE TECHNOLOGY
FIG. 5 - - B u c k l i n g p r e s s u r e v s a n g l e ~.
FIG. 6 - - P r e d i c t i o n of c o l l a p s e p r e s s u r e e n h a n c e m e n t factor.
LO AND ZHANG ON COLLAPSE RESISTANCE MODELING 105
FIG. 7 - - E f f e c t of ~ on c o l l a p s e e n h a n c e m e n t factor.
-I- Unsymmetricol Mode e SymmetricQI Mode F I G . 8--Enhancement f a c t o r vs ~ .
106 BURIED PLASTIC PIPE TECHNOLOGY
d i a m e t e r . T a b l e 1 s h o w s t h e a v e r a g e s t i f f n e s s a n d w a l l t h i c k n e s s o f t h e e n c a s e d p i p e s a m p l e s a n d T a b l e 2 s h o w s t h e c o l l a p s e p r e s s u r e t e s t r e s u l t s . T h e c o l l a p s e p r e s s u r e s for t h e c o r r e s p o n d i n g u n s u p p o r t e d p i p e s w e r e c a l c u l a t e d u s i n g E q (7) a n d a r e s h o w n in T a b l e 3. T h e r e s u l t i n g c o l l a p s e p r e s s u r e e n h a n c e m e n t f a c t o r s w e r e c a l c u l a t e d a n d a r e s h o w n in T a b l e 4.
T A B L E 1 - - E n c a s e d p i p e m o d u l u s a n d w a l l t h i c k n e s s .
R e s i n - A R e s i n - B R e s i n - C
a v e r a g e a v e r a g e a v e r a g e
M o d u l u s T h i c k n e s s M o d u l u s T h i c k n e s s M o d u l u s T h i c k n e s s
(GPa) (mm) (GPa) (mm) (GPa) (mm)
2 . 5 5 1 5 . 5 4 5 3 . 2 2 7 5 . 6 8 9 3 . 4 0 6 5 . 5 1 4
T A B L E 2 - - C o l l a p s e p r e s s u r e of e n c a s e d p i p e .
R e s i n - A R e s i n - B R e s i n - C
S a m p l e C o l l a p s e S a m p l e C o l l a p s e S a m p l e C o l l a p s e
# p r e s s u r e # p r e s s u r e # p r e s s u r e
(kPa) (kPa) (kPa)
1-3 5 5 8 . 5 2-7 6 2 0 . 6 3-2 6 8 9 . 5
1-5 5 5 1 . 6 2 - 1 4 6 5 5 . 0 3 - 4 6 2 0 . 6
i-I0 5 5 8 . 5 2 - 1 7 6 4 1 . 2 3-9 5 8 6 . 1
1-14 5 7 2 . 3 . . . 3 - 1 5 4 6 8 . 9
1-19 5 3 7 . 8 . . . 3 - 2 0 6 5 5 . 0
A v e r a g e 5 5 5 . 7 A v e r a g e 6 3 8 . 9 A v e r a g e 6 0 4 . 0
T A B L E 3 - - C o l l a p s e p r e s s u r e of u n s u p p o r t e d p i p e .
R e s i n - A R e s i n - B R e s i n - C
C o l l a p s e P r e s s u r e C o l l a p s e P r e s s u r e C o l l a p s e P r e s s u r e
(kPa) (kPa) (kPa)
3 6 . 9 6 5 0 . 6 1 4 8 . 5 4
T O c o r r e l a t e t h e t e s t r e s u l t s w i t h p r e d i c t i o n s f r o m t h e m o d e l s , t h e s i z e o f t h e r a d i a l g a p b e t w e e n t h e e n c a s e d p i p e s a m p l e s a n d t h e s t e e l p i p e s w a s c a l c u l a t e d . T h i s r a d i a l g a p w a s d e v e l o p e d u p o n c o o l i n g a f t e r c u r i n g in t h e e n c a s e d p i p e i n v e r s i o n p r o c e s s d u e t o d i f f e r e n t i a l t h e r m a l e x p a n s i o n b e t w e e n t h e e n c a s e d p i p e a n d t h e s t e e l p i p e . S i n c e t h e c o e f f i c i e n t of t h e r m a l e x p a n s i o n of t h e e n c a s e d p i p e m a t e r i a l is
s u b t a n t i a l l y h i g h e r t h a n t h a t of t h e s t e e l p i p e , t h e i n i t i a l s i z e of t h e r a d i a l g a p c a n b e c a l c u l a t e d as
A,=~0RIA T ( 16 )
w h e r e ~o is t h e l i n e a r t h e r m a l e x p a n s i o n c o e f f i c i e n t o f t h e e n c a s e d p i p e
LO AND ZHANG ON COLLAPSE RESISTANCE MODELING 107
m a t e r i a l a n d A T is t h e d i f f e r e n c e b e t w e e n t h e g l a s s t r a n s i t i o n
t e m p e r a t u r e a n d t h e t e s t t e m p e r a t u r e . T a b l e 5 s h o w s t h e v a l u e s of ~ a n d A T for t h e t h r e e d i f f e r e n t e p o x y r e s i n s y s t e m s .
W h e n h y d r o s t a t i c p r e s s u r e is i n t r o d u c e d i n t o t h e r a d i a l g a p d u r i n g t h e c o l l a p s e t e s t , t h e g a p w i l l i n c r e a s e in s i z e u n d e r p r e s s u r e . T h e a m o u n t of i n c r e a s e c a n b e e s t i m a t e d b y u s i n g t h e f o l l o w i n g e q u a t i o n [9]
A2=__~E(I_v2)Rao (R2o2+(R2o-t) 2 __V ) (17)
R2o 2 - (a20- t) 2 l-v
T h e t o t a l g a p s i z e is e q u a l t o t h e s u m of A l a n d A 2 a n d t h e y are s h o w n i n T a b l e 6.
F i g u r e 9 s h o w s t h e c o l l a p s e t e s t r e s u l t s s u p e r i m p o s e d o n t h e p r e d i c t i o n s b y t h e m o d e l s . T h e c o r r e l a t i o n b e t w e e n t h e t e s t r e s u l t s a n d t h e p r e d i c t i o n s is q u i t e good. It is i n t e r e s t i n g t o n o t e t h a t t h e t e s t r e s u l t s fall w i t h i n t h e p r e d i c t i o n r a n g e of t h e t w o m o d e l s . S i n c e t h e s e t w o m o d e l s a r e b a s e d o n t w o d i f f e r e n t c o m p e t i n g c o l l a p s e m o d e s , t h e y c a n b e u s e d t o e x p l a i n t h e s c a t t e r in t h e t e s t r e s u l t s o b t a i n e d in t h e c o l l a p s e t e s t of e n c a s e d p i p e s .
T A B L E 4 - - C o l l a p s e e n h a n c e m e n t f a c t o r s o f e n c a s e d p i p e .
R e s i n - A R e s i n - B R e s i n - C
S a m p l e E n h a n c e m e n t S a m p l e E n h a n c e m e n t S a m p l e E n h a n c e m e n t
# F a c t o r # F a c t o r # F a c t o r
1-3 1 5 . 1 1 2-7 1 2 . 2 6 3-2 14.2
1-5 1 4 . 9 2 2 - 1 4 1 2 . 9 4 3 - 4 1 2 . 7 8
i-I0 1 5 . 1 1 2 - 1 7 1 2 . 6 7 3-9 1 2 . 0 7
1 - 1 4 1 5 . 4 8 . . . 3 - 1 5 9 . 6 6
1-19 1 4 . 5 5 . . . 3 - 2 0 1 3 . 4 9
A v e r a g e 1 5 . 0 3 A v e r a g e 1 2 . 6 2 A v e r a g e 1 2 . 4 4 T A B L E 5 - - T h e r m a l e x p a n s i o n c o e f f i c i e n t a 0 a n d t e m p e r a t u r e d i f f e r e n c e AT.
M a t e r i a l u0 A T ( ~ )
R e s i n - A 9x105/~ 4 5 . 0
R e s i n - B 7.2xi0"5/~ 8 8 . 8 9
R e s i n - C 7 . 2 x i 0 4 / ~ I 0 0 . 0
F i g u r e i0 s h o w s t h e c o r r e l a t i o n b e t w e e n t h e p r e d i c t i o n s b y t h e m o d e l s a n d t h e c o l l a p s e p r e s s u r e t e s t r e s u l t s p r o v i d e d b y [i0]. In t h i s
case, t h e i n i t i a l r a d i a l g a p s i z e ( b e f o r e t h e a p p l i c a t i o n o f e x t e r n a l h y d r o s t a t i c p r e s s u r e ) w a s e s t i m a t e d f r o m m e a s u r e m e n t o f t h e e n c a s e d p i p e a n d t h e s t e e l p i p e d i a m e t e r s . T h e f i n a l g a p s i z e w a s o b t a i n e d b y a d d i n g t h e c o n t r i b u t i o n f r o m e x t e r n a l h y d r o s t a t i c p r e s s u r e c a l c u l a t e d u s i n g E q
(17). It c a n b e s e e n f r o m t h e f i g u r e t h a t g o o d c o r r e l a t i o n is a g a i n o b t a i n e d w i t h t h e t e s t r e s u l t s o v e r a w i d e r a n g e of g a p s i z e ratios. In a d d i t i o n , t h e t e s t r e s u l t s c l e a r l y d e m o n s t r a t e t h a t as t h e g a p s i z e
108 BURIED PLASTIC PIPE TECHNOLOGY
ratio increases, t h e enhancement in t h e collapse p r e s s u r e r e s i s t a n c e decreases. However, it is important to point out t h a t a large initial g a p size seldom exists in pipe r e h a b i l i t a t i o n . The t e s t results shown in Figure i0 w i t h large g a p size ratios are o n l y p r o v i d e d to c l e a r l y
d e m o n s t r a t e the effect of gap size r a t i o on c o l l a p s e r e s i s t a n c e of e n c a s e d pipes.
TABLE 6--Radial qap b e t w e e n t h e e n c a s e d p i p e and host pipe.
Sample # A l (mm) A 2 (ram) (A,+A2)/R , (%)
1-3 1-5 1-10 1-14 1-19 2-7 2-14 2-17 3-2 3-4 3-9 3-15 3-20
0.6172 0.6172 0.6172 0.6172 0.6172 0.9754 0.9754 0.9754 1.0973 1.0973 1.0973 1.0973 1.0973
0.7696 0.7595 0.7696 0.7874 0.7391 0.6553 0.6934 0.6782 0.7137 0.6426 0.6071 0.4851 0.6782
0.909 0.903 0.909 0.921 0.890 1.069 1.093 1.083 1.186 1.139 1.116 1.037 1.162 D I S C U S S I O N
T h e two models d e v e l o p e d in this study are a i m e d at d e v e l o p i n g a simple m e t h o d to d e t e r m i n e the c o l l a p s e resistance of e n c a s e d pipes.
E v e n t h o u g h r e a s o n a b l e c o r r e l a t i o n has b e e n o b t a i n e d b e t w e e n p r e d i c t i o n s by t h e m o d e l s and some available test results, m o r e t e s t data are needed to fully e s t a b l i s h t h e v a l i d i t y of t h e models. D e s p i t e this, t h e effect of radial g a p size on t h e collapse r e s i s t a n c e of e n c a s e d pipe is c l e a r l y demonstrated. I m p r o v e m e n t to the m o d e l s and/or n e w m o d e l s are e x p e c t e d as f u r t h e r k n o w l e d g e and u n d e r s t a n d i n g on e n c a s e d p i p e collapse r e s i s t a n c e are accumulated.
As expected, t h e collapse p r e s s u r e e n h a n c e m e n t factor a s s o c i a t e d w i t h t h e symmetric c o l l a p s e m o d e is h i g h e r t h a n that a s s o c i a t e d w i t h the u n s y m m e t r i c a l c o l l a p s e mode. At small g a p size ratios, t h e d i f f e r e n c e in t h e p r e d i c t i o n s b y the m o d e l s is r e l a t i v e l y large. At larger g a p size ratios, t h e d i f f e r e n c e b e t w e e n t h e p r e d i c t i o n s diminishes. A l t h o u g h t h e o b s e r v a b l e final c o l l a p s e mode of almost all of the e n c a s e d p i p e s t e s t e d in t h e l a b o r a t o r y shows an u n s y m m e t r i c a l collapse mechanism, t h e
t e n d e n c y to d e v e l o p symmetric c o l l a p s e m o d e b e f o r e final collapse occurs has b e e n o b s e r v e d in some collapse tests. Hence, t h e collapse p r e s s u r e m e a s u r e d in the c o l l a p s e test m i g h t not be associated w i t h only t h e u n s y m m e t r i c a l collapse mode. It could also be a r e f l e c t i o n of t h e c o m b i n e d effect of t h e s e two c o m p e t i n g collapse mechanisms. This m i g h t h e l p to e x p l a i n t h e scatter in t h e t e s t d a t a o b s e r v e d in the c o l l a p s e test. The p r e d i c t i o n s on enhancement factor given in F i g u r e 6 can, therefore, be r e g a r d e d as the b o u n d i n g v a l u e s for t h e c o l l a p s e p r e s s u r e .
30.0
,~, 25.0-
20.0"
~ lS.O- 10.0"
z <
z 5.0"
LO AND ZHANG ON COLLAPSE RESISTANCE MODELING
/
... { ... r ... ] . . . --'~ + - - R e s i n - A
/ i l / A--Resin- B
\ i ! / ~ 1 7 6 C
y _ ~ I s,m~,o~,,,o,o t l ... I . . .
. . . /
t
0 . 0 . . . . i . . . . . . . . ~ . . . . . . . .
0.0 2,0 4.0 6,0 8.0 10.0
GAP/RI(%)
F I G . 9 - - M o d e l p r e d i c t i o n c o r r e l a t e s w i t h t e s t d a t a .
109
30
~ 2 5 "
20"
O ,~ 15"
10' z <
,.r- z 5 -
0 0
... ... ! . . . + - - R e s i n - D i ,
\ i i O--Resin-E '1
\ i ~ - - R e s i n - F I
it i i
, i , , i i i , , I v i i , I ' i J i i i , ,
2 4 6
GAP/RI(%)
F I G . 1 0 - - M o d e l p r e d i c t i o n c o r r e l a t e s w i t h t e s t d a t a .
110 BURIED PLASTIC PIPE TECHNOLOGY
In a d d i t i o n to the short t e r m c o l l a p s e resistance, the long t e r m collapse r e s i s t a n c e of e n c a s e d pipes is also an important d e s i g n concern. W h e n subjected to sustained h y d r o s t a t i c pressure, the radial gap b e t w e e n the encased pipe and the host pipe will increase in size with time due to creep in the encased pipe. Hence, a c c o r d i n g to the m o d e l s d e v e l o p e d here, the enhancement factor and the c o r r e s p o n d i n g collapse p r e s s u r e will d e c r e a s e with time due to creep. This p r e d i c t i o n is q u a l i t a t i v e l y consistent w i t h e x p e r i m e n t a l observations. However, further t h e o r e t i c a l and e x p e r i m e n t a l w o r k are needed to e s t a b l i s h the u t i l i t y of the m o d e l s for assessing the e n c a s e d pipe's long t e r m collapse resistance.
R E F E R E N C E S
[!] Kyriakides, S. and Youn, S.-K., "On The collapse of C i r c u l a r C o n f i n e d Rings Under External Pressure," International Journal of Solids Strutures, vol. 20, No. 7, 1984, pp 669-713.
(2__] Kyriakides, S. and Li, F.-S., "On The R e s p o n s e and Stability of Two Concentric, C o n t a c t i n g Rings U n d e r External Pressure,"
International Journal of Solids Structures, Vol. 27, No. i, 1991, pp 1-14.
[_3] E1-Bayoumy, L., "Buckling of a C i r c u l a r Elastic R i n g C o n f i n e d t o a
U n i f o r m l y C o n t a c t i n g Circular Boundary," ASME Journal of A p p l i e d Mechanics, 1972, pp 758-766.
[_4] Pian, T. H. H. and Bucciarelli, L. L., "Buckling of R a d i a l l y C o n s t r a i n e d Circular Ring Under D i s t r i b u t e d Loading,"
International Journal of Solid Structures, Vol. 3, 1967, pp 715- 730.
[s_] Bucciarelli, L. L. and Pian, T. H. H., "Effect of Initial Imperfections on the Instability of a R i n g C o n f i n e d in an Imperfect R i g i d Boundary," ASME Journal of A p p l i e d Mechanics, 1967, p p 979-984.
[6] Zagustin, E. A. and Herrmann, G., "Stability of an Elastic Ring in a R i g i d Cavity," ASME Journal of A p p l i e d Mechanics, 1967, p p 263- 270.
[2] Timoshenko, S. P.and Gere, J. M., "Theory of Elastic Stability,"
McGraw-Hill, New York, 1961.
[8] LO, K. H., Chang, B. T. A., Zhang, Q., and Wright, W. J.,
"Collapse R e s i s t a n c e of C u r e d - I n - P l a c e Pipes," N o r t h A m e r i c a n Society for Trenchless Technoloqy, N o - D i q '93, M a y 2-5, 1993.
[9] Venkatraman, B., and Patel, S. A., "Structural M e c h a n i c s with I n t r o d u c t i o n to E l a s t i c i t y and Plasticity," McGraw-Hill, New York, 1970.
[i0] Private C o m m u n i c a t i o n w i t h Insituform Technologies, Inc.
Laboratory Testing
David W. Woods' and S t e v e n R. F e r r y '
COMPRESSIVE BUCKLING OF HOLLOW CYLINDERS: IMPLICATIONS FOR PRESSURE TESTING OF PLASTIC PIPE