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Reference 19 as:



^ 2 r2



(B-5)




0.05 0.10 0.15 0.20

Radial Displacements, w (in.)



Figure B-25. Radial displacement behavior
Table B-6 summarizes the along length of Group 3 cylinders

having t/D^ = 0.037 and simple

ultimate radial displacements and support at a pressure load of o.95P,n,.

calculated strains. It is interesting
to note that although w, is typi-
cally several times the magnitude of

w , the calculated strains e and s, are nearly equal. At implosion
m m b J ^ f

the strains at the flat-spot location on the inside wall experienced slight
tension while on the outside wall strains were on the order of 4,000
pin. /in. compression.



69



Table B-6. Ultimate Radial Displacements and Calculated Strains



Group


No. of
Specimen


n


ARj

(in.)


(in.)


"'m
(in.)


Wb
(in.)


■^m
(Min./in.)


Eb
(^in./in.)


Strains at I'iat Spot on -


biside Wall,

f m ^ <^b

(Min./in.)


Outside Wall,

fm + fb
(Min./in.)


1
2
3


1
2
4


3
2
3


0.050
0.057
0.042


0.200
0.508
0.185


0.043
0.051
0.052


0.207
0.514
0.175


1,670
2,040
2,080


1,560
2,240
2,040


+ 110

-200

-40


+ 3,230
+4,280
+4,120


Average




-110
tension


+ 3,880
compression



ANALYTICAL RESULTS AND DISCUSSION

Analysis Description

A structural analysis was performed on the experimental specimens
using a finite element method called NONSAP-A that incorporated an
advanced constitutive relation subroutine for the concrete. The analy-
sis was conducted by Chen, Chang, and Suzuki (Ref 20) without the
benefit of the test results . Information on specimen geometry (includ-
ing the out-of- round geometry, boundary conditions, and material
properties) was supplied. It was desired to computationally model the
test specimens as realistically as possible and then determine the accu-
racy of the predictions .

Constitutive Model . The constitutive model was developed in three
parts - elastic, plastic and fracture - for concrete under general stress
states .

For elastic concrete, it was assumed that, initially, concrete is an
isotropic homogeneous linear elastic material and its stress-strain rela-
tions are described completely by two elastic constants, Poisson's ratio,
V, and Young's modulus, E. For the present analysis, v = 0.19 was
used, and E = 3.66x10^ and 4.19x10^ psi (25.2 and 28.9 GPa) were



70



determined from Figure B-7. The elastic limit envelope in general
stress space was obtained by scaling the fracture envelope down to a
size where uniaxial yield point corresponded to about 43% of the uniaxial
strength .

For plastic concrete, a strain-hardening plasticity model as pro-
posed previously in Reference 21 was used to describe the nonlinear
irreversible stress-strain response of concrete material. The plastic
incremental stress-strain relationship based on the normality flow rule
in the theory of plasticity are developed in detail in Reference 22.

For fracture, the concrete failed when the state of stress reached
a certain critical value. Two different types of fracture mode are
defined here.

(1) "Cracking" Type - When the principal stresses are
either in the tension-tension state or tension-
compression state and their values exceed the limit
values.

(2) "Crushing" Type - When the principal stresses are in
the compression-compression state and their values
exceed the limit values . When concrete cracks , the
material is assumed to lose only its tensile strength
normal to the crack direction but to retain its strength
parallel to the crack direction. On the other hand,
when concrete crushes, the material element loses its
strength completely.

In the present analysis, a dual representation of fracture criterion
was expressed in terms of both stresses and strain and specifies .the
limit value under multiaxial state of stresses or strains in the following
forms :



71



(1) Stress Criterion

f(a..) = J_ +4a I, + a I,^ = t ^ (B-6)

ij 23ullu



where A and t are material constants and where a is equal to zero
when the principal stresses are in the compression state and equal to
-1/6 when in the tension-compression or tension-tension state. The
first invariant, I , corresponds to the mean stress component of the
stress state. The term J„ is the second invariant of deviatoric
stresses.



(2) Strain Criterion



A/e\ r, / e \2

u / u \ .^ , 2



;(^j) = J2"y^l^)i^ = ^- iT^l (B-7)



Maximum of the Principal Strains = e (B-8)

in which I' corresponds to volumetric strain and J' is the second invari-
ant of deviatoric strains . The terms £ and e specify the maximum
ductilities of concrete under uniaxial compressive and tensile loading
conditions, respectively. Herein, the compressive cylinder strength
was assumed as 7,000 and 8,000 psi (48 and 55 MPa); and maximum
compressive strain, e , was 3,500 pin. /in. The tensile strength, f,

was assumed to be 0.09 f; and maximum tensile strain, e^, was 800

c ' t

|jin./in. When the stress state in the concrete satisfied either the
stress criterion (Equation B-6) or the strain criteria (Equations B-7



72



and B-8), fracture of concrete was assumed to occur. If the fracture
stress state lies in the tension-compression or tension-tension zone, a
crack was assumed to occur in a plane normal to the direction of the
offending principal tensile stress or strain .

Finite Element Program . In the present work all the analyses were
performed using NFAP program (Ref 23) on computer system IBM model
370-158. NFAP is a modified and extended version of NONSAP-A pro-
gram (Ref 24), which is a modified version of the NONSAP program
originally developed by Bathe, Wilson, and Iding (Ref 25). The pre-
sent concrete constitutive model has been incorporated as a subroutine
in the NFAP program. The average computing time for the two-
dimensional (plane strain or axisymmetric) problems was about 5 minutes
for each case. The average computing time for each three-dimensional
analysis was about 62 minutes.

Geometry of Analysis . The eight cases as listed in Table B-7 were
analyzed using isoparametric shell elements .

(1) Cases 1 and 3 were modeled as axisymmetrical problems
with simple- support end -condition.

(2) Cases 5 and 7 were modeled as plane strain, axisym-
metrical problems .

(3) Cases 6 and 8 were modeled as plane strain, asymmet-
rical problems. Out-of-roundness in the form of n = 2
(see Table 2) was included in the analysis.

(4) Cases 2 and 4 were treated as three-dimensional pro-
blems with large displacement. Out-of-roundness in
the form of n = 3 was included in the analysis .



73



dT


- u

"e


1 °°
d


d


1 0^

d



1 q




dT


1 °°
d


1 "^

d


1 "^
d


OS

1 °°
d


rt


u IS
B S


H "rt


13 "3


"3 "3


e s


U 1)

S B


1 i


dT


- u


o ^

o o
d d


so


d d


(SI On

0\ ■*



d d


so r^

d d




i-"


o -*

o o

d d


q q

d d


00 OS


d d


01

d d


to


00 u->




^ so
so m


00

Csl 00

rn q^


Oh ^


00 u->
OS r^


.-^ 00
m


00 - ^


so
ir. so

'^


M- O.


o o
o o
9, 9,
oo" oo'





q^




q, q.




P. 9.
00' 00




<




1 ^.


• S B
3 '3

CI. G.


■3 'rt


3
U


i 8


1 g

u


1


1

(U


t/5


■5. G,

B B






(U


^ 1 o




d d



d d



d d


so SO



d d




t-H (N


ro ^


in so


t - 00



74




Wall thickness = U,,„ -0.08 in.



Figure B-26. Idealized initial out-of-round shape for cylinders
with n = 3.




AR„ = -0.10 in.



Wall Thickness = ta„a -0-08 in.



Figure B-27. Idealized initial out-of-round shape for cylinders with n = 2.



75



Cross-sectional geometry
for the cylinders, which
includes data on idealized
out-of- roundness, is shown in
Figures B-26 and B-27 with
geometry values given in
Table B-8.

Implosion Results . Table

B-7 summarizes the results of

the analyses in terms of

implosion pressures that were

controlled by strain failure

criteria, (P. ) , and stress
mi e

failure criteria, (P. ) . The
' mi a

implosion strength is given by
the nondimensional ratio of



Tabic B-8. Cross-Sectional Geometry for Cylinders















Numbers


Case


RavgQO


AR


tQO


t60°


^90°


of


No.


(in.)


(in.)


(in.)


(in.)


(in.)


I.obcs,



1


26.345





1.31


-


-





2


26.345


0.04


1.31


1.23


'-


lb


3


26.015





1.97


-


-





4


26.015


0.04


1.97


1.89


-


3^


5


26.015





1.97


-


-





6


26.015


0.06


1.97


~


1.89


2^'


7


25.305





3.39


-







8


25.305


0.06


3.39


-


3.31


2''



45°



'avgQO + ARCOS2

0° < < 45°

•(S -45°) 45° < < 90°



P. /f ' . The analytical implo-

mi c ^ ^



30°



< 30°



(0 - 30°) 30° < e < 60°



See Figure B-26.
■^Scc Figure B-27.



sion strength is compared to
the experimental strength by
the ratios shown in the last
two columns of Table B-7.

The experimental specimens were out-of-round cylinders so a true
comparison between analysis and experiment is only for out-of-round
cylinder cases (Cases 2, 4, 6, and 8). The average ratio of strain-
controlled implosion strength to experimental implosion strength was
0.89 and for stress -controlled implosion strength to experimental implo-
sion strength was 0.93.

The stress criterion failure mode predicted implosion with better
accuracy than the strain criterion method. Looking more closely at
individual cases. Case 2 was an instability failure mode and analysis
predicted implosion 15% lower than experimental. Cases 4, 6, and 8
were material failure modes, and analysis predicted implosion only 4%
lower than experimental.



76



Table B-9. Reduction in Implosion Strength Due to
Out-of-Roundness



Interestingly, the strain criteria that controlled in all cases,
except Case 2, was a tensile strain limit of 800 |jin./in., and not a
compressive strain limit. The limiting tensile strain occurred in the
radial direction of the wall (increase in wall thickness) at midlength for
the free-support specimens and at a distance of | - 0.4 from the end
for the simple- support specimens. Tensile strain had an influence on
faOure because the wall thickness would laminate and facilitate a shear-
compression type of material failure of the wall. Evidence of wall
lamination has been observed in fragments of thick-walled spheres
under hydrostatic loading (Ref 26) but was not observed in the frag-
ments of cylinder specimens.

The effect of out-of-
roundness in reducing the
implosion strength of a per-
fectly circular cylinder is
shown in Table B-9. Cylin-
der t/D influenced the out-
o

of-roundness effect con-
siderably. Cases 1 and 2 are
thinner specimens than Cases
3 and 4, but all have a
simple-support end-condition;
the thinner specimens showed
a 44% reduction due to out-
of- roundness, whereas the thicker specimens showed a 16% reduction.
A similar observation is made between Cases 5 and 6 which are thinner
than Cases 7 and 8, all having a free-support end-condition.

The influence of end-condition on out-of-round effect can be

observed with Cases 3 and 4 and 5 and 6, all of which have t/D' of

o

0.037. Cases 3 and 4 are simply supported and showed a reduction of
16%; whereas. Cases 5 and 6 are freely supported and showed a reduc-
tion of 46%.





Perc


:nt Reduction in Implosion Strength


Failure-
Criteria




Between Case Numbers" —














1 & 2


3&4


5 &6


7 &8


Average


Strain
Control


31


4


38


9


20


Stress
Control


44


16


46


18


31



First case number designates perfect cylinder, second
case number designates out-of-round cylinder.



77



The effect of cylinder length can be observed from Cases 3 and 4

and 5 and 6, all of which have the same t/D ratio of 0.037 but differ-

o

ent effective lengths. Cases 3 and 4 had an L/D ratio of 2.35, and
Cases 5 and 6 had an L/D ratio of infinity. For the out-of-round
cylinders (Cases 4 and 6), the shorter cylinder had a predicted
increase in implosion strength of 53% over that of the infinitely long
cylinder. Experimentally, the increase in strength was 41%.



Displacement Behavior

The predicted deflected shapes for free-support and simple-
support specimens are shown in Figures B-18 and B-19. For the free-
support cylinder (Figure B-18), the predicted shape is a fair approxi-
mation of the experimental shape. It should be noted that the pressure
level for the experimental shape is near implosion at 400 psi (2.8 MPa)
where the analytical shape is at implosion at 346 psi (2.4 MPa). For
the simple-support cyhnder (Figure B-19), the comparison is good.

The predicted radial displacement behavior as a function of pres-
sure is shown in Figures B-20 and B-22. Comparison of the experi-
mental to analytical behavior is quite good. For the out-of-round
cylinders, note that the predicted implosion pressures using the strain
or stress criteria are approximately the same.

A large difference in ultimate radial displacement was observed
between perfect and out-of-round specimens. For cylinders of t/D -
0.037 (Figure B-21), the experimental out-of-round cylinder showed
w = 0.508 inch (13 mm), while the perfect cylinder had w = 0.08 inch
(2 mm) - a 6.4-fold increase. For specimens having the same t/D
ratio of 0.037 but different end-support conditions (Figures B-21 and
B-22), the free-support cylinders showed an ultimate displacement of
w = 0.508 inch (13 mm) compared to the simple- support cylinders of
w = 0.185 inch (5 mm) - a 2.7-fold increase.



78



Radial displacement behavior along the length of the cylinder is
shown in Figures B-23 through B-25. The effect of the simple-support
is vividly shown in Figures B-23 and B-25. The compliance of the
actual ring stiffener in the experimental tests can be observed in Fig-
ure B-25 where approximately 0.02 inch (0.5 mm) of radial movement
occurred.

For the free-support cylinder (Figure B-24), the difference
betvireen experimental and analytical behavior appears great. However,
this same difference is shown in Figure B-21, where the comparison
appears better. Experimentally, the free-support end-condition using a
rubber gasket modeled the ideal free- support quite well.



FINDINGS

1. Analytically, using the finite element program NONSAP-A with
an advanced constitutive material model, the behavior of the cylinder
specimens was predicted with good accuracy. The implosion pressures
were predicted 7% lower than actual when a stress criterion controlled
failure. It was found experimentally that specimens of L/D of 2.35
had an implosion strength 41% greater than specimens of infinite length
(long cylinders); analytically, the increase in strength was predicted as
53%.

2. Out-of- roundness was an important parameter in implosion
strength and radial displacement behavior. Analytically, the effect of
out-of-roundness was to reduce the implosion strength of perfect cylin-
ders by 16% to 46% depending on t/D ratio and end-support condition.
The ultimate radial displacement for the free- support experimental
specimens of t/D = 0.037 was 0.508 inch (13 mm), which was 6.4 times
the displacement for a perfect cylinder. The need to model out-of-
roundness to obtain accurate analytical predictions was found important.



79



3. Radial displacement data for the specimens showed that the
deflected shape for the free-support cylinders had two lobes (n = 2)
and for the simple- support cylinders had three lobes (n = 3). The
membrane and bending radial displacements were determined, and esti-
mates of strain were calculated at the failure location. It appeared that
at the worst flat spot the strain level at failure was slight tension on
the inside waU and about 4,000 pin. /in. compression on the outside
wall.



80



LIST OF SYMBOLS



Outside diameter

Initial elastic modulus

Secant elastic modulus

Tangent elastic modulus

Uniaxial concrete com-
pressive strength

Material strength factor
for cylinder structures

Material strength factor
for spherical structures

Cylinder length

Number of lobes

External pressure

Implosion pressure

Analytical implosion pres-
sure controlled by strain
criteria

Analytical implosion pres-
sure controlled by stress
criteria

Average radius

Outside radius

Average wall thickness

Minimum wall thickness

Radial displacement from
initial to deflected shape

Bending radial displace-
ments

Membrane radial displace-
ments



w„ Total radial displacement

(see Figure B-17)



AR



Deviation in radius



AR. Inside deviation from

average radius

AR Outside deviation from

average radius

At . t - t .

mm mm

£ Ultimate strain

u

£, Bending strain

e Membrane strain

m

n Empirical plasticity reduc-

tion factor

Angular coordinate (see

Figures B-26 and B-27)

6^ , e„ Angular coordinates of
failure zone

6„ Angular coordinate of

center of failure zone

V Pois son's ratio

4 Nondimensional distance

along cylinder length
(see Figure B-4)

a Wall stress

a. Wall stress at implosion



(a. )t5 Wall stress at implosion
predicted by Bresse's
equation (Equation 5)

(a. )j. Wall stress at implosion
predicted by Donnell's
equation (Equation 4)

<|) Ratio of radial displace-

ments between end and
middle of cylinder



81



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Trondheim: P,S. Hafskjold. Oslo: R. Sletten. Oslo: S. Field, Oslo: S'iv Ing Knut Hove, Oslo
OFFSHORE POWER SYS (S N Pagay) Jacksonville, FL
PACIFIC MARINE TECHNOLOGY Long Beach. CA (Wagner)


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Online LibraryH. H HaynesDesign for implosion of concrete cylinder structures under hydrostatic loading → online text (page 5 of 6)