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-r-f,' #-3'7 UI LU-ENG-70-103
~~NGINEERING STUDIES STRUCTURAL RESEARCH SERIES NO. 367
RECEIVEu
C. E. REHR£NG£ aOOM
~ ~~ .. - --
MATERIAL BEHAVIOR CHARACTERISTICS FOR
REINFORCED CONCRETE SHEllS STRESSED BEYOND THE
ELASTIC RANGE
~.'! ?~!:; :::1 . ~-.' :_'.~~'S::ctll}el1t by
M. J. Mikkola W. C. Schnobrich
ISSUED AS A TECHNICAL REPORT OF A RESEARCH
PROGRAM SPONSORED
by
The National Science Foundation Grant No. GK 11190
UNIVERSITY OF ILLINOIS URBANA, ILLINOIS
AUGUST 1970
MATERIAL BEHAVIOR CHARACTERISTICS
FOR
REINFORCED CONCRETE SHELLS STRESSED
BEYOND THE ELASTIC RANGE
by
M. J. Mikkola
w. C. Schnobrich
Issued as a Technical Report of a Research
Program Sponsored by
The National Science Foundation Grant No. GK 11190
University of Illinois Urbana, I 1 1 i no is
August 1970
TABLE OF CONTENTS
LIST OF FIGURES
ACKNOWLEDGMENTS . i i
1. INTRODUCTION . 1
1.1 General • ••• 1
2. MATERIAL PROPERTIES.
2. 1 Cone rete
...... 4
· ... 4
2.2 Reinforcement
2.3 Reinforced Concrete
2.3.1 Initial elastic behavior
2.3.2 Elastic behavior after cracking ..
2.3.3 Plastic behavior after cracking.
2.3.4 Plastic behavior: concrete yielding
· . 9
· . 9
· .10
.12
· .14
in biaxial compression .......... 16
2.4 Unloading
3. THE F!N!TE ELEMENT SOLUTION
APPENDIX
REFERENCES
3.1
3·2
The Elastic State
The State of Cracking and Yielding.
.20
.22
... 22
· .24
... 29
· ... 33
i i
ACKNOWLEDGMENTS
The results reported herein were developed in a research study
supported by the National Science Foundation under Grant NSF GK-11190
At the time this report was written Dr. M. J. Mikkola was at the University
of 111 inois on a Fulbright Post Doctoral Fellowship. The support of the
Committee on International Exchange of Persons which coordinates the Fulbright
Program was indispensable in the conduct of this project.
The cooperation and suggestions of Dr. H. K. Hilsdorf are grate
fully acknowledged.
1 • I NTRODUCT ION
1 .1 Genera 1
Structural design of shells is usually based on 1 inear elastic
analysis with simple assumptions regarding both loading and support con
ditions. The effect of cracking and non1 inear material behavior are normally
neglected because their influence has not been investigated sufficiently to
delineate their effect. However, the determination of load-deformation
characteristics and an understanding of the behavior of concrete shells
after cracking are necessary for the formulation of criteria and guidel ines
for economical and re1 iab1e design.
Some experimental data (1), ... (15) obtained from tests on micro
concrete models show a considerable deviation of the behavior from that pre
dicted by elastic theory. Bouma (1) conducted an extensive "test series using
micro-concrete models of cyl indrical shells. These tests were performed
to estab1 ish the behavior of cyl indrica1 shells and to provide design
information. Bouma's tests did demonstrate that for acyl indrica1 shell the
behavior up to the load level comparable to the design load magnitude,elastic
design predicted values reasonably well. For other shells this may not be the
case. Tests have also been performed on an umbrella form of a hyperbol ic
paraboloid including both a plastic and a mortar model (2). These tests
demonstrated that the behavior of the shell after initial cracking can in
volve a significant change in the load carrying mechanisms. For such cases
cracking should exercise a strong influence on the design of the shell.
For the structure in question both an elastic analysis and the results of a
2
plexiglass model test predicted that the structure could not withstand
the appl ied external load without excessive thickening in regions of
high moment and a corresponding excessive increase in reinforcement.
However, the micro-concrete model indicated that in fact the shell, even
with reduced reinforcement,was capable of resisting not only the design
load but a substantial overload. The disagreement between the elastic
analysis and the real behavior of the shel 1 as found from the micro-concrete
model is due to the alterations in the load carrying mechanisms resulting
from a downgrading of the bending action compensated by a more active
participation of membrane action even in the edge zones after the cracks
begin to form.
The use of models for obtaining design information is desirable
but is very expensive and requires skilled experimental investigators to
properly interpret the results. Furthermore, a new model .must be con
structed for each new influence to be studied. Hence, there is an apparent
need for analytical or numerical procedures for the prediction of the
behavior of the concrete shells beyond the elastic range. Analytica!
studies confirmed by comparisons with experimental results represent an
economical and expedient way to obtain the needed information.
Li~it analysis provides an elegant way for the determination of
the carrying capacity of shells. Olszak and Sawczuk (16) give an extensive
I ist of investigations on the 1 imit analysis of shells. However,! imit
analysis can only give an estimate for the collapse load but tells nothing
about the loaa-deformation characteristics before reaching the limit state.
The advent of digital computers has facil itated the stress analysis
of complex structures (16) , ... ,(25). The approaches which have been used
3
for elastic-plastic analyses of shells are the finite difference method (18),
the lumped parameter method (19) and the finite element method (20) , ... (25),
the last one being the most popular. Most analyses use the Mises yield
criterion and the associated flow rule. In (23) also the Mises-Hi11 yield
criterion for orthotropic materials is employed as well as some hardening
rules.
General forms of constitutive relations of elastic-plastic shells
have been.developed in (21) and (25) although only isotropic relations were
used in numerical calculations. However, all these procedures are inapp1 i
cable to reinforced concrete structures which exhibit different strength
characteristics in tension and compression as well as the unstable cracking
phenomenon. In the present study equations are developed to simulate the
actual behavior of reinforced concrete and, in particular, to take into
account effects due to cracking and plastic deformations. Biaxial state
of stress is considered for a material composed of concrete and steel bars.
Steel is assumed to have elastic perfectly plastic properties. For the
concrete a cracking criterion based on the octahedral shearing stress theory
is employed and in biaxial compression a similar yield criterion is chosen.
The composite concrete-steel material has anisotropic properties due to the
presence of the steel bars, especially, after the formation of cracks. The
stress-strain relations will subsequently be used in finite element or
lumped parameter analyses of reinforced concrete plates and shells. At
this ~tage it is convenient to choose a sandwich or layered system to model
the actual plate or shell. Each layer of the system is assumed to be in a
state of plane stress. The numerical results will be compared to available
experimental data.
4
2. MATERIAL PROPERTIES
2. 1 Concrete
The failure criteria of plain concrete have been the object
of extensive research in recent years. Unfortunately, various results,
e.g. (26) , ... , (37) sh?w considerable deviations so that no generally
accepted failure criterion for multiaxial stress states exists at the
present time. The most propounded failure criteria are the octahedral
shearing stress criterion suggested by Nadai (38) p. 225
'T = f(p) oct
1 2 2 2 2 'T = 3[ (°11 - (22) + (°22 - (33) + (°33 - (11) + 6(°12 + °23 + (31)] oct
1 + °22 + (33) (2.1) p = 3(°11
and the Mohr criterion (38) p. 214
° 11 + °33 F ( ) ;
2 (2.2)
A modification of the t10hr criterion which takes into account the inter-
mediate principal stress 02 is
'. -~ - ...... ,.... -~ ., .... ... ... -.~ ~ - J..:. '.,.J' ..... ........
........ '._. 4 __ • •••• _.;;:
f .... - ·c: ;'-. -: __ '-.-J""".~,
5
For brittle fracture the criterion of ~aximum tensile strain
has been frequently used.
s . s max' max (0.01 - 0.03)% (2.4)
Experimental results are usually 1 imited to failure under
particular stress combinations, e.g. biaxial compression, compression-
tension, biaxial tension, etc. Bresler and Pister (29) and McHenry and
Karni (32) studied the strength only in biaxial compression-tension con-
ditions. Kupfer, Hilsdorf and Rusch (31), Robinson (34), Vile (35), and
Weigler and Becker (36) investigated the biaxial stress states, while
B a 1 me r ( 27), Bella my ( 28), Han nan tan d F red e ric k ( 30), Ric h art, Bra n d t z a e g
and Brown (33) were concerned with triaxial states of stress. Investigations
reported in (29), (31), (32), (36) and (37) i nd i cate that the i ntermed i ate
principal stress has a considerable influence on the critical values in the
compression-tension and in the biaxial compression states. On the other
hand, the results obtained in (2J) and (33) have been explained to support
the Mohr criterion (30).
In selecting a failure criterion the following factors are to
be considered:
1) The criterion should provide a reliable prediction of the
failure stresses for those combinations which can occur in the structure.
To judge the reliabi lity of the criterion, it should be confirmed by test
resul ts.
2) The criterion to be used should be as simple as possible,
certainly no more compl icated than is warranted by its relation to the
6
supporting experimental data and to the other hypotheses to be used in the
analys is of the structure.
In view of the last factor the octahedral shearing stress criterion
looks most attractive because of its invariant character. Further, if the
criteria described by the Eqs. (2.1 - 2.4) are considered as surfaces in
3-dimensional (01'02'03) space, then it is found that the octahedral shearing
stress criterion corresponds to a smooth surface while the surfaces repre
senting the other criteria show certain discontinuities, in the form of
corners, along the intersections of the forming surfaces.
In this study we are dealing with biaxial states of stress. Con
sequently, the fai lure criterion is represented by the intersection of the
3-dimensional failure surface with one of the coordinate planes, say the
01 02 - plane. Experimental results for this case have been recently reported
by Kupfer, Hilsdorf, and Rusch (31) (Fig. 1). Using the octahedral shearing
stress criterion a fairly good match is obtained by two 1 inear expressions
of the fo rm
T oct a - bp (2.5)
One equation is val id for biaxial compression, while a second expression
is applicable in the compression-tension and the biaxial tension regions
(Fig. 1). For the triaxial case these two linear expressions represent
ci rcular cones in the stress space with a common axis 01 = 02 = 03 and
intersecting in the plane 01 + 02 + 03 = -fc · A 1 inear expression for
biaxial compression-tension cases has been previously used by Bresler and
Pister (29).
7
For the determination of the coefficients in Eq. (2.5) it is
suitable to use the characteristic strength values of the concrete:
( 1 ) uniaxial compressive strength -f (f > 0) c c
(2) uniaxial tens i 1 e strength f t = af c' a ;; 0.1
(3 ) biaxial compressive strength °1 = ° = -f = -Sf S ;; 1. 16 2 c2 c'
where the a- and S- values are in accordance with the experimental
results given in (31). Using the data described above the fo1 lowing
expressions are obtained:
T + 12 1 -a p _ 212 ~ f oct l+a 3 l+a c 0; (° 1 > 0 or °2 > 0) (2.6a)
Toct + 12 ~~~l P - ~ 2:-1 fc = 0; (° 1 < 0 and °2 < 0) (2.6b)
As can be seen in Fig. 1, the linear form of the Mohr criterion
a l+a fc
and the maximum tensile strain criterion also provide a reasonably accurate
prediction of the occurrence of tensile cracking.
From the preceding discussion, it follows that in biaxial com-
pression the octahedral shearing stress criterion provides a good approxi-
mation for the failure stress. In case of compression-tension or biaxial
tension any of the discussed criteria could be used. For consistency, the
octahedral shearing stress criterion was chosen in this study.
8
As to the di rection of the cracks, there is experimental evidence
( 3 1 ), ( 36) t hat the c r a c k d ire c t ion i s pe r pen d i c u 1 art 0 the d ire c t ion 0 f
the maximum tensile stress, except in the case of uniform biaxial tension
where no preferred direction exists.
Little is known about the deformations of concrete under mufti-
axial states of stress. Strain measurements in the biaxial case have been
performed by Kupfer, Hilsdorf, and R~sch (31) and by Weigler and Becker (36),
and nonlinear behavior similar to uniaxial case was observed. The final
fai lure occurred along a plane incl ined about 20-30 degrees with respect
tot he dire c t ion 0 f the 1 a r g e r comp res s i ve s't re s s ( Fig. 2).
In this study, concrete is assumed to behave 1 inearly and iso-
trop.ically up to cracking or yielding, i.e.
O •• IJ
(2. 7)
The di rection of cracks is taken as perpendicular to the di rection of the
maximum tensile stress. For plastic deformations in biaxial compression
the associated flow rule of the theory of plasticity ds~. = Adf/do .. is I J I J
assumed to be applicable. Consequently, from the yield criterion (2.5)
fo 11 ows
ds~. IJ
~(Oij _p_ 3
- 0.. + o .. b) Toct I J Toct I J
(2.8)
9
2.2 Reinforcement
For the steel bars the uniaxial elastic, perfectly plastic
ideal ization is used
o
r (E s, lsI < 0 IE - p st ~ st
~Op sign s, lsI> 0 IE t P s
2.3 Reinforced Concrete
In constructing a model to describe the mechanical behavior
of reinforced concrete, many simpl ifying assumptions have to be made, even
if the behavior of the components, concrete and steel, were completely
known. The final justification of the simpl ifications made can be obtained
by the usefulness of the constructed model, i.e. by comparison its pre-
dictions with experimental results.
Here, the following basic assumptions are made:
1) deformation is uniform, i.e. concrete and reinforcement have the
same strains or full bond is maintained
2) up to cracking or yielding concrete is isotropic linearly
elastic
3) cracking occurs when the cracking criterion Eq. (2.6a) is
satisfied; after cracking, concrete has no tensile strength in the direction
per pen die u 1 art 0 the c r a c k ; i nth e c rae k d ire c t ion co ncr e t e be h a v e sun i -
axially. No bond sl ip is assumed to occur even over the finite region of
the material which corresponds to the region of cracking
10
4) yielding of concrete occurs when yield criterion (2.6b) is
satisfied; for plastic strain increments the associated flow rule is
accepted; yielding of concrete is not affected by reinforcement.
During the loading process various kinds of material behavior
can occur. This is demonstrated in Fig. 3. Figure 4 describes qualit~tively
the "yield criterion ll of the reinforced concrete model.
2.3.1 Initial elastic behavior
We consider a concrete plate of thickness t. This plate contains
two systems of reinforcing steel denoted by Rl and R2 , whose directions
make angles ¢l and ¢2 with respect to the xl-axis as shown in Fig. 5.
Denoting by A the steel area per unit width of section, the relative amounts
of reinforcing steel are ~l = Alit and ~2 = A2/t, respectively. Thus, the
relative amount of concrete is ~O = 1 - ~l - ~2· Denoting the stresses in
the various components or constituents of the plate by O~6' O~6' and O~6'
respectively, the pseudostress, i.e. the average stress through the thick-
ness of the plate is
{a} 012 Wo {o } + ~1 {o } + ~2 {o } (2. 10)
The stresses and strains in a coordinate system xl x2 rotated
by an angle ¢ with respect to the xl x2-system are effected by the fol-
lowing transformation rules:
{O} [T] {o}, {C} T ] {s} (2.11)
11
where the stress and strain vectors are
I ~ I t..ll
{a} {s}
and the transformation matrix and its inverse are
2 . 2¢ 2cos¢ sin¢ cos ¢ sin
[T] . 2¢ 2 -2 cos¢ sin¢ (2.12a) sin . cos ¢
-coscpsin¢ COS¢sin¢ 2¢ . 2¢ cos - sin
2 . 2¢ -2cos¢ sin¢ cos ¢ sin
[T] [T] -1 . 2¢ 2 2cos¢ sin¢ (2.12b) sin cos ¢
-cos¢sin¢ 2 . 2¢ cos¢sin¢ cos ¢ - sin
We assume that the strain is uniform, i.e. that the strain is
the same both in concrete and the reinforcing steel. Hence, the pseudo-
stresses can be related to the strains by formula
{a} [c] {s}
where the elasticity matrix is
[c] 012 1-10 [c ] + 1-11 [c ] + 1-12 [c ]
(2. 13)
(2. 14)
12
The material property matrices of the individual constituents or
components are
\J o 1
[CO] E --2 \J
l~V J l-\J
; 0 0 L
(2. l5a)
o
o (2. 1 5b)
o o
o
o (2.l5c)
o
where [Tl 1 and [lJ2
mean the transformation matrix (2.l2b) evaluated at
¢ = ¢l and ¢ = ¢2' respectively.
Equations (2.10-2.15) establ ish a model for the anisotropic
elastic behavior of reinforced concrete.
2.3.2 Elastic behavior after cracking
Cracking will occur if the following conditions are satisfied:
~ 13
1) one of the principal stresses of concrete cr~ or cr~
is positive,
where
T oct
2) the cracking criterion (2.6a)
l-a 212 To c t + 12 1 +a p - 3
a ~--f
+ a c o
The direction of the crack is taken as perpendicular to the
di rection of the maximum tensile stress. Let us denote the angle of
the crack direction with respect to the Xl-axis by ¢c. After cracking
the concrete behaves uniaxially in ¢ -direction. Hence, the stiffness c
matrix of concrete after cracking is as follows
E o o
cn c o o o (2.16)
o o o
where [r] is the transformation matrix (2. l2b) evaluated at ¢ = ¢ . c c
The elastic behavior after cracking is determined by Eq. (2.13)
where now [CO] is substituted in place of the isotropic properties of c
Eq. (2.15a).
It is possible, of course, that the concrete could be stressed
into a second cracking system. This occurs if the concrete stress in the
14
¢c- di rection also reaches the 1 imiting value ft
= af , c
i . e.
-0 0 2 rh 0 . 2rh 2 0 . f ( ) ° 011 cos ~c + 022 sin ~c + 012 cos¢c sln¢~ a c 2.17
After the formation of a second crack system only the steel
reinforcement is effective. In which case the effective material property
mat ri xis
[c] 1 2 11 1 [c ] + 112 [c ] (2. 18)
2.3.3 Plastic behavior after cracking
In cracked concrete the following possibil ities for plastic
behavior exist:
1) Reinforcement yields in tension or compression
2) Reinforcement 2 yields in tension or compression
3) Concrete yields in uniaxial compression in the crack direction.
The corresponding yield and loading criteria are
Rl:
and
R2 :
and
- 1 - (1 2rh 1 . 2rh 2 1 rh' rh ) 0 + a - 0p = ~ 011 cos ~l + 022 sin ~1 + 012 cos ~1 sln~l - 0p =
-2 + a - ° == + p
(2.19a)
( 2 2rh 2. 2~ 2 2 rh' rh ) 0 11 cOS ~2 + 022 sin ~2 + 012 cos~2 sln~2 - 0p o
(2.19b)
15
(The upper sign corresponds to yield in tension, the lower sign to
yield in compression).
Concrete: _0° - f c
( ° 2tf, 0. 2tf, 2 ° .) f °11 cos ~c + °22 sin ~c + °12 cosCPc slncpc - c
and
The fol lowing notation is introduced
< 0 1> < cos2
CPl ' · 2cp cosCPl s i nCP1) (2.20a) ° sin l' p p
< 02>p <cos2
CP2' · 2¢ cosCP2 sinCP2> (2.20b) = ° sin 2' p
< oOlp f < cos 2cp c' · 2¢ cos¢2 sin¢c > (2.20c) sin , c c
Plastic behavior of a cracked region is possible in one of the
following combinations:
1) Cracking in one direction
la) One component, Rl, R2 or concrete yields, other two
remain elastic
1b) Two components yield while the third remains elastic
1c) All three yield
2) Cracking in two directions
2a) Reinforcing yields in one direction while the other
remains elastic
2b) Reinforcing in both directions yields
°
16
In case la), for example, when the steel Rl is yielding, the
stress-strain relations are
{a} o 2 1 (~o [C ] + ~2 [C ]) {s} ~ {o }p
(2.21a)
{do} o 2 (~o [C ] + ~2 [C ]) {ds}
Similarly for case 2b)
{do} {a}
The relationships for the other combinations are obtained in an analogous
manne r.
In case of unloading, i.e. when the loading criteria in (2.19)
are not satisfied, the elastic relations should be used.
2.3.4 Plastic behavior: concrete yielding in biaxial compression
The case where concrete is yielding in biaxial compression
probably occurs very seldom in actual two-dimensional structures. Concrete
is assumed to yield jf the yield criterion (2.6b)
1 oct I2_S_f ( ) 3 2S-l c = 0 °1 < 0 and °2 < 0
17
is satisfied as well as the loading criterion (3f/30 . . )do .. = O. A IJ IJ
more suitable form for the loading criterion is given later. Further,
it is assumed that the yielding process of concrete is not· affected by
the re i nforc i ng.
The relations between stress and strain increments are deter-
mined by the normal ity law of the theory of plasticity. For this end,
the three-dimensional relationships have to be derived first. The
elastic strain increments and stress increments are related by Hooke's
law
[8] {do} (2.22a)
The plastic strain increments are given in (2.8), from which the vector
form fo 11 ows
(2.22b)
Here, the vectors are
< ~~ >= °33- P 2° 12 2° 23 2
T
0 31 ) + b, + b, -T--' -T--'
Toct oct oct oct
The total strain increment is the sum of elastic and plastic parts
18
It is desirable to find the inverse relations
{do} [AJ {dE}
This has been done in Appendix. The elements of the matrix [AJ are
according to (A12)
A .... I I I I
A .... I I J J
A ... k I I J .
A .... I J I J
=
coo
-
p + ~b)2 I I
] 2c[ ~ - 'Tact 1- 2v ~
(i=1,2,3, no sum) (2.24a) 1-2v l+v b2) 3 (1 +-
G
2G 3
I .i-
2G 3
I
1- 2v
(0 .. - P 1 + 9 (0 .. - P
II +_v_b JJ 'T 1-2v \'T oct ' oct (i+j ;j=l ,2,3, no sum)
(2.24b)
(0 i i - P 1 +V b-\ 0 j k 'T + 1-2v -:J oct 'Tact U+k; i,j ,k=l ,2,3) (2.24c)
2 (~ij Y
] oct (i+j; i ,j=l ,2,3) (2.24d) 3 l+lJ b2
+ 1=-2
(2.24e)
19
Equation (2.23) with coefficients (2.24) represents the three-dimensional
constitutive relations. Since we are dealing with two-dimensional case
the relationship suitable for plane state of stress
should be found. From the condition
follows (Notice that d€23 = dS 31 = 0)
From this we conclude that
(a, S 1 ,2) (2.26)
Thus, the incremental stiffness matrix for the plane state of stress is
.1- .'- .1-
A;' 111 A;' 122 A;' 112
.1- .1- .1- .'-[A" ] A;' 122 A;222 A;212 (2.27)
.'- .1- .1-
A;' 112 A;212 A;'212
20
The loading criterion (A14) can be manipulated into a form
suitable for plane state of stress
{.:.. 0, loading
< 0, unloading
where [CO] is the matrix given in Eq.(2.15a), {dE} the incremental
strain vector in plane state, and
+ b, + b, ° :~12 /
oct
For the reinforcing either the elastic relations are val id or the rein-
forcing is yielding in compression. Referring to (2.20) the stresses
are in case of yielding
(2.29 )
2.4 Unloading
Two different cases of unloading can occur:
1) unloading after plastic yielding. In this case the elastic
relations are appl icable.
2) unloading where cracks are closing. This will occur if com-
pressive stresses tend to develop perpendicularly to the crack direction.
r" _____ .. __ .l..1~.
\.,url:::>eyuerILIY, ~~ .L..L __ _ I I Lller e is cracking in one ..J: ___ .... : ~_ ~~ 1.·,
U I It::\.... L I VII VII I Y ,
crack wi 11 close if
21
E + VE= < 0 (2.30) 1
where E1 denotes the strain perpendicular and E the strain parallel
to the crack. If there are cracks in two (perpendicular) directions,
then one crack will close if
E < 0 1-
(2.31)
After closing of a crack the behavior of concrete is the same as before
the opening of that particular crack.
22
3. THE FINITE ELEMENT SOLUTION
3.1 The Elastic State
In the elastic region the finite element procedure follows
the common pattern (cf. (39)). We assume that a certain element type
and arrangement have been chosen. The displacement field {u} is deter
mined by the equation
{u}
where {r} is the nodal displacement vector and [~] the shape function
matrix. The strain vector is obtained by differentiation
{c} [D] {u} [8] {r}
where
[8] [D] [ep]
Stresses and strains are related by the equation
{a} [c] {c}
23
where [C] is the pertinent elasticity matrix (2.14). The finite element
equation for the solution of the nodal displacements is
[K] {r} {R}
where [K] is the st i ffness matri x of the system and is determi ned from
[K] J [B] T [C] [B] dA A
and {R} the vector of nodal forces
{R} J [B]T {F} dA + J [B]T {T} dS A S
The vector {F} defines the load field and the vector {T} the prescribed
boundary forces.
24
3.2 The State of Cracking and Yielding
Once cracking and/or yielding has started, the load must be
appl ied in increments in order to trace the nonl inear load~deformation
behavior of the structure. The unstable cracking phenomenon, in particular,
causes abrupt changes in element stiffnesses and, in this way, also in the
system stiffness. Small increments of load enable a closer modeling of
the true behavior of the structure and stabil ize the iteration process
to be used in the solution.
The incremental iteration procedure adapted for this study
has similar features with those used in (40) and (41). The change in
the elastic stiffness matrix due to cracking at· each load increment is
taken into account. Further, the stresses released by cracking and the
plastic deformation developed in the system are transformed into nodal
pseudo-loads to be distributed through the structure using the current
elastic stiffness matrix. Unloading cases (closing of cracks or reveral
of plastic strains) can be handled in a similar way.
Assume that a stable equil ibrium position has been found by
iterations pertaining to the load increment {~ lR}. The corresponding n-
total load and displacement vectors are denoted by {Rn- 1} and {rn- l },
respectively. The elastic stiffness matrix [K 1 J relating to this n-,
configuration makes allowance for the changes due to the cracking which
has occurred up to this point.
of the external load is appl ied.
In the (nth' step, the increment {~ R} n
As starting values for the iteration
the final values from the previous step are chosen:
25
The first displacement increment of the nth step is
{6 r} n
[K ]-1 {6 R} nOn
These displacements are then processed through the analysis procedure
including computation of new stiffness matrix forming an iterative cycle.
A typical step in this iteration cycle is as follows:
(The subscript n is omitted for convenience)
1) Determine the ith displacement increment
{6r} . I
where {LP}. 1 is the pseudo-load corresponding to cracking and plastic 1-
defornation in the (i-l)th iteration cycle.
2) Determine the strain increment
{6S}. I
[B] {6r}. I
and the corresponding elastic stress increments of the distinct constituents
where the elasticity matrices [C k] are given by Eqs. (2.15-16), (For
tota lly cracked concrete [CO] = [0]).
3) Evaluate the litrueil stress increments
k {60 }. 1
k [C ] i-l {6S} (k=0,1,2)
The incremental elasticity matrix [Ck]. 1 is determined as follows:
1-
a) If a material component in an element is elastic, elastic
relat ions (2.15-16) are used.
b) If a component is yielding, i.e. k f({o }i-l) = ° and the
pertinent loading criterion is satisfied, the elasto-plastic relations,
Eq. (2.21) or (2.27) are used.
c) If a component is plastic, i.e. k f{o }i-l) = 0, but unloading
occurs, the elastic properties are val id
4) Evaluate the stress which has to be supported by body forces
{60}~ 1
and compute the corresponding pseudo-load
{r }. n 1
{6P}~ J [8]T {60}~ dA I
A
5) Store the total quantities so far
{r}. 1 + {6r}., {s}. n 1- 1 n 1
k {s }. 1 + {6S}. l' {o }. n 1- 1- n 1
{ k} {"ok}. o . 1 + u-n 1 - 1
27
6) For uncracked elements check whether the cracking criterion
(2.6a) or (2.17) at the stress {aO
}. is satisfied or violated (for a cracked I
element the closing criterion (2.30) or (2.31) is to be checked). In case
of cracking, a pseudo-load vector
{6P}~ {a}~ dA I I
is evaluated, where
{o}~ = I i
Sin2cp l c I
2 I cos cP j
I-cos</> CSin </> L. c c
is the stress released in cracking. a~ is the maximum tensile stress
of concrete and the angle cP defines the crack direction. For cracked c
elements the elasticity matrix (2.15a) is replaced by that of cracked
concrete (2.16) (in case of second crack concrete stiffness is nil).
In this way, a new approximation of the elastic stiffness matrix [K]. is I
obtained.
Check whether the stress {ok}. satisfies the pertinent I
yield criterion. If f({a k}.) > 0, then the stress vector is brought back I
to the yield surface. For reinforcing steel or concrete 'in uniaxial
compression the corrected stress is simpiy the yield stress (2.20). For
concrete in biaxial compression, the correction is achieved by the formula
o {a }i, corrected
o {a }. I
This kind of correction keeps the stress vector always in 011022012-
subspace.
8) Compute the increment of the pseudo-load
{6P}. = {6P}~ + {6P}7 I I I
If it is not small enough, go back to 1).
where
The yield criterion is
T oct
29
APPENDIX
o
The associated flow rule is
ds~ . IJ
\ ~ = ~ (0 i j _ _p _ + ) 1\ d 3 0.. o .. b ° i j T oct I J Toct I J
For elastic strain increment Hooke1s law
ds~. IJ
(A 1) .
(A2a)
(A2b)
is assumed to hold. The total strain increment is the sum of elastic
and plastic parts
ds .. IJ
ds~. + ds~. I J I J
(AS)
30
The inverse of (A4) is
dO' •• I J
Us i ng (A5) and (A3) one can wr i te
dO' .. IJ
At yielding state the relation
holds, so that
3f ~
IJ dO' •.
IJ
3f -",- dO'.. 0 00'. . I J
IJ
C ~ d~ - \ C 3f df i j k 1 d0 i j c-k 1 /\ i j k 1 d0 i j d0 k 1
From this equation A can be solved
C ~d~ "IJ"kl "\ c-kl 00" "
IJ A = -------
(A6)
(A7)
o
31
Inserting the value of A into (A7), the relationships
are obtained, where
Ai j k 1 C i j k 1
do .. fJ
c .. f J pq
-C pqrs
c af af rsk 1 acr acr pq rs
af af acr acr pq rs
(A9)
(A 10)
By using the actual expressions of Cijk1 from (A6) and af/aoij
from (A3)
the relations
af C. . a
fJrs 0rs
C af
pq rs ao pq
3.3
G(oij - 0 .. _p_+ 0 ~b) L oct f J L oct i j 1-21-1
(A 1 1 a)
(A 1 1 b)
are found. Substituting these into (A10) the stiffness coefficients
32
(A 12) o ..
(_IJ __ 8 _p_ + T iJ' T
o 8 .. ~ b) (~-
IJ 1-2").l T t 8 -P-+8 ~b)
k1 T kl 1-211 2 3
oct oct
are finally obtained.
The loading criterion is
af -"\- do .. 00. • I J
IJ
oc oct I-"
o for loading i j
-<, (A 13) l < 0 for unloading
Since the coefficient A must be nonnegative, the formula (A8) suggests that
the expression
(~ 0 for loading af d ~-~ skl.
IJ L< 0 for unloading
(A 14)
could be used as a loading criterion.
33
REFERENCES
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2. Private Communication between A. L. Parme of Portland Cement Assoc. and W. C. Schnobrich of the University of Illinois.
3. Abovsk ii, V. P., Abramov i ch, K. G., G 1 e i zer, M. A. and Ku 1 i ush in, A. M., Eksperimenta1 1 nye iss1edovania sbornykh zhe1ezobetonnykh obo10chek. Krasnoiarskoe knizhnoe izdate1 Istvo g.Krasnoiarsk, 1966.
4. Adler, F., and Lusher, J. K., liThe Analysis and Design of Wrexham Swimming Pool Shell Roof ,II lASS Congress Internat iona1 Sobre 1a Ap1 icacion de Estoucturas Laminares en Arquitectura, Mexico D. F., 1967.
5 • Betero, V. and Cho i, J., IIChem i ca 11y Prestressed Concrete Hyperbo 1 i c Paraboloid Shell ModellJ, Proceedings, World Conference on Shell Structures, San Francisco, Calif., 1964, National Academy of Sciences, Washington, D. C.
6. Griggs, P. H., "Buck1ing of Reinforced Concrete Shell Structures,1J Report R68-85, Dept. of Civil Engineering, MIT, November 1968.
7. Jones, L. L., IITests on a One-Tenth Scale Model of a Hyperbolic Paraboloid Shell Roof ,II Technical Report TRA/334, Cement and Concrete Association, London, August 1960.
8. Jones, L. L., IITests on a One-Sixth Scale Model of a Hyperbolic Paraboloid Umbrella Shell Roof,11 Technical Report TRA/347 Cement and Concrete Association, London, January 1961.
9. Jones, L. L., G. D. Base, B. J. Corcoran and G. Somerville, Tests on a 1/12 Scale Model of an Ell iptica1 Paraboloid Shell for Smithfield Market, Cement and Concrete Association, London, 1964 (7).
10. Long, J. E., l'Experimenta1 Investigation of the Effect of Edge St iffening on a Square Hyperbol ic Paraboloid Shell ,II Technical Report TRA 400, Cement and Concrete Association, London, December 1966.
11. Munro, J. and Ahuja, B. M., I~n Investigation of the Strain Distribution in Reinforced Concrete Shallow Thin Shells of Negative Gaussian Curvature," Proceedings, Symposium on Shell Research, Delft, 1961., North Holland Publishing Co., Amsterdam.
12~ Panas, G. E. V., l'Experimental Investigation of Stresses in R. C. North Light Shell Roof,11 M.Sc. Thesis, Imperial College, London, 1956.
34
RE FERENCES (Cont i nued)
13. Rowe, R. E., IITests on Four Types of Hyperbol ic Paraboloid Shell ,II Proceedings, Symposium on Shell Research, Delft, 1961, North Holland Publ ishing Co., Amsterdam.
14. Thuramn, A. G. and Herman, G. J., IIModel Studies of a Concrete Hype r b 0 1 i cPa r abo 1 0 i d , I I J 0 urn a 1 0 f S t r u c. D i vis ion, AS C E, Vo 1. 88., No. S T6, Dec. 1962, pp. 161 -181 •
15. Yu, C. Wand Kriz, L. B., IITests of a Hyperbol ic Paraboloid Reinforced Concrete Shell ,II Proceedings, World Conference on Shell Structures, San Francisco, 1964, National Academy of Sciences, Washington, D. C ..
16. 01szak, W., and Sawczuk, A., IIInelastic Behavior in Shells,11 P. Noordhoff Ltd., Groningen, 1967.
17. Mendelson, A., and Manson, S. S., IIPractical Solution of Plastic Deformation Problems in the Elastic-Plastic Range,I' NASA TR R-28, 1959.
18. Stern, P., "Elastic-Plastic Analysis of Shells of Revolution by a Finite Difference Method,I' LMSD-288183, Lockheed Missiles and Space Division, Lockheed Aircraft Corporation, January 1960.
1 9 . Shoe b, N. A. and S c h nob ric h, W. C., I 'A n a 1 y sis 0 f E 1 a s t 0 - P 1 as tic She 1 1 Structures,'1 Civil Engineering Studies, Struc. Research Series No. 324, University of III inois, August 1967.
20. Fowler J. N., IIElastic-Plastic Analysis of Asymmetrically Loaded Shells of Revolution,11 M.S. Thesis, Dept. of Aeronautics and Astronautics, MIT, 1967.
21. Khojasteh-Bakht, M., '~nalysis of Elastic-Plastic Shells of Revolution Under Axisymmetric Loading by the Finite Method,I' SESM Report 67-8, Structural Engineering Laboratory, University of Cal if., Berkeley, April, 1967.
22. Marcal, P. V., 'ILarge-Deflection Analysis of Elastic-Plastic Shells of Revolution,I' Proceedings, 10th. ASME/AIAA Structures, Structural Dynamics and Materials Conference, New Orleans, April 1969~
23. Whang, B., "Elasto-Plastic Analysis of Orthotropic Plates and Shells,I' Research Report R68-83, Dept. of Civil Engineering, MIT, 1968.
35
REFERENCES (Cont i nued)
24. Witmer, E. A., and Kotanchik, J. J., "Progress Report on Discrete-Element Elastic and Elastic-Plastic Analyses of Shells of Revolution SUbjected to Axisymmetric and Asymmetric Loading," Proc. 2nd. Conf. on Matrix Methods in Struc. Mech., Wright-Patterson Air Force Base, Ohio, 15-17 October, 1968.
25. Yaghma i, S., I II ncrementa 1 Ana lys is of Large Deformat ions in Mechan i cs of Solids with Applications to Axisymmetric Shells of Revolution," SESM Report No. 68-17, Structural Engineering Laboratory, University of Cal ifornia, Berkeley, 1968.
26. Freudenthal, A., "The Inelastic Behavior and Failure of Concrete," Proceedings, First U.S. National Congress of App1 ied Mechanics, 1951, pp. 641 -646.
27. Balmer, G. G., "Shearing Strength of Concrete Under High Triaxial Stresses," Laboratory Report No. SP-23, U.S. Department of the Interior, Bureau of Reclamation, October 1949.
28. Bellamy, C. J., "Strength Under Combined Stress," ACI Journal, Proceedings, Vol. 58, No.4, October 1961, pp. 367-381.
29a. Bresler, B. and Pister, K.S., "Failure of Plain Concrete Under Combined Stresses ," Transact ions, ASCE, Vol. 122, pp. 1049-1068.
29b. Bresler, B. and Pister, K. S., "Strength of Concrete Under Combined Stresses," ACI Journal, Proceedings, Vol. 55, No.3, September 1958, pp. 321-345.
30. Hannant, D. J. and Frederick, C. 0., "Failure Criteria for Concrete in Compression," Magazine of Concrete Research, Vol. 20, No. 64, September, 1968, pp. 137-144.
.. 31. Kupfer, H., Hi 1 sdorf, H. K., Rusch, H., "Behav i or of Concrete Under
Biaxial Stresses,"ACI Journal, Proceedings, Vol. 66, No.8, August 1969, pp. 656-666.
3 2 • M c Hen r y, D. and Ka r n i, J., "S t r eng tho f Comb i ne d Ten s i 1 e and Com pre s s i ve Stress," ACI Journal, Proceedings, Vol. 54, No. 10, April 1958, pp. 829-839.
33. Richart, F. E., Brandtzaeg, A., and Brown, R. L., IIA Study of the Failure of Concrete Under Combined Compressive Stresses," Bulletin No. 185, University of Illinois, Engineering Experimental Station Nov. 1928.
36
REFERENCES (Cont i nued)
34. Rob i nson, G. S., "Behav i or of Concrete Under B i ax i a 1 Compress ion, II J. Struct. Div. ASCE, Vol. 93, No. ST1, February 1967.
35. Vile, G. W. D., "The Strength of Concrete Under Short-Term Static Biaxial Stress," Proceedings of an International Conference on the Structure of Concrete, London 1965, Cement and Concrete Assoctation, 1968.
36. Weigler, H. and Becker, G., "Uber das Bruch-und Verformungsverhalten von Beton bei mehrachsiger Beanspruchung," Der Bauingenieur, Vol. 36, Heft 10, pp. 390-396, October 1961.
37. Wast1und, G., Nya ron anagaende betongens grundlaggande hal1fasthetsegenskaper," Betong (Stockholm), Vol. 3, 1937.
38. Nadai, A., Theory of Flow and Fracture of Sol ids. Vol. 1 2nd Edition. McGraw-H i 11, 1950.
39. Zienkiewicz, O. C. and Cheung, Y. K., The Finite Element Method in Structural and Continuum Mechanics. McGraw-Hill, 1967.
40. Zienkiewicz, 0., Va11iappan, S. and King, I. P., "Elasto-P1astic Solutions of Engineering Problems; 'Initia1 Stress l
, Finite Element Approach," Int. J. Num. Meth. Engng. Vol. 1, 75-100, 1969.
41. Frankl in, H. A., "Non1 inear Analysis of Reinforced Concrete Frames and Panels," SESM-Report No. 70-5, Univ. of California, Berkeley, March 1970.
42. Cervenka, V., "Inelastic Finite Element Analysis of Reinforced Concrete Panels Under In-Plane Loads," Ph.D. Thesis, University of Colorado, 1970.
37
Mohr
-1.2
--+~~--+-----+----f--t-----t----j;~t-- -1.2-t-----
Fig. 1 Biaxial Strength of Concrete.
38
--~ -------,.....--
Fig. 2 Fai lure Mode for Concrete in Biaxial Compression.
39
Initial Elastic
Behavior
C rack in One Direction
Elastic Behavior
Concrete Yields in
Biaxial Compression
Cracks in Two Directions E las t ic 8ehovioi
......... ~.....( --v
Re inforcement
Yields
Concrete Yields in
Uniaxial Compression
Re inforcement Yieids
Fig. 3 Flow Diagram of Stages in Behavior of Reinforced Concrete Material
Concrete Uniaxial
40
2nd. Cracking of Concrete
\ \ I
Yielding i~/\ ~concrete Yielding i~ Compression \ Biaxiai Compression
o i 0"22,fLj0"22
/ . 2
fL20"P Sin 4>2
R2 Yield ing
RI Yielding
I sf. Cracking of Concrete
Fig. 4 IIYield criterionl'of reinforced concrete. Intersection of the yield surface with the 01l022-plane.
4\
~\
F i 9' S
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