-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

1. Bouma, A. L., et a1., "Investigatior.~ on Models of Eleven Cylindrical Shells Made of Reinforced Concrete and Prestressed Concrete,11 Proceedings of the Symposium on Shell Research, North Holland Publ ishing Co., Delft, 1961.

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