AD-A091 738 CONSTRUCTION TECHNOLOGY LABS SKOKIE IL F/G 13/13DES1GN CRITERIA FOR DEFLECTION CAPACITY OF CONVENTIONALLY REINF--ETC(U)OCT 80 N IGSAL, A T DERECHO N6830S-79-C-0009

UNCLASSIFIED CEL-CR-80.026 NL

,ENOMONEEEEuLluulumlin

llElhllEEE~lI-

ElllEEllEEI-.mE...'.-ll

LEVELon CR 80.026

CIVIL ENGINEERING LABORATORYNaval Construction Battalion CenterPort Hueneme, CA

Onsored byAVAL FACILITIES ENGINEERING COMMAND

DESIGN CRITERIA FOR DEFLECTION CAPACITY OFCONVENTIONALLY REINFORCED CONCRETE SLABS, . ,PHASE I - STATE-OF-THE-ART REPdRT "

October 1980

An Investigation Conducted by

CONSTRUCTION TECHNOLOGY LABORATORIESStructural Analytical Section5420 Old Orchard RoadSkokie, Illinois 60077

c3 N68305-79-C-0009

LLU_.J

Approved for public release; distribution unlimited

______ 80 11 04 012* *;'J . .. J -

Unclassified9/SCU1 Lt6,~PCTIO14 OF THIS PAGE (When Dat1.1-d.r~)

READ INSTRUCTIONSREOTDOCUMENTATION PAGE BEFORE COMP ETiNG FORM,Cf. N an 2. GOVT ACESO . CIPIENT'S CATALOG NUMBER

t1T1 '- .. ~- *-~~ S.TYPE OF REPORT A PERIOD COVERED4 ESIGN fRITERIA FOR.DEFLECTI0NJ;APACITY OF6 NEN 1ONLYENOCD~NRT§A Final Jan 4979 - Aug 1979

PHASE I STATE-OF-THE-ART REPORT. .PROMN RG EOTNME

7. AUTNOR(eJ 8. CONTRACT OR GRANT NUMDEW.)

Mohammadliqbai AEAOAOESt L.IISElL2!TIJC.TS

-)CONSTRUCTION TECHNOLOGY LABORATORIES AREA& OKUINUES

A Division of the Portland Cement Association FC53 94 .135420 Old Orchard Rd., Skokie, Illinois 60077 _______,_______

11. CONTROLLING OFFICE NAME AND ADDRESS _ 2RIW1,A -

Civil Engineering Laboratory OctNUUE O8PGE

Port Hueneme, Ca. 93043r.NUBROPAE14. MONITORING AGENCY NAME A AOONESSti dille-#n Itent C-1-1.,I10.8 Off(..) 15. SECURITY CLASS. ("E *7

Unclassified

('P1 . &~9/ .SCHEDULE

is. OISYRtIUTION STATEMENT (.1 dol.Pp#

Approved for public release; distribution unlimited.

17. DISTR18UTION STATEMENT (of th. .b.I,.cI .IeH 0. Blockh 20. If dil.,.I hen R.P..I)

S. SUPPLEMENTARY NOTES"

Reinforced Concrete Tensile Membrane Action

Reinforced Concrete Slabs

AEIST RACT (C-90-P., -*~ M-- I id* it end 141-IItr 6V Week -6-1..I)

results. A method for predicting incipient collapse deflection, when tensile membrane resis-* tance can be developed, is recommended for further study in Phase 1I of a three phase study.

The recommended relatfionship gives incipient collapse deflection as a function of the Ishort span of the slab and the rupture strain of the reinforcing steel.A

DD JN7 1473 EgorTOm OF 1 Nov s OS OLETE Unclassified\.,SECURITY CLASSIFICATION OW rHIS PAGE (01401 007110f"90)~

TABLE OF CONTENTS

Page

1. INTRODUCTION 1

*1.1 Objectives and Scope 11.2 Background 21.3 Definition of Incipient Collapse Deflection 2

K2. METHODS OF SLAB ANALYSIS FOR DESIGN PURPOSES 3

2.1 Elastic vs. Plastic Theory 32.2 Slab Design Using Plastic Theory 3

2.2.1 Yield-line Theory 42.2.2 Lower-bound Approach 42.2.3 Membrane Action in Reinforced

Concrete Slabs 62.3 Load-Deflection Relationship 6

3. COMPRESSIVE MEMBRANE ACTION IN TWO-WAYRESTRAINED SLABS 9

3.1 Review of Previous Investigations 93.2 Ockleston's Work 93.3 Wood's Work 11

3.3.1 Experimental Investigation 113.3.2 Analytical investigation 133.3.3 Load-Deflection Relationship 16

3.4 Sawczuck's Work 173.5 Park's Work on Compressive Membrane Action 19

3.5.1 Experimental Investigation 193.5.2 Analysis of Compressive Membrane

Action 243.6 Hopkins and Park's Work 273.7 Morley's Work 303.8 Work at Rutgers University 303.9 Work at the Massachusetts Institute of

Technology 37H3.10 Moy and Mayfield's Work 41

3.11 Datta and Raimesh's Work 413.12 University of Illinois Tests 463.13 Desayi and Kulkarni's Work 49

*4.* TENSILE MEMBRANE ACTION IN TWO-WAY RESTRAINED SLABS 51

4.1 Park's Work on Tensile Membrane Action 514.2 Keenan's Work 524.3 Work at U.S. Army Engineer Waterway

Experiment Station 554.4 Herzog's Work 614.5 Hawkins and Mitchell's Work 65

TABLE OF CONTENTS (Continued)

Page

5. MEMBRANE ACTION IN TWO-WAY SIMPLY-SUPPORTED SLABS 67

5.1 Wood's Work 675.2 Taylor, Maher, Hayes and Morley's Work 715.3 Work at U.S. Army Engineer Waterways

Experiment Station 785.4 Sawczuk and Winnicki's Work 805.5 Kemp's Work 805.6 Brotchie and Holley's Work 855.7 Desayi and Kulkarni's Work 87

6. MEMBRANE ACTION IN ONE-WAY SLABS 90

6.1 Christiansen's Work 906.2 Robert's Work 906.3 Park's Work 936.4 Other Investigations 93

7. NONLINEAR FINITE ELEMENT MODELS FOR REINFORCEDCONCRETE SLABS 98

7.1 Review of Finite Element Models 98

7.2 Use of Program ADINA 103

8. PLASTIC METHOD TO DETERMINE DEFLECTION CAPACITY 104

8.1 Idealized Load-Deflection Behavior of aRestrained Strip 104

8.2 A Comparison with Experimental Results 110

9. DEVELOPMENT OF DESIGN CRITERIA 113

9.1 Introduction 1139.2 Restrained Two-Way Slabs 1139.3 Simply-Supported Slabs 1139.4 One-Way Slabs 1149.5 Parameters Affecting Slab Behavior 115

9.5.1 Short Span of Slab 1159.5.2 Lateral Movement of Slab Edges 1159.5.3 Span-Depth Ratio 1159.5.4 Combined Short Span-Steel Breaking

Strain Effect 115

-ii-

I If l

TABLE OF CONTENTS (Continued)

PAge

9.6 Comparison of Existing Design Methodand Test Data 120

9.7 Selection of Approach to DetermineIncipient Collapse Deflection 123

10. SUMMARY AND RECOMMENDATIONS 125

ACKNOWLEDGMENT 127

REFERENCES 128

TABLES 136

IIAce7

~C 13B

JUticati on

fllStrlbuti

1. INTRODUCTION

1.1 ObJective and Scope

The primary objective of this investigation is to develop designcriteria for conventionally reinforced concrete slabs understatic uniform load based on the incipient collapse condition.Major emphasis is placed on the deflection capacity associatedwith incipient collapse. This involves a reexamination of therelevant design criteria contained in NAVFAC P-397, *Structuresto Resist the Effects of Accidental Explosions" (1), in thelight of experimental and analytical data that have becomeavailable since the publication of the manual in 1969.

The investigation has been subdivided into three phases. UnderPhase I, a definition of incipient collapse for conventionallyreinforced concrete slabs is proposed. Then, based on an eval-uation of available analytical and experimental data, recommen-dations on an appropriate analytical method of estimating thisincipient collapse deflection of reinforced concrete slabs understatic uniform load is presented.

Phase II of the investigation will mainly involve a parametricstudy of a number of variables to identify the most significantamong these in terms of their effect on incipient collapse.Design criteria that will account for the major design param-eters will then be developed and presented in useful format.

Minimum design and construction requirements necessary to devel-op the tensile membrane behavior at incipient collapse will alsobe developed.

The work under Phase III will consist mainly in summarizing thework under Phases I and II in the form of a supplement toNAVFAC P-397.

The scope of this investigation is limited to one-way and two-way slabs under uniformly distributed static load near incipientcollapse.

Work accomplished during the first phase of the investigationis presented in this report. Of the two major objectives ofthis report, the first consists of a literature review of exper-imental and analytical work on reinforced concrete one-wayslabs, two-way slabs, and flat slabs with drop panels. Slabswith and without lateral and rotational edge restraints areconsidered. Particular emphasis is placed on studies consider-ing tensile membrane action. This is followed by a review ofcurrent design criteria for estimating the incipient collapsedeflection of the reinforced concrete slabs. Based on the re-view, a method for determining the incipient collapse deflectionof conventially reinforced slabs under static uniform loads isrecomended.

1.2 Background

NAVFAC P-397 (1) is a government standard for designing struc-tures subject to accidental explosions. Although this standardis simple to apply, it loes not take into account the influenceof slab geometry, section properties, boundary conditions, mate-rial properties, and load distribution on incipient collapsedeflection of reinforced concrete slabs. There is a need tore-examine the approach used in the the manual by providingrealistic design critera for the incipient collapse deflectionof conventionally reinforced concrete slabs. If NAVFAC P-397is overconser vat ive, then a significant reduction in the cost ofprotective structures for certain specific uses can be achieved.

The use of yield-line theory (2) for calculating the collapseload of reinforced concrete slabs is prescribed by NAVFAC P-397.Yield-line theory, which considers only flexural action inslabs, gives collapse load values that are theoretically upperbounds, i.e., "on the unsafe side." However, experimental in-vestigations show that the actual maximum load, and in manycases, the collapse load, are usually higher than those calcu-lated using yield-line theory. This enhancement in strength isattributed to membrane action. Several analytical and experi-mental researches have been reported in the literature. How-ever, design criteria for reinforced concrete slabs near incip-ient collapse have not been presented.

1.3 Definition of Incipient Collapse

Incipient collapse for conventionally reinforced concrete slabsis defined here as that state of a slab characterized by a dropLn the load capacity following mobilization of tensile membraneaction. The collapse condition is associated with tensile rup-ture of the flexural reinforcement. It is assumed that theslab is properly designed to preclude premature bond or shearfailuire. it is further assumed that concrete is effectivelyconfined within the reinforcing mesh so that no major gaps occurin the slab as a result of concrete fragments falling off.

-2-

2. METHODS OF SLAB ANALYSIS FOR DESIGN PURPOSES

2.1 Elastic vs. Plastic Theory

A slab system may be designed using either elastic or plastictheory. These two theories serve different purposes. Accordingto elastic theory, when a slab is loaded with small loads withinthe elastic region, stresses are proportional to strains. Plas-tic theory, on the other hand, considers the behavior of a slabwhen loaded well into the inelastic range.

An advantage of elastic theory is that it provides informationunder the action of permissible loads. It may thus be used tocalculate deflection and stress distribution under such loads.A slab analysis using elastic theory necessarily involves astudy of flexural and torsional moments at several points in aslab. For irregularly shaped slabs, it is often laborious and

sometimes impossible to apply effectively.

With the widespread use of electronic computers, the finiteelement method has become the most important tool for analyzingcomplex structures. Use of simplified elastic models, however,can only give an approximate description of structural behavior.Results are limited to load levels within the elastic range.

When structural safety is of prime importance, information be-yond the elastic limit is essential. Consequently, there hasbeen an increasing interest in understanding the behavior ofreinforced concrete in the inelastic range. Inelastic methodsof reinforced concrete design have been accepted in severalcodes.

Plastic theory provides a relatively simple means for calculat-ing the capacity of slabs and for determining design momentsthat result in a suitable safety factor against failure. Itdoes not give a unique solution, but an infinite number of solu-tions. For example, it is possible to decrease the amount ofreinforcement at one section if a corresponding increase isintroduced at another section. These solutions are not equiva-lent for design purposes, because they lead to differences indeflection, crack width, and construction costs. When plastictheory is accepted in a building code, some restrictions on itsapplications are required to prevent the selection of unsuitablesolutions.

2.2 Slab Design Using Plastic Theory

At present, there are four methods for designing slabs usingplastic theory,. These are:

1. Yield-Line Theory

2. Lower-Bound Approach

Strip Theory

-3-

I _____________A

"_________________-.x

3. Upper Bound Method including the effect of membraneaction.

2.2.1 Yield-Line Theory. Present knowledge of the yield-line design of reinforced concrete- slabs is based on Johansen'swork (2), first published in 194 3 . The theory has provedbroadly successful in predicting initial hinging load in rein-forced concrete slabs with negligible membrane forces.

Yield-line theory is based on the premise that a certain charac-teristic pattern of cracks (yield-lines) is formed that leadsto failure at ultimate load. Along these yield-lines the plas-tic moment capacity of the slab cross section is assumed tohave been reached thereby transforming the slab into a mecha-nism. Yield-line theory is based on the pure moment capacityof a slab section in the direction of the reinforcement, and assuch uses a principal moment yield criterion. The deformationof the slab takes place due to rotation of slab segments alongyield-lines. The portion of the slab between yield-lines isassumed to remain ri.gid. In addition, all elastic deformationsare neglected, as shown in Fig. 1.

in the early 19601s, yield-line theory came to be recognized asbeing only a part of the more general limit analysis of plates.Solutions obtained from it were known to give an upper bound tothe ultimate load. Collapse loads calculated using yield-linetheory were then considered essentially as unsafe solutionssince the true collapse load was thought to be less than or

* equal to that calculated from the theory.

2.2.2 Lower-Bound Approach. For a proper estimate of thecollapse load, an upper bound solution itself is insufficientand a corresponding lower bound solution should be available.Lower bound solutions are those that satisfy equilibrium andboundary conditions and provide a strictly admissible momentfield without violating yield conditions anywhere in the slab.This is essentially a safe solution since the collapse load maybe greater than or equal to the calculated value. In addition,lower bound solutions provide valuable information on the re-quired distribution of positive and negative reinforcement (3).A unique solution is obtained when yield-line theory and thelower bound solution provide identical collapse loads. Unfor-tunately, very few lower bound solutions have been found toagree closely enough with the corresponding upper bound solu-tions. Those that are available are restricted to relativelysimple cases (3,6).

The Strip Method. The strip method, developed by Hiller-borg (7), is based on the lower bound approach. Usually a slabis designed to have reinforcement in orthogonal X and Ydirections. Hillerborg considered it appropriate to deliber-ately eliminate the twisting moment, Mxyol from the plateequilibrium equation. Thus the total load carried by the slab

-4-

Momert

My

Curvature

M yield

Fig. 1 Moment-Curvature Relationship for ReinforcedConcrete Slab Assumed in Yield-Line Theory

or

Fig. 2 Arching Action in Restrained ReinforcedConcrete Slabs (from Ref. 13)

-5-

is split into two parts. Part of the load is assumed carriedby strips in the X-direction and the remainder by strips in theY-direction. The strip method provides a simple and powerfultechnique for the design of two-way slabs. It is particularlyiseful for design of flat plate structures (8,9). The approachis allowed in the Swedish Code (10), but is practically unknownin North America.

Wood and Armer (9) examined Hillerborg's strip method criti-cally and found it a powerful alternative to yield-line theory.Armer (L1) tested seven half-scale model slabs designed by thestrip method and found the method to be safe and satisfactory.

2.2.3 Membrane Action in Reinforced Concrete Slabs. Asmentioned earlier, both yield-line theory and lower bound solu-tions are based on the pure moment capacity of the slab cross-section and do not take into account in-plane forces. The pres-ence of in-plane forces results in an increase in the ultimateload to a magnitude beyond that predicted by the yield-linetheory.

Efforts to understand and utilize the considerable reserves ofstrength in reinforced concrete slabs, have steadily intensifiedsince 1955 when Ockleston (12) tested to destruction a slab ina dental hospital building in Johannesburg. It was noted thatthe interior panel of the underreinforced floor system, whichacted as a restrained slab, carried more than double the loadpredicted by the yield-line theory. In a later paper,Ockleston (13) showed that the unexpected results could not beascribed to strain hardening of the reinforcement or to theeffect of the tensile strength of the concrete. Nor couldcatenary action due to tensile membrane stresses account forthe observed behavior. It was concluded that the large increasein slab capacity was due to the development of in-plane com-pressive forces, termed "arching" or "dome action".

For underreinforced slabs, a substantial shift occurs in theneutral axis position in the post-cracking range. This createsa tendency for the slab edge to move outwards as slab deflection

increases. If the outer edges are restrained against movement,compressive forces are induced in the slab, as shown in Fig. 2.Arching action occurs because the compressive force at the cen-ter of slab acts above the slab mid-depth, while along the edgesit acts below the slab mid-depth. Due to arching action, theload-carrying capacity of a restrained slab is increased sub-stantially above that predicted by yield-line theory. The bene-ficial effect of compressive forces on slab yield is the secondfactor leading to the enhancement of slab capacity (14).

2.3 Load-Deflection Relationship

The load-deflection relationship of uniformly loaded reinforcedconcrete slabs is significantly influenced by the boundary con-ditions along the slab edges, as shown in Fig. 3. The dashed

-6-

Laterally Restrained

Simply SupportedF

< ~ D i ' Incipient CollapseLa

0

Johansen.... Load. E Restraint W

I bc nar Reitne-'

SI " I

A 8O 8 uit.

CENTRAL DEFLECTION, B

Fig. 3 Load-Deflection Relationship for Two-Way

Reinforced Concrete Slabs

-7-

t

W .-! . . ..

:3I

curve in the figure shows that a simply-supported slab deflectselastically and then elasto-plastically as the load is increasedfrom A to B. Near load stage B, a yield-line pattern developsand the slab deflects at a faster rate. Beyond this stage, theslab acts as a tensile membrane until reinfurcement ruptures at

When the slab edges are restrained against lateral movement,slab capacity is enhanced due to arching (compressive membrane)action, as shown by point D on the solid curve in Fig. 3. Be-yond D, the load carried by the slab decreases rapidly becauseof a reduction in the compressive membrane force. As point Eis approached, membrane action in the central region of theslab changes from compressive to tensile. Beyond E, the slabcarries load by the reinforcement acting as a plastic tensilemembrane with cracking penetrating the slab thickness. Theslab continues to carry greater load with increasing deflectionuntil the reinforcement ruptures at F.

In both simply-supported and restrained slabs, rupture ofreinforcement precipitates collapse. Alternately, failure ofbond between reinforcement and concrete may trigger prematurecollapse.

-8-

3. COMPRESSIVE MEMBRANE ACTION IN TWO-WAY RESTRAINED SLABS

3.1 Review of Previous Investigations

During the last three decades since Ockleston (12) publishedhis test results, extensive research into limit behavior andstrength of reinforced concrete slabs has been completed. Anumber of slabs have been tested. A brief review of theseefforts is given below.

3.2 Ockleston's Work

Ockleston (12) conducted two tests on full-scale two-way slabson the second floor of an existing three-story dental hospitalbuilding in Johannesburg. The slabs in both tests were151-10h" x 131-6" in plan and 4-1/2" thick. They were boundedby transverse main beams spaced 16 ft apart, and by secondarybeams spaced equally on either side of the longitudinal center-line of the floor, as shown in Fig. 4. Table 1 lists the dimen-sional and geometric properties of the slabs investigated.

Both slabs were tested under uniform load. In the first test,load was applied to only one of the interior slabs. In thesecond test, two adjacent slabs were loaded simultaneously. Inboth tests the slabs behaved in a similar manner.

Results showed that at working load level the slabs behavedelastically. Deflection and steel stresses at these low loadswere much less than predicted by the usual design methods. Atthe maximum resistance level, WD (corresponding to point D inFig. 3) the slabs developed yield-line patterns in reasonableagreement with those predicted by Johansen's yield-line theory.However, the loads at which a decrease in loading capacityoccurred (WD) were higher than those calculated using yield-line theory. As shown in Table 1, the ratios of observed tocalculated loads were 2.55 and 2.73. Maximum crack width atload WD was about 0.1 inch, with cracks extending rightthrough the slab thickness. The deflection 6D was about 2-1/2inches. This corresponds to an edge rotation of 1.54 degrees.

Ockleston also tested small-scale single-panel slab models andshowed that the observed load increased by arching action dueto development of compressive membrane forces. Arching actionis caused by a substantial shift in the neutral axis accompany-ing cracking of concrete and yielding of the tensile reinforce-ment. This creates a tendency for the slab edge to move out-wards as slab deflection increases. If the outer edges of theslab are restrained against any movement, compressive forcesare induced in the slab, as shown in Fig. 2. As a result, theload-carrying capacity of the slab is substantially increasedand a load greater than that predicted by yield-line theory isreached. Another factor contributing to the enhancement of the

-9-

In

U,1

CE

I4-)

-

.Po_

04

4-4

U

-- I-4

0, . .I

0

.4-)

"0

.,-)

loads is the beneficial effect of compressive forces on themoment resistance of the slab cross section.

The effect of arching action is most noticeable in underrein-forced slabs in which cracking causes large movement of theneutral axis. Arching can occur onlyi if the deflections aresmall and the horizontal spreading at the support is restrained.If the deflection becomes sufficiently large, arching actionwill disappear and may, with further increase in deflection, bereplaced by tensile membrane action.

Ockleston's unique test of the continuous floor panels in anexisting building is difficult to interpret. First, variablereinforcement in the slabs makes the analysis difficult. Sec-ond, the supporting beams deflected considerably under load.Third, partial restraint against spreading along outer edgesdid not permit development of full membrane action. Also, theincipient collapse deflection level of the slab was not reachedduring the test.

At about the same time in 1956, Powell (15) tested nine small-scale rectangular isotropic slabs with fully restrained edges.The reinforcement in each slab was varied, as shown in Table 1.Results showed that the peak resistance of a slab, WD (cor-responding to point D in Fig. 3) is significantly influenced bythe slab steel ratio. For a slab with a steel ratio of 0.25%,the measured peak resistance, WD, was 8.2 times higher thanthat given by yield-line theory.

Powell's test results further confirmed the major role that mem-brane forces play in enhancing the capacity of a slab. Further,it was shown that the effect of membrane forces on slab strengthis greater in the range of lower steel ratios. However, noattempt was made to analyze the results of the experiment.

3.3 Wood's Work

3.3.1 Experimental investigation. Wood (9) tested severalsingle-panel slabs under 16-point loading. Among these, fivetests are of special interest when examining the nature of mem-brane action in slabs. As Table 1 shows, slab Specimens FS12and F513 were fully clamped along all four edges. Specimens G5and G6 were supported on four encased steel beams, and SpecimenL2 had two opposite edges free and the other two simply sup-ported. Specimen L2 acted essentially as a one-way slab.

The first slab, Specimen FSl2, showed no sign of cracking atthe Johansen load, W~. Deflections were very small, and dueto compressive membrane action, the slab carried a load ofnearly three times W1 with just a tiny diagonal crack appear-ing on the tension side. Except for this crack, the slab atthis stage showed no sign of distress, as indicated in Fig. 5.

11

e., 4 J

F 4 J

(n 0

"-4 lQ4-

-12--

At a load over f ive times Wj, cracks near the center wereobserved. At this load, the deflection was only 10% of theslab thickness. As the load increased to 10.9 times Wj, thecenter deflection increased to about half the slab depth, withno signs of punching shear around the edges. This was followedfirst by a rapid increase of deflections hnd then sudden col-lapse, as shown in Fig. 6. The associated yield-line pattern,shown in Fig. 7, indicates distinct cracking along the diagonalsand around the periphery of the slab. This yield-line patternverifies Johansen's hypothesis that the energy in a slab nearthe limit of flexural capacity is concentrated in the bands ofcracking along the diagonals and periphery of the slab.

On reloading, a central tensile membrane developed, accompaniedby increasing diagonal crushing and cracks extending throughthe slab thickness. Unlike compressive membrane action, tensilemembrane action is stable. Both moment and stretch redistribu-tions occurred, but because of the favorable change in geometry,the deflections remained controllable. The test was terminatedwhen the deflection was about 6 in., corresponding to a load of3.6 times Wj. The load versus deflection relationship isshown in Fig. 6.

The second slab, Specimen 1S13, had both top and bottom steelbut with a reduced concrete strength. Behavior of this specimenwas similar to that of 7S12, but the incipient collapse load of7513 was 4.38 times Wj.

After evaluating the above tests and others, Wood concludedthat, for a clamped slab, the important feature is not the nega-tive support reinforcement but rather the restraint againstlateral expansion. In a clamped slab, a peripheral compressiondevelops and induces "self -prestress" in the slab, thereby re-ducing cracking e~nd deflection. As a result, yield moment andthe corresponding peak load increases. Wood argued that archingaction is not a good enough description for compressive membraneaction. Rather, it is the favorable increase in moment resist-ance due to compressive forces in the slab that enhances theslab capacity.

3.3.2 Analytical Investigation. Wood (4) presented meth-ods to determine the strength of circular, clamped slabs, usinga yield criterion including both bending and membrane stresses.

*1 He also noted that yield criteria for circular slabs are alsoapplicable to clamped square slabs. This led to his proposingthe following simplified equations for determining the collapseload of a clamped, square slab:

WD a U 6 Eq. (1)

The above equation applies to reinforced slabs where the amountof steel is light, i.e. with a steel percentage p *0.0020.

-13-

4J)

C> -f

0o~ En rU

4~

40

F- C)

NIJZc 20 0 00

~U /0 inq i

z

0

N P- avoi~u NwNHO'0

-14-

> 9

u) M -4

444J 0~~~ it,0 0

U) 0 4*. 4,4

-4 4J -

0)*>-'IVo 0 a

0 0a

XZ -

.0 wJ IU4 0) f

0)r

04 2c 0 ) 0U

f o 0

-15-4 (

For hea vily reinforced ,Iabs with p = 0.0080, the equationbecomes:

W = Wj (I + 0.3 -) Eq. (2)

D h .

In these equations,

WD = peak strength of the slab (corresponding to pointD in Fig. 3)

W. = Johansen's yield-line load

6D = deflection corresponding the point D in Fig. 3

h = slab thickness

Wood's design rule incorporates the observation that slab loadenhancement increases as the percentage of slab reinforcementis reduced. Equations I and 2 require that a reasonable valuebe used for 6 D/h. When 6 D/h is limited to 0.5, then only a15% increase is allowed when p = 0.0080.

The extreme conservatism in Eqs. 1 and 2 was justified by Woodas follows:

1. Compressive membrane action reduces deflection andcracking in a slab. This reduces incipient collapsewarning. Because of the change in behavior near col-lapse from a "slow" to a sudden condition, the loadfactor should be raised.

2. Creep buckling is possible.

3. The subject is still in its infancy and much more re-search is needed.

4. 4 recommended load factor of 4 or 5 is intended not somuch to guard against instability as to allow for thefact that the plastic theories are not on firm groundwhen concrete crushing is expected.

3.3.3 Load-Deflection Relationship. Wood (4) was thefirst to analyze reinforced concrete slabs for compressive mem-brane action. He used large-deflection plate theory and assumedthe material to be rigid-plastic. Due to the assumption thatmaterial behaves in a rigid-plastic manner, a rather surprisingload-deflection relationship was obtained. As shown in Fig. 6,the maximum calculated load for clamped Slab PS12 occurs atzero deflection. In reality an appreciable deflection occurrelbefore sufficient compressive forces were induced to increasethe slab capacity beyonl the Johansen load.

-16-

If compressive membrane action is to be utilized in the designof slab-beam floors, lateral restraint at the edges of eachpanel must be provided by the surrounding beams and panels.Lateral stiffness available has to be examined very closelybecause the development of membrane action is dependent on therestriction of very small horizontal translations. Furthermore,large horizontal forces are involved. wood (16) shoved thatsupport stiffness, air gaps and prestressing significantly af-fect slab behavior. As shown in Fig. 8, slab failure can occurwith little increase of load above that of a simple slab when arestraint of relatively low stiffness is provided. A very stiffsurround might cause an enormous increase in load aboveJohansen's load and a corresponding decrease in slab deflection.An air gap showed an initial Johansen-type failure followed byrecovery and delayed snap-through failure. Prestressing mightincrease the slab strength, but too much prestressing coulddestroy the arching action.

3.4 Sawczuk's Work

Sawczuk (17,18) applied the theory of plasticity to the anal-ysis of compressive membrane action in reinforced concreteslabs. The slabs were restrained against lateral movement, butunclamped against rotation at the edges. Assuming that theenergy of a slab near failure is concentrated in yield lines,and that the energy in the yield lines is a combination of bend-ing and membrane stresses, Sawczuk put in mathematical form theenergy equation for a slab, as follows:

nf pwdA= ij (MLil i + NLi i i) Eq. (3)

F A

where p -intensity of uniformly distributed loads

w =rate of deflection

N - membrane stress

M bending moment

w rate of rotation at yield-line

Li:a yield-line length

n a number of yield lines

1 srate of horizontal extension at yield lines

dA w element of slab area, A.

Equation 3 states that total energy due to plastic motion is

equal to the sum of bending and membrane energy dissipated in

-17-

Very Stiff Surround

Very Weak Surround6-

Stiff Specimen (smallL/0 h %

~~ 4 Less Stiff Specimen, (large Lh

< U)

Vr-ery eak_ Surround

-- - - - - -Johansen

Load

0 I23DEFLECTION

SLAB THICKNESS

Fig. 8 Effect of Stiffness on Specimen andSurround -Wood (from Ref. 16)

the yield lines. Using this energy approach, Sawczuk gave aload-deflection relationship:

Wi(- &(l-i1V - 2)(3 -2)

62 [ n+(3 -2 n) 21 Eq. (4)

where Wi - Johansen's yield-line load

11 W A Sfy/f~d

6 D= deflection corresponding to peak load

3a2 +l1 _

a - a/b, ratio of slab sides.

Due to the tedious calculations involved, this method receivedlittle acceptance in slab design. However, the analysis ishighly significant due to the innovative approach used in deter-mining the collapse load of slabs including membrane effects.Hung and Navy (19) extended this method to slabs with differentboundary conditions. This is discussed later in the text.

3.5 Park's Work

One of the most notable contributions to the understanding ofmembrane action in reinforced concrete slabs is due to Park(20-23). Park conducted extensive tests on reinforced concreteslabs and attempted to analyze two-way rectangular slabs forcompressive membrane action.

3.5.1 Experimental investigation. The geometric and mate-rial properties of four single panel reinforced concrete rec-tangular slabs tested by Park under uniformly distributed loadare given in Table 1. The test frame used is shown in Fig. 9a.The slabs, which were to be fully fixed against rotation andtranslation, were clamped to the frame as shown in Fig. 9b.Hold-down studs prevented rotation and horizontal screws bearingagainst steel plates at the slab edges prevented horizontalspread.

Uniformly distributed loading was applied upwards using a rubberbag placed underneath the slab and filled with water at therequired pressure. The pressure was measured using a Bourdonpressure gauge. Load-deflection curves for all four specimensare shown in Figs. 10a and l0b. Examples of slab yield-linepatterns after testing are shown in Figs. 11 and 12.

The test results confirmed the observation of earlier investi-gators that for restrained slabs the peak strength under mono-

-19-

Ste Bern . ~ l I" .ia Ste Studs

I" Dio Honzuit-

Scew Fw 60" ItrI

Fig. Stee TestFrameand de SpoteetdsStePBark (fomRt.e0

-20-4 Cane

U- 40-n~4 ____ Experimental

.... Theoretical

0

A3~20-

Johansen s Yield-Linei 0 LoadAI:A 1

0U-z I

0 0.5 1.0 1.5 2.0CENTRAL DEFLECTION

SLAB THICKNESS

I- 40]- Experimental

a. " Theoretical

%%J(A 40

_j A40 2.N."2-

CO ohansen's Yield-10 Line Load

-J

0. 0.5 1.0 L5 2.0CENTRAL DEFLECTION

SLAB THICKNESS

Fig. 10 Load-versus-Central Deflection Curves for Slabswith All Edges Restrained - Park (from Ref. 22)

-21-

EI 4-4

4-

0

U)

-c w

4J

4C(-) -J

4 1

-)

-a2M

* .V

(a) Loaded Face

(b UnoddFc

Fig.~~ ~ ~ 12 Yil ieIfaRetann lba

Failure ~ ~ ~ - ak(rmRf 2

-23-

tonic loading is consiierably greater than the Johansen load.It was noted that the maximum slab deflection is never greaterthan 1/500 of the short span for loads within one-third of themeasured slab capacity. Slab cracking did not become visibleuntil at least 32% of the measured loading capacity, W , wasattained for partially-hinged : ;abs and 42% for fully-clampedslabs.

On the basis of his own tests and those of others, Park madethe following observations:

1. The load calculated for a central deflection equal toone-half the slab thickness can be taken as the maximumresistance of the slab, WD -

2. Percentage of reinforcement plays an important role inthe enhancement of strength beyond the yield-lineload. Maximum enhancement of slab strength over thatgiven by yield-line theory was obtained for the lowestpercentage of steel. This was true for all supportconditions.

3.5.2 Analysis of Compressive Membrane Action. In 1964,Park (20) presented an analysis of fully-restrained two-wayrectangular slabs for compressive membrane action in the rangeDE of Fig. 3. He approximated the two-way slab by strips run-ning along the short and long directions as shown in Fig. 13.Using rigid-plastic approximation, the extra compression atyield sections was obtained from the geometry and equilibriumof rigid strips as shown in Fig. 14. It was assumed that thesum of elastic, creep, and shrinkage axial strain was zero. Acomparison of Park's theory and experimental work is shown inFigs. 10a and 10b.

Park's theory is quite simple and straightforward. The theo-retical curve is similar to the region DE of the curve shown inFig. 3. However, it does not correspond to the complete load-deflection curve (ADEF of Fig. 3) of a real two-way slab. Inaddition, an assumed value for deflection 6D , such as 0.5times the slab thickness, is necessary to estimate the slabstrength, WD. This assumption, based on limited test data,may not be applicable in all cases.

On the other hand, Park (24) argues that the assumption isconservative and that great precision in determining the de-flection 6 D is unnecessary. Iwankiw and Longinow (25) com-pared results obtained by Park's method with those using a non-linear finite element computer program. They concluded thatPark's method is quite satisfactory. A special advantage isits relative simplicity. Recently, Park (24) extended hismethod to include effects of both lateral movements and slabaxial strains. He derived coefficients for computing the reduc-tion in compressive membrane action due to these effects.

-24-

0 0

(0I4z4

.4

0 0-

LO

.4-

00

-<%

.

DW0

-25-

PiC'StIC Hinges

GL OL

Fig. 14 Mechanism of Restrained Strip(from Ref. 20)

SA A-IL ' . " 121

7/L/// Loaded Area (Interior panel)- Diameter Roller Supports

Uniform Lai N

Exterior iInterior ExteriorPanel Panel Panel

Fig. 15 Details of Loading and Supports ofSlabs Tested by Park (from Ref. 22)

-26-

Park (22) also investigated the degree of restraint stiffnessrequired at the periphery of an individual panel of a multi-panel slab-beam system (Fig. 15) to ensure full enhancement inload-carrying capacity due to compressive membrane action. The

1. Thesurrounding panels should be almost square, fordevelopment of full membrane action.

2. Tie reinforcement continuous around the edges of theinterior panel and around the outside edges of thefloor, as shown in Fig. 16, is essential to developcompressive membrane action.

3. Stretching of the tie reinforcement results in outwarddisplacement of the panel edges. This should be in-cluded in calculating the collapse load.

4. To mobilize compressive membrane action in a slab-and-beam floor system requires more steel as ties in thebeams than that saved in the panels.

3.6 Hopkins and Park's Work

Hopkins and Park (27) tested a 1/4-scale, nine-panel reinforcedconcrete slab-beam floor system. This is shown in Fig. 17.The system was designed on the basis of equations developed byPark (22) for estimating the peak resistance of slab, WD.The design load-carrying capacity was 800 psf. The design re-quired an enhancement factor (WD/Wj) of 2.00 for the inte-rior panel, 1.35 for the center edge panels and 1.00 for thecorner panels. The panels were lightly reinforced, the top andbottom steel of all panels being 0.16% and 0.15% of the grossconcrete area, respectively. Steel had a yield strength of 52ksi.

From the known steel quantities in the panels and the requiredslab strength, the maximum allowable lateral movements at thepanel edges were estimated and the maximum compressive membraneforces in the panel were calculated. These membrane forceswere then considered as uniformly distributed in-plane forcesacting outward on the surrounding beams and panels. The beamswere designed for the required strength for bending due to grav-ity loads and for tension due to membrane forces. Lateral de-formations of the floor due to axial stretch of the beams under

*1 tension as well as bending and shear deformations of the edgepanels under membrane forces were estimated. Beam sizes andreinforcement were then adjusted iteratively until the outwardmovement of the panel boundaries was approximately equal to themaximum allowed.

The f loor behaved well both at service load and at the peakload levels. The load-versus-central deflection relationship

-27-

34 in. Square

Interior Pnelfr _ 14in Squre

Pig. 16 Position of Two Rinq Bars Essentialto Develop Compressive MembraneAction in Interior Panel - Park

(from Ref. 22)

All Panels IL Thick

rIIL _ a n ~ e j tJ _ _Corner IAll Beams

!I Centr4,, IEdeI 1I 32- wide

-Panel II Panel III

44 Beam Centres

Fig. 17 A 1/4-Scale Beam-Slab System - Hopkinsand Park (from Ref. 27)

-28-

1,200

LL

a.800

0

n400 I

Z

* Test Terminated at This PointVi i I ! _ I

0 2.0 4.0 6.0

CENTRAL DEFLECTION, in.

Fig. 18 Measured Load-Central Deflection Relationshipfor Interior Panel of Nine-Panel Slab-BeamSystem - Hopkins and Park (from Ref. 27

-29-

I __ ___ ____ __ ~~~~~~~~~~* .,~~~.~ * - *.~

'I

f-r the interior panel, shown in Fig. 13, indicates a peak loadof 850 psf. The .- sociatod centrat deFlection was ilmost equalto tht- panel thickn-. ;. itgui o 1 show!- top and hottom viewsoF the floor after tho tst. The fol lowing conclusions werodr.iwn from the test rsul ts:

1. Designing to take advant:ige of the enhancement in loadcapacity due to compressive membrane action is possibleprovided an adequate safety margin is incorporated inthe design procedure.

2. Lack of information concerning the long-term behaviorof the slab may limit the applicability of compressivemembrane ict'on in design.

3.7 Morley's Work

The conventional vield-line theory of two-way reinforced con-crete ;labs was extended by Morley (28) to allow for membraneaction and moderately large deflections. Tn this approach,compressive membrane forces are calculated from a considerationof displacement rates in the assumed collapse mechanism and tlein-plane eqilibri-am of compressive membrane forces. The load-deflection relationship is established by the principle of vir-tual work. This method is also based on the rigid-plasticapproach and gives a load-deflection relation similar to thatgiven by Wood (4) and Park (20). An empirical value for thedeflection corresponding to the maximum load, 60, is necessaryto estimate the collapse load. With an assumed value of deflec-tion, 6T), equal to 0.5 times the slab thickness, the resultscompared very well with those of Park (201. This theory islimited to isotropic slabs. Secondary effects like lateralmovement or elastic shortening cannot be included easily.

3.8 Work at Rutgers University (Naw_ and Associates)

Over 100 two-way slab specimens were tested by Nawy and hisassociates (18,29-31) at Rutgers University. The objective ofthese investigations was to gain better understanding of slabswith different boundary conditions and reinforcement ratios.

Of special. interest are the twelve isotropic, fully restrainedslib specimens tested under uniformly distributed load by Hungand Nawy (19). Geometric and material properties of thesespecimens are listed in Table 1. The loading system is shownin Fig. 20.

During early load stages, the deflected shape of the slab wasthat of a parabola. As the load increased, the center deflec-tion increased more rapidly than elsewhere, and the deflectedshape of the slab became more marked. All tests were terminatedonce a reduction in slab capacity was observed. The load-deflection relationship beyond this stage was not recorded. The

-30-

2 A- I

;-oided Surt-ice

Fig. 19 Slab and EBcam Flooi After Test - HopkinsaxPark (from Ret. 27,

0 LU

w LU

Z - LJ >I?

Ir z zL U

0D E

000

-i L)

UU)

00(r a: w

6- LJ z< C)

yield-line mechanism, shown in Figs. 21a through 21c, followedJohansen's yield lines. Load-deflection curves of the testslabs are shown in Fig. 22. In general, all curves have similarcharacteristics. The curves show that the value of deflectionassociated with the peak resistance, 6D , is not constant at0.5 times slab thickness as assumed by Park (20) but variesbetween 0.6 to 1.0 times the slab thickness. Park attributesthis difference to the effect of restraint stiffness (21). Acomparison of measured peak loads with those predicted byJohansen's yield-line theory is listed in Table 1. This indi-cates that the mean experimental load is 1.80 times that pre-dicted by Johansen's yield-line theory.

Hung and Nawy (19) extended Sawczuk's (17) energy approach forcomputing the load-carrying capacity of restrained slabs. Byidealizing the slab as shown in Fig. 23, the equation for thepeak load of isotropically reinforced concrete slabs restrainedon all four edges is obtained as follows:

M2 M -( 2 Eq. (5)

where 1 1 - m(X1 +X 2)

C1 3 48 1 - + X+2) 1 8 A3 (i -X 3 )m J

48 3 +( 2 3X (l A 3)M

2 4 [3 m(X1 + "2)] 3 + 1 Xm (1- X3

M1 - fd2[ 0.24 (1 + + q (1 0.07

<f~d 2[q (1 + f) -0+ 0.07]

m4 - fd 2 [0.12 (1 + f) 0.02]

As = area of tension reinforcement per unit width ofslab

d - distance from extreme compression fiber to cen-troid of tension reinforcement

d' - distance from extreme tension fiber to centroidof reinforcement

-33-

A

IbI

Fig. 21 Y ield-Lin( Pat ternfonr Rect angulIa rSlabs Tested a tRutqers Universit%,

(from R~ef. 1.9)

c)

-34-

6,000-0.. x - Test terminated o this point

00

; -J- 4,000 4

Cn 2,0000?t C I- Series

.. oM

cc

IL Iz 0 0.5 1.0 1.5

CENTRAL DEFLECTIONSLAB THICKNESS

U_in 6,0000.o x - Test terminated at this point4

oC 4,000:-_O 2

n- C 4- Series0U-

0 0.5 1.0 1.5CENTRAL DEFLECTION

SLAB THICKNESS

Fig. 22 Load-Deflection Relationships for SlabSeries Cl and C4 Tested at Rutgers

University (from Ref. 19)

-35-

L LX3L~ X3 Ly

FX1 L-+-XL-i-1 KX I LH HX2 LiC I- SERIES C 4 - SERIES

NOTE! X1, X2, X. Define the Ideoilized Yield-line Geomeiry

Fig. 23 Idealized Segments of Slab Panels at Failure-Hung and Nawy (from Ref. 19)

-36-

V = compressive strength of concreteCf = yield strength of reinforcementy

m - ratio of short span to long span for two-wayaction slabs

q = A f /f'd = reinforcement indexs ych = total thickness of slab (in.)

WD = load/unit area corresponding to point D in Fig. 3

X 2' 3 parameters defining the yield-line geometry(Fig. 23)

6D = deflection corresponding to point D in Fig. 3

Hung and Nawy (19) further extended this approach to slabs withthree edges fixed and one edge hinged as well as to slabs withtwo edges fixed and the other two hinged. As in the case ofother methods, this method suffers from the drawback of hav-ing to assume a value of deflection, 6D in order to determinethe slab capacity, WD.

Nawy and Blair (29) tested to failure ninety two-way slab spec-imens to investigate flexural cracking behavior of slabs. Thegeometric and material properties as well as test results forfifty-one slab specimens restrained along all four edges arelisted in Table 1.

3.9 Work at the Massachusetts Institute of Technology(Jacobson, Brotchie and Holley)

Jacobson (32), and Brotchie and Holley (33) at M.I.T. testedforty-five 15-in. square two-way restrained slabs to investigateelasto-plastic behavior. The length-to-depth ratio was variedfrom 5 to 20 and the reinforcement ratio from 0 to 3%.Jacobson (32) was the first to measure the restraining forcealong the slab boundary. Typical plots of the experimentalload-versus-central deflection, load-versus-average restrainingforce along the boundary, and crack patterns are given in Figs.24a through 24c, respectively. In the first, elasto-plasticstage, the restraining force increases almost linearly withdeflection. Gradually, both restraining force and load capacityapproach a peak value. The slab subsequently fails either byinstability or in compression. In the second, plastic stage,the slab capacity and the restraining force decrease as thedeflection increases. The upper limit of deformation for thisstage corresponds to point E in Fig. 3, just before the onsetof tensile membrane action. In the tensile membrane stage, nocompressive restraining force is provided by the support.

-37-

(a)

OEFLEGTICN

(b) a

APPLIEO LCAO

(c)

(A) SPREA,;JG (B) IEC ANS. (C) TENSILFNP.ECHANISM (PLASTIC) MEMBRANL

Fig. 24 Typical Load-versus-Deflection, Load-versus-Restraining Force, and Cracking Pattern(bottom of slabs) for an Under-Reinforced

Slab with Edge Restraint Showing ThreePhases of Behavior:

(a) Loading Phase, Elastic or Elasto-Plastic,(b) Unloading Phase, Elastic or Elastic-Plastic,

and(c) Reloading Phase, Tensile Membrane

(from Refs. 32 and 33)

-38-

.. ......

Results of Jacobson's and Brotchie and Holley's experiments aresummarized below:

1. Load-deflection curves for restrained slabs with dif-ferent reinforcement ratios and with span-depth ratiosof 20 and 10 are shown in Figs. 25 and 26, respec-tively. For laterally restrained, unreinforced slabswith a span-to-depth ratio of 20, the peak load, WD ,was approximately 5,000 lbs/sq ft. For thicker slabs,i.e., slabs with lower span-to-depth ratio, the capac-ity increased more rapidly than the square of thethickness, e.g., for a span-depth ratio of 10, thepeak load was 22,000 lb/sq ft.

The increase in peak load with reinforcement ratio isless marked and added strength due to increased rein-forcement is less than the initial capacity resultingfrom external restraint alone. Magnification of loadcapacity due to the restraining effect decreases asthe reinforcement ratio increases.

2. The magnitude of the restraining force developed variesessentially linearly with thickness of the slab andwith applied loading up to the maximum arching peakload.

3. Reinforcement ratio has only a slight effect on magni-tude of the maximum restraining force. The effect ofedge restraint differs from that of reinforcement inthat it is sensitive to deflection.

4. Cracking is significantly reduced in restrained slabs,up to the peak arching load.

5. For thin slabs, with span-to-depth ratios of 10 ormore, it is necessary to provide essentially full re-straint against displacement if the full increase inload capacity is to be attained. For thicker slabs,the restraining force is still required, but additionaledge and restraint displacement may be tolerated with-out significantly reducing load capacity.

6. Tensile membrane action is significant only at largedeflections.

Jacobson (32) analyzed the test results for elastic, elasto-plastic and rigid-plastic cases. The rigid-plastic analysisprovided an uppey, bound solution and is similar in approach tothat of Morley. 82o)

Based on the studies of Jacobson (32) and other work at M.I.T.,Brotchie and Holley (33) presented an analysis for compressivemembrane action. They gave the following simplified expressionsfor the maximum (peak) unit load carried by arching action,

-39-

&gloom

120

j SP-AN-- P0DE' [H

U)

o 80

0 Reinforcement

Ld-

g40

0%

0 1.0 2.0 3.0DEFLECTION, in.

Fig. 25 Load-Deflection Relation for Restrained Slabs withVarious Reinforcement Ratios - Brotchie and Holley

(from Ref. 33)

400- SPAN 10DEPTH

L Reinfnrcement

-J

0 1.0 2.0 3.0

DEFLECTION 'in.

Fig. 26 Load-Deflection Relation for Restrained Slabs withVarious Reinforcement Ratios - Brotchie and Holley

(from Ref. 33)

-40-

- 2 _-j- - ' " - .. : i

L. ,',

WD- 6f () 2 - 0.00133 (- Eq. (6)

The central deflection corresponding to WD, and fc, is given by:

6 0.001 (h

And the maximum restraining force, Cmax, is given by:

C max = Ech 1 - 0.00033 (I!Eq. (7)

where c - 0.85f + 16p d fy 6D/L2

h = slab thickness

p = positive steel percentage

pt - negative steel percentage

f= - cylinder concrete strength

A comparison of experimental and theoretical load capacity,shown in Fig. 27, shows good agreement. However, the agreementbetween theoretical and experimental deflections and restrain-ing forces, shown in Figs. 28 and 29 is, less satisfactory.

3.10 Moy and Mayfield's Work

Moy and Mayfield (34) applied Massonet's general elastic-plastic membrane theory (35), along with a proper yield cri-terion to determine the effect of membrane action in reinforcedconcrete slabs. They applied finite difference techniques tosolve the equilibrium equations and the yield criterion curvefor a restrained slab. The same investigators also conductedexperiments to verify the analysis. However, comparison betweentheoretical and experimental results was unsatisfactory. Thiswas primarily due to the fact that the analysis did not includethe influence of concrete cracking, a mechanism that has amajor effect on the load-deflection characteristics of slabs.

3.11 Datta and Ramesh's Work

Datta and Ramesh (36,37) investigated the effect of compressivemembrane action on isolated slab-beam floors. Their analysis

-41-

0.4- @0.03

0.020 0

0.2-0.0P

a.1-P

0.05- 0.01

0.025-0.15-PC.3 Experimental

PZO- Theoretical

0 0.05 0.1 0.2- I THICKNESS

SPA N

Fig. 27 Comparison of Experimental and TheoreticalPeak Load Capacities - Brotchie and Holley

(from Ref. 33)

0. -1 0.02.0.01 00.03.0 0.005 00.01

00.

040.03010.0020.03 00.02

0.3- 00.03

o 00.005E ~

0.2-

0 Experimental0.1 - Theoretical

0 0.05 0.1 0.2

THICK N ES SSPJAN

Fig. 28 Comparison of Experimental and TheoreticalMaximum Restraininq Forces -Brotchie and

Holley (from Ref. 33)

-42-

1.0

0.8- 0 Experimental6 0.8- Theoretical

40.6-

0.4- 60001D h

LLL

0ig 29Cmaio20 xeietlan hoeia

8

Lo SPAN = 12DEPTH

6 Reinforcement Ratio =0.155%

0

a z< W4-

z0 0.67

2 0.5

0 0.25 0.50 0.75 1.00

CENTRAL DEFLECTIONSLAB THICKNESS

Fig. 30 Relation between 6 D/h and WD/Wj for VariousDegrees of Edge Restraint -Ramesh

and Datta (from Ref. 36)

-43-

considered the edge beam lateral displacement due to in-planecompressive forces in the slab. Using Park's (20) strip ap-proach, Ramesh and Datta (36) developed an expression for thein-plane compressive force as a function of edge beam lateraldisplacement. The in-plane force was calculated by prescribingedge beam boundary conditions. Knowing the axial force, stressresultants on the yield lines were calculated. The load-deflec-tion relationship was then established using the principle ofvirtual work.

Figure 30 shows the load-deflection curves obtained from thetheory for different values of, C., the degree of edge restraintprovided by the surround of an isotropically reinforced squareslab. The parameter is a non-dimensional quantity defined as

If -lE Eq. (8)

where k = coefficient determining the compressive force inconcrete in Hognestad (38) stress-block

L = sides of a square slab panel

I= moment of inertia of edge beam (for lateral de-flection)

E c = modulus of elasticity of concrete

h = slab thickness

f= cylinder concrete strengthC

Figure 30 shows that, for degrees of edge restraint belowC= 0.67, tensile membrane action predominates over compressive

membrane action. For C >, 4, a state of full compressive mem-brane action is attained.

Datta and Ramesh (37) also tested nineteen single-panel,square, isotropic slab-beam specimens. Figure 31 shows generaldetails of the test specimens. Ten out of nineteen slabs werecast with the slab located at mid-depth of the beams to avoidT-beam action. The remaining specimens had the slab cast atthe top of the beams. Load was applied equally at 64 pointsdistributed uniformly on the slab surface. Variables includedreinforcement ratio, degree of edge restraint, i, and influenceof T-beam action.

The experimental load-deflection curves are similar to thoseshown in Fig. 22. The following conclusions were drawn fromthis study:

-44-

wwj

Varies CornerBlock

-4-0"

LVaries

PLAN

Vae I f 4'0 - Varies

(a) SLAB AT THE END OF THE BEAM

WVriel-' 4'-0 '" 1 "fVaries

(b) CENTRE OF SLAB COINCIDING WITHCENTRE OF BEAM

SECTION

Fig. 31 General Details of Slab-Beam SpecimenTested by Datta and Ramesh

(from Ref. 37)

-45-

1. The def lect ion cor respond i ng ti the peak load, 1 Dincreases wi th a decrease in the degree of edgerestraint.

2. Enhancement in slib c.ipacity due to compressive mem-brane action decreases with increasing reinforcementratio.

3. For slab-beam panels wit!i partial horizontal restraintalong their edges, i.e., with 0.67, there was prac-tically no enhancement in slib capacity and the slabsacted as ten3ile membranes.

4. Slab-beam panels with -* 4 behave, as if it they werefilly restrained.

3.12 University of Illinois Tests (Girolami, Sozen, and Gamble)

Six reinforced concrete panels were tested at the University ofIllinois by Girolami et al (39). All specimens were subjectedto both transverse and in-plane loading. A number of pointloads were applied over the panel surface to simulate uniformvertical loading. Equal horizontal loads were applied at fiveequally spaced points on each side of the slab to simulate mem-brane forces. For simplicity, membrane loads were held constantduring the test.

Each panel was 6-ft square and 1.75-in. thick. The slab wascast monolithically with 6-in. deep by 3-in. wide doubly rein-forced spandrel beams, as shown in Fig. 32. Vertical loadswere also applied to cantilever extensions of the spandrel beamsto maintain restraint at the corners. Two types of boundaryconditions were considered. In the first three specimens, thepanels were supported at the corners and the beams were per-mitted to deflect freely. In the remaining three cases, thespecimens were supported at several points along the spandrelbeams to simulate simple edge support conditions.

The three corner-supported structures initially developed diag-onal yield-lines in the panel, hut finally failed in the yield-lines parallel to the panel edges. The beams participated inthe failure mechanism, as shown in Fig. 33. For the three spec-imens with nondeflecting beams, failure occurred within thepanel. The measured panel load capacity, WD, was 1.7 to 2.1times the corresponding Johansen load, Wj. The correspondingdeflection was approximately half the slab thickness.

For slabs with nondeflecting beams, the equilibrium equation atthe peak load may be written as

W L mL L (m + m') - 6 Eq. (9)

-46-

z--V

4 r A, _

--Fl

+ '10

PLAN L

I 15II "-'~

SECTION

Fig. 32 Typical Geometry of Slab-Beam SpecimensUsed in University of Illinois Tests

(from Ref. 39)

-47-

"-- -

i, .-- 7-- i '- -t -- -=-'='-i

Fig. 33 Top View of a Slab After Testat University of Illinois

(from Ref. 39)

-48-

where W = total load on the panel

L = panel span

m, m' = positive and nejative slab moments/unit width

= slab central deflection

N = total applied membrane force on one side of thepanel

In Eq. 9, deflections were assumed to vary parabolically acrossthe slab so that the mean slab deflection along a positive yieldline is given by 26/'3. Since both W and 6 are unknown in Eq. 9,an iterative procedure was used to calculate the theoreticalultimate load. The effect of membrane forces was included.Each step in this iterative procedure involved estimating andthen calculating W. The procedure was repeated until the loadgiving the estimated deflection, calculated using elastic platetheory with fully cracked sections, agreed with the load givenby Eq. 9. Tha ratio of experimental to theoretical load capac-ities estimated by this procedure varied between 1.02 and 1.17.This indicates that the load capacity of a panel can be esti-mated accurately if the membrane forces acting on the panel areknown.

3.13 Desayi and Kulkarni's Work

Desayi and Rulkarni (40) presented a method to determine theload-deflection relationship corresponding to the portion ADEof the solid curve shown in Fig. 3. The analysis is carriedout in two stages. In the first stage, a semi-empirical methodis used to calculate deflections from zero load to Johansen'syield-line load. In the second stage, an analysis consideringmembrane action is used to find the load-deflection relationshipbeyond Johansen's load.

In the first stage, results of classical plate theory are used.Cracking of concrete and yielding of steel are accounted for bysuitably modifying flexural rigidity. Changes that occur insupport conditions due to possible yielding are also considered.In the second stage of analysis, Park's (20) strip method isused with some modifications. Desayi and Kulkarni comparedcalculated ultimate loads and deflections with those obtainedfrom test results of sixty-seven slab specimens. Satisfactoryresults were obtained. Load-deflection curves obtained by thismethod are compared with Hung and Nawy's (19) experimentalcurves in Fig. 34.

-49-

4,000 les;t lermninnied (it This Point

010

S3,000-

_j P

0 2,000-

Exeieto uv

0 0.5 1.0 1.5 2.0 2.5 3.0DEFLECTION ,in.

Fig. 34 Comparison of Experimental and CalculatedLoad-Deflection Relationships

(from Ref. 50)

-50-

4. TENSILE MEMBRANE ACTION IN TWO-WAY RESTRAINED SLABS

For a slab with very stiff edge restraint, as point E in Fig. 3is approached, the membrane forces change from compression totension in the central region of the slab. At this point, theboundary restraints begin to resist inward movement of the slabedges. Initially, the outer regions of the slab act with therestraint as part of the compression ring supporting the tensionmembrane action in the inner region of the slab. With furtherdeflection beyond point E, the region of tensile membrane actiongradually spreads throughout the slab. As this occurs, loadcarried by the yielding reinforcement acting as a tensile mem-brane (with full depth concrete cracking) increases until thesteel starts to fracture at point F. For restrained slabs,point F represents the condition of incipient collapse.

4.1 Park's Work on Tensile Membrane Action

In analyzing tensile membrane action in orthotropic, restrainedslabs, Park (41) made the following assumptions:

1. All the concrete has cracked throughout its depth andis incapable of carrying any load,

2. All reinforcement has reached the yield strength andacts as a plastic membrane,

3. No strain hardening of steel occurs,

4. Only reinforcement that extends over the whole area ofthe slab contributes to membrane action.

Using standard membrane theory (4), Park (41) presented thefollowing load-deflection relationship of the plastic tensilemembrane:

wL 2 3

Hn-l Eq. (10)

n=,3,5 n n 1TL~

where w = uniform load/unit area

6 . central deflection of membrane

T x, Ty = yield force of the reinforcement/unit width inx and y direction, respectively

Lx, Ly aslab side length in x and y direction (Lx Lyy

-51-

2/To :;impLify the use of Eq. 10, values of wL ,T for variousvalue! (f X/LY and T. /T havP been plotted Yin YFig. 35. OnlyT ev T for slabs wit >,T was considered since more steelwill generally be required in the direction of the short spanthan in that of the long span.

A comparison of Eq. 10 with experimental data is shown in Fig.36. It may be noted that Eq. 10 gives a linear relationshipbetween w and 6. it provides a conservative estimate forportion EF of the experimental curve due to the followingreasons:

1. A pure plastic tensile membrane did not develop overthe whole slab. Tn lightly reinforced slabs, the loadis carried mostly by a combined bending and tensilemembrane action.

2. The assumption of no strain hardening of steel alsomakes the theory conservative.

Ba3ed on his own work and that of Powell (15), Park concludedthat a conservative value of the central deflection that can beassociated with the development of full tensile membrane actionwould be 0.1 of the short span. Any greater deflection mayresult in bar fracture. The associated loading can exceedJohansen's load when the amount of reinforcement is largeenough.

For a square slab, Eq. 10 becomes:

hfw = k(p + p') 2 Eq. (11)

L2

where f = steel stress

p, p' = positive and negative steel ratios, respectively

k = 13.5, determined from Fig. 35

Equation 11 was derived assuming a pure tensile membrane action(Tx = Ty). However, tests (41) showed that pure tensilemembrane action did not develop. Keenan (42) allowed for thisdiszrepancy in Eq. 11 by modifying the value of k.

4.2 Keenan's Work

Keenan (42) tested six fully clamped slabs under uniform pres-sure. All slab specimens were square, with a clear span of 72in. The slab thickness ranged from 3 to 6 in. and reinforcementfrom zero to 1.33%, as shown in Table I. In all six specimens,tension cracks first became visible at a load corresponding toover 70% of the Johansen load. The slabs hinged initially in

-52-

14

12 , ri r

10 L

9-:

8 t5

0

"F , t I I I ,

1.0 1.5 2.0 2.5 3.0 3.5 4.0Lx

LY

Fig. 35 Load-Deflection Relations from Eq. 10for Uniformly-Loaded Plastic TensileMembranes with Rigid RectangularBoundaries - Park (from Ref. 41)

-53-

4r_ __ _

D -

0o OL-9 Eaa 2 . 6 -

4 J 'T

Z 0 4-4

L 0U)) 0

m~ LC)J

-LJ

C -J< 0

4-)

cc

0 0

ISd GCIO-1 G31(-.91dLSIG AM OINn 0

-54-

flexure, followed by total collapse at a much greater deflec-tion. Complete load-deflection relationships for the specimensare shown in Figs. 37 through 39.

The shapes of the load-deflection curves are similar for all* slabs. However, the loss of resistance in transition from com-

pressive to tensile membrane action is least for the slab with*highest reinforcement ratio, as shown in Fig. 39. This finding

agrees with the M.I.T. (32,33) test results shown in Rigs. 25and 26.

Collapse corresponded to rupture of the reinforcement in ten-sion, with the slab acting as a membrane. The central deflec-tion at collapse was more than 2.5 times the slab thickness forthe thinner slabs.

Based on his and other's work, Keenan (42) developed a semi-empirical method to determine the load-deflection relationshipfor square, full restrained slabs. The method involves solutionof eight equations covering different ranges of slab behaviorfrom elastic through tensile membrane action.

To obtain better agreement with his test results, Keenan modi-fied the factor k in Park's pure membrane formulation (Eq. 11).He determined that a value of k equal to 20 yielded a bettercorrelation with the measured value of the deflection at sec-ondary resistance, Ss. It was assumed that the associatedsecondary resistance, W., is equal to Johansen load, WjBy setting k equal to 20, the ratio T/T beoeslsst*unity. This implies that reinforcement in the two directionsdo not yield simultaneously.

Keenan further determined that a safe maximum value for thecentral deflection associated with tensile membrane action isone-tenth of the span. This confirms Park's (40) recommenda-tion mentioned earlier. No attempt was made to extend hismethod to rectangular slabs.

4.3 Work at U.S. Army Engineer Waterways Experiment Station

Denton (43) and Black (44) tested small-scale slabs at the U.S.Army Engineer Waterways Experiment Station. Both simply-supported and restrained slab specimens were subjected to uni-form pressure.

Geometric and material properties of four restrained slabstested by Black (44) are listed in Table 1. The specimens,square in shape, were tested in a Small Blast Load Generator(SBLG), shown in Fig. 40.

The load-deflection relationships, shown in Figs. 41 through 44indicate behavior similar to that observed by earlier investi-gators. The dual peak shape of the load-deflection curves

-55-

50- SPAN 24 Steel RupturedTHICKNESS at Supr ine

40-

C. 30 -

< 2

20 Water leak /

0 0.5 1.0 1.5 2.0 2.5 3.0DEFLECTION THICKNESS RATIO, _t

Fig. 37 Load-Deflection Relationship for 3-in. Thick SlabsTested by Keenan (from Ref. 42)

100-

80SPAN 152

09

60

0_j 40

20

0 0.25 0.50 0.75 1.00 1.25 1.50DEFLECTION THICKNESS RATIO,-

Fig. 38 Load-Deflection Relationship for 4.75-in. Thick SlabTested by Keenan (from Ref. 42)

CYa

V))U))

W

U)

0) U

U)U. 0)

0) -)

U)0)0

Lj wU

x O4).-4-4

4

.- 4I~d 'C'4-I

-57-

SRLG PRESSVRE BONNET--

TEST SLA) ____ 6 BL

4-9"/SPACER

-\DEF LECTIONGAG E

SCALE' IN EC

0

Fig. 40 SBLG Test Set-up for Slabs Tested by Black(from Ref. 44)

-58-

30L

30- Restrained

II Simply Support

20a.

(4

9 (0.79% Steel)Herzog Method

l .Sab IS-I1(0.9 % Steel)

02 4 6MIDSPAN DEFLECTION, in.

Fig. 41 Load-Deflection Relationship for Slab IS-iTested by Black (from Ref. 44)

30- Restrained

Simply Support

20- lab S-2 44) --,-Wire Ruptures

F'g a-(0.79 % Steel) Computed Bw1ysing

r Herzog Method

010-

- (43)SlbIS- 2

(0.79 % Steel)

04MIDSPAN DEFLECTION, in.

Fig. 42 Load-Deflection Relationship for Slab IS-2Tested by Black (from Ref. 44)

-59-

30-

S Computed 8,1using

Herzog Mlo

(Eq. 170 2 4 6

MIDSPAN OFFL-ECTION, in.

Fig. 43 Load-Deflection Relationship forRestrained Slab IS-3 Tested

by Black (from Ref. 44)

40) RestrainedSimply Support

Wire Ruptures

20-

Computed 8, using

Herzog Method

02 4 6MIDSPAN DEFLECTION, in.

Fig. 44 Load-Deflection Relationship forSlab 115-1 Tested by Black

(from Ref. 44)

-60-

characterizes two failure mechanisms: compressive membrane andtensile membrane. Maximum deflection at the end of tensilemembrane action, 6 ult, was approximately one-fifth of thespan length for the first series of slabs. Deflection 6 ultfor the second series was about one-sixth of the span length.Slab IISI, which had 50 percent more reinforcement than theSeries I slabs, showed considerable increased strength. It wasnoted that the increased steel percentage made the slabstiffer. Cracking patterns for Slabs IS2 and IISl are shown inFigs. 45 and 46.

Black determined that the use of k = 20 in Eq. 11 is justified.He noted that Park's (41) and Keenan's (42) estimates of themaximum value for central deflection at the end of tensilemembrane action, 6 ult , are conservative. Based on his tests,Black proposed the following relationship for 6 ult:

6 ult = 0.15L Eq. (12)

Equation 12, in conjunction with Eq. 11, yielded results in

good agreement with Black's tests.

4.4 Herzog's Work

Herzog (45) presented approximate procedures for calculatingincipient collapse deflection capacity, ult, of restrainedtwo-way slabs. He determined that ult depends on two param-eters: slab span and reinforcement elongation at rupture.Herzog derived his equation using a one-way tension member inthe following manner.

Consider a cable of horizontal length, L, under a load uniformlydistributed in the horizontal direction, as in Fig. 47.. From aconsideration of cable equilibrium, it can be readily shownthat the length of the stretched cable, LI, to a first approx-imation, is given by:

8 62L . L + 8 6 Eq. (13)1 3 L

In other words, the cable elongation,

8 6 2EL - L1 - L = Eq. (14)

In limiting case, when the cable is at the point of rupture,

62SL 8 ult Eq. (15)u 3 L

6ult = Eq. (16)

-61-

* - ,.. -

(a) T'op Face

1~~ I' -

(b) Bottom Face

Fi,,. .15 Crackini Pattern for Slab 1S2 Tested by Black(from Ref. 44)

-62-

(a) Top Face

(bbotcoin Face

riq. I :.c.-il~q fctrnfor Slab IlSI Tested by Black(from Ref. 44)

-63-

....... - --- i- -- -. -.--v-a-.--

rN

L

Fig. 47 Parabolic Deflected Profile of a CableUnder a Uniformly Distributed Load

TL (lb./in.)

Fig. 48 Equilibrium of Rectangular Tensile MembraneHawkins and Mitchell (from Ref. 49)

iix

in Y Direction - Hawkins and Mitchell

(from Ref. 49)

-64--

i ........ .l Wl ,,, - .

,

Herzog accounted tor he difference between the uniform straindistribution associated with pure tensile membrane action andthe irregular strain distribution in the slab reinforcement byassuming as average ultimate strain equal to one-fourth cu.The assumption appears reasonable in view of the nonuniformstrain distribution along the bar length (46). Thus, Herzogobtained for the midapan ultimace deflection after tensile mem-brane action,

ault = Ly 36 u Eq. (17)

where L = short span of slaby

= 3teel strain at ruptureu

4.5 Hawkins-Mitchell Method

Most recently, Hawkins and Mitchell (47) developed simplifiedexpressions for tensile membrane action in two-way flat slabs.Assuming that the membrane takes a circular deformed shape,they developed the load-deflection equation in the followingmanner.

Cqnsider a rectangular slab shown in Fig. 48. The slab is sub-jected to a uniform load w, resisted by edge tensions per unitlength of Tx in the x-direction and Ty in the y-direction.The deformed shape assumed for the slab in x-direction is shownin Fig. 49.

Vertical equilibrium gives

WL L = 2Tx sin 0 + 2Ty Lx sin 6 Eq. (18)xy x x x y

and geometry gives:

L x = 2Rx sin 6 Eq. (19)x x

Ly = 2Ry sin 0y Eq. (20)

Substituting Eqs. 19 and 20 in Eq. 8 yields

T Tw = R + Eq. (21)

x y

The loading lengthens the membrane from the straight line ABCin Fig. 49 of length 2RX sin 0 to a circular arc AB'C oflength 2R Y By approximatirlg sin 0 y by the first twoterms of t e corresponding Taylor series expansion,

-65-

2AB'C - ABC Eq. (22)

y ABC 60 2 /6

Similarly, = 02/6~x

Thus the deflections 6x and 6 y aeRy

6 = - 2 = 3R E Eq. (23)x 2 x x x

and 6 = -Y 02 = 3R y Eq. (24)y 2 y y y

Since 6 = 6 at the center of the membrane,y

E = £y (Ly /L x)2 Eq. (25)

Substituting Eq. 25 into Eq. 21,

2T sin sinT+ 2Ty s E-- Eq. (26)w L -L

9 y

For a rectangular slab with length Lx greater than L , theshort span elongation E increases faster than sx. Kt the

yincipient collapse deflection, Cy equals Eu and thus Equa-tion 14 becomes:

6ul t = 3Ry Eu Eq. (27)

Expressing Ry in terms of and u ,

ilt = 1. u Eq. (28)

sin 6 u

it should be noted that the properties of the short span steelare sufficient to compute the incipient collapse deflectioncapacity of a slab.

-66-

5. MEMBRANE ACTiuN IN TWO-WAY SIMPLY-SUPPORTED SLABS

in the case of simply-supported reinforced concrete slabs, mem-brane action develops as the slab defl-ects. This occurs inuniformly loaded Blabs at relatively large deflections, whenthe slab regions at the edges t~end to move inward but are re-strained from doing so by the adjacent outer regions. The re-sult is an outer ring of compression supporting tensile membraneforces in the inner (central) region of the slab. The compres-sive forces have a beneficial effect on the yield strength ofthe concrete, resultin~g in an increase in slab resistance asthe deflection increases.

The effect of membrane action on the load-deflection character-istics of s imply- supported slabs has been investigated by sev-eral researchers. A summary of these works is given below.

5.1 Wood's Work

Wood (4) tested three s imply- supported slab specimens. Twospecimens, denoted by G5 and G6 were large-size models, while athird specimen, L~2, was a small-scale one. The specimen prop-erties are given in Table 2.

Specimens G5 and G6 were identical in all respects except fortheir boundary conditions. Specimen G5, shown in Fig. 50, wassimply-supported but had encased steel beams along its edges.Specimen G6, also s imply- supported, had no edge beams. Thebeams in Specimen G5 were sufficiently strong to prevent com-bined beam-slab type collapse. Also, beam deflections werenegligible.

F-igures 50 and 51 show the yield-line pattern for Specimen G5under 16-point loading. Slab G6 exhibited similar behaviorduring testing. In both cases, cracking initiated at 56% of theload predicted by Johansen's theory. For Slab G5, the rein-forcement yielded at 1.15 times Johansen's load. The corres-ponding value for G6 was 1.0. There was no definite yield loadfor either specimen. The slab load increased continuously dueto tensile membrane action. The tests were suspended when slabloads for both specimens were at least 50% higher thanJohansen's load, as shown in Table 2.

The behavior of small-scale Specimen L2 was very similar tothat of the large-scale Specimens G5 and G6.

For s imply- supported reinforced two-way slabs, compressive mem-brane action is non-existent and t~nsile membrane action comesinto play at large deflections, as shown by the experimentalcurve in Fig. 52. Also shown in Fig. 52 is the theoreticalload-deflection curve suggested by Wood.

-67- -

N-4,*

Fig. 50 Underside of a Square Slab G5, Supported onFour Encased Steel Beams, After CollapsingUnder 16-Point Loading. The beams were suf-ficiently strong to prevent the compositebeam-and-slab mode of collapse. -Wood

(from Ref. 4)

-8

Ii14

-6---

21 1w11

Fig. 51 U~pper Sui-face of Slab G5 Sh 'x'~n ii vFig. 50.'No-te the toi-sional effects c je ccrincrgivinq rise to crushing ail c csk i n a tright anq1t.,s. As the ceritc-r of the slabis ap,-roached ii:e crushing onl the diago-rnals changes to tensile crac .s which cutthrough the slah. This is ~4 ricev.,denceof mernbrar.Q action V ronRef. 4)

2.0

,0 1.5 Experimental0o" 1C0 Theoretical Curve00

T-,0.5

|II I0 0.5 1.0 1.5 2.0

Central DeflectionThickness

Fig. 52 Typical Theoretical and ExperimentalLoad-Deflection

Curves for a Simply-Supported Two-Way Slab(from Ref. 4)

0-70-

* - -V0.5

There is a discrepancy between the initial portions of theexperimental and theoretical curves. This is primarily due tothe fact that the analysis neglects curvatures that occur priorto Johansen's load. In addition, the theoretical curve shows acontinuous increase in load with deflection. In practice, rup-ture of the reinforcement and/or the loss of its bond with con-crete will result in loss of slab capacity.

Wood suggested that compressive membrane action can be inducedin s imply- supported slabs by using prestressing. This leads todesign of slightly thinner slabs. However, excessive compres-sion may lead to violent collapse.

5.2 Taylor, IMaher, Hayes and Morley's Work

Taylor, et al (48), suggested a possible basis for incorporat-

Figure 53 illustrates a uniformly loaded simply-supported squareslab with large deflections after the yield-line pattern hasformed. In the central region of the slab the cracks will havepenetrated the entire slab depth at this stage.

From the equilibrium condition of Segment A (see Fig. 53), itis evident that the total tension in the reinforcement shown inelevation must be balanced by the total compression in the com-pression zones at each end of the segment. These forces willbalance each other if the entire system of yield-lines is con-sidered.

However, they will not be in equilibrium

along each

part of the yield-line as assumed in Johansen' s yield-linetheory. In other words, the lower region of the yield-line isin tension and the upper, outer region in compression. Theeffect of the change of geometry due to deflections is simplyto increase the effective lever arm of the internal forces.

Consider a square slab of side L carrying a uniformly distrib-uted load of w per unit area. If the equilibrium of Segment Ais considered by taking moments about the support line, the fol-lowing equation can be written for a particular slab deflection:

w2 . ZT Eq. (29)

where T = total tensile force in reinforcing steel

y = distance from centroid of steel to centroid ofconcrete compression

Taylor obtained load-deflection curves from calculations basedon Eq. 29. These agree well with experimental data after thedevelopment of the yield-line pattern, as shown in Fig. 54.The figure shows the theoretical curve starting from theJohansen load. This is because the method is based on a rigid-plastic approach and curvatures prior to Johansen's load are

-71-

Simply ASupported

EdgesRegion of/ Overall

I I Tension

Cracks Penetratingto top face of Slab

PLAN of SLAB

2. Level ofCompression Zone- Leve of

Centroid ofCompressed Concrt Compression

Tno.Lever Arm,yLevel ofTension Reiforcement ,...LvloCentroid of

ELEVATION of SEGMENT 'A' Tension

Fig. 53 Tensile Membrane Action in Uniformly LoadedSimply-Supported Slab - Taylor

(from Ref. 49)

-72-

48-

32- Experimental Curvefrom Ref. 49 (Slab SO))

Taylr,'s Theory1(48)

16 tCollapse Load by Yield-Line Theory

-',Test Terminated at This Point

0 1 2 35 4

CENTRAL DEFLECTION, in.

Fig. 54 Comparison of Theoretical and ExperimentalRelationships for a Simply-SupportedSquare Slab -Taylor (from Ref. 48)

-73

16iColpe odb YedLneTer

neglected. Also, for the most part, the expertimental curvesgive a higher load carrying capacity, tlie largely to the strainhardening of the reinforcement at large deflections.

Taylor, Maher and Hayes tested ten two-way, simply-supportedslabs to investigate tne effects of reinforcement arrangementand :;lab-depth ratio on slab behavior. The slabs were designedto have the same maximum flexural 'esistance under uniform load-ing. Slab properties are given in Table 2. As rig. 55 shows,Slabs S1, S7, and S9 had uniformly spaced reinforcement parallelto the slab edges. Slab $6 had reinforcement placed diagonallyand1 the remnaininq siK slabs had variahle reinforcement.

The slabs were loaded at sixteen uniformly spaced points bymeans of small hydraulic jacks. The loading of specimens wasterminated because of excessive deflections at the center androtation of the supports. As the load-deflection curves inFig. 56 indicate, the specimens withstood appreciable deflectionwithout any noticeable decrease in slab capacity. The testswere terminated because of excessive deflections at the centerand rotations at the supports. Often there was no fall-off inload-carrying capacity of the slab specimen. The incipientcollapse def~lction capacity was not recorded. Associatedcracking patterns are shown in Fig. 57.

In the 2-in. thick slab specimens, Sl-S6, deflections remained

small up to initiation of cracking. The deflections then in-

creased more rapidly. The variably reinforced slabs were3lightly stiffer than the uniformly reinforced slabs. However,the stiffness of slabs with stopped-off bars deteriorated rapid-ly after formation of a square yield pattern following approxi-mately the line of the ends of the stopped-off bars. This ispattern C shown in Fig. 57. In the 1-1/2-in, thick SpecimensS7 and S8, deflections increased more rapidly than in the 2-in.thick slabs, Sl-S6. Again, the variably reinforced Slab S8 wasalso observed to be slightly stiffer than the uniformly rein-forced Slab S7.

In the two 3-in. thick slabs, the cracks did not occur untilthe design capacity had been exceeded. Because of the low per-

centage of reinforcement, the cracks widened very rapidly andextended to the top surface in the central region. Load-deflec-tion curves for the two slabs were very similar and the maximumloads recorded were the same, as shown in Fig. 56.

Taylor, Maher and Hayes drew the following conclusions for sim-ply-supported square slabs:

1. In slabs designed by the yield-liie method, the use of

variable spaced bars wil.l lead to minimal economy, ifany, compared with uniformly-spaced bars. Stopping-off bars, however, will effect some economy.

-74-

3 in. 1.3in 3im. (6in) (12 in) spoc"' sm.iar to S2

4.5 in

ISin.

U99

et al. On Son-urtro

renocmn is. symmetrical.n

Dimensions~ iniaesing

ofbas

hn

rnigfo

-nimbr;itoppe f to boto exetfr1

2.5 3Iin spaci2in4. ng.(fro -ef. 49

-75- II

(1) 0 ) (1aC) 0 0 0

CfC

U))

a-4-4

U)

W (0 4-4

00

0

ELO > 0

C)

0 0

z

.U 1

(101

-~~~~~~~

cc)

0__

- - --

- -- - ---

SI, S9, SIO S2, S4, S7,S9

(a) (b)

S3, S5 S6

(c) (d)

Fig. 57 Different Yield-Line Patterns forSlabs SI-SI0 Tested by Taylor,Maher and Hayes (from Ref. 49)

-77-

17

2. Variably-reinforced slabs are slightly stiffer thanuniformly-reinforced slabs over the initial load rangeup to the Johansen load.

3. All slabs sustained loads higher than that predictedby yielri-line theory, partly because of tensile men-brane action. The enhancement depends on the particu-lar yield-line pattern by which the slab achieves theJohansen load. For slabs with stopped-off bars, theenhancement is small.

4. The use oF variable reinforcement does not lead tohigher enhancement of load-carrying capacity.

5. The use of variable reinforcement reduces crack widthin the central region of -labs, hit increases the crackwidths in the corner regions.

5.3 Work at U.S. Army Engineer Waterways Experiment Station

Geymayer and McDonald (51) at the U.S. Army Engineer WaterwaysExperiment Station tested seven simply-supported, thin, square,s;lab specimens under uniformly distributed loading. The objec-tive of this investigation was to determine the influence ofreinforcing details on the yield-line pattern and load-carryingcapacity of the slabs. Variables selected were geometry andnumber of bent or terminated bars, corner reinforcement, andcolumn strip width. Specimens had a clear span of 60 inchesand a uniform overall thickness of 1 inch. The mid-span rein-forcement ratio was 0.8%. The specimen properties are listedi-n Table 2.

The tests3 were stopped well before the incipient collapse de-flection capacity of the specimens was reached. Maximum d'flec-tion recorded was about 4 in., i.e., 4 times the slab thickness,as shown in Fig. 58. The ccrresponding maximum edge rotationwas about 8 degrees. The folLowing conclusions were drawn:

1. Small variations in reinforcing ietails, although nor-mally not considered in analysis, may significantl vaffect the load-carrying capacity and yiel-i-line pat-tern of reinforced concrete slabs. In general, cornerreinforcement increases and bent bars decrease slabcapacity.

?. Formation of a yield-line system does not result inimmediate collapse.

3. Variations in reinforcing details within the range ofthe ACT recommendations (26) do not significantlyaffect slab deflections under working loads.

-78-

a:a

4)4

0) 00.~c p1 c 4nW r-4 -z4

(~~w I. -

0.) II

0-4 4

"-44-4

C

E (1) (

0)04-

4- O

00

o0 0

;Sd'M' aVO-l

F -79-

5.4 Sawczuk and Winnicki

Sawczuk and Winnicki (18) tested three types of two-way simply-supported slabs, using two identical specimens for each type.Reinforcement and aspect ratio were the two variables investi-gated. Geometric and material properties of the slab specimensand some test results are listed in Table 2. The load-deflec-tion relationship, Figs. 59 through 61, show the significantinfluence of tensile membrane action on both strength and defor-mation capacity of slabs. The tests were stopped before theincipient collapse deflection capacity was reached. Membraneaction is accompanied by large permanent deflections, as shownin Fig. 62.

Sawczuk and Winnicki observed that tensile membrane action islocalized in yield-line zones. Membrane action causes steel togo into the strain-hardening range. This enhances the load-carrying capacity of slab. The influence of strain-hardeningon slab capacity is relatively easy to calculate by using thebreaking strength in place of the yield strength in thecalculations.

To determine the complete load-deflection relationship of slabspecimens, Sawczuk and Winnicki presented a kinematical approachfor the analysis of membrane action in simply-supported slabs.Based on kinematically admissible collapse modes, dissipationfunctions were established. Axial forces and moments at theyield sections were considered dependent on deflections. Theresulting load-deflection relationships were linear, as shownby the dashed curves in Figs. 59 through 61.

In computing total deflection, elastic deflections were addedto post-yield deflections. This method of simple addition isnot fully justified and represents an approximation. In com-puting elastic deflection, it was assumed that up to the yieldlimit, Wj, a slab is linearly elastic. Beyond this stage theslab was assumed to be perfectly plastic. The upper theoreticalcurve corresponds to the ultimate steel stress, f,, while thelower theoretical curve corresponds to the steel yield stress,fEY. As Figs. 59 through 61 show, there is good agreementbetween the theoretical and experimental curves.

5.5 Kemp's Work

Kemp (14) suggestel that the increase in strength in simply-supported slabs arises partly from tensile membrane action pro-duced in the central region of the slab and partly from theincreased yield moment in the outer regions where compressivemembrane action occurs. Kemp presented an upper bound solutionfor simply-supported slabs that accounts for the effect of mem-brane action. The approach follows Wood's (4) method for cir-cular isotropic slabs and is limited to square isotropic slabs.

-80-

Reinforcement Ratio =0.91%

- = 1.45 ,9':

6-

a-s Thi Pa , 0 f int

4

0

_j Experimental

2 - Theoretical

Test Terminated at This Point

0 2 4CENTRAL DEFLECTION

SLAB THICKNESS

Fig. 59 Comparison of Experimental and TheoreticalLoad-Deflection Relationships for Slabs 12,

with Lx/Ly = 1.45 and ReinforcementRatio = 0.91% (from Ref. 18)

-81-

6Reinforcement Ratio 0.45 %

X 2.0

Ly

_ 4 fs- fu .0,wo , ,

OWO

d .. . , ,v . " 'f S f< y

J

Experimental.- Theoretfcal

;' Test Terminated at This Point

II I0 2 4

CENTRAL DEFLECTIONSLAB THICKNESS

Fig. 60 Comparison of Experimental and TheoreticalLoad-Deflection Relationships for Slabs Ii,

with Lx/Ly = 2.0 and ReinforcementRatio = 0.45% (from Ref. 18)

-82- "

6 Reinforcement Ratio =0.91% 0

L 44 0.1

X 00

2

.0,

- Experimental

,.--Theoretical

*', Test Terminated at This Point

I I I0 2 4

CENTRAL DEFLECTIONSLA8 THICKNESS

Load-Deflection Relationships forSlabs IT1, with Lx/Ly 2.0 and

Reinforcement Ratio 0 0.91% (from Ref. 18)

- C

'.

Fig. 62 Permanent Deflection of a Slab Due to

Membrane Action and Bending - Sawczuk

and Winnicki (from Ref. 18)

-84-

Assuming the slab to be made up of rigid-plastic material,stress resultants were obtained from the geometry of rigidregions and their horizontal equilibrium under membrane forces.it was concluded that for low percentages of reinforcement,enhancement in slab strength was pronounced and that a savingin reinforcement was possible by allowing for membrane action.

5.6 Brotchie and Holley's Work

Brotchie and Holley (33) tested fokir s imply- supported slabs.All slabs were square, spanned 15 in., and were uniformlyloaded. Slab specimens were reinforced near the bottom onlywith smooth steel wire uniformly and equally distributed ineach direction. Material and geometric properties of thespecimens are listed in Table 2. Reinforcement ratio and span-depth ratio were the two variables investigated.

Measured load-deflection relationships are shown in Figs. 63and 64. The figures show that reinforcement ratio has an impor-tant effect on slab behavior. As Fig. 63 illustrates, the'fully reinforced' slab (p = 0.03) shows a peak in resistancefollowed by instability. With further increase in slab deflec-tion, the slab resistance increases as a result of tensile mem-brane action. The lightly reinforced slab (p = 0.01), on theother hand, exhibits a relatively flat post-yield slope with nostrength increase.

The question of the applicability of the results of these small-scale slab tests to full-scale slab systems was considered.Brotchie maintained that the results should apply to full-scaleslabs provided that the same geometric proportions, reinforce-ment ratios, and material strengths are used.

&-yjes (50) presented an equilibrium approach to allow for themembrane action in reinforced concrete slabs. The method isquite similar to that of Sawczuk and Winnicki (18) except thatHayes used an equilibrium approach instead of an energyapproach. He assumed that in-plane plastic hinges are formedon the long side, and that in-plane forces exist along the yieldlines. The magnitudes of axial and shear forces were calculatedby using in-plane equilibrium of the rigid portions between theyield-lines. Moments of the forces were taken about the in-plane plastic hinges. The load-deflection relationship thusobtained was linear and similar to that shown in Fig. 52. Anal-ysis showed that the membrane forces were independent of deflec-tion, a conclusion that does not appear to be reasonable.

Hayes (50) also tested slabs to verify the analytical work.However, no details are available on the geometry or materialproperties of the slab specimens.

-85-

100

DEPTH

CL P O.03

o 50

0-J O

0 2

DEFLECTION, in.Fig. 63 Load-Deflection Relation for Simply-Supported

Slabs Tested by Brotchie and Holley(from Ref. 33)

200

SPAN 10

DEPTH

p=0.03

Q_ 4 -- Slab 15

S(00o Slab 12

0.

I I ,I. I I

0 I 2 3DEFLECTION, in.

Fiq. 64 Load-Deflection Relation for Simply SupportedSlabs Tested by Brotchie and Holley

(from Ref. 33)

-86-

5.7 Desayi and Kulkarni's Work

Desayi and Kulkarni (52) presented the most notable work on theload-deflection behavior of simply-supported reinforced concreteslabs. Their investigation included both analytical and experi-mental work. The analysis to determine the load-deflectioncharacteristics was carried out in two stages. In the firststage, a semi-empirical method was used for calculating deflec-tion up to the Johansen load. In the second stage, membrane ac-tion was taken into consideration in predicting load-deflectionbehavior beyond Johansen's load.

Desayi and Kulkarni followed the CEB method (53) for calculat-ing the load-deflection relationship up to the Johansen load.Figure 65 shows the three-segment idealized load-deflectioncurve used in the analysis. The reduction in flexural rigiditywith increasing load signifies concrete cracking and steelyielding in the slab. Empirical constants were introduced todetermine the efective flexural rigidity in Segments 2 and 3 inFig. 65.

To calculate load-deflection behavior beyond Johansen load,Kemp's approach (14) for square slabs was used with two modi-fications. First, the method was generalized for rectangularorthotropic slabs. Second, effects of deflection prior to theJohansen load on the neutral axis depth and the membrane forceswere considered. To simplify the analysis, yield lines wereassumed to make a 450 angle with the edges, as shown in Fig. 66.

Twelve slabs were tested by Desayi and Kulkarni to verify theiranalytical approach. The three variables investigated wereaspect ratio, span-depth ratio, and coefficient of orthotropy.The coefficient of orthotropy represents the ratio of the designyield moments along the two principal directions. Details ofslab properties are listed in Table 2.

The experiments were stopped before reaching incipient collapsedeflection capacity of the specimens. The main reason was theexcessive slab deflection and instability of loading system atthe large deflection levels. Computed and experimental load-deflection curves for Specimen S4 are shown in Fig. 67. Thetwo computed curves shown in the figure correspond to the twoassumed values of steel stress at the Johansen load, namely,yield stress, f , and breaking stres3, f The figure alsoshows the theoretical load-deflection relationships obtained byMorley (28) and Hayes (49).

It was concluded in this investigation that the use of steelbreaking stress instead of yield stress provided better agree-ment between analytical and experimental curves. No data isavailable on experimental load deflection relationship at theincipient cellapse-.de fectin-capacity. "

-87-

tw

-< w

0 cr

0 B

CENTRAL DEFLECTION

Fiq. 65 Assumed Load-Deflection Behavior forSimply-Supported Slabs Up to Johanser

Load - Desayi and Kulkarni(from Ref. 52)

x

ly

Fig. 66 Assumed Collapse Mechanism at Yield-LineLoad - Desayi and Kulkarni

(from Ref. 52)

-88-

C,

.4-..4-.

4wNIn - In

0W 44

0n

I -i

a))

0 4Ja0

I It

0.0

L %0

a)i

-89

.n .0

6. MEMBRANE ACTION IN ONE-WAY SLABS

6.1 Christiansen's Work

Christiansen (54) tested four one-way slabs restrained alongparallel edges. The strips were 3-in. thick, 6 in. wide, had aclear span of 5 ft, and were loaded with a concentrated load attheir midspan. Also tested was an identical set of four slabstrips that were simply-supported. The purpose of these testswas to gain greater understanding of arching action in two-wayslabs.

The ratio of slab capacity, WD, to Johansen's yield-line loadva-ied between 1.42 and 3.83. To show the effect of archingaction on slab strips, Christiansen calculated load-deflectioncurves for simply-supported slabs of identical dimension usingbending theory. Experimental load-deflection curves for a pairof restrained and unrestrained slab specimens are shown in Fig.68. The vertical distance between the two curves representsthe load carried by arching.

By considering the outward support movement, slab shorteningdue to axial force, ani slab lengthening due to hinge rotation,Christiansen determined the depth of concrete in compression."'he resultant compressive forces at the support and midspanwere assumed to induce arch action in the slab. Maximum loadthat could be carried by the arching action was calculated bymaximizing the moment due to arching forces. Using this ap-proach, a good comparison between experimental and theoreticalresults was obtained. However, analysis was not generalizedfor two-way slabs.

Since the tests were not carried into the tensile membraneregime, no information on the incipient collapse deflectioncapacity of one-way strips can be derived from this experimen-tal investigation.

6.2 Roberts' Work

Roberts (55) tested 36 strips representing one-way slabs. Thepurpose here was to gain an understanding of strip action as abasis for explaining compressive membrane action in two-wayslabs. The strips were restrained against longitudinal expan-sion in a specially designed frame, shown in Fig. 69. Thestrips were loaded by several point forces to simulate uniformlydistributed loading. Properties and test results of all thirty-six strips are listed in Table 3. A representative load-deflec-tion curve is shown in Fig. 70. The following conclusions wereArawn from the test results:

I. The ratio of peak load to tihat given by 7ohansen'syield-line theory varies from approximately 17 forstrips with high concrete strength and a low percentage

-90-- O t

_____ ___ 4

3,500Laterally Restrained

3,000-

- 2,5006wl_

Laterally Unrestrained

1,500

1,000-

500 30 -H

* Test Terminated at This PointI I .

0 0.1 0.2 0.3 0.4 0.5

CENTRAL DEFLECTIONSLAB THICKNESS

Fig. 68 Details and Load-Deflection Curves forOne-Way Slab Strips Tested

by Christiansen(from Ref. 54)

-91-

AD-A091 738 CONSTRUCTION TECI4NOLOG IY LABS SKOKIE IL F/6 13/13DESIGN CRITERIA FOR DEFLECTION CAPACITY OF CONVENTIONALLY REINF--ETC(U)OCT 80 M IGBAL. A T DERECHO N68305-79-C-0009

UNCLASSIFIED CEL-CR-80.026 NL; uuuuuuuuuumlllllEnmmunmmunnuuuuuummuEEEllhElhhEEE-llllllll

A 0

Surround Slab

LAN (inserted)

COne-Way Slab

B

SECTION 'A-A'

|:L''e~ hf " steel-- plates

DETAILS - II

Fig. 69 Details of Surrounded and StripSupport - Roberts (from Ref. 55)

6

5 Mechanism

"4 --- First Beam0_J

-- ~0C0

2

0 0.2 0.4 0.6 0.8 1.0Central Deflection

Beam Depth

Fig. 70 Experimental Load-Deflection Curvesfor Beams RB23 Tested by Roberts

(from Ref. 55)

-92-

of reinforcement, to 3 for beams with low concretestrength and a high percentage of reinforcement.

2. Deflection at maximum load is not a fixed proportionof the strip thickness. The average value of 6D/hwas 0.27 for 2-in, thick strips and 0.16 for the 3- n.thick beams.

3. It is not necessary for the restraint to have enormousstiffness to develop enhanced peak loads. Theoreti-cally, the load is increased by 10% when the restraintstiffness, initially equal to that of the beam, isincreased eleven times.

4. Comparison between theoretical and experimental deflec-tions was not satisfactory due to the neglect of elas-tic curvatures in the analytical model.

6.3 Park's Work

Park (24) calculated load-deflection curves corresponding tothe strips tested by Roberts (55). One such comparison, shownin Fig. 71, indicates the peak load predicted by the theory tobe conservative. It will be noted in Fig. 71 that the load onthe actual slab decreases more rapidly than that predicted bythe theory. Also, deflection at Johansen's yield-line load is-zero. Park explained these discrepancies as follows:

1. The theory assumes concrete strength to be the uniaxialvalue. In the test, the concrete at the strip endswas confined transversely by the friction between thestrip end and the restraining frame. Roberts (18)showed experimentally that concrete strength at thestrip ends was about 2,000 psi greater than the cyl-inder strength.

2. The theory assumes that the concrete compressive stressblock parameters remain at the ACI (26) values. How-ever, at high strains the stress block parameters willchange and concrete cracking will occur.

3. The theory assumes that concrete stress-strain proper-ties are reversible as the neutral axis decreases. inreality, permanent set occurs on reversal of strain.

4. Initial strip behivior is mostly elastic. Plastictheory neglects this portion of the test.

6.4 Other Investigations

Geymayer and McDonald (51) at the U.S. Army Engineer WaterwaysExperiment Station tested two one-way slab strips. The stripswere s imply- supported and loaded equally at the middle-third

-93-

.1 .. .

5

Mechanism

C

~z30 LL

Experimental, Strip RB 18 from Ref. 55Pork Method (24 )

0 0.1 0.2 0.3 0.4 0.5

CENTRAL DEFLECTION 8STRIP DEPTH h

Fig. 71 Comparison of Experimental and TheoreticalLoad-Deflection Relationships for One-Way

Slab Strip (from Ref. 24)

-94-

points. The strips were quite slender, with a span-depth ratioof 60. Material and geometric properties are given in Table 3.

The experimental load-deflection curves, shown in Fig. 72,indicate a considerable loss of slab stiffness first at theonset of cracking and then at steel yield. Observed slabstrength was higher than that predicted by Johansen'syield-line theory, as indicated in Table 3. The loading wasstopped when the strip deflected 2 to 2.5 times the stripthickness.

Iqbal (54) investigated the effect of steel properties onload-deflection behavior of one-way slabs. Two strips weretested. One was reinforced with hot-rolled steel having adefinite yield plateau and the other with cold-rolled steelwith no definite yield point or plateau. The strips wereloaded symmetrically at their middle-third points. The stripproperties are given in Table 3.

Observed load-deflection curves are shown in Fig. 73. Bothspecimens showed a resistance slightly greater than thatpredicted by Johansen's yield-line theory. In the post-yieldrange, the strip reinforced with steel having a definite yieldplateau sustained the ultimate load with large deflections.However, the resistance of the other strip dropped due to lossof bond between the steel and surrounding concrete. Permanentdeflection of one strip after unloading is shown in Pig. 74.

Birke (57) developed a relationship between ultimate moment dueto compressive membrane action and the span-to-thickness ratio.The relationship is applicable to slabs with a single load atmidspan. The edge restraint may be full or partial.

Komoro (58) developed a method to compute the load-deflectionrelationship of one-way reinforced concrete slabs under asingle load at midspan. Sixteen slabs were tested to confirmthe accuracy of the approach. The main conclusions from thetests are as follows:

1. Magnitude of the compressive membrane force at sup-ports changes with slab deflection.

2. Arching effect due to the compressive membrane forceincreases as span-depth ratio decreases.

3. The load-deflection curve for one-way slabs loaded atmidspan, from zero load to the onset of tensile mem-brane action (point E in Fig. 3), can be calculatedusing Komoro's method.

-95-

I " lS ,,

6O0-

Slab Strip A400-

20 2d 20o 200-

II I I I I02 3

MIDSPAN DEFLECTION,in.Fig. 72 Load-Deflection Relationship of One-Way

Simply-Supported Slab Strips Tested byGeymayer and McDonald (from Ref. 51)

2 Slab Strip Reinforced with Steel Having No Definite Yield Point

Slob Strip Reinforced with Steel Having Definite Yield PointU,

0 _JP P

0 I 2 3 4MIDSPAN DEFLECTION, in.

Fig. 73 Load-Deflection Relationship for One-WaySlab Strips Tested by Iqbal

(from Ref. 56)

-96-

Pig. 74 Permanent Deflection and Cracking Patternof Simply-Supported Slab Strip at

Termination of Test -Iqbal (from Ref. 56)

-97-

7. NONLINEAR FINITE ELEMENT MODELS FORREINFORCED CONCRETE SLABS

7.1 Review of Finite Element Models

With recent advances in computational methods, the finite ele-ment technique has become a powerful analytical tool. Severalapproaches have been used in modelling reinforced concrete.Bazant, Schnobrich and Scordelis (59) provide an excellent sum-mary of work in this area.

Analysis of the behavior of reinforced concrete beams and slabshave received considerable attention from investigators (60-85).Two basic approaches have been used. These are the modifiedstiffness approach and the layered element approach.

An early application of the finite element method to reinforcedconcrete was carried out by Ngo and Scordelis (60) . Theydeveloped an elastic two-dimensional model of reinforced con-crete beams with defined crack patterns. Bond slip betweenconcrete and steel bars was modeled by finite spring elementsdesignated as bond links spaced along the bar length. Crackingwas modeled by separation of nodal points and a redefinition ofstructural topology.

Nilson (61) extended Ngo and Scordelis' work by including non-linear properties. This approach has not achieved popularitydue to the difficulties encountered in redefining the structuraltopology after each load increment. Mufti, Mirza, McCutcheonand Tioude (62) used the same model, but without modifying thetopology. They deleted the cracked element from the overallstiffness. Forces in the cracked element were redistributedduring the next cycle.

Rashid (63) introduced another approach in which the crackedconcrete was treated as an orthotropic material. Steel elementswere assumed to be elastic/perfectly plastic. The von Misesyield criterion and the Prandtl-Reuss flow equations were usedto define the behavior of the steel in the range of plasticdeformation. This approach proved to be more popular and manyinvestigators have used it with variations in material proper-ties and modes of failure.

Isenberg and Adham (64) introduced a nonlinear orthotropicmodeL and demonstrated its use on tunnel problems. The non-linear stress-strain behavior of concrete and steel was ideal-ized with bilinear stress-strain curves. Bond and the effectof lateral confinement on compressive and tensile strength wereconsidered.

Valliappan and Doolan (65) studied the stress distribution inreinforced concrete beams, haunches, and hinges using an elasto-

-98-

.-. -

plastic model for steel and concrete. The concrete was repre-sented as a brittle material in tension.

Nam and Salmon (66) compared the constant stiffness and thevariable stiffness approaches for nonlinear problems. Using acombination of isoparametric elements and bar elements, theyfound the variable stiffness approach to be far superior forproblems involving the prediction of cracking in reinforcedconcrete structures.

Bell and Elms (67) presented a method for computing deflectionsand crack patterns of reinforced concrete slabs for the entirerange of loading, i.e., from zero to ultimate. Triangular bend-ing elements and the method of successive approximations wereused. Cracking normal to the principal moment direction wasaccounted for by using a reduced stiffness. However, theiranalytically derived displacement curves did not agree wellwith selected experimental data. They also developed a par-tially cracked element, but found that analysis using this ele-ment was neither as accurate nor as well behaved as one basedon either an elastic or a totally cracked element.

Jofriet (68,69) used a quadrilateral plate bending element withfour corner nodes and three degrees of freedom at each node.Cracking on normals to the principal moment directions wasaccounted for by using a reduced stiffness for beams. Theirresearch did not take into account load history or post-yieldbehavior.

Scanlon (70) presented a finite element analysis to determinethe effects of cracking, creep, and shrinkage on reinforcedconcrete slabs. The finite elements consisted of a series oflayers, each with a different plane stress constitutive rela-tionship. Cracks progressed through the thickness of the ele-ment, layer by layer, parallel or perpendicular to the orthog-onal reinforcement. The concrete was modeled as a linear elas-tic material in compression and an elastic brittle material intension. The modulus in tension, after cracking, was obtainedusing a stepped stress-strain diagram. Stiffness of a layerwas evaluated by superposing the stiffnesses of steel and con-crete. The shear modulus of a layer, whether cracked or un-cracked, was taken to be that of an uncracked plain concretelayer. It was found that tensile stiffening of concrete betweencracked zones resulted in a significant redistribution of mo-ments. Comparison made with the experimental data and theoret-ical results of Jofriet (68) showed good agreement, as indi-cated in Fig. 75. it will be noted that the range of slabdeflection shown in Fig. 75 has a maximum value equal to 11 ofslab span. This is considerably less than the experimentalincipient collapse deflection capacity which ranges between 10and 15 percent of the slab span.

Lin and Scordelis (71) extended the work of Scanlon to includeelasto-plastic behavior for the steel in tension and compression

-99-

MOA

0 00

o 4) 4-4

0 0'-4-

I 00

oUI4-

~ I z0 Er-4 0)

4-6 z

0 0 _jI

(0 ' r-0

0Qf (s< Qovo

-100-E10

.4 4

and for the concrete in compression. For tension in concrete,they replaced the actual curve by a triangular-shaped curvewith a descending slope after initial tensile failure. Thepost-yield behavior was defined by von Mises' yield criterion.Incremental loading was used with iteration within each incre-ment. A comparison of analytical results with Joffriet's (68)experimental load-deflection curve is shown in Fig. 75.

Hand, Pecknold and Schnobrich (72) used a layered element todetermine the load-deflection history of reinforced concreteplates and shells of uniform thickness. The nonlinear behaviorof steel and concrete was considered in the analysis. Steelwas modeled as elasto-plastic; concrete was assumed to be elas-tic brittle in tension and to have a bilinear stress-strainrelationship up to yield in biaxial compression. They used thestrength envelope obtained by Kupfer, Hilsdorf and Rusch (73)as a yield criterion. A shear retention factor was introducedto provide torsional and shear stiffness after cracking. Thelayered finite element allowed the material properties to varythrough the element depth. Bending and membrane forces wereconsidered and a doubly curved rectangular shallow shell ele-ment with twenty degrees of freedom was used in the analysis.The authors stated that their numerical results were as goodor better than the modified stiffness approaches, as shown inFig. 75.

Wanchoo and May (74) introduced a layered model with concretein compression and steel following the von Mises criterion.Concrete was elastic-brittle in tension. A rectangular finiteelement was used. Fig. 75 shows a comparison of the theoreticalrelationship with an experimental curve taken from Taylor,Maher and Hayes (49). The experimental slab was simply-sup-ported and had a reinforcement ratio of p = 0.005 and an aspectratio of 36. It is seen that the agreement is excellent bothin linear elastic range and in the subsequent cracking range.However, the comparison is limited to a maximum deflection equalto 1.4% of the slab span and does not include larger deflectionlevels where effects of membrane forces are notable.

Wanchoo and May also obtained theoretical load-deflection rela-tionships for clamped slabs. These relationships pertained tosmall deflections (range AD in Fig. 3) and did not cover therange where tensile membrane action is predominant. Also, nocomparison was made with experimental results.

Kabir (79-80) modeled the reinforced composite section as alayered system of concrete and "equivalent smeared" steellayers. Perfect bond is assumed to exist between concrete andsteel layers. Stiffness properties of an element are then ob-tained by integrating the contributions from all the layersacross the section. Concrete behavior under the biaxial stateof stress is represented by a nonlinear constitutive relation-ship that incorporates tensile cracking at a limiting stress,

-101-

ZIA-

tensile stiffening between cracks, and strain-softening phenom-enon beyond the maximum compressive stress. Reinforcement isrepresented by a bilinear, strain-hardening model exhibitingthe Bauschinger effect. The constitutive relations are basedon small displacement theory. This may represent a seriousshortcoming when applying the method to the analysis of slabbehavior in the tensile membrane range.

Most recently, Bathe and Ramaswamy (81) developed a three-di-mensional concrete model and incorporated it in the computerprogram ADINA (82). In this model, concrete is treated as ahypoelastic material based on a uniaxial stress-strain rela-tionship that is generalized to take biaxial and triaxial stressconditions into account. Tensile cracking and compressioncrushing conditions are identified using failure surfaces.Figure 75 shows the load-displacement relationship predicted byADINA for Jofriet's (68) test slab. It is seen that the ana-lytical and experimental displacements compare reasonably well,particularly at the higher load levels. No results using ADINAare available for slabs tested to the point of incipientcollapse. Bathe and Ramaswamy (81) point out that significantfurther studies, evaluations and improvements of the model areneeded.

Another difficulty in using a finite element model to computethe entire load-deflection relationship )f a restrained slabarises due to presence of an unstable region (DE in Fig. 3) inthe load-deflection relationship. In this region, the loaddecreases, with increasing deflections. No finite element studyis available which attempts to predict the behavior in thisunstable region.

Analysis of reinforced concrete systems including cracking,nonlinear geometric and material properties involves complexproblems of finite element modeling. Because of the nature ofsuch nonlinear problems, even the speed and storage capacitiesof today's large digital computers are sometimes insufficientto provide solutions at reasonable costs. Despite some notablebreakthroughs in the use of nonlinear finite element methods,no computer program is available that can be used to calculatethe entire load-deflection relationship up to incipient col-lapse. In fact, no computer program has been found to ade-quately reproduce the load-deflection curve for a restrainedslab even up to point E in Fig. 3.

3efore the finite element method can be expected to predictwith reasonable accuracy the actual response of reinforced con-crete slabs at large deflections, additional basic experimentalresearch must be conducted to develop the necessary stress-strain and load-deflection relations to be used with the finiteelement model. The required information relates to theFollowing:

A. constitutive relations and fail'ire criteria for con-crete under combined stresses

-102-

b. bond stress-slip relationship

C. tension stiffening effect of concrete between cracks

d. aggregate interlock, and

e. dowel shear.

7.2 Use of Program ADINA

It is mentioned earlier in the text that no computer program isavailable that can be used to calculate the entire load-deflec-tion relationship up to the incipient collapse. The main reason

-' for this is the presence of an unstable region (DE in Fig. 3)in the load-deflection relationship of restrained slabs. Sincethe primary concern in this study is the behavior in the ten-sile membrane range, it was thought that with the use of appro-priate boundary conditions, it might be possible to bypass theportion of the curve to the left of point E in Fig. 3. Theintent was to remove any restraint to horizontal edge displace-ment during the early part of the response and thus prevent thedevelopment of compressive membrane action. It was hoped thatthis could be accomplished without significantly affecting thecalculated incipient collapse deflection. With this rationalein mind, the computer program ADINA was implemented on North-western University's Computer Center. However, further effortsto employ the program ADINA were abandoned in view of dissatis-faction of other users with ADINA. It was learned that equi-librium problems arise when the concrete model in ADINA is usedto analyze reinforced concrete systems where cracks transversean embedded reinforcing bar (86). Since this project is con-cerned with slab behavior under conditions where cracks pene-trate the entire slab thickness, the program ADINA in its pres-ent form does not appear to be a useful tool.

-103-

8. PLASTIC HINGE METHOD TO DETERMINE DEFLECTION CAPACITY

&Since no computer program considering both geometric and mate-rial nonlinearities is available to reasonably determine theeffect of various parameters on deflection capacity, an approx-[mate analytical approach was developed. The method considersend conditions, force equilibrium, strain compatibility and thedeflected shape of a slab strip. The approach follows closelythe procedure used by Park (20) and Keenan (42) and extendsthis to take into consideration spalling in the hinging regions.

This chapter describes a step-by-step development of the ap-

proach and examines the analytical results obtained.

8.1 Idealized Load-Deflection Behavior of a Restrained Strip

A fixed-end slab strip with flexural mechanism developed isshown in Fig. 76. The strip is initially of length L and isfully restrained against rotation and translation at the ends.The strip is considered to have symmetrically positioned plastichinges as shown in the figure. The following assumptions aremade:

1. At each plastic hinge, the tensile steel has yielded.

2. The compressed concrete has reached its strength withthe stress distribution idealized as an equivalentrectangular concrete stress block as defined by ACI318-77 (26).

3. The tensile strength of concrete can be neglected.

4. Top steel areas (per unit width) at opposite supportsare equal.

5. Bottom steel is constant across the strip span, butthe amount of top and bottom steel may be different.

6. Segments of the top strip between critical (plastichinges) sections are assumed to remain straight.

7. The axial tensile strain, £, is constant along thestrip span, corresponding to a constant membrane forcealong the length.

8. Cover concrete has spalled off.

Portion AB of the strip ABCD has been enlarged in Fig. 77 toshow the relationship between the depths to the neutral axis, cand c', and the geometry of the strip under the verticaldeflection,

-104-

":): .: . ...

plastic hinges

\PL

Fig. 76 Mechanism of Restrained Strip

h-c) tan8

Setin A and BfFi.7

I --,W__Co.h

RL o- 0.54(-201L

Fig. 77 Portion of Strip Between YieldSections A and B of Fig. 76

-105-

Because of axial tensile strain,E, elongation of the middle portion

BC of the strip = CL(I - 2 )

Due to symmetry, the distance by

which the ends of portion BC willmove away from the center of strip = 0.5 EL(t - 2a)

Since there is no outward horizontaldisplacement at supports, the hori-zontal distance from each end ofportion BC to the adjacent support = BL - 0.5 EL(l - 28)

Owing to axial tensile strain, 6,the lengths of end portion AB andCD = ( + ) 8L

From geometry, this dLstance A'B' ={ L - 0.5 CL (1 - 28)}sec6

= (I +E:) aL + (h - c) tan0 - c' tan0 (30)

where 0 = angle that segment AB makes with the horizontal

c = distance of neutral axis from topmost fiber

c' = distance of neutral axis from bottom fiber

Equation 30 can be rearranged as Follows:

h - c - c= 2 L Sin 2 0/2 B8 L Cose - 0.5(1 - B)L (31)Sin

For this equation, since e and 0 are small

6Sin0 = 2 Sin 0/2 -- , and CosO 1.0

6 L2c + c' = h - - + - (32)

2 26

Also, for equilibrium, the membrane forces acting on Sections Aand B of segment AB of the strip are equal. Therefore:

C' + CI - T' = Cc + C s - T (33)

where C' = concrete compressive force at Section Ac

C' = steel compressive force at Section As

T' = steel tensile force at Section A

Cc = concrete compressive force at Section B

-106-

C .ste.. ... .T - steel compressive force at Section BT a steel tensile force at Section B

Using ACI concrete compressive stress block (Fig. 78) the con-crete compressive forces for a strip of unit width can be writ-ten as:

C'o -0.85 f' 01 (c' -d) (34)Cc c1

Cc - 0.85 f 01 (c- ds) (35)

where f', - concrete cylinder strength

01 - ratio of the depth of equivalent rectangularstress block to the neutral axis depth, as de-fined by ACI 318-77 (26)

i.e., 81 * 0.85 for fc' 4 ksi, and

a 0.85 - 0.05 (f' - 4) V > 4 ksi

c - distance of neutral axis from topmost fiber

as = depth of spalled concrete

Substituting Eqs. 34 and 35 into Eq. 33 and rearranging gives:

-T' T - CS + Cs= .85 fs1 (36)

c1

By solving Eqs. 32 and 36 simultaneously, the neutral axisdepths at the critical sections are given as:

C' . h - + aL 2 T' - T Cs + C s (

S2 4 +4 1.7 fc 1

h 6 +CL 2 T' - T -C + CAEL C (38)2 4 4 1.7 fc, 01

Figure 78 shows conditions at a positive moment yield sectionof unit width. The forces at the section, Cc, Cs and T arestatically equivalent to the tensile membrane force nu, actingat mid-depth, and the resisting moment mu . Therefore, for astrip of unit width:

nu T -Cc C T - Cs -0.85f8 01 (c- ds ) (39)

-107-

..--

u U

n0

coo

4- 4J

ot

@10 0

4JJ

-108-

a O .85f4 {c (0.Sh -0.5 01 c) d da (0.5h - 0.5d8)

+ C s(0.5h - d') + T(d - 0.5h) (40)

For a negative moment yield section of unit width, a' is givenby an equation similar to Eq. 40, and nu' = nu for equilibrium.

For each of the segments AB and CD of the strip, the moment sumsum about one end is m' + a + n 6. Shear forces have been ne-glected since their neA coAribhion to the analysis by virtualwork will be zero. On substituting c' and c for Eqs. 36 and 37into the equation for au , mu and nu , it is found that:

mu + mu + nu8 - 0.425f' 01c(h 01c) - 2ds(h - ds )

c'(h - B0c') + (C5 + Cj)(0.5h - d') + T + T'(d - 0.5h) +

T - C 8 - 0.85f. 01 (c - d s)) (41)

The value of unit elongation, c , required in Equation 41 can bedetermined as follows:

n un = (42)

where 9 - effective modulus of elasticity of strip

Substituting values of nu and c from Equations 38 and 39, c maybe written as follows:

T-C8 ~~T -0.Sf -1 . T C, + C1.7f c '1 s

hE( 0.2125 fc 81 OL2 (43)

.1 + 6

If portion AB or CD of the strip is given a virtual rotation A6,the virtual work done by the actions at the yield sections ofthe portion is:

(m + m + n 6 )AOu u u

Work done by the actions at the yield section of the strip por-tions given by Eq. 44 may be equaled to the work done by theloading on the strip in undergoing the virtual displacement.From this, an equation can be obtained which relates the stripdeflection to the load carried.

For illustration, consider a fixed-end reinforced concrete stripof length L, carrying a uniformly distributed load per unit

-109-

K,

length, w. In this case, value of a is 0.5. If end of thestrip is given a virtual rotation AO about the support, thevirtual work done by the loading on each end portion iswL LAOw- x LAO. Hence the virtual work equation may be written as

wL LAOx--4 (mu +m + n 6) O

wL2 = ml + m + n 6 (45)

where the right-hand side of Eq. 27 is given by Eq. 41. Thus,the load-deflection relation of the strip may be arranged asfollows:

W = 3.40 , 8 &c (h - - - id + c'(h - c)

+ C~ + C')(0.5h -d') + (T + T')(d -0.5h)}

+ !(T C5 - 0.85f' a l(c -ds) (46)

The required values of c', c and E are given by Eqs. 37, 38 and43, respectively. The values of forces T, T', C , and C' areobtained iteratively in order to satisfy compatiility.

It should be noted that the load-deflection relationship thusobtained assumes that critical sections have reached theirstrength from the onset of deflections. Therefore, the derivedload-deflection relationship is not applicable at small deflec-tions when the critical sections are acting elastically or par-tially plastically.

8.2 A Comparison with Experimental Results

To evaluate the reasonableness of the assumptions used in theapproach, a small computer program was developed. A load-deflection relationship was determined for a single-reinforcedstrip tested by Roberts (53). A comparison of the experimentalload-deflection curve with that determined using the plastichinge method is shown in Figure 79. Due to simplifications in-troduced in the analytical method calculated deflections atsmall deflections exhibit a trend that is unrealistic when com-pared to the experimental results. As mentioned, Eq. 46applies only when critical sections in the slab have developedtheir full flexural (plastic) strength. For these reasons, thecalculated values at small deflections have not been shown inFig. 79.

-110-

ii.~

Iz

a.,-

w 0

z 0i

ULU0 0

00

q j- 4J)

Sr LL. 0-

CLi/ 0

-. 0 0

o 0.

00*w

L0 'a

d' -- .

z $4

N 0

OVOI N3SNVHoravo-'

of primary interest is the behavior of the slab at moderatelylarge deflections as predicted by Eq. 45. The analytical curvemarked "p'/p - 0" in Fig. 79, shows that the slab capacitydrops to zero as bath axial thrust and depth to the neutralaxis diminish. In the absence of any flexural capacity, themodel represents a slab acting as a linkage deforming underpure tension.

As a further application of Eq. 46, another strip was analyzed.The strip was identical to the one analyzed above in all butone aspect. This slab was doubly reinforced, with top and hot-torn steels having identical areas. As the curve marked "p'/p1.0" in Fig. 79 indicates, the load drops from A to B and then

rises to 0. At level 1), the associated depth of the neutral

axis is zero. Beyond D, the strip acts as a tensile membrane.After the moment capacity in the plastic hinges drops to zero,the link model considered in Eq. 46 degenerates into a truss,with its members subjected to tension only. At this stage, amodel that assumes the defl.ected shape of the slab to take theform of a catenary or parabola would be more realistic. How-

H ever, as will 13e shown in the next chapter, the use of a para-bol.ic or similar deflected shape to represent the -3lab in thetensile membrane range yields values of the incipient collapsedeflection considerably greater than those observed in experi-ments. Tn using such an approach, investigators have found itnecessary to apply an empirical constant to the analytical re-sults in order to bring them into agreement with experimentaldata. Thus, the plastic hinge approach developed here, as re-flected in Eq. 46, has limited utility with respect to shed-ding light on the behavior of a slab in the tensile membranerange, and particularly with reference to the incipient collapsedeflec~tion. The approach is more appropriate in the range ofbehavior when the slab flexural capacity is intact and has notbeen lost as a result of cracks penetrating the entire slabthickness.

9. DEMELOPMENT OF DESIGN CRITERIA

9.1 Introduction

The primary objective of this Investigation is to develop designcriteria for conventionally reinforced concrete slabs understatic uniform load based on the incipient collapse con Ation,with emphasis on deflection capacity.

The literature review shows that considerable data exist on thebehavior of slabs beyond the Johansen Load. It was noted thatthe load-deflection relationship of uniformly loaded reinforcedconcrete slabs is significantly influenced by the restraintconditions along the edges, as shown in Fig. 3.

Available data indicate that most test specimens were not loadedinto the tensile membrane regime. instead, the tests were ter-minated once the specimen showed a decrease in load-carryingcapacity. This would be just after the stage represented bypoint D in Fig. 3. Tests carried to a deflection level aqualto or near that corresponding to incipient collapse are of par-ticular interest in determining deflection capacity of conven-tionally reinforced concrete slabs.

This chapter summarizes the experimental work on restrainedreinforced concrete slabs. It is followed by a comparison ofexperimental values of incipient collapse deflection with ana-lytical predictions using formula proposed by Park (41), Keenan(42), Black (44), Herzog (45), and Hawkins-Mitchell (47).Finally, the data on experimental incipient collapse deflectionicapacity is examined to determine the influence of vari.ous

9.2 Restrained ToWySlabs

The review of two-way restrained slab test data indicates thatboth compressive and tensile membrane actions, though occurringat different deflection stages, enhance slab load-carrying ca-pacity. When sufficient lateral restraint exists at boundariesof a slab, slab capacity is increased to several times thatpredicted by Johansen's yield-line theory, as shown in Table 1.The ratio of slab deflection at the peak load, 6 D, to theslab thickness, h, for a restrained slab does not have a con-stant value of 0.5 as suggested by Park but varies between 0.11and 0.97, as summarized ini Table 4. Major parameters affectingthe ratio 6D/h are degree of edge restraint, span-depth ratio,and reinforcement ratio.

Near the end of the compressive membrane action range, corre-sponding to point 3 in Fig. 3, the large stretch in the centralregion of the slab surface causes cracks there to penetrate the

-113-

entire thickness of the slab. At this stage, load in the cen-tral region is carried mainly by the reinEorcing bars acting asa tensile membrane.

For a fully restrained iab, v poi nt r in I'iq. 3 is -Ipproached,the membrane forces change (rom compression to tension in thecentral region of the slab. Beyond this stage, the boundaryrestraints begin to resist inward movement of the slab edges.Initially, the outer regions of the slab will act with the edgerestraint as part of the compressive ring supporting the tensilemembrane action in the inner region of the slab. With furtherdeflection beyond point B, tensile membrane action graduallyspreads throughout the slab. Subsequently, the load carried bythe yielding reinforcement increases until the steel starts tofracture at point F. Point F represents the condition of incip-ient failure for restraint slabs.

Knowledge of the region DE is important since the load will dropsuddenly as soon as point D is reached unless the slab is duc-tile enough to "catch" the load. Thus, tensile membrane ac-tion is useful in preventing a catastrophic failure. This as-sumes that a resistance greater than that corresponding to pointD can be developed. Test data show that for heavily reinforcedslabs the collapse or ultimate load at point F in Fig. 3 cansignificantly exceed the peak load at point D. The maximum loadassociated with tensile membrane action tends to increase withincreasing reinforcement ratio, as shown in Figs. 25 and 26.

It is important to note that most restrained slab specimensreviewed in the preceding chapters were not loaded into thetensile membrane range. Instead, the tests were terminatedonce a slab showed a decrease in load-carrying capacity. Thiswould usually be just after the stage represented by point D inFig. 3 is reached. Available test data on tensile membraneaction indicate that the ultimate deflection before rupture ofsteel lies between 10 and 15 percent of the slab span (Table 5).The associated maximum edge rotat'-n ranges between 11 and 16degrees. Herzog (45) , and Hawkins and Mitchell (47) pointedout that the breaking strain of steel, in addition to spanlength, influences ultimate deflection.

The methods used for computing slab behavior in the compressiveand tensile membrane action ranges have been examined. Nosingle method of analysis is available for determining the en-tire load-deflection relationship up to incipient failure. Theavailable methods are semi-empirical in nature and some mayhave only limited predictive capacity.

9.3 Simply-Supported Two-Way Slabs

For two-way slabs with simply-supported edges, the geometry ofdeformation permits development of some membrane forces in theslab. This occurs in uniformly loaded two-way slabs at rela-

-114-

1

tively large deflections when the slab regions at the edgestend to move inwards, but are restrained fromt doing so by th.adjacent outer regions. The result is an outer ring~ of co~m-pression resisting tensile membrane forces in the inner regionof the slab, as shown in Fig. 50. A representative load-deflec-tion curve for a s imply- supported two-way slab is shown as adashed curve in Fig. 3.

Test results of 65 two-way simply- supported slab srpecirren6 showthat slab capacity is always greater than Lhat predicted byJohansen's yield-line theory. Measured maximum edge rotationsrange from 2.2 to 12.4 degrees, as listed in Table 2. A mlajorreason behind such scatter is that not all test specimens wereloaded to incipient collapse. Some tests were terminated ear-lier either due to the loading system bein~g inoperable at largedeflections, or disinterest in slab behavior in the region wheretensile membrane action predominates.

9.4 Restrained One-Way Slabs

The section on behavior of one-way slab strips presents dataand test results on forty-four restrained and s imply- supportedspecimens. These are listed in Table 4. A significant numberof tests were carried out using two equal loads at the mniddle-third points.

Test results on restrained slab strips indicate a maximum edgerotation close to one degree. The reason for this low edgerotation is that the tests were stopped before tensile membraneaction developed. Measured maximum edge rotation for simply-supported slab strips ranged from 3.8 to 10.5 degrees. A com-parison of test results by Geymayer and McDonald (49) and byIqbal (54) clearly shows that measured deflection capacity issignificantly influenced by the investigator's objective intesting a specimen.

No data is available for one-way strips tested under uniformlydistributed load.

9.5 Parameters Affecting Slab Behavior

The major objective in examining the available data has been toget a better understanding of slab behavior near incipient col-lapse and to identify the most important parameters affectingthe deflection capacity at incipient collapse. Park (413,Keenan (42), and Black (44) suggested that the short span of aslab is the only parameter affecting incipien-t collapse de-flection capacity. Hlerzog (45), and Hawkins and Mitchell (47)hypothesized that, in addition to slab span, the steel breakingstrains affect deflection capacity. There has been a tendencyto believe that the deflection capacity may be dependent onother slab parameters such as: span-depth ratio, aspect ratio,size of specimen, boundary conditions, etc. An examination of

-115-

the correlation between selected parameters and incipient col-lapse deflection .1sing available experimental data is giveni below.

As mentioned, most tests were not carried to deflection levelsequal to or near the incipient collapse deflection capacity.Thus, in this parametric study, not all test results listed inTables 1-3 can be employed. Only those carried to the incipientcollapse deflection are used.

9.5.1 Short Span of Slab. The effect of short span lengthon the slab deflection capacity is quite significant, as shownin Fig. 80. The incipient collapse deflection, 6ult , in-creases almost linearly with an increase in slab's short span.The relationships

S=0.1 L, andult y

ult 0.1

provide almost lower and upper bounds for the experimental data,covering slab spans ranging from 15 inches to 72 inches. Themedian 6ult-span ratio was 0.13, as shown in Fig. 81.

9.5.2 Lateral Movement of Slab Edges. A comparison betweenrestrained and simply-supported two-way slab test results showedthat the deflection capacity-span ratio of simply-supportedslabs is slightly greater than that of restrained slabs. Theaverage 6 ult-span ratio of simply-supported slabs is 0.18whereas the 6Qlt-span of restrained slabs is 0.14. This isindicated in Fig. 81.

9.5.3 Span-Depth Ratio. A plot of incipient collapse de-flection, 6'Ilt, and short span-depth of restrained two-wayslabs shows a wide scatter (Fig. 82). This implies that shortspan-depth has no notable affect on 6ult . This observationlends support to the hypothesis that a slab acts essentially asa cable net in the tensile membrane action range.

9.5.4 Combined Short Span-Steel Breaking Strain. Two ap-proaches are available to relate 6 tiut with a combined effect ofshort span and breaking strain of the reinforcement. Herzog(45) hypothesized that

ult (

Hawkins and Mitchell (47) suggest that

L c6 a y uult

sin V-1U

-116-

.1.

UU>

C)

~~(o

00 0W c

N4 M

00 u

C. <N Ci

CL -5 .-

oq -

o oo 0

CL I oC

-117-

00

C rQ

0l.

-C) 0

To E-4

a).a)-

0

o Z m

LU a

< cr0

()

0n )

- ~ 4-0

4)

00

CN 0- N ~ 0 0 (24

SNBV3flDBdS ISi. 10 'ON

.90

'-4 -t

0p

4J

V) 0)

cl. 1>

o0 00 oDo.

0 44-

4JU

0 00

44 4-4

0 ic

0. 39 CL 4 0

44 0.0 0 rCD

Cu!) *I'NOIJ.31J3G 3SdV-1-1O IN31dlDNI

-119-

Figures 83 and and 84 show plots of the above relationships.Both show strong positive relationship with ult" The scatterseems identical in both cases. The experimental data availablefor which this combined effect can be considered is quite lim-ited. The reason is that few investigators have reported thebreaking strains of reinforcement used in their respective testspecimens. However, it should be noted that the data cover awide range of specimens sizes. The smallest slabs were 29 in.square while the largest was 72 in. square.

9.6 Comparison of Existing Design Methods and Test Data

Park (41) and Keenan (42) determined empirically the safe max-

imum value for central deflection of restrained slabs, ult,in tensile membrane actions to be%

ult = 0.(30)

where L = short span of slaby

Later, Black (44) determined Eq. 30 to be too conservative anestimate of the deflection capacity. Black suggested the de-flection capacity, 6ult, to be:

6ult = 0.15 Ty (31)

A comparison of Eqs. 30 and 31 with available test data indi-

cates that Eq. 30 yields a lower bound whereas Eq. 31 yields anupper bound on the test data.

Herzog (45) determined that the incipient collapse deflection

capacity, 6 ult , depends on two parameters: short span of slab,

and steel elongation at rupture. Assuming the slab to take theshape of a parabolic cable, taking into account the irregular

strain distribution in slab reinforcement, Herzog obtained forthe midspan deflection after tensile membrane action:

6 = Ly - - 0.31 Ly JFu (32)6ult y 2

where Ly = short span of slab

u = steel strain at rupture

A comparison of Eq. 32 with available test data, shown in Fig.83, indicates that the equation gives a reasonable estimate ofthe incipient collapse deflection capacity of restrained slabs.For large specimens, Eq. 32 appears to be slightly on theunsafe side.

-120-

!A

*EJ* ci

ii .4J 04u CLC) 0

N ~4U NP

c0 la

o.M 4-4

1c

44.

a. x. co

0 V) a,, o~

-121-

Pi crA

0 a

U)

4-O

(aO

1-

-W 4J

m. C>

00

o 000 O

4-J

w 041

co0)

I I

!ut fu~ a)LL313 (aV1O .~I

CL-1 2-t

L En91E

Hawkins Mitchell (47) assumed that the tensile membrane takesthe shape of a circular arc and developed the following expressfor 6ul t

1.5 L VE U (6ult - sin, 6(33)

Figure 84 shows that Eq. 33 grossly overestimates The incipientcollapse deflection of the test slabs considered. A probablereason for this discrepancy is the assumption that the slabbehaves merely as a cable net, and that there is no rigidityprovided by concrete enclosed by the steel mesh.

9.7 Selection of Approach to Determine Incipient Collapse

Deflection

A comparison of the test data with the existing approaches

indicates that two reasonable approaches exist to determine theincipient collapse deflection of slabs, 6ult. One approachassumes that 6ul t depends only on short span of the slab.The second approach stipulates that both short span of slab andsteel strain at breaking point are needed to determine 6 ult"

When only the short span of a slab is used to determine 6ult ,Eq. 30 provides a conservative estimate. However, it seemsmore realistic to include both breaking strain of steel andshort span of slab in predicting 6ul t , Both Eqs. 32 and 33use these two parameters in predicting 6 ult , but neither pro-vides a safe estimate. It is proposed that the following equa-tion be used to determine the incipient collapse deflection:

6ult - 0.25 Ly/ u (34)

A comparison of Eq. 34 with the test data is shown in Fig. 83.It should be noted that the Eq. 34 is slightly more conservativethan Herzog's equation and provides a practical lower bound tothe test data.

There are several points that should be noted in relation tothe proposed Eq. 34. First, the test data on which it is basedinclude both square and rectangular slabs. The short span ofthe slabs range from a low value of 29 in. to a high value of72 in., more than twice the shortest span. Finally, the testresults are from the work of not just one investigator but ofthree different investigators. In spite of the scarcity ofdata, the fact that the proposed relationship (i.e., designcriterion) represents a reasonable lower bound on data coveringa wide range of conditions provides some assurance of itsreliability.

As mentioned earlier, the incipient collapse deflection of sim-ply-supported slabs was, on the average, higher than that of

-123-

restrained slabs. Since no data was available on their steelbreakincj %train, no separate relationship is proposed. It isbelieved that Eq. 34 can provide a safe estimate for two-waysimply supportel1 slabs. A~s Eq. 34 was derived using a singlecable, its use for one-way slabs seem realistic.

it should be borne in mind that incipient collapse as definedhere is assumed to be initiated by rupture of flexural steel.It is further assumed that the slab is designed so that prema-ture failure due to bond or shear does not occur. The designconstruction requirements to develop tensile membrane behaviorso that incipient collapse will occur by tensile rupture offlexural steel will be examined in the report on Phase IT ofthis investigation.

-124-

10. SUMMARY AND RECOMMENDATIONS

This report, which covers the work on Phase I of the inves-tigation, presents a review and evaluation of literature onanalytical and experimental work on conventionally reinforcedconcrete slabs. Included are simply-suppcrted as well asrestrained one-way and two-way slabs under static loading.

Primary attention in the investigation was focused oa theincipient collapse deflection capacity of conventionally rein-forced concrete slabs under static loading. For t'- purpose

of this study, incipient collapse is defined as that state ofa slab characterized by a drop in the Icad capacity followingmobilization of tensile membrane action. The collapse condi-tion is associated with tensile rupture of the flexural rein-forcement. Emphasis wa3 thus placed on the tensile membranebehaviur of slabs.

The main objective of work in Phase I has been the evaluationof proposed analytical methods for predicting incipient col-lapse deflection of slabs. Comparison of analytically pre-dicted deflections with available experimental data was usedas the principal basis for determining the reliability of ananalytical method.

The literature review showed that although a large number oftests on slabs have been done, very few tests have beencarried out to the point of incipient collapse. This reflectsthe limited interest that slab researchers as a group have hadin the behavior of reinforced concrete slabs in the rangeapproaching total collapse.

In terms of analytical prediction, some approaches to deter-mining the load-deflection carve for simply-supported slabshave shown reasonably close agreement with measured curves.However, none of these methods was developed for predictingincipient collapse deflection capacity. Also, no single,

rigorous analytical method is available for predicting theentire load-deflection relationship of restrained two-wayslabs. Proposed methods for predicting load-deflection curvesfor restrained two-way slabs have consisted essentially intrying to predict the general trends of separate segments ofthe overall curve, without clearly defining the endpoints ofthese segments. Available methods for predicting incipientcollapse deflection of two-way slabs are approximate and basedon the assumption of pure membrane action, i.e., on theassumption that a typical slab strip behaves as a cable in thetensile membrane range.

The assumption of pure membrane action implies a uniformstrain distribution along the length of the slab reinforce-ment. This results in a predicted collapse deflection

-125-

--- __ . .r%,- .

(corresponding to rupture of the r-inforcement) consi.lerablygreater than that observed in tests. Tt is obvious thatbecause of cracking, the strain listribution in the flexuralreinforcement of an actual slab, even in the tensile membranerange, is non-uniform. The magnitude of the strains atvarious points along the reinforcement may also be affected byprevious flexural response histor,.

Wrom a correlation of the basic expression for pure membraneaction and available experimental data, an expression whichprovides a reasonable lower bound to the test ,ata is pro-posed. This expression for the incipient collapse deflectionof two-way reinforced concrete slabs under static uniform loadis given by

6 ult 0.25 Ly Vi'

where

Ly = short span of slab

E = breaking strain of flexural steel.

The above expression indicates that the incipient collapsedeflection is primarily a function of two parameters, namely,the short span of the slab and the rupture strain of the rein-forcement. As indicated in Fig. 83, the above equation providesa reasonably safe estimate of the incipient collapse deflectioncapacity of two-way restrained slabs. The equation is slightlymore conservative when applied to simply-supported slabs. Ttis therefore recommended that the same expression be used forboth restrained and simply supported slabs.

The proposed expression for 6ult implies an angle of rotationat the support (in the direction of the short span) of about 10degrees for an Eu = 0.11 or about 13 degrees for C = 0.20.

In recommending the above expression for estimating the incip-lent collapse deflection of conventionally reinforced concreteslabs subjected to static uniform loads, it is implicitlyassumed that the slab will be properly designed to precludepremature failure due to shear or bond.

-126-

ACKNOWLEDGMENT

This work was undertaken as part of the activities of the Struc-tural Analytical Section, Engineering Development Division,Portland Cement Association. The draft of this report wasreviewed and valuable suggestIons were provided by Dr. W.G.Corley, Divisional Director, Engineering Development Department.

r

-127-

REFERENCES

I. Anonymous, "Str(ctures to Resist the Effects of AccidentalExplosions," Department of Navy Publication, NAVFAC P-397,June 1969.

2. Tohansen, K.W., arudlinieteorier, Copenhagen, 1943; EnglishEdition, "Yield-Line Theory," Cement and Concrete Associa-tion, London, 1962.

3. Timoshenko, S.P. and Kriegter, W., "Theory of Plates andShells," 2nd Edition, McGraw-Hill, 1959.

4. Wood, R.H., "Plastic anJ ELastic Design of Stabs andPlites," Thames and Hudson, L961.

5. Kemp, K.O., "A Lower Bound Solution to the Collapse of anOrthogonally Reinforced Slab on Simple Supports," Magazineof Concrete Reset ch, Vol. 14, No. 41, July 1962, pp. 79-84.

6. Vijayarangan, S., "Lower Bound Solutions for ContinuousOrthotropic Slabs," Journal of Structural Division, ASCE,Vol. 99, No. ST3, March 1.973, pp. 443-452.

7. Hil.lerbor,3, A., "A Plastic Theory for the Design of Rein-Forced Concrete Slabs," Prel. Publication, 6th Congress ofthe International Association of Bridige and StructuralEngineering, 3tockholm, 1960.

8. Crawford, R.E., "Limit Deqign of Reinforced ConcreteSlabs," Journal of the Engineering Mechanics Division,ASCE, Vol. 90, No. EM5, October 1964, pp. 321-342.

9. Wood, R.H. and Armer, G.S.T., "Theory of the Strip Metholfor the Design of Slabs," Proc. of the Institution ofCivil Engineers (London), Vol. 41, October 1968, pp.285-311.

10. Swedish State Concrete Committee, "Massive Concrete Slabs,Specifications for Design Methods, Etc.," Stockholm, 1958.

I1. Armer, G.S.T., "Ultimat- Load Tests of Slabs Designed bythe Str;p Method," Proc. o F the institution of CivilEng[.leers (London), Vol. 41, October 1968, pp. 313-331.

12. Oc-leston, A.I., "Load Tests on a Three Storey ReinforcedConcrete Buil ding in *ohannesberg," Structural Engineer,Vol. 33, No. 10, October 1955, pp. 304-322.

i. Ockleston, A.I., "Arching Action in Reinforced ConcreteSlabs," Structural Engineer, Vol. 3;, No. 6, June 1958,pp. 197-201.

-1 2A-

AFFERENCES0,o-it'd)

14. Kemp, K.O., "1'ivt d o a Square Reinforced Concrete Slab onSimple Support:- lIowing for Memhrane Action," StructuralEngineer, Vol. 45, No. 7, Jul", 167, pp. 235-240.

15. Powell, D.S., "The UIltinate Strength cf Concret< Parne.sSubjected to Un.iforiIy Distributed L o;K -7 ar'WridqeUnivecsit7 Thesis, 1956.

16. Wood, R.R., "Discus:io.- o i the PapeLf Ultimate .trengthof Rectanaular 'c n-.Aete Slabs ui.der Short -rc.-m UriformLoading with Edges Restrained Againrt Latetal Movement,"Proc. Institution of Civil Engineers (Lcc:dcn), Vol. 32,September 1965. pp. 97-102.

17. Sawczuk, A., "M.'nbrane Action in Flexuze of RectangularPlates with R'trained Edges," Flexural ?echanics of Rein-forced Coitcrete, ACI/ASCE, SP-12, Detroit, 1965, pp.347-358

18. Sawczut, A. and Winicki, L., "Plastic Behaior of SimplySupported Plates at Fiodcrately Large Deflections," Inter-national Journal of Solids and Structures, Vol. 1, No. 1,February 1965, pp. 97-111.

19. Hung, T.Y. and Nawy, E.G., "Limit Strength and Service-ability Factnrs in Uniformaly Loaded isotropicallyReinforced Two-Way Slabs," Cracking, Deflection andUltimate Load of Concrete Slab Systems, ACI, SP-30,Detroit 1971, pp. 301-311.

20. Park, R., "Ultimate Strength of Rectangular Concrete SlabsUnder Short Tecm Uniform Loading with Edges RestrainedAgainst Lateral Movement," Proc. Institution of CivilEngineers (London), Vol. 28, June 1964, pp. 125-150.

21. Park, R., "The Ultimate Strength and Long-Term Beinaviourof Uniformly Loaded Two-Way Concrete Slabs with PartialLateral Restraint at All Edges," Magazine of ConcreteResearch, Vol. 16, No. 48, September 1964, pp. 139-152.

22. Park, R., "The Lateral Stiffness and Strength Required toEnsure Membrane Action at the Ultimate Load of a Rein-forced Concrete Slab and Beam Floor," Magazine of ConcreteResearch, Vol. 17, No. 50, March 1965, pp. 29--38.

23. Park, R., "Furthec Test on a Reinfojrced Concrete FloorDesigned by Limit Poced',res," Cracking, Deflection andUltimate Load of Concrete Slab Systems, ACI SP-30, 1971,pp. 251-270.

- 129-

REFERENCES

(Cont'd)

24. Park, R. and Gamble, W., "Reinforced Concrete Slabs," JohnWiley, New York, to be published.

25. Twankiw, N.R. and Longinow, A., "Structural Response of

Floor Assemblies to Fire," preprint ASCE Annual Conven-tion, October 1978, Chicago, Illinois.

26. ACT Standard, "Building Code Requirements for ReinforcedConcrete," ACI 318, 1977.

27. Hopkins, D.C. and Park, R., "Test on a Reinforced ConcreteSlab and Beam Floor Designed with Allowance for MembraneAction," Cracking, Deflection and Ultimate Load of Con-crete Slab Systems, ACT, SP-30, Detroit, 1q71, pp. 251-269.

28. Morley, C.T., "Yield Line Theory for Reinforced ConcreteSlabs at Moderately Large Deflections," Magazine of Con-crete Reserch, Vol. 19, No. 61, December 1967, pp. 211-221.

29. Nawy, E.G. and Blair K., "Further Studies on FlexuralCrack-Control in Structural Slab Systems," Cracking,Deflection and Ultimate Load of Concrete Slab Systems, ACISP-30, 1971, pp. 1-41.

30. Nawy, E.G., et al, "Crack Control, Serviceability andLimit Design of Two-Way Action Slabs and Plates," RutgersUniversity, Engineering Research Bulletin 53, 1972.

31. Nawy, E.G., "Crack Control Through Reinforcement Distri-bution in Two-Way Acting Slabs and Plates," Journal ofACT, Vol. 69, No. 4, April 1972, pp. 217-219.

32. Jacobson, A., "Membrane Flexural Modes of RestrainedSlabs," Journal of Structural Division, ASCE, Vol. 93,No. ST5, October 1967, pp. 85-112.

33. Brotchie, J.F. and Holley, M.J., "Membrane Action inSlabs," Cracking, Deflection and Ultimate Load of ConcreteSlabs Systems, ACI, SP-30, Detroit, 1971, pp. 345-377.

34. Moy, S.S.J. and Mayfield B., "Load Deflection Characteris-tics of Reinforced Concrete Slabs," Magazine of ConcreteResearch, Vol. 24, No. 81, December 1972, pp. 209-218.

35. Massonet, C., "General Theory of Elastic-Plastic MembranePlates," Paper presented at Conference on EngineeringPlasticity, Cambridge University, March 1968.

36. Ramesh, C.K. and Datta, T.K., "Ultimate Strength of Rein-forced Concrete Slab-Beam Systems," The Indian ConcreteJournal, Vol. 47, No. 8, August 1973, pp. 301-308.

-130-

(Cont' d)

37. Datta, T.K. and Ramesh, C.K., "oie Exper7.a:nt. LOvuli.c-on a Reinforced Concrete Slab-Peam Sy.'t.erm," Ma9 aZire oLConcrete Research, Vol. 27, No. 91, June rq5. tl,. iil-'20

36. Hognestad, E., Hanson, N.W. anA ch i. i . t rStress Distribution in Ultimate StrentL ;; Pr-ceedings ACI, Vol. 52, 1955-1956, pp. %55-'Y

39. Girolami, M.A. , Soze"., M.A. n d Gmle, .r:,1: .

Strength of Reinforced Corncot.n S'abs w t: -

Applied In-Place Forces," and Gamble, W.1 . ,uug, H. iSozen, M.A., "Strength of Slab subjeced ':o Multlaxia!Bending and CoDmpression," Civil Engineering Stua-es,Structural Research Se-i.e. No. 3 6;,. Urive: t., nfIllinois, Urbana, 1970.

40. Desayi, P. and Kulkarni, A.B., "ioad-UefIectiRestrained R/C Slabs," Journal of Structural DiV. nASCE, Vol. 103, No. ST2, February 1977, pp. 405-419.

41. Park, R., "Tensile Membrane Behavior of Uniformr-.' L( :,ecRectangular Reinforced Concrete Slabs wit'- FuL> Restr. *.,-,e(;Edges," Magazine of Concrete Research, V 1l. 16, No. 45,March 1964, pp. 39-44.

42. Keenan, W.A., "Strength and Behv if.r P, RestraindReinforced Concrete Slabs Under Static an6 Dynamic oa,Technical Report R621, U.S. Naval. Ci. , nginer

Laboratory, Port Hueneme, California, April 1969.

43. Denton, D.R., "A Dynamic Ultimate Strength Study r)f S:mpivSupport Two-Way Reinforced Concrete Slabs,' U.S. Ar'Engineer Waterways Experiment Station, Vickqb r%,Mississippi, Technical Report 1-789, July 1967.

44. Black, M.S., "Ultimated Strength Study of Tw---Way Conc:eteSlabs," ASCE Proc., Journal of Structural Division, Vol.101, No. ST1, January 1975, pp. 311-324.

45. Herzog, M., "The Membrane Effect in Reinforced ConreteSlabs According to Tests," Beton and Stahlbetonbhau, Vel.(71), 1976, pp. 270-275 (in german).

46. Gilbert, R.I. and Warner, R.F., "Tero. stirfrni inReinforced Concrete Slabs," Jou;r, al of zS.-uctural D:i:-sion, ASCE, Vol. 104, No. ST12, Decemner i97.

47. Hawkins, N.M. and Mitchell, D., "Progressive Collapse ofFlat Plate Structures," ACI Journal, Vol. 76, No. 7, juv1979, pp. 775-808.

-131-

- .... .

REFERENC3S

(Cont' d)

48. Taylor R., "A Note on a Possible Basis- for a New Method ofUltimate Load Desiqn of Reinforced Concrete Slabs," Maga-zine of Concrete Research, Vol. 1.7, No. 53, December 1965,pp. 183-186.

49. Taylor R., Maher, D.R.H. and Hayes B., "Effect of theArrangement of Reinforcement on the Behavi r of ConcreteSlabs," Magazine of Concrete Research, Vol. 18, No. 55,June 1966, pp. 85-94.

50. Hayes, B., "Allowing for Membrane Action in the PlasticAnalysis of Rectangular Reinforced Concrete Slabs," Maga-zine of Concrete Research, Vol. 20, No. 65, December 1968,pp. 205-211.

51. Geymayer, H.G. and McDonald, J.E., "Influence of Rein-forcing Details on Yield-Line Pattern and UltimateLoad-Carrying Capacity of Reinforced Concrete Slabs,"Miscellaneous Paper No. 6-911, U.S. Army EngineerWaterways Experimental Station, Vicksburg, Mississippi,August 1967.

52. Desayi, P. and Kulkarni, A.B., "Load-Deflection Behaviorof Simply-Supported Rectangular Reinforced ConcreteSlabs," International Association for Bridge and Struc-tural Engineering, Proceedings P-11/78, February 1978.

53. Comite Europeen du Beton, "Recommendations for an Inter-national Code of Practice for Reinforced Concrete," ComiteEuropeen du Beton, Paris, 1964.

54. Christiansen, K.P., "The Effect of Membrane Stresses onthe Ultimate Strength of the Interior Panel in a Rein-forced Concrete Slab," Structural Engineer, Vol. 41, No. 8,August 1963, pp. 261-265.

55. Roberts, E.H., "Load Carrying Capacity of Slab Strips Re-strained Against Longitudinal Expansion," Concrete, Vol. 3,No. 9, September 1969, pp. 369-378.

56. Iqbal, M., "Behavior and Strength of R. C. Plates Rein-forced with Hot and Cold-Rolled High Strength Steels,"M.S. Thesis, Middle East Technical University, Ankara,Turkey, August 1969.

57. Birke, H., "Arch Action in Concrete Slabs," Meddelande NR108, Institutionen for Byggnadsstatik, KTH, Stockholm,Sweden, 1975. (In Swedish with English Abstract.)

-132-

~-4,A

REFERENCES(Cont'd)

58. Komori, K., "Studies on Strength and Veflect.'on fReinforced Concrete One-Way Slabs (Part ')0. Transactionsof the Architectural Institute of Japan, Vol. 269, July1978, pp. 61-72. (In Japanese with English k'st.:act.)

59. Bazant, Z.P., Schnobrich, W.C. arid rcordeis, A.C.,"Finite Element Analysis of Reinforced Concrete Strkc'-tures," Fratelli Presenti Specialty Course on Reariforce6Concrete Construction, Milan Po~ytechnic, Malan, !taly,June 1978.

60. Ngo, D. and Scordelis, A.C., " %inite Analysis of Rein-forced Concrete Beams," Journal of the American ConcreteInstitute, Vol. 64, No. 3, March 1967, pp. 152-160.

61. Nilson, A. H., "Nonlinear Analysis of Reinforced Concreteby the Finite Element Method," Journal of the AmericanConcrete Institute, Vol. 65, No. 9, September 1968.

62. Mufti, A.A., Mirza, M.S., Cutcheon, J.O. and Houde, J., "3Study of the Nonlinear Behavior of Structural ConcreteElements," Proceedings of the Specialty Conference? ofFinite Element Method in Civil Engineering, Mon,:ea;,Quebec, Canada, June 1972.

63. Rashid, Y.R., "Ultimate Strength Analysis of PrestressedConcrete Pressure Vessels," Nuclear Engineering Dksiqn,Vol. 7, p. 334, 1968.

64. Isenberg, J. and Adham, S., "Analysis of OrthotropicReinforced Concrete Structure," Journal of the StructuralDivision, ASCE, Vol. 96, No. ST12, December 1970, pp.2607-2624.

65. Valliappan, S. and Doolan, T.F., "Nonlinear Stress Anal-ysis of Reinforced Concrete," Journal of the StructuralDivision, ASCE, Vol. 98, No. ST4, April 1972, pp. 885-898.

66. Nam, C.H. and Salmon, C.G., "Finite Element Analysis ofConcrete Beams," Journal of the Structural Division, ASCE,Vol. 100, No. ST12, December 1974, pp. 2419-2432.

67. Bell, J.C. and Elms, D.G., "Nonlinear Analysis of Rein-forced Concrete Slabs," Magazine of Concrete Research,Vol. 24, No. 79, June 1972, pp. 63-70.

68. Jofriet, J.C., "Flexural Crackiig of Concrete FlatPlates," Journal of ACI, Vol. 70, No. 12, December 1973,pp. 805-809.

-133-

REFERENCES(Cont'd)

69. Jofriet, J.C., "Short Term Deflection of Concrete FlatPlates," Journal of Structural Division, ASCE, Vol. 99,No. STI, January 1973, pp. 163-182.

70. Scanlon, A. and Murray, D.W., "An Analysis to Determinethe Effects of Cracking in Reinforced Concrete Slabs,"Proceeding Conference on Finite Element Methods in CivilEngineering, McGill University, 1972, p. 841.

71. Lin, C.S., Scordelis, A.C., "Nonlinear Analysis of RCShells of General Form," Journal of Structural Division,ASCE, Vol. 101, No. ST3, March 1975, pp. 523-538.

72. Hand, F.A., Pecknold, D.A. and Schnobrich, W.C., "LayeredFinite Element Procedure for Inelastic Analysis of Rein-forced Concrete Slabs," Civil Engineering Studies, Struc-tural Research Series No. 389, University of Illinois,Urbana, Illinois, August 1972.

73. Kupfer, H.B., Hilsdorf, d.K. and Rusch, H., "Behavior ofConcrete Under Biaxial StLesses," Journal of the AmericanConcrete Institute, Vol. 66, No. 8, August 1969, pp.656-668.

74. Wanchoo, M.K. and May, G.W., "Cracking Analysis of Rein-forced Concrete Plates," Journal of Structural Division,ASCE, Vol 101, No. STI, January 1975, pp. 201-215.

75. Backlund, J., "Finite Element Analysis of NonlinearStructures," Chalmers Tekniska Hogskola, Gotenberg,Sweden, 1973.

76. Berg, S., "Nonlinear Finite Element Analysis of ReinforcedConcrete Plates," Norwegian Institute of Technology,University of Trondheim, Norway, 1973.

77. Knopfel, H., "Calculation of Rigid Plastic Slabs by FiniteElement Method," Institut for Baustatik, ETH, Switzerland,Bericht No. 47, August 1973.

78. Bashur, F.K. and Darwin, D., "Nonlinear Model for Rein-forced Concrete Slabs," Journal of Structural Division,ASCE, Vol. 104, No. ST1, January 1978, pp. 157-170.

79. Kabir, A.F., "Nonlinear Analysis of Reinforced ConcretePanels, Slabs, and Shells for Time Dependent Effects,"University of California, Report SESM 76-6, December 1976.

80. Kabir, A.F., "Analysis of RC Shells for Time DependentEffects," preprints, ASCE Annual Convention held inBoston, April 1979.

-134-

REFERENCES(Cont'd)

81. Bathe, K.J. and Ramaswamy, S., "On Three-DimensionalNonlinear Analysis of Concrete Structures," MassachusettsInstitute of Technology, Cambridge, Massachusetts, 1978.

82. Bathe, K.J., "ADINA-A Finite Element Program for AutometicDynamic Incremental Nonlinear Analysis,' MI' Reports82448-1 and 82448-2, 1976.

83. Jain S. and Kennedy, J., "Yield Criterion for ReinforcedConcrete Slabs," Journal of Structural Division, ASCE.Vol. 100, March 1974, pp. 631-644.

84. Suidan, M. and Schnobrich, W.C., "Finite Element Analysisof Reinforced Concrete," Journal of Structural Division,ASCE, Vol. 99, No. STl0, October 1973, pp. 2109-2122.

85. Murray, D.W. and Wilson, E.L., "Finite-Element LargeDeflection Analysis of Plates," Journal of EngineeringMechanics Division, ASCE, Vol. 95, No. EMl, February 1969,pp. 143-166.

86. Personal Communication with the Naval Civil EngineeringLaboratory Staff, Port Hueneme, California, June 1979.

-135-

-SO

I! P.

I I r

ofto ..

.:136-

0~. A 0 -1 64 1. 3 I

'-i ! i

m" - :7 J: _ L F... .!'- .+ [-T): i';H-, ,.... ....io

E-

0 'o

., I 5

--- -- -.. -I - r-

.I 71

71, L1, h ,; * . .....:

I

a :LLL 2 2 2 k99*w Q!~ C! ". . a

la,-

8 .1 3 a a S a i a a a S! 1

4 4 1 1 . 4 . .

"

Ot CI -t t. -

9 0P9 ....

I

-t,- - .

-137-

1 £~~~~~~~ lI a n, a a a l I.~

El!i

li~~~~ ~ ~ ~ C; C;C ;C ;0 04 ;4 ;

I I! C!I

310 1__._F_ 1 ,1 1 "

g- Li i ii* I - . ' o . .

L- I 2;

-: 0)138-

V II A.! I

4-4 :3 ,

I i l I I

,i r -I - -, . . .i . , . .'-'-4-

A4 _ I !_ 0

, o : ;1" -I-I0- I • 1- - .I .0 i -

, i. ... . . . ... t

,- ' ,. I- .- ., . . . . I ll .

" ! "' '-" ' . :. . . . i

" I. I "Il

A A]

0 -- - -_ _

-139- C - i

~ * 0 0-00%

uia*l

:I •

E-

9~~~. ° t00 !Q!C

I.

C! G! G! C! I * a 4

-4

S 40-j

I i-n -

0 _

o) - -s1 a 0 ' me . em- .

C) -:,zi°!i li -± f.o I I I~ I: ]

1 -140 -

I s.

1:- 'i - -

U) gih g

E- .43 s I 3 2 A

0 o a o S . o

o ~ ~ -Z ___

*~~~1 .x

. . . . .

--- 4 ] . . - C S i C

I ;:i :

-4b

2.1 2

0 Q) ; i . o , ,,

!i-I-

- 4

a it! IM

WU 0

En _ __ ~

I 0*24~-.- - - --. -

(.12 KE-4@0 . @'~~e l

I142

Tible 3 Properties and Test iResu!Ls fo): :DW1 1,)

as Lb.,.

0~3 0p.

I 2. is I- ~ - 4- -- -306 --- 3 3.

133 "2 womo.1 2.1 i 3 /1- a.42 U 730 1e00 m.m 1.17 270 1 0.911 ....rL

via- - __sel wvw3 lc 0.34 U.0 3010 1.01 0.2) 3.ite s t.4

sel:1*001J 1.001 "A -4-0 0.. .... 1,2as 1.-- 1---- 0.2 t.f.,rek t.@.Al, 1.0 .3047 1" IOL 271 ~ .q.

-; 70. 315 .0 2010 0.49 10

313 34'm. is sli 0.74~ 35.0 30 0.34 0..34

mu1 m~oO2 0.3 5.0 03. 2A 0

0e) M se M.74 37.403 0."1 .:1 ......... 4 0.2

1111) Wp ull 9 G. 0.9 0.24

0534 200.2 4.053 3 1muK f20 1 1 a." __

.1.!!___ _ .0 0.3

mS1 D-- 1."3.0

1110 30e9.3 21 0.92 030 .3M 1.0 3.45 0.146

027 WW-3 36 0.93 080 0.60 I.22 3_~ .27 0.34

ate4 %Whog3 191 0.30 3139 0.431 0.87 4.7 0.1

"it 0.30 30000 13

059 9 0.83000.34 1.10 3.33 0.A

_____ 0.303 i 3330 0.30 1.01 6.78 0.37

am2 0."2 210 0.41 410.111 3.17 0.14

_ _ _.9 3.330 330.1__ _

ii NS 1] 0.921 1 3 0o.47 0.93 3.03 0.14

ma23 $609 191 0 1 ~ 030 0.51 3.03 S06 01

0523 60 t .2 0413 03 0

19001 "Ip0 7" 0.29 1.19 303 0.13us2 "out.I 39 0 . l 2000 0.43 1.72 0.14

'.t.... 4 3 .23 07 0 U **fft 0. .0 3130 03 .03 .9 7031 1.036~~

set *%ft r*1 G"0 210 I 4.1 1.01 M 17.30 2.14 tid oe.

MAI I It.73. 2 24 S 0.4 1 .3 It37" 37 3. 3 0.6,63 1.02 2.11 0** 6t

3.hT .4 3e. 0 .1 03.1 2770 18 4. 0 3.6 0.3 130 3.02 1.2' s4..1 . 0.-

-14 3-

Table 4 Measured Central Deflection/Slab Thickness atUltimate Load of Uniformly Loaded Laterally

Restrained Slabs

Investigator (s) o L/Ly Ly/h Range of D/.

Wod(4) 2 1.0 30 0.50 - 0.70

Ockleston (1 2 ) 2 1.20 36 0.56

Powell(15) 15 1.75 16 0.33 - 0.44

Hung and Nawy(1 9) 7 1.0 24 0.81 - 0.89

" 5 1.43 17 0.62 - 0.74

Park( 2 0 - 2 3 ) 5 1.5 20 0.39 - 0.50

" 1 1.5 27 0.48

U 3 1.5 40 0.37 - 0.50

Nawy and Blair( 2 9) 28 1.0 20 0.39 - 0.89

12 1.4 17 0.55 - 0.91

Brotchie and Holley(33 ) 9 1.0 20 0.36 - 0.57

" 8 1.0 10 0.10 - 0.22

Keenan(4 2) 4 1.0 24 0.33 - 0.51

1 1.0 15 0.20

"1 1.•0 12 0.18

Black( 4 4) 4 1.0 33 0.34 - 0.71

Total No. of Slabs: 107

-144-

- - .

Table 5 Measured Maximum Deflection/Span Patio forTwo-Way Slabs Acting as Ten;ile Membrane

Boundary L L ma x i;um

Conditions Number x _1 Def2zctionInvestigator (s) of Slabs of Slabs LShort Span

Wood (4) Restrained, 1 -Single Panel 1 10 2.

Park(22) Restrained. 1.5 (3. 0.10-0.12Single Panel _

Hopkins and Interior PanelPark(27) in 9-Panel Slab- 1 1.0 36.0 0.09Beam Floor

Sy s tern

Brotchie and Restrained, 3 1.0 20.0 0.14-0.17

Simply-Supported 4 1.0 20.0 0.14-0.17

Restrained, 3 1.0 10.0 0.13-0.14

Simply-Supported 3 1.0 10.0 0.14-0.17

Keenan(4 2 ) Restrained 4 1.0 24.0 0.111 2.0 15.2 0.09

1.0 12.0 0.06

Black (44 ) Restrained 4 1.0 33.0 0.14-0.16

Geymayer and simply-Supported 5 1.0 60.0 0.06-0.07

McDonald (49 )

Sawczuk and Simply-Supported 2 1.45 37.0 0.08-0.lI

WInnickil 1 2.0 33.0 0.15

Taylor, Maher Simply-Supported 1-6 1.0 36.0 0.03-C.04

and Hayes (46) 7-8 1.0 41.0 0.059-10 1.0 24.0 0.05

Desayi and Simply-Supported 1 1.2 20.0 0.08

Kulkarni (50) 1 1.2 26.7 0.06

-145-