1990 Punching shear strength of reinforced concrete flat

University of WollongongResearch Online

University of Wollongong Thesis Collection University of Wollongong Thesis Collections

1990

Punching shear strength of reinforced concrete flatplates with spandrel beamsMasood FalamakiUniversity of Wollongong

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Recommended CitationFalamaki, Masood, Punching shear strength of reinforced concrete flat plates with spandrel beams, Doctor of Philosophy thesis,Department of Civil and MIning Engineering, University of Wollongong, 1990. http://ro.uow.edu.au/theses/1260

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PUNCHING SHEAR STRENGTH OF REINFORCED CONCRETE FLAT PLATES

WITH SPANDREL BEAMS

A thesis submitted in fulfilment of the requirements

for the award of the degree of

Doctor of Philosophy

from

THE UNIVERSITY OF WOLLONGONG

°y

MASOOD FALAMAKI, CPEng., B.Sc, M.Sc,

MIEAust., MAPEA

DEPARTMENT OF CIVIL AND

MINING ENGINEERING

1990

"In the Name of God, the Beneficent, the Merciful"

(ii)

DECLARATION

I declare that this work has not been submitted for a degree to any university or

such institution except where specifically indicated.

Masood Falamaki

April 1990

(iii)

ACKNOWLEDGMENTS

Grateful acknowledgment is made to my thesis adviser, Associate

Professor Y. C. Loo, for the close supervision, fruitful discussions and guidance

he has given for many years. The author also greatly appreciates the beneficial

training in research skills given by him during the course of this study; in fact,

without him this thesis would not have been possible.

The author also wishes to express his sincere gratitude to the Ministry

of Culture and Higher Education of the Islamic Republic of Iran for the scholarship

and the grant received during 1984 - 1988. In the past sixteen months the author

has worked as a research staff under a grant provided by the Australian Research

Council.

The author also acknowledges the generous material and personnel

support given by the following companies and institutions in Wollongong and

Sydney:

Acrow Pty Ltd.

Anitech, N.S.W.

A R C Engineering Pty. Ltd.

A S C , Austral Standard Cables Pty.

Ltd.

Baines Concrete Pumping

Cable Makers (ACT) Pty. Ltd.

Cleanaway

School of Civil and Mining Eng.,

Sydney University

G o Hire

- K G R Fabrications

- Kenweld Constructions Pty. Ltd.

- Newtek Electronics

- Nippy Crete Concrete

- Vernier Engineering Pty. Ltd.

- Wollongong TAFE, in particular the

Departments of Carpentry, Fitting

and Machining, Hydraulics and

Welding, and the Steel Store

(iv)

Sincere thanks are due to the technical staff of the Department of Civil

and Mining Engineering, University of Wollongong, in particular Messieurs. R. H.

Webb, A. G. Grant, G. K. Caines, F. Hornung, C. Allport, and I. N. Bridge for

their help in the experimental work.

Special acknowledgment is due to Dr. R. Kohoutek for his fruitful

discussion during the early period of the study.

Mr G. A. Aly was given a very difficult word-processing task but was

able to finish it with distinction in limited time. H e also skilfully produced most of

the drawings, with M r M . Habibnejad, and M r M . Hamedi assisting in the

process.

Special thanks are also due to Mr H. Vakili for his moral support and

friendship during the course of this study.

Special acknowledgment is due to his wife, Shahin, for her constant

support and encouragements given throughout the whole period of this study.

Finally, the author is indebted to all other members of his family in Iran

for the understanding and suffering during the rather long period of this study.

(v)

ABSTRACT

Research on the punching shear strength, Vu, of slab-column connections

of reinforced and prestressed concrete flat plates with spandrel beams has received

considerable attention by the engineering profession in recent years. In the case of slab-

column-spandrel connections of flat plates at the edge- and corner- positions there is still

no reliable procedure for the determination of Vu. Thus the main objective of the present

study is to develop an analytical method for the prediction of V u for these types of

connections. Needless to say, the development of a sound analytical method requires the

test results from large-scale models with proper boundary conditions.

Tests up to failure were carried out on five cast-in-situ flat plate models,

four with spandrel beams of different depths and steel ratios, and one without any

spandrel. Representing two adjacent panels at the comer of a real structure these half-

scale models were tested under simulated uniformly distributed loads. For ease of

construction, instead of concrete columns, each flat plate model was supported on six

prefabricated steel columns (with equivalent stiffnesses). The three reaction components

at each of the hinged column supports were measured by means of specially designed

load cells. Strain gauges were also attached to the reinforcing bars of the slab. The strains

and other electrical signals were logged using a Hewlett Packard 3054A data acquisition

control system via a Hewlett Packard 9826 computer.

In conjunction with the experimental work a theoretical study was carried

out. This led to the development of a prediction procedure for the punching shear

strength, V u, of reinforced concrete flat plates with spandrel beams. Details of the

theoretical work are presented herein. Applicable to the analysis of failures at the corner

and edge-column positions, the proposed procedure takes in to consideration the

following parameters:

(vi)

(1) the overall geometry of the connection,

(2) the concrete strength,

(3) the size and location of flexural reinforcement of the slab,

(4) the slab restraint on the spandrel, and

(5) the enhanced strength of the slab-column connection due to membrane effects.

Based on the model test results from the present study and those obtained

by other authors, a comparative study is carried out. The proposed analytical procedure

is found to be superior to the alternative approach recommended in the new Australian

Standard for Concrete Structures (AS 3600-1988). While the proposed procedure is more

accurate and consistent in its prediction, the Australian Standard approach suffers, at

times, the serious drawback of considerably overestimating the value of Vu, especially for

failure at the comer-column positions.

(vii)

TABLE OF CONTENTS

TITLE PAGE i

DECLARATION ii

ACKNOWLEDGEMENTS iii

ABSTRACT v

TABLE OF CONTENTS vii

LIST OF FIGURES xiv

LIST OF TABLES xx

NOTATION xxi

1 INTRODUCTION 1

1.1 The Problem 2

1.2 Existing Analytical Methods and Experimental Data 4

1.3 Codes of Practice 6

1.4 Size of the Model Structures 7

1.5 Objectives 8

1.6 Outline of Thesis 10

2 TRANSFER OF FORCES IN SLAB-COLUMN CONNECTIONS

OF FLAT PLATES 13

2.1 General Remarks 14

2.2 Spandrel Beams and Slab Behaviour 15

2.2.1 One-way slab action 15

2.2.2 Distribution of forces along the spandrels 17

2.3 Spandrel Beam and Modes of Failure 18

2.4 Equilibrium Equations 19

2.4.1 Definitions 19

2.4.2 Formulas 21

2.5 Distribution of Shear Force 23

2.5.1 Assumptions 23

2.5.2 The procedure 24

(viii)

3 INTERACTION OF TORSION, SHEAR AND BENDING IN

SPANDREL BEAMS 34


3.2 Slab Restraint and the Enhanced Strength of Spandrels 36

3.2.1 Slab restraint on the elongation of spandrels 36

J2>.1.1 Slab restraint on the rotation of the spandrels 38

3.2.3 Effects of column width 39

3.3 Interaction Surface for Isolated Reinforced Concrete Beams

(Truss Analogy) 40

3.3.1 Historical review 40

3.3.2 The interaction equation 41

3.4 Proposed Interaction Equation for Spandrels 44

4 DETERMINATION OF MOMENTS FROM REINFORCEMENT

STRAINS 54


4.2 Theoretical Moment-Strain Relationship 56

4.3 Bending Test on Isolated Reinforced Concrete Beams 59

4.4 Analysis of Results 60

4.5 Summary 61

5 EXPERIMENTAL PROGRAMME AND GENERAL BEHAVIOUR

OF THE FLAT PLATE MODELS 71

5.1 The Half-Scale Models 72

5.2 Analysis and Design of the Model Stractures 74

" 5.2.1 Analysis of forces using idealized frame method 74

'5.2.2 Design of the flat plates and the spandrels 76

y5.2.3 Design of the steel columns (of equivalent

stiffnesses) 77

(be)

5.3 Materials 77

5.4 Construction of the Models 78

5.4.1 Formwork 78

5.4.2 Reinforcement details 79

5.5 Casting and Curing 81

5.6 Instrumentation and Testing Procedure 81

5.6.1 Loading system 82

5.6.2 Testing procedure 83

5.6.3 Reaction measurements 83

5.6.4 Strain measurement 85

5.6.5 Measurement of deflections 86

5.6.6 Measurement of the angle of twist of the spandrels 87

5.7 Cracking and the Ultimate Load 88

5.8 Concluding Remarks 90

6 RESEARCH SCHEME FOR THE DEVELOPMENT OF THE

PREDICTION PROCEDURE FOR Vu 152


6.2 Outline of the Research Scheme 153

6.3 Determination of the Internal Forces within the Slab-Column-

Spandrel Connections 155

6.3.1 Semi-empirical equations for M j and Yl 155

6.4 Prediction of V u 156

6.4.1 Calibration of the proposed interaction equation 157

6.4.2 The prediction formulas 157

6.4.3 Comparison and discussion of results 158

(x)

DISTRIBUTION OF MOMENT AND SHEAR ALONG THE

CRITICAL PERIMETER 163


7.2 Measurement of the Internal Forces at Slab-Column Connections 165

7.2.1 Slab strip moments 165

7.2.2 Slab moments M l and M m 166

7.2.3 Shear force VI 167

7.3 Distribution of the Total Bending Moment 168

7.3.1 Edge column positions 168

7.3.2 Corner column positions 170

7.3.3 Accuracy of results 171

7.4 Distribution of the Total Shear Force 172

7.4.1 Theoretical background 172

7.4.2 The formulas 173

7.4.3 Edge column positions 174

7.4.4 Comer column positions 175

7.4.5 Comparison of results 176

7.4.6 Discussion 177

PREDICTION FORMULAS FOR SPANDREL PARAMETERS 185

8.1 Scope 186

8.2 Theoretical Consideration and Spandrel Parameters 186

8.3 Detenriination of the Spandrel Parameters 189

8.3.1 Measurements of co0 and \|/ 189

8.3.2 Prediction formulas for \j/ and X 192

8.3.3 Comparison and discussion of results 193

8.4 Slabs with Torsion Strips Without Closed Ties 194

8.5 Slabs with Deep Spandrel Beams 196

8.6 Summary 197

(xi)

9 PREDICTION PROCEDURE FOR PUNCHING SHEAR

STRENGTH Vu 205


9.2 The Proposed Procedure 207

9.2.1 Background 207

9.2.2 Assumptions and applicability of the procedure 208

9.2.3 Corner connections 209

9.2.4 Edge connections 212

9.2.5 Connections with torsion strips 213

9.2.6 Connections with torsion strips without closed ties 214

9.2.7 Connections with deep spandrel beams 214

9.2.8 Accuracy 215

9.3 The AS 3600-1988 Procedure 216

9.3.1 Background 216

9.3.2 Formulas 217

9.3.3 Accuracy 219

9.4 Comparison of the Procedures .....220

9.5 Summary 221

10 CONCLUSIONS 227

10.1 Failure Mechanisms 227

10.2 Moment and Shear Transfer Between Slab and Columns 228

10.3 Proposed Interaction Equation for Spandrel Beams 228

10.4 Prediction of V u 229

10.5 Versatility and Accuracy of the Proposed Procedure 230

10.6 Recommendations for Further Study 230

REFERENCES 232

(xii)

Appendix I Design Ultimate Load and Membrane Effects 239

Appendix II Interaction Equations and Prediction Formulas for Vu 244

Appendix III Strain Data of Flat Plate Models at the Ultimate State 249

Appendix IV The Critical Perimeter 253

Appendix V Reinforcement Details 257

Appendix VI Load-Strain Diagrams for the Reinforcing Bars 270

Appendix V H Measured Slab Strip Moments at the Ultimate State and the

Calculated Yield Moments of the Critical Slab Strips 277

Papers Published Based on This Thesis 285

LIST OF FIGURES

(xiv)

LIST OF FIGURES

FIGURE PAGE

1(1) Typical flat plate structure with spandrel beams 12

2.2(1) Typical flat plate model after failure 27

2.2(2) The top surface crack pattern of a one-third scale flat plate

model tested by Symmonds (1970) 28

2.2(3) Typical soffit crack pattern of the exterior panels of a

flat plate model tested by Rangan and Hall (1983) 29

2.2(4) Typical flat plate floor loaded on alternate spans 30

2.2(5) Theoretical variation of moment, shear and torsion along

the spandrel beams 31

2.4(1) Free-body diagram for edge- and comer-connection 32

2.5(1) Variation of moments along the slab strips 33

3.2(1) Induced compressive force, p, in the spandrels due to the

slab restraint 48

3.2(2) Effects of the column width on the slab restraining factor, 49

3.3(1) Forces in the vicinity of a comer column 50

3.3(2) Failure surface (compression zone in bottom of the beam) 51

3.3(3) Typical skew failure surface at the column position W4-C,

at the ultimate state 52

3.3(4) Failure surfaces for isolated reinforced concrete beams 53

4.1(1) Moment-strain relationship 63

4.2(1) Flow chart of the computer program used for the calculation

of moments from strain data 64

4.3(1) Details of the beams specimens Bl, B2, and B3 65

4.3(2) Formwork for the beam specimens 66

4.3(3) Flexural failure of beam specimen B1 67


(XV)


4.4(1) Comparison of typical theoretical and experimental

moment-strain relationships. 70

5.1(1) Half-scale flat plate system 92

5.1(2) Model W l 92

5.2(1) Distribution of the width of the idealized frame into the

affected and remaining widths 93

5.2(2) Plan view of flat plate Models W 2 to W 5 93

5.2(3) Design of spandrel / torsion strip - slab connections of

Models W l to W 5 94

5.2(4) Typical pre-fabricated steel column 95

5.2(5) Typical slab-column connection 96

5.3(1) Typical load-strain diagram 97

5.4(1) The steel pedestals 98

5.4(2) The adjustable support system for the prefabricated steel

columns 99

5.4(3) Typical support system 100

5.4(4) General view of the formwork 101

5.4(5) Typical arrangement of the slab reinforcement 102

5.4(6) Details of the U-bars 103

5.4(7) Connection of adjacent meshes 104

5.4(8) Effects of welding on the strength of the welded fabrics 105

5.4(9) Reinforcement details for column positions W l - A and

Wl-F 106

5.4(10) Reinforcement details for column positions W l -B and

W l - G 107

5.4(11) Reinforcement details for column positions W l - C and

W l - H 108

5.4(12) Typical reinforcement details 109

(xvi)

5.6(1)

5.6(2)

5.6(3)

5.6(4)

5.6(5)

5.6(6)

5.6(7)

5.6(8)

5.6(9)

5.6(10)

5.6(11)

5.6(12)

5.6(13)

5.6(14)

5.6(15)

5.6(16)

5.6(17)

5.6(18)

5.6(19)

5.6(20)

5.6(21)

5.6(22)

5.6(23)

5.6(24)

5.6(25)

5.6(26)

5.6(27)

5.6(28)

Plan view of the reaction frame

The reaction frame, general view

The loading system

Details of the hydraulic system

Construction details for a typical load cell

Moment direction for flat plate slabs

Torsion, shear and bending at column positions W l - A

Torsion, shear and bending at column positions Wl-B

Torsion, shear and bending at column positions W 2 - A

Torsion, shear and bending at column positions W 2 - B

Torsion, shear and bending at column positions W 2 - C


Torsion, shear and bending at column positions W 3 - B





Torsion, shear and bending at column positions W5-B


Typical strain data of Models W l to W 5

Measurement of the slab deflections

Load-deflection diagram for Model W l

Load-deflection diagram for Model W 2




Installation of the dial gauges

Location of the dial gauges used for the measurement

of the angles of twist of the spandrels

110

111

112

113

114

115

116

117

118

119

120

121

122

123

124

125

126

127

128

129

130

131

131

132

132

133

134

135

(xvii)

5.6(29) Angles of twist of the spandrel beams of models W l ,

W 2 , W 3 and W 5 in relation to the edge column B 136

5.7(1) Punching shear failure at column position W l - A 137

5.7(2) Punching shear failure at column position Wl-B 138

5.7(3) Flexural failure of slab - spandrel connections of Model W 2 139

5.7(4) Punching shear failure at column position W 3 - A 140

5.7(5) Punching shear failure at column position W3-B 141

5.7(6) Punching shear failure at column position W 3 - C 142




5.7 (10) Punching shear failure at column position W5-B 146


5.7(12) Soffit crack pattern of Model W 2 148

5.7(13) The top surface crack pattern of Model W 2 149

6.2(1) Research scheme for development of the prediction

procedure for V u 159

6.2(2) Freebody diagrams for edge-and comer-connections 160

6.3(1) Determination of forces and moments in slab-column

connections of the flat plates 161

6.3(2) Measurement scheme for M i and Vi 162

7.2(1) Size and location of the slab strips of Models W l to W 5 179

7.3(1) Measured moments, M i , versus the corresponding

yield moments of slab-edge positions 180

7.3(2) Measured moments, M i , versus corresponding yield

moments of slab-comer positions 180

7.4(1) Variation of moment along the critical slab strip 181

7.4(2) Calibration of the parameter gedge for the edge-column

positions (see Eq. 7.4(5)) 182

(xviii)

8.2(1) Research Scheme for the determination of co0 and \j/ 200

8.3(1) Relationship between \y and the corresponding

spandrel strength parameter-comer locations 201

8.3 (2) Relationship between \j/ and the corresponding

spandrel strength parameter-edge locations 202

8.4(1) Typical details at discontinuous edges of flat plates

with torsion strips without closed ties 203

9.2(1) Flowchart of the proposed procedure for the prediction

ofV u 223

9.3(1) Transfer of forces between slab and edge columns

(Rangan, 1987) 224

9.3(2) Flowchart for the AS3600-1988 procedure for the

prediction of V u 225

AI(1) D o m e effects in flat plates (Nielsen etal, 1988) 242

AIII(l) Ultimate strains of the top steel bars of flat plate Models

W l to W 5 along the measuring station adjacent to

column line B C 250

AIII(2) Ultimate strains of the top steel bars of flat plate Models


column U n e G H 251

AIII(3) Ultimate strains of the top steel bars of flat plate Models


column line F G 250

AIV(l) Differences between the proposed critical perimeter and

the critical shear perimeter suggested by AS3600-1988 256

AV(1) Reinforcement details of column positions W 2 - A and W2-F 258




(xix)

AV(5) Reinforcement details of column positions W2-B and W2-G 262




AV(9) Reinforcement details of column positions W2-C and W2-H 266


AV(ll) Reinforcement details of column positions W4-C and W4-H 268


AVI(l) Load-strain diagram for 5 mm hard drawn wires (type 1) 271

AVI(2) Load-strain diagram for 6.3 mm hard drawn wires (type 1) 271

AVI(3) Load-strain diagram for 8 mm hard drawn wires (type 1) 272

AVI(4) Load-strain diagram for 4 mm hard drawn wires (type 1) 272

AVI(5) Load-strain diagram for W6.3 mm wires (type 2) 273

AVI(6) Load-strain diagram for W8 mm wires (type 2) 273


AVI(8) Load-strain diagram for W6.3 mm wires (type 3) 274


AVI(10) Load-strain diagram for Y12 mm wires (type 1) 275

AVI(11) Load-strain diagram for F62 meshes (type 1) 276

LIST OF TABLES

(xx)

LIST OF TABLES

TABLE

5.3(1) Concrete strength of Models Wl to W 5

5.3(2) Reinforcement properties

7.3(1) Measured and predicted values of Mi and M m

7.4(1) Measured and predicted values of Vl

8.3(1) Measured and predicted values of \j/

9.4(1) Measured and predicted values of V u

AI(1) Spandrel's size, membrane effects and the load carrying

capacities of the slabs

AVTI(l) Slab strip moments per strip width at the ultimate state

AVH(2) Slab strip moments per strip width at the ultimate state

(contd.)

AVJT(3) Slab strip moments per strip width at the ultimate state

(contd.)

AVII(4) Slab strip moments per strip width at the ultimate state

(contd.)

AVII(5) Yield moments of the critical slab strips per strip width

AVH(6) Yield moments of the critical slab strips per strip width

(contd.)

PAGE

150

151

183

184

204

226

243

279

280

281

282

283

284

NOTATION

(xxi)

NOTATION

Ais total area of longitudinal steel

At area of the rect.angle defined by the longitudind bars

in the comers of the closed ties

Aws cross-sectional area of the bars from which ties are

made

b i width of the spandrel beam

Ci, C2 column dimensions

d effective depth of slab averaged over the critical

perimeter

di effective depth of the spandrel beam

Dx overall depth of the spandrel beam

Ds slab thickness

fly. fwy yield strength of longitudinal and web steel,

respectively

Lc clear span, measured face-to-face of supports

Mi negative yield moment of the slab over the front

segment of the critical perimeter

Miy yield moment of the slab over the front segment of

the critical perimeter

Mc 1, Mc2 the total unbalanced moment transferred to the column

centre in the main and transverse moment directions

respectively.

(xxii)

M m positive yield moment of slab at midspan of the

critical slabs strip (see Fig. 7.2(1))

Mmy yield moment of the critical slab strip at midspan (see

Fig. 7.2(1))

s spacing of closed ties

T2 torsional moment over a side segment of the critical

perimeter

ut perimeter of the area At

Vi shear force over the front segment of the critical

perimeter

V2 shear force over a side segment of the critical

perimeter

Vu punching shear strength

a longitudinal steel ratio of the spandrel (see Eq. 3.4(6))

p transverse steel ratio of the spandrel (see Eq. 3.4(9))

\j/ slab restraining factor

X column width factor

co0 additional transverse strength of the spandrel (see

Section 3.2.2)

co transverse strength of the spandrel (Awsfwy/s)

8 spandrel strength parameter (see. Eq. 3.4(11))

CHAPTER 1

INTRODUCTION

2

CHAPTER 1

INTRODUCTION

From an architectural and constructional view point, the flat plate frame

(which is a column and slab system without drop panel and column capitals) is an

ideal structural form. Flat plate framing requires relatively simple formwork. The

overall depth of the flexural members is a minimum and columns can often be

buried in the wall. This form of structure is popular in most countries.

In the design of reinforced concrete flat plate structures, the regions around

the columns always pose a critical design problem. Experimental data on the

performance of slab-column connections at the edges and comers are very limited,

especially for slabs with spandrel beams. Fig. l(l)t shows a typical reinforced

concrete flat plate structure with spandrels beam. It may be seen that at the edges,

the slab load is transferred to the exterior columns through the spandrels.Thus they

are subjected to large torsional moments in addition to bending moments and shear.

The strengths of the spandrel beams have a significant effect on the punching shear

strengths and mechanisms of failure of the slab-column connections at the edges

and corners of building floors. However the strength behaviour of these

connections is not well understood and it calls for further research.

1.1 The Problem

Determination of the punching shear strength, Vu, of the slab-column-

spandrel connections of flat plates, at the edge- and comer-positions, has received

considerable attention by the engineering profession in recent years. A reliable

t Figures are given at the end of each chapter followed by tables.

3

method for the prediction of Vu, requires a general analytical method for a slab-

column-spandrel connection that can predict both the punching shear strength of the

connection and the mechanisms by which the load is carried. This problem may

also be expressed in terms of the following questions:

How would the size and location of the slab reinforcement affect the distribution of

moments and shears at the edge and comer-column positions?

How would the strength of the spandrel beams affect the magnitude of Vu?

Other relevant questions that might arise in the process of solving the above

problem may be listed as follows.

(i) What are the effects of the spandrel strength on the mechanisms of failure?

(ii) What is the most suitable critical perimeter?

(iii) What are the governing equilibrium equations?

(iv) How to quantify the restraining effects of the slab on the elongation and

rotation of the spandrel beams?

(v) How do torsion, bending and shear interact in the spandrel beam in the vicinity

of the connection?

Information regarding the behaviour of the slab-column-spandrel

connections near failure is reviewed and some of the assumptions of the existing

analytical methods are assessed in terms of how well they conform to the observed

behaviour. This is described in the next section.

4

1.2 Existing Analytical Methods and Experimental Data

Extensive reviews of the existing knowledge have been given previously

by the A C I - A S C E committee 426 (1974) in a state-of-the-art report, by Hawkins

(1974), and by Regan (1981). These literature reviews indicate that there have been

three different approaches to the problem. That is the existing analytical methods

for the prediction of the punching shear strength, Vu, may be classified as follows :

(i) methods based on a linear distribution of shear stress on some critical

perimeters, which do not consider the effects of reinforcement and its applicability

in the post-cracking stage.

(ii) methods based on elastic plate theory. This classification includes the finite

element analysis which may account for cracking and plastic behaviour. However

these methods do not account for any distribution of the stress caused by the

cracking of concrete and yielding of the steel bars. Finally,

(iii) methods based on beam analogies, which describe a slab-column connection as

the junction of orthogonal beam elements contained within the slab. Each beam is

assumed to be able to develop its ultimate bending, torsion and shear, making due

allowance for interaction effects, at the critical sections near the column faces. The

strength of the connections is calculated by summing the contributions of the

strengths of the beams.

From 1981 onward, and especially in the last three years, the following

contributions have been made by other researchers on the prediction of the

punching shear strength for slabs without spandrel beams.

5

Regan (1981) developed an equation for the calculation of Vu. Regan's

shear perimeter for rectangular columns was a "rounded rectangle" located 1.25d

out from the column. Jiang et al. (1986) developed a theoretical solution for the

punching shear strength of concrete slabs. In this approach the problem is treated as

a three-dimensional axisymmetrical one,and the material assumed to be rigid-

plastic. Chen (1986) developed a procedure for the prediction of the punching

shear strength of flat plates without shear reinforcement while transferring shearing

force only. Solanki and Sabinis (1987) presented a simple design approach for the

calculation of V u for the curved/shell concrete structures. Rankin and Long (1987)

developed a method for the estimation of V u from rational concepts of the various

modes of failure. This method is an extension of the method proposed by Long

(1975) for the prediction of Vu. Bazant and Cao (1987) were primarily concerned

with size effects, but they did propose a formula for the prediction of Vu. Gilbert

and Glass (1987) proposed a method for predicting Vu, which is based on the shear

criterion of failure. This method was then extended to cover the use of shear head

reinforcement by redefining the critical-area term. Alexander and Simmonds (1987)

in their paper proposed that punching shear failure could be represented by a truss

analogy and that failure is due to the concrete cover failing to contain the out-of-

plane components of force between the reinforcement and the concrete compression

stmts. Gonzalez et al. (1988) based on a nonlinear finite element analysis,

developed an analytical method for the prediction of V u , in which failure is

governed by the tensile strength of the concrete. Moehle et al. (1988) proposed an

expression for shear strength in the absence of significant moment transfer, as well

as three alternative procedures for the computation of the strength under combined

shear and moment transfer.

All of the above prediction procedures are for the case of slab-column

connections of flat plates without spandrel beams. Thus none of these works has

6

any direct relation to the present study which concentrates on flat plates with

spandrel beams.

A review of the existing publications also indicates that experimental data

on the performance of the slab-spandrel-column connections of flat plate slabs are

very limited. Hatcher, Sozen and Siess (1961) tested a multi-panel flat plate

containing spandrel beams. However, the punching shear failure occurred at a

column away from the corners and edges. Rangan and Hall (1983) tested a series

of four half-scale models with spandrel beams. In their models "3" and " 4 "

punching shear occurred at an edge column. N o corner column failure data were

available from their work. Rangan (1987) published a method for the prediction of

Vu. This method also allows for the prediction of the punching shear strength of

the slab-column connections with spandrel beams.

1.3 Codes of Practice

The design provisions incorporated in the various building codes are a

direct result of the empirical procedures derived from experimental studies.

However in the U.K., U.S.A. and Australia the development of the design

recommendations have followed different routes. The British code (BS8110-1985)

is based primarily on the work of Regan (1974), the American code (ACI318-83),

on the work of M o e (1961), and the new Australian Standard (AS3600-1988), on

the work of Rangan (1987). Note that the recommendations of the European code

(CEB-FIB-1978) and that of the Canadian code (CSA A23.3-M84) are in general

similar to those proposed by ACI318-83.

Among the abovementioned codes only AS3600-1988 provides a

prediction procedure for the punching shear strength, Vu, for slab-column-spandrel

connections at the edge- and comer-column positions. However an early

7

examination of the code procedure (Falamaki and Loo, 1988) indicated that these

proposed formulas overestimate the punching shear strength values, especially at

the comer positions.

In a separate report (Falamaki and Loo, 1990) the inadequacy of the code

formulas has been attributed to the use of : (i) incomplete set of equilibrium

equations, (ii) inadequate interpretation of the restraining effects of the slab on the

strength of the spandrel beams, and (iii) inadequate assumptions for the

distribution of shear force along some critical perimeters. These Australian

Standard formulas also do not consider the effects of the size and location of the

slab reinforcements on the magnitude of Vu. Further, the effects of bending

moment are not included in the assumed interaction equation for the spandrel beam.

1.4 Size of the Model Structures

To investigate the punching shear strength of the slab-spandrel-column

connections at the edge- and comer-positions theoretically or experimentally it is not

practical to deal with the whole building. Thus a localized portion in the vicinity of

the connections is considered. Of course the localized model should be adopted in

such a way so as to ensure that the distribution of the total unbalanced moment and

the shear force transferred from the slab to the column is the same as in the whole

building. O n the other hand, in an experimental study, adoption of a larger region

of the structure may require a smaller model and size effects may then be a problem,

which is one of the salient aspects of fracture mechanics.

According to fracture mechanics (Bazant and Cao, 1987) size effects

decreases as the structure size increases. Therefore by the adoption of large scale

model structures, the problem of size effects can be eliminated. It is important to

note that the strength of the beam and slab elements at the various sides of the

8

connection is affected by the deformational restraints provided by the surrounding

slabs of the building. Thus the model structure should be large enough to cover

the full length of these elements.

Regarding the above discussion, a sound analytical model not only should

be based on physical behaviour and test data of large scale test models with proper

boundary conditions but also account for the variation in each of the following

parameters:

(1) the overall geometry of the connection,

(2) the concrete strength,

(3) the size and location of flexural reinforcement of the slab,

(4) the slab restraint on the spandrel, and

(5) the enhanced strength of the slab-column connections due to membrane effects.

1.5 Objectives

The existing analytical methods for the prediction of Vu have been

summarized in Sections 1.2 and 1.3. For the case of slab-column-spandrel

connections of flat plates at the edge- and corner-positions there is still no reliable

procedure for the prediction of Vu. Thus the main objective of the present study is

to develop an analytical method for the prediction of V u for these types of

connections. Needless to say, the development of a sound analytical method for the

prediction of V u requires the test results from large-scale models with proper

boundary conditions. Experimental work of this nature is a highly expensive and

labour intensive task.

9

The objectives of the experimental phase of the present investigation are to

observe the behaviour of flat plate slabs with spandrel beams of different depths

and steel ratios, and to obtain essential data to use for the establishment of the

prediction procedure for Vu. A total of five cast-in-situ half-scale flat plate models

representing two adjacent panels at the corner of a real structure have to be tested up

to failure, under a uniformly distributed vertical load. Also to accelerate the

construction, in the design of flat plate models, instead of concrete columns,

prefabricated steel sections (with equivalent stiffnesses) may be used. The

instrumentation and test procedure ought to be designed in such a way as to provide

the required data for the analytical phase of the investigation.

The analytical studies which led to the development of the prediction

procedure for V u are mainly based on the behaviour and the experimental results

obtained from the present five half-scale models plus those tested by Rangan and

Hall (1983).

To establish the prediction procedure for Vu, the tasks for the analytical studies are:

(i) determination of the total unbalanced moment and total shear force distribution

along some critical perimeters at the edge- and comer-positions, and

(ii) determination of the strength of the spandrel beam and slab elements joined to

the different faces of the edge- and comer-columns, with the aid of semi-empirical

formulas.

Note in (ii) that for the determination of the forces and moments in the

spandrel beam an interaction equation is to be developed for the combined effects of

torsion, shear and bending. Also for the determination of the strength of the slab

the effects of the size and location of the slab reinforcement, clear span of the slab

10

(in a direction perpendicular to the slab edge) and the in-plane forces in the slab are

to be considered. It is worth mentioning that for development of the interaction

equation for the spandrel beams, the restraining effects of the slab on the rotation

and elongation of the spandrels are to be studied first.

Thus the prediction equations for Vu that may be obtained from the above

study will then cover the cases of the slab-column-spandrel connection under axial

force and biaxial bending moments, at the edge- and comer-positions. This study

also investigates the effects of column width on the magnitude of V u at the comer

positions.

It is important to note that the abovementioned study is mainly for the case

of the slab-column connections with spandrel beams. However to obtain a better

picture for the effects of the spandrels on the behaviour of the slab, the last of the

five half-scale models is designed as a slab with torsion strips (but without closed

ties) at its edges.

1.6 Outline of Thesis

One of the requirements of the analytical study for the development of the

prediction formulas for V u is the determination of the shear force distribution along

some critical perimeter within the slab. This is presented in Chapter 2. The second

requirement is the determination of the torsion, shear, and moment interaction for

the spandrel beams which is discussed in Chapter 3. The last requirement is the

development of a computational procedure for the calculation of the slab bending

moments from the flexural reinforcement strain data. This is expanded in Chapter

4. In Chapter 5 the experimental programme is described in detail. Behaviour of

the test models and modes of failure are also presented herein.

11

A n outline of the research scheme for the prediction of the punching shear

strength, Vu, is presented in Chapter 6. The contributions of the discussions in the

other chapters in relation to the development of the proposed prediction procedure

for V u are also discussed in this chapter.

In Chapter 7, some semi-empirical formulas are developed. These

formulas may be used for the determination of the distribution of moment and shear

along the critical perimeter. The bases of the formulas are the analytical and

experimental studies carried out in Chapters 2,4 and 5.

The results of the experimental study presented in Chapter 5, have also

been used for the calibration of the interaction equation developed herein for the

spandrel beams. This is discussed in Chapter 8.

The formulas of the proposed prediction procedure for Vu are presented in

Chapter 9. The prediction method recommended in the AS3600-1988 is also

included in this chapter, where in the light of the experimental results reported

herein a comparative study is carried out. Finally, conclusions and

recommendations for further study are given in Chapter 10.

It should be noted that for each of the chapters, the figures are given at the

end of the text followed by the tables (if they exist).

12

t

Walls

Jf

• • •

•

•

• D D •

a) Plan view

Spandrel Beams

Yr I T Yr it Jil Jyt Jjui

t

ii

b) Section 1-1

Fig. 1(1) Typical flat plate structure with spandrel beams

CHAPTER 2

TRANSFER OF FORCES IN SLAB-COLUMN CONNECTIONS

OF FLAT PLATES

CHAPTER 2

TRANSFER OF FORCES IN SLAB-COLUMN

CONNECTIONS OF FLAT PLATES

2.1 General Remarks

The question of the transfer of shear force .and bending moments between

a slab and the column of a flat plate and their distribution along some critical

perimeter has always been a design problem, especially at the edge- and comer-

locations. In order to quantify the distribution of these forces and moments the

behaviour of the slab at the exterior panels should be investigated first. This

chapter expands the fundamentals of the slab-column-spandrel behaviour and

derives the useful equilibrium equations.

In Section 2.2 the effects of the size of the spandrel beams as well as the

loading pattern on the deflected shape of the exterior panels of the flat plates are

discussed. According to this discussion, at the ultimate state and under certain

specified conditions the deflected shape of the exterior panels of the flat plates may

be assumed similar to that of the one-way slabs. The effects of the spandrel's

strength on the distribution of forces in the vicinity of the slab-column connections

are investigated on the basis of this assumption.

The strength of the spandrel beams also affects the failure mode of the

slab-column connections.This is described in Section 2.3. Based on the expected

failure mechanism for the slab-column connections with shallow spandrels the

equilibrium equations of both the edge - and corner-connections are derived in

Section 2.4.

15

The above hypothesis then leads to the development of a new technique for

the determination of the actual distribution of the total shear force between various

faces of the edge- and comer-columns. This is detailed in Section 2.5. It should be

noted that this new technique is verified in Section 7.4.

2.2 Spandrel Beams and Slab Behaviour

2.2.1 One-way slab action

The behaviour of the present Models Wl to W5 which represent the two

adjacent panels at the comer of a typical flat plate floor is reported in Section 5.7.

The deflected shape of the slabs as well as the slab crack patterns all indicated a

one-way slab action at the ultimate state. Fig. 2.2(1) shows the deformed shape of

a typical slab of the present model structures after failure. Note that all the model

structures (Wl to W 5 ) failed under a uniformly distributed load. They all had

spandrel beams at the free edge*, except Model W 5 . This model also exhibited one

way slab behaviour.

Further, Simmonds (1970) tested a one-third scale model of a flat plate

structure. It consisted of square panels and rectangular columns with cross sections

elongated in one direction. H e found that the model behaviour changed from

essentially two-way to one-way slab action. Fig. 2.2(2) shows the top crack pattern

of this slab.

Furthermore, Hatcher et al. (1961) studied a quarter-scale reinforced

concrete flat plate model. The structure consisted of nine square panels with

spandrel beams at the discontinuous edges. Tests up to failure were also conducted

* Edges not stiffened by walls or other bracings (see Fig. 1.1)

16

by Rangan and Hall (1983) on half-scale models simulating the edge panels of flat

plate floors with spandrel beams. The bottom crack patterns of all the above flat

plate models indicate a one-way slab action in the exterior panels. A typical bottom

crack pattern is depicted in Fig. 2.2(3).

The above observations indicate that the comer and edge panels of the flat

plate slabs with rectangular panels would have a one-way slab action at the ultimate

state. This is true provided that the rotational stiffness of the slab-column

connections in one direction is higher than that in the other direction. This

condition may be attained by :

(i) using rectangular columns with cross sections elongated in one direction ( See

Fig 2.2(2)),

(ii) provisions of spandrel beams at the free edges, or

(iii) loading the slab on alternate panels to provide maximum unbalanced moments

at the slab-column connections ( See Fig. 2.2(4)).

Note that in (iii) at the ultimate state, the higher rotational stiffness of the

uncracked (adjacent) slab would help to create one-way action in the failed slab (see

the behaviour of Model W 5 in Section 5.7).

In summary, at the ultimate state, under certain specified conditions the

exterior panels of flat plate floors would have a one-way slab action. This type of

behaviour can be used as a basis to investigate the effects of the strength of the

spandrel beams on the distribution of forces in the vicinity of slab column

connections.

17

2.2.2 Distribution of forces along the spandrels

Fig. 2.2(5)a shows a one-way slab, treated as a series of narrow

individual slab strips spanning in a direction perpendicular to the spandrel, in which

the slab resistance against twisting is ignored. The slab is under a uniformly

distributed vertical loading and is assumed to be cast monolithically with the

supporting columns. It is further assumed that the column bases are fixed and the

vertical deflections of the spandrel beam are small .and may to be neglected.

Theoretically, at the ultimate state, the magnitudes of the bending moment

and shear force of each slab strip (at a section located at the face of the supporting

spandrel) is proportional to the magnitudes of the strains in the top steel bars of the

slab (in the corresponding section) in a direction parallel to the slab strips. The

magnitudes of the steel strains are in turn proportional to the angle of twist of the

spandrel with respect to the exterior columns supporting the spandrel.

In slabs with deep spandrel beams and very rigid columns, both the

spandrel and the columns provide near full bending restraint for the connecting

slab. Therefore the angle of twist of the spandrel in relation to the columns reduces

to zero and the slab will deform in the same manner all along the spandrel. In this

case provided the slab reinforcements are designed for a practical ultimate load, the

reinforcement strains at the face of the spandrel would all attain their maximum

values (or yield strains). Consequently a uniform distribution of bending moment

along the spandrel is expected. This is illustrated in Fig. 2.2(5)b. It may be seen

that the variation of the slope of th? bending moment diagram (i.e. the shear force),

and that of the torsional moment (as a result of the above bending moment and the

shear force) are both straight lines.

18

W h e n the spandrel is shallowt , the bending restraint provided by the

spandrel for the connected slab is less than that provided by the columns (which is

assumed to be rigid). Therefore the bending of the slab tends to rotate the spandrel

beam with respect to the columns. Fig. 2.2(5)c shows the effect of the spandrel

twist on the distribution of the slab reinforcement strains along the spandrel. In

other words (depending on the strength of the spandrel) the full bending restraint

provided at the column face reduces as w e get closer to the panel centerline.

Therefore a non-uniform variation of the bending moment and the shear force

(similar to that of the steel strains (See Fig. 2.2(5)c) would be expected. In this case

the variation of torsional moment will not be a straight line, but increase sharply

near the columns.

For slabs with no spandrel or (torsional strip), a variation similar to that of

the slabs with shallow spandrels is expected, but with a higher concentration of the

moment and shear in the vicinity of the columns (see Fig. 2.2(5)c).

In summary the angle of twist of the spandrel and its adjacent edge- and

comer-column depends on the strength of the spandrel beams. This observation is

used to investigate the possible mechanisms of failure (see Section 2.3).

2.3 Spandrel Beam and Modes of Failure

As discussed in Section 2.2, the effects of the strength of spandrel beams

on the failure mechanisms of the slab-column connections may be expressed in

terms of the angle of twist of the spandrel and its adjacent edge- and comer-

columns. For slabs with deep spandrels the angle of twist tends to be zero and

t The differences between the shallow and the deep spandrel beams are discussed in Section 8.5

19

consequently, at the ultimate state a negative yield line would occur along the face

of the spandrel and the slab-spandrel connection fails in negative bending.

For shallow spandrels, again as discussed in Section 2.2, due to the full

bending restraint provided by the (rigid) column a yield line would first develop at

the ultimate state across the front face of the edge - and comer-columns. Further

increases in loads increase the angle of twist of the spandrel in relation to its

adjacent columns. This continues until the spandrel-column connection fails. In

this process, because of the concentration of torsion and shear at the side face(s) of

the column, failure occurs by the formation of inclined spiralling cracks in the

spandrel. Similar failure mechanisms prevail in the case of connections without

spandrel or torsion strip.

2.4 Equilibrium Equations

2.4.1 Definitions

Fig. 2.4(1) shows the freebody diagrams of typical slab-column

connection of flat plates with spandrel beams. The following features should be

noted.

(i) The critical perimeter for the direct transfer of the slab bending moment and

shear force to the column is also shown in Fig. 2.4(1). The front segment of the

critical perimeter is located at a distance 0.5d from the front face of the column,

where d is the effective depth of the slab. The side segments of the critical perimeter

are located at the column side face(s). Note that the present definition of the critical

perimeter, instead of the critical shear perimeter prescribed by AS3600-1988, leads

to better predicted results for Ml and Vx. This is discussed in Appendix IV.

20

(ii) The point of contraflexure at the edge- and comer-columns is assumed at a

distance L 2 from the center of the slab (see Fig. 2.4(l)c). In this figure Fh and V u

are respectively the horizontal and vertical column reactions at the contraflexure

point.

(iii) The forces and moments of the spandrel beam at the left and right sides of the

slab-column connection are respectively shown as V 2 L T 2 > L and M 2 > L , and V 2 R,

T 2 R and M 2 R where V denotes shear, T denotes torsion, and Mdenotes moment.

It is important to note that in this study V2L and V2R are assumed to be

equal to V2, and T 2 L and T 2 R equal to T2. When both the slab panels adjacent to

the edge connection are similar, M 2 L is equal to M 2 R and consequently the total

unbalanced moment in the transverse direction, M C 2 , is zero. When one panel is

slightly stiffer, the bending moments M 2 L and M 2 R would no longer be equal. In

this case the unbalanced bending moment with respect to the centroid of the

spandrel is designated as M 2 and the corresponding total unbalanced bending

moment as M C 2 . Note that M C 2 is obtained by taking moments with respect to point

O t (See Fig. 2.4(l)c).

The horizontal column reaction, Fh, that is to be resisted by the slab's

inplane forces is shown in Fig. 2.4(1). By considering the equilibrium of forces at

the slab-column connections in the horizontal direction, it is obvious that, part of Fh

is to be resisted within the width C 2 at the front face of the column and the

remainder within the width bounded by the panel center line(s) adjacent to the

column. In the derivation of the equilibrium equations, the portion of Fh resisted

within the width C 2 is ignored. This is because at the ultimate state, the

development of a negative yield line over the front width of the column (see section

2.3) would cause the formation of a wide crack across this width.

21

Based on the above definitions and discussions the derivation of the

equilibrium equations for the slab-column connections of flat plates with spandrel

beams is carried out in the next section.

2.4.2 Formulas

The definitions given in Section 2.4.1 can now be used for the derivation

of the equilibrium equations. The freebody diagram of the slab-column connections

at the edge- and comer-locations are shown in Fig. 2.4(1). In a comer connection,

the equilibrium of forces in the vertical direction (at the ultimate state) may be

expressed as

VU = V2 + V! 2.4(1)

Enforcing the equilibrium of forces with respect to the center of the

spandrel (point O ) , in the main and transverse moment directions, while

incorporating the above definitions gives:

M C 1 = T 2 + M 2 + yx(°l2 d ) + M T 2-4(2)

"<*'' D M ? - D ^ " ^

2Li

Similarly for the edge-connections we have

VU = 2V2 + V! 2.4(4)

MC1 = 2T2 + Mx + v/bl2+ dl + MT 2.4(5)

22

M2

**=' D l - D r 2A^

' 2Lj

In the above equations, M C 1 and M C 2 are respectively the total moments in

the main and transverse moment directions (see Fig. 2.4(1)) with respect to point

Oj; V u is the total shear at the column centerline; T2, V 2 and M 2 are respectively the

torsion, shear and bending moment at the side face of the critical section; M1 and Vj

are respectively the bending moment and shear force at the front face of the critical

section. And finally,

MT'Tff^'l+Vj-V^l 2.4(7)

Note that in Eq. 2.4(6) M 2 is the unbalanced moment with respect to the

centroid of the spandrel. Obviously, M C 2 = 0 if the two panels adjacent to the edge

connection are identical.

For the general case in which the width of the spandrel is the same as that

of the column, Eq. 2.4(7) reduces to :

MT ^(^^JMci 2-4(8)

Obviously MT = 0 if the depth of the spandrel and the slab are similar.

Also for the particular case in which the spandrel beam is projecting upward, M T

becomes

M T ..^Bl^syvf-l^X) 2.4(9)

23

2.5 Distribution of Shear Force

2.5.1 Assumptions

Flat plate is an indeterminate system. Therefore the measurement of the

forces in the vicinity of the slab-column connections requires a sophisticated

analytical process and proper instrumentation. Magnitude of the total shear force at

the column centerline, V u, which is the ultimate shear strength may be measured

directly by means of vertical load cells in the support system. A fraction of V u is

resisted along the front segment of the critical perimeter by V j (see Fig. 2.4(1)),

and the remainder by V 2 at the side face(s). Therefore by the development of an

experimental method for the measurement of Vl 5 the shear force V 2 may be readily

calculated.

The proposed procedure for the measurement of V1 is mainly based on the

assumption that at the ultimate state, and under certain specified conditions (see

Section 2.2.1) the edge- and corner panels of flat plate slabs have a one-way slab

action. The other assumptions used may be expressed as follows. (Note that to

clarify the understanding of the procedure some of the assumptions of Section

2.2.1 are repeated here).

(i) Similar to one-way slabs, the flat plate is treated as a series of narrow individual

slab strips, spanning in a direction perpendicular to the spandrel (see Fig. 2.5(l)a),

in which the slab restraint against twist is ignored.

(ii) The slab is under a uniformly distributed vertical loading and is assumed to be

cast monolithically with the supporting columns.

24

(iii) The column bases are fixed and the vertical deflections of the spandrels are

small and therefore negligible.

(iv) Variation of the bending moment along each slab strip is parabolic.

2.5.2 The procedure

Details of the proposed procedure for the measurement of the shear force

Vj are described below.

(i) According to assumption (iv) of Section 2.5.1, for the slab strip i, the bending

moment per strip width is equal to

Mi = AfX2 + BiX + Q 2.5(1)

in which the magnitude of the bending moment at each section of the strip is

proportional to the slab reinforcement strains in the corresponding section.

(ii) For slab strip i, the magnitude of the shear force per strip width (i.e. the slope

of bending moment diagram) is also a function of the slab's steel strain. Therefore

the first derivative of Eq. 2.5(1) gives the shear force

^ = 2AiX + Bi 2.5(2)

Eq. 2.5(2) indicates that the magnitude of the shear force per strip width at X = 0 is

equal to B{ (see Fig. 2.5(l)b).

(iii) To determine the three parameters Ai5 B^nd Q of Eq. 2.5(1) for slab strip i,

three measured slab strip moments are to be substituted into that equation.

25

Therefore the three measuring stations 1, m, and 3 respectively at distances X = 0,

X = Xjn and X = X 3 were considered for the determination of moments with the aid

of experimental strain data. Note that X = 0 corresponds to a distance equal to d/2

from the face of the spandrel beam (see Fig. 2.5(l)a), where d is the effective depth

of the top steel bars of the slab.

(iv) According to step (iii), to measure the slab strip moments at the measuring

stations 1, m , and 3, strain gauges have to be attached to selected slab

reinforcement at the corresponding distances X = 0, X = X m and X = X3. The

slab bending moments may then be obtained, using the measured strains of the slab

steel bars, with the aid of an established moment-strain relationship.

(v) Substituting the coordinates of the three measured moments of the measuring

stations 1, m and 3 of each slab strip i, namely (0, M H ) , ( X m , M 3 i) and ( X3,M3i)

into Eq. 2.5(1) and solving for Bj gives

Bi = xm(x^3- xm)

[Mmi + Mli] + x3(x^m xm)

[M3i"Mli] 2'5(3)

where according to (ii) above, Bj is the shear force per unit width of strip i at X = 0

(i.e. at the measuring station 1); and M H , Mmi and M 3 i are the absolute values of

moments per strip width. Note that the subscript i stands for the strip numbers and

subscripts l,m and 3 respectively refer to the measuring stations l,m and 3.

(vi) Magnitude of the total shear force at the center of column support, Vu, may be

measured directly by means of the vertical load cells. Subtracting from the Vu, the

self weight of the column and the portion of the slab (including the spandrel) that is

represented by the shaded area in Fig. 2.5(l)a, gives the magnitude of the total

shear force along the critical section 1. This force may be designated as Vul. Note

26

that the shaded area, as shown in Fig. 2.5(l)a is bounded by the panel centerlines

from two sides, and the measuring station 1 (i.e. lines pq) from the third side.

(vii) Vul may also be determined with the aid of the data obtained from the strain

gauges attached to the slab reinforcement. Thus dividing the same region of the slab

defined in step (vi) into n slab strips perpendicular to the spandrel (see Fig.

2.5(1 )a), w e have

Vi = ZBi 2.5(4) i = l

where Bj is the measured shear force (using Eq. 2.5(3)) at X = 0.

(viii) The portion of the total shear force Vu which is resisted along the front

segment of the critical perimeter may now be calculated as

Vi = -^l-Vui 2.5(5)

IBi i= l

where Bj is the measured shear force (using Eq. 2.5(3)) of the slab strip located in

the front of the column, with a width C2.

Eq. 2.5(5) indicates that the proposed procedure for the measurement of

V j is based on the data obtained from the strain gauges attached to selected slab

reinforcing bars, and the vertical load cells at the column supports. The reliability of

this measuring system is discussed in Chapter 7.

27

Or

J .

s .3 a 03

"3 -a o

s "a es

"is

a

Cui

• PN

28

Positive yield line

Columns

Loading points

Fig. 2.2(2) The top surface crack pattern of a one-third scale flat plate model tested by Symmonds (1970)

29

(NUMBERS ON THE CRACKS ARE LOAD IN kN/m2)

Fig. 2.2(3) Typical soffit crack pattern of the exterior panels of the flat plate models tested by Rangan and Hall (1983)

30

a) Plan View

b) Elevation

Fig. 2.2(4) Typical flat plate loaded on alternate spans

Slab strips

Spandrel beams

a) Typical one way slab

Column face

Variation of:

Beam

^nferMne Column face

Moment

Column face

Variation of

Shear

Torsion

Moment

Shear

Torsion

Beam center,,ne

Column face

b) Deep spandrel beams c) Shallow spandrel beams

+ i.e. strain in top steel bars of the slab in the main moment direction

Fig. 2.2(5) Theoretical variation of moment, shear and torsion along the spandrel beams

32

c 'uS cu cu

^cu

"S >

c o

c

cu

c C o cu

•a W

c "u*uJ

cu Or

e e o cu

r-

C r»

o eg

cu cu

c C o u->

I r-

Or

c J-

o cu •a

c es •

cu WO T3 CU r-

E ea r-

Wj

es r>.

o cu cu u [a.

IN WD • —

33

Interior Face of the Spandrel Beam

Panel Centerline Measuring Station 1

Strip 1

Spandrel Beam ^^«

Slab Edge;—&°

Shaded Area

&•;•*?».

Column

a) Plan view

d/2

Critical Slab Strip

Panel Centerline strip n

Measuring tation m

•Measurin Station 3

Column

Typical Slab Strip i

Bj

b) Elevation

,

t VmHUHIfft

M li

M li

X=0

c) Moment diagram

Fig. 2.5(1) Variation of moment along the slab strips

CHAPTER 3

INTERACTION OF TORSION, SHEAR AND BENDING

IN SPANDREL BEAMS

35

CHAPTER 3

INTERACTION OF TORSION, SHEAR AND

BENDING IN SPANDREL BEAMS

3.1 General Remarks

In slab-column connections of flat plates with spandrel beams, the

spandrels are under the combined effect of torsion, shear and bending. To quantify

these forces an interaction equation needs to be developed. However, because of

the slab restraining effects and consequently increase in the strength of the

spandrels, the calibration of any semi-empirical interaction equation requires a

substantial amount of test data.

A theoretical investigation of the restraining effects of the slab on the

elongation and rotation of the spandrel beams is described in Section 3.2.

According to this investigation, the slab restraining effects may be expressed in

terms of an increase in the longitudinal and transverse steel bars of the spandrel.

By the determination of the restraining effects of the slab on the strength of the

spandrels (compared to the isolated beams) the following procedure m a y be used

for the development of the interaction equation for the spandrel beams.

It is believed that the most complete interaction surface for isolated beams

under the combined effects of torsion, shear and bending was developed by Elfgren

et al. (1974). In Section 3.3 the applicability of this interaction surface for the

spandrel beams is investigated. To do so, the deformational restraint provided by

the slab m a y be ignored. Also on the basis of the physical. observations it is assumed

that the skew failure surface occurs on the sides and top of the beam while the

compression zone is located at the bottom. The analysis leads to the determination

36

of an interaction surface for a beam with the same loading condition as the

spandrels.

In Section 3.4 the slab restraining effects as discussed in Section 3.2 are

incorporated into the interaction equation proposed in Section 3.3. This led to an

interaction equation for spandrel beams, Eq. 3.4(15). The reliability of this

equation is discussed in Chapter 8.

3.2 Slab Restraint and the Enhanced Strength of Spandrels

Theoretically when compared to isolated reinforced concrete beams the

restraining effects of the slab provide a higher strength for the spandrels. This

increase in strength is a result of the slab restraint on i) the elongation and ii) the

rotation of the spandrel beams. Also it is expected that the spandrel beams joining a

slab-column connection with a wider column, yield a higher punching shear

strength for the connection. These are elaborated in the following sections.

3.2.1 Slab restraint on the elongation of spandrels

In 1972, Onsongo and Collins reported on the results of the tests on a

series of longitudinally restrained reinforced concrete beam elements subjected to

torsion. According to their results, any longitudinal restraint on the beam

elongation, increases its torsional capacity. This increase in strength may be

computed by expressing the restraint in terms of an equivalent area of additional

longitudinal steel.

The enhanced strength of the spandrel due to the restraining effects of the

adjoining slab was first reported by Rangan and Hall (1983). In their report based

on the work of Onsongo and Collins (1972) Rangan and Hall analysed the spandrel

37

beams of some half-scale flat plate models and found that the the torsional strength

of the spandrel increases by a factor of 4 to 5 when compared to isolated beams.

Rangan (1987) later revised the above conclusion and suggested that the restraining

effects of the slab increase both the shear and torsional strengths of the spandrel by

a factor of 4.

Rangan's proposal which has been incorporated in the new Australian

Standard for Concrete Structures (AS3600-1988), is not supported by any test data

on slab-column connections of flat plates with realistic spandrel beams. This fact

helps to explain the shortcomings of the Australian Standard approach (see Section

9.4). It also calls for the development of a more general procedure for the

determination of the slab restraining factor, \j/.

To investigate qualitatively the restraining effects of the slab on the

elongation of the spandrels, let us examine the behaviour of the flat plates with

spandrel beams at the ultimate conditions. A n isolated beam increases in length

when subjected to torsion. A spandrel beam in a building floor will also tend to

increase in length under load. This tendency causes a tensile force, P, to develop in

the adjoining slab, at the face of the spandrel. The reaction, therefore, is a

compressive force (equal to P) in the spandrel itself (see Fig. 3.2(1)). This

compressive force reduces the expected magnitude of the tensile forces in the

longitudinal bars of the spandrel. Thus it is similar to increasing the strength of

these bars from A, f, to \\fA, f, . O n the other hand, the magnitude of the induced

compressive force, P, may be expressed as a function of the angle of twist of the

spandrel beam in relation to its adjacent column. Increase in the strength of the

spandrel provides more bending restraint to the rotation caused by the adjoining

slab and consequently reduces the spandrel rotation with respect to its adjacent

columns. This decrease in rotation reduces elongation of the spandrel and

consequently the induced compressive force, P. Therefore, with the assumption

38

that deep spandrel beams provide near full bending restraint for their adjoining

slab, the induced compressive force, P, would then reduce to zero. Consequently

the slab restraining factor \|/, tends to unity.

In summary, the slab restraint on the elongation of the spandrel beam

enhances its load carrying capacity. This enhanced strength decreases as the

strength of the spandrel beam increases. The enhanced strength m a y be expressed

in terms of the strength of the longitudinal steel bars of the spandrel beam, i.e.

Aj fj increases to \|/A|f, .

3.2.2 Slab restraint on the rotation of the spandrels

In normal design practice, the center of twist of the spandrel is below the

horizontal centroidal axis of the slab. W h e n the spandrel beam is twisted, the

horizontal displacement of the top portion of the spandrel will be restrained by the

large horizontal stiffness of the slab. The spandrel rotation will also produce a

vertical displacement at the spandrel-slab interface. This vertical displacement will

be restrained by the vertical stiffness of the slab.

In other words, the restraining effects of the slab on the rotation of the

spandrel increases its strength in the transverse direction. This enhanced strength

m a y be considered similar to the provision of more transverse reinforcement in the

spandrel beam. Therefore it m a y be assumed that the transverse strength of the

spandrel beam, co''" is increased to co + coo, where co0 is defined as the additional

transverse strength of the spandrel beam.

f co = Awsfwy/s, where fwy and A w s are respectively the yield strength and the cross-sectional area of the bars from which lies are made. The spacing of the closed ties is s.

39

It is expected that an increase in the area of the spandrel-slab interface

(which is proportional to the overall depth of the slab) increases the magnitude of

C0o. However for the present study the slab depth is constant. Note that the

investigation of the effects of abovementioned interface area is beyond the scope of

this thesis.

3.2.3 Effects of column width

Fig. 3.2(2) shows schematically the effects of column width on the slab

restraining factor, \i/. Fig. 3.2(2)a illustrates the plan view and the angles of twist

of the two corner columns A and C of dimensions 2a x b, and 2c x b

respectively (where c > a). Note that in this figure 9 A and 0c are respectively the

angles of twist of the spandrel beams (at critical sections located near the face of the

columns) in relation to the centerline of columns A and C.

According to Fig. 3.2(2)a, as the distance of the above mentioned critical

sections from the column centerline increases, the angles of twist of the spandrel

cross-section in relation to the column also increase. In Section 3.2(1) it has been

concluded that increase in the angle of twist of the spandrel in relation to its adjacent

columns increases the induced compressive forces in the spandrels and

consequently, the slab restraining factor y.

Fig. 3.3(2)a shows that the angle of twist of column C is larger than that

of column A. Therefore, the slab restraining factor for column C (i.e. \|/c) is greater

than that of column A (i.e. \J/A).

Based on the observation made in Section 3.2.1 the slab restraining factor,

y, decreases as the spandrel strength increases. This is schematically presented in

Fig. 3.2(2)b for the case of column positions C and A. It may be seen that the

40

relationship between \|/A and \|/c can be expressed as \j/c = X + \J/A , where X is

defined as the column width factor. Calibration of X is described in Section 8.3.2.

3.3 Interaction Surface for Isolated Reinforced Concrete Beams

(Truss Analogy)

3.3.1 Historical review

A study of the interaction of torsion with bending and shear may be based

on truss analogy. The pioneering work on reinforced concrete members subjected

to torsion was carried out by Rausch(1929). H e assumed that a concrete member,

reinforced with longitudinal and transverse reinforcement, acts like a tube, so that

the applied torsional moments is resisted by the circulatory shear flow in the walls

of the tube. Furthermore, the tube is assumed to act like a space truss in resisting

this circulatory shear flow.

The space truss analogy has been generalized by Lampert and Thurlimann

(1969) for members subjected to torsion or to combined torsion and bending.

Since in their analytical model the angle of the concrete struts was not restricted to

45°, they called their theory the variable-angle truss model. This trass model was

further applied by Elfgren (1972) to members subjected to torsion, bending and

shear.

A review of the existing literature by Hsu (1984) indicates that the most

general and complete interaction surface for the isolated reinforced concrete beams

under the combined effects of torsion, bending and shear is the one developed by

Elfgren et al. (1974). They observed that for rectangular beams with closed ties:

41

(i) The ultimate strength in combined torsion, bending and shear, after some

simplifying assumptions, can be evaluated from a study of the equilibrium of

external and internal forces on the inclined failure surfaces.

(ii) The concrete compression zone can form in the top, in the bottom, or in one of

the vertical sides of the beam. This leads to three different modes of failure (i.e.

modes t,b and s). Then corresponding to each mode of failure they developed an

interaction surface.

(iii) The interaction surfaces for the three modes together form an interaction surface

which governs the load-carrying capacity of a beam.

The interaction surface established by Elfgren et al. (1974) is used herein

as a basis for the derivation of the interaction equation for the spandrel beams. This

is discussed in the ensuing sections.

3.3.2 The interaction equation

Fig. 3.3(1) shows a typical slab-column connection at the comer of a flat

plate with spandrel beams, under a uniformly distributed vertical loading. It may be

seen that the spandrel is under the combined effects of torsion, bending and shear.

The resulting skew failure surface for the spandrel beam, under the above

loading condition, at the ultimate state is shown in Fig. 3.3(2)a. The corresponding

internal forces at the spandrel support, i.e. the torque T2, the bending moment M 2 ,

and shear force, V 2 are depicted in Fig. 3.3(2)b. D u e to the different diagonal

tensile stresses in the different faces of the beam, the inclination of the cracks and

that of the concrete compression struts will vary from face to face. According to

Fig. 3.3(2) the failure surface on three sides is defined by an inclined spiralling

42

crack and on the fourth side, the bottom of the beam, the ends of the cracks are

joined by a compression zone. The above failure surface is defined on the basis of

the observed behaviour of the slab-column connections of the present half-scale

flat-plate models, at the edge- and corner-positions, at the ultimate state. A typical

punching shear failure of the present model structures at the corner position, W4-C,

is shown in Fig. 3.3(3).

To investigate the interaction surface due to the internal forces and

moments T2, V2, and M2, we may first ignore the effects of the adjoining slab on

the strength of the spandrel. This allows us to compare the spandrel beam with an

isolated reinforced concrete beam for which the failure surface on the top and side

faces is defined by an inclined spiralling crack, while a compression zone occurs on

its bottom.

A comparison between the spandrel's mode of failure (as discussed above)

and the modes of failure suggested by Elfgren et al. (1974) (see Section 3.3.1),

indicates that the spandrel failure mode is similar to the proposed failure mode t.

The corresponding interaction formulas for this mode of failure may be derived

from the following equilibrium equation:

2M2 +rT2V s ut +fV2Y s 2di _1 3 3(1) Aisfiydi lv2AtJ Awsfwy Akfiy |^2diJ Awsfwy Aisfiy

where At and ut are respectively the area and the perimeter of the rectangle defined

by the longitudinal bars in the comers of the closed ties; fjy and Au are respectively

the yield strength and total area of the longitudinal steel bars, fwy and Aws are

respectively the yield strength and cross-sectional area of the ties; and s is the

spacing of the ties. Note that in this equation the vertical distance between the

43

longitudinal steel bars is assumed to be equal to d b where di is the effective depth

of the spandrel beam.

From Eq. 3.3(1) the load carrying capacity of the isolated reinforced

concrete beams can be evaluated for pure bending, Mu s, for pure torsion, Tus, and

for pure shear Vus. They are

Mu-.-jAj-fjyd! 3 3 ( 2 )

Tus = 2A lo> v/^fix utco •V 3.3(3)

v-=^»V?F" ^ where

w ~ s 3.3(5)

Substituting M u s , Tus, and V u s respectively from Eq. 3.3(2) to 3.3(4) into Eq.

3.3(1) gives

us J M u s TusJ "IVusJ +W:=1 3.3(6)

Eq. 3.3(6) corresponds to mode t of the interaction surface proposed by Elfgren et

al. (1974). This is shown in Fig. 3.3(4) schematically .

Fig. 3.3(4) also shows the straight-line shear-torsion, shear-bending, and

torsion-bending interaction. While simple to use, this straight-line variation appears

44

slightly conservative for isolated reinforced concrete beams. The corresponding

interaction equation may be expressed as

?^ + ^=l 3.3(7) xus vus iVius

It is important to note that in this analysis, the first order interaction

equation, Eq. 3.3(7), is used as a basis for the derivation of the interaction equation

for the spandrel beams and consequently for the prediction of the punching shear

strength, Vu. This is done simply because this equation, as compared to the second

order interaction equation, (Eq. 3.3(6)), leads to more accurate and consistent

values for Vu. This is discussed in Appendix H

3.4 Proposed Interaction Equation for Spandrels

The interaction equation for the spandrels may now be obtained by the

incorporation of the restraining effects of the slab in the first order interaction

equation for isolated beams, Eq. 3.3(7). That is the strength terms YAlsfiy and

co + co0 are respectively substituted for the corresponding terms Alsfiy and co into

Eq. 3.3(3) and 3.3(4) (see Sections 3.2.1 and 3.2.2). Expressing the load carrying

capacities of the spandrel beams for pure bending, pure torsion and pure shear by

Msp, Tsp, and Vsp respectively, we have

M s P = ifAlsflydl(^) 3'4(1)

lsp — 2 A ,(CO + C O 0 ) A / - ^ ^ - 3.4(2)

45

_ n Uu (co+ co0) A / —

\ ut(

Vsp = J2cLu7 (co + co0) \l ¥ lsly 3.4(3)

' "t(CO + co0)

Substituting M s p, Tsp, and V s p respectively for Mu s, Tus, and V u s in Eq. 3.3(7)

gives

T2 V2 M2

Eq. 3.3(4) may now be considered as the interaction equation for the

spandrel beams. This equation may also be expressed in terms of the spandrel

parameter, co0, and the slab restraining factor, \\f. To do so the following

definitions are proposed.

(i) Isolated reinforced concrete beam with minimum reinforcement - This is a

reinforced concrete beam with minimum practical transverse and longitudinal

reinforcement, i.e. one Y12 longitudinal bar at each comer, and 4 mm hard-drawn

wire stirrups at a maximum spacing as specified in Clause 8.3.8(b) of AS3600-

1988. Note that the Y12 designation is for a 12 mm diameter deformed bar. The

tensile test results on the hard-drawn wires and deformed bars vary between 400 to

550 MPa. In Australia the average strengths of the 4 mm and 12 mm bars are

assumed to be equal to 480 and 450 MPa respectively.

(ii) Longitudinal steel ratio, a - This is the ratio of the spandrel longitudinal

strength, A, f, , to that of an isolated reinforced concrete beam with minimum

longitudinal strength. That is

A f ass7KT^ 3-4(5)

(, ls ly^min

46

Substituting for (Aj £") from definition (i) into Eq. 3.4(5), gives ^ •''min

mm

Alsfl a "200000 3-4<6)

(in 1 Transverse steel ratio, p - This is similar to the definitions for a which may be

expressed as

ut(co + COQ) P =-? x 3.4(7)

(utco) . v /min

or _ut(co + co.) 3^4(8)

utf Aws

wy s I min

where ut is the perimeter of the rectangle defined by the longitudinal bars in the

comers of the closed ties; fwy and Aws are respectively the yield strength and the

cross-sectional area of the closed ties, and s is the spacing of the ties. Substituting

for | ut fwv ~~~ 1 . from definition (i) into Eq. 3.4(8) gives ^ ' s ^min

ut(co + COQ) 13 _ 50000 K }

(iv) Spandrel strength parameter (o) - This is the product of the longitudinal steel

ratio, a, the transverse steel ratio, (3, and the ratio of di/d, where di and d are

respectively the effective depths of the spandrel and the slab. Thus

8 = ccP^ 3.4(11)

where a and P respectively are defined in Eqs. 3.4(6) and 3.4(9).

47

The interaction equation for spandrel beams, Eq. 3.4(4), may now be

expressed in terms of co0 and y with the aid of the above definitions. Assuming

0.5 Akfiydi as the yield moment of the top steel bars of the spandrel at the face

of the column support, M y , and substituting Alsfly and ut(co + co0) respectively

from Eqs. 3.4(6) and 3.4(9) into Eqs. 3.4(2) and 3.4(3) give

Msp = ¥My 34(12)

Tsp = 200 000^(apx|/)1/2

3>4(13)

Vsp = 200 00o|âpV)1/2

3.4(14)

Substituting Eqs. 3.4(12) to 3.4(14) into Eq. 3.4(4), we have

^ + =200000 f Y'V,,, " . =200 000 P^ At\ fch I \j/ U J V2ut

\y-%) •

Note that Eq. 3.4(15) contains the undetermined co0 and co which are to be

calibrated experimentally (see Chapter 8).

48

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49

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Spandrel beam

Column centerline

Critical section near the side face of column A

Critical section near the side face of column C

Distance

a) Angles of twist of the spandrel in relation to the corner columns A and C

¥ C = ^ + ¥ A

¥

b) Variation of V and the corresponding spandrel strength

Fig. 3.2(2) Column width and the slab restraining factor, ¥

50

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CHAPTER 4

DETERMINATION OF MOMENTS FROM REINFORCEMENT STRAINS

55

CHAPTER 4

DETERMINATION OF MOMENTS FROM

REINFORCEMENT STRAINS

4.1 General Remarks

To predict the punching shear strength, Vu, one of its components, Vi,

acting along the front segment of the critical perimeter should be predicted first. To

calibrate the semi-empirical equations for Vi, it is necessary to calculate the slab

bending moments from the corresponding strain data. Thus a suitable moment-

strain relationship is to be established.

Hatcher et al. (1961) used a bi-linear moment-strain relationship to convert

the steel strains to bending moments (bending moment per unit width in the case of

the slab, see Fig. 4.1(1)). This approach was then adopted by Rangan and Hall

(1983) in their analysis. However these results are not applicable in the present

study, simply because the stress-strain characteristics of the reinforcements used in

their slab models were also bi-linear. In the present study the stress-strain diagrams

of the steel bars used are non-linear. Therefore a new moment-strain relationship is

proposed.

In Section 4.2 the proposed moment-strain relationship and the relevant

formulas are discussed. The formulas are based on the constitutive laws of

mechanics.

56

To check the accuracy of the proposed procedure, the experimental

moment-strain diagrams (obtained from bending tests on three isolated cantilever

beams) are compared with the theoretical moment-strain diagrams. The

experimental phase of the investigation is described in Section 4.3 and the accuracy

of the results are discussed in Section 4.4

4.2 Theoretical Moment-Strain Relationship

The basic flexure theory for cracked and uncracked reinforced concrete

sections indicates that the resisting bending moment of a section increases as the

reinforcement strains increase.

Defining es as the reinforcement strain, ecr as the cracking strain (i.e. the

steel strain at which the first crack occurs), and esy as the yield strain of the steel

bars, the steel strain may fall within one of the following categories: es < ecr,

Ecr < es < esy and es > esy. Thus the calculation procedure for the bending moment,

M, may be classified into three groups.

(i) For low values of strain, i.e. es < ecr, the section is still uncracked and the

bending moment, M, may be obtained directly from

IesEc 4 2 ( 1 )

(d - x)

in which

(n- l)(dAs + dA's) + b - ^

X = , — 4.2(2) (n- 1) (As +A's) + b D s

and

57

3

I = -jf- + b D / x - + (n - 1)AS(X - d')2 + (n - l)As(d - X ) 2 4.2(3)

where I is the moment of inertia; n is the modular ratio (i.e. Es/Ec); Ec and Es are

respectively the concrete modulus and modulus of elasticity of steel; X is the neutral

axis position of the uncracked section; Ds is the overall depth of the section, As and

A's are respectively the areas of the tensile and compression reinforcement; b is the

width of the section; and finally d and d' are respectively the depths of the tensile

and compression reinforcement in relation to the extreme compressive fibre.

Note that the cracking strain, ecr, may be obtained from the following

equation

where

M„ = S 4.2(5,

in which f is the concrete compressive strength, D s is the overall depth of the

section, and Mo- is the cracking moment.

(ii) When e^ < es < e^, the section is cracked and the bending moment, M, may be

obtained as

M = ZAsfs 4.2(6)

58

where fs is the stress in the tensile steel bars, magnitudes of which may be readily

calculated from the measured steel strains, es, using the corresponding stress-strain

relationship; and Z is the lever arm of the section which may be defined as

4.2(7)

in which

(

X c = *V [n •d

[np + (n-l)p'] + A / [np + (n-l)p']2 + 2[np + (n - l)p-j

4.2(8)

i . i

where p (= Ag/bd) and p (= A s/bd) are respectively the tensile and compression

steel ratios, and X c is the neutral axis position of the cracked section. It is

important to note that M > M c r .

(iii) At the ultimate state, if es > ey, the following equation may be used for the

calculation of M :

M = ZA sf s y 4.2(9)

where

Z = l-0.59^1d 4.2(10)

in which fsy is the yield strength of the tensile reinforcement.

59

The flowchart of the computer program developed for the calculation of M

is shown in Fig. 4.2(1). This program incorporates the theoretical moment-strain

relationship for the prediction of moment from the reinforcement strain data.

Before the application of the theoretical moment-strain relationship to the

half-scale flat plate models for the analysis of the slab moments, its accuracy has to

be examined. The experimental work presented in the following section produced

the experimental moment-strain relationships for comparison with the proposed

theoretical one.

4.3 Bending Test on Isolated Reinforced Concrete Beams

The experimental investigation consists of tests on three reinforced

concrete cantilever rectangular beams, simulating the slab strips. The beams were

reinforced in tension only, with steel ratios identical to those used in the test slabs.

The depth of the beams was 100 m m , equal to the nominal thickness of the slab,

while the width was a variable. The effective depth of the reinforcement was about

86 m m , corresponding to the effective depths used in the test slabs. The cross

sections and other design details of these beams are shown in Fig. 4.3(1). The

isolated reinforced concrete slab strips were fixed at one end (simulating the

spandrel slab connection) and were loaded by a concentrated load, up to failure.

The span of the cantilevers is 620 m m . A 5 tonne hydraulic jack was used for

applying the load. To measure the applied load the jack was connected to an

electrical load cell. For the measurement of steel strains electrical resistance strain

gauges were attached to the tensile reinforcement near the cantilever support. These

strains were recorded continuously during each test.

60

The materials used in the fabrication of the isolated beams were identical to

those used in the slab structures. The concrete was a ready mix design with a

maximum aggregate size of 10 mm and a strength of f'c = 26 MPa. The

reinforcement consisted of 8 mm diameter hard-drawn wires similar to those used

as the slab reinforcement, with a yield strength of 550 MPa. Also 4 mm diameter

hard-drawn wires were used to provide nominal shear reinforcement

To avoid bond slip of the top steel bars of the beam, the steel bars were

taken right up to the back edge of the beam at the support, then bent down and

around to provide sufficient anchorage length. This formed a U-shape loop.

The beams were cast in plywood forms in groups of three. Plastic chairs

tied to the formwork provided support for the reinforcement and maintained proper

cover. Dowels (with a diameter of 18 mm) were screwed to the formwork (from

underneath) to provide at least 4 holes in each beam which allowed the installation

of the beam as a cantilever (see Fig. 4.3(2)).

The concrete was placed with the aid of an internal vibrator. The beams

were covered by wet hessians and polyethylene plastic sheets which were removed

after 2 weeks. The curing condition for the beams and cylinders were similar to

that for the half-scale flat plate models. All the beams were under-reinforced and as

expected their ultimate strengths were governed by the yield strength of the

reinforcement (see Figs. 4.3(3) to 4.3(5)).

4.4 Analysis of Results

The computational procedure developed in Section 4.2 is used for the

analysis of the test results of the beam specimens Bl, B2 and B3. In this

procedure, for each test specimen, the strain data corresponding to different stages

61

of loading (i.e. before cracking, after cracking and at the ultimate state) are used for

the calculation of bending moments. This leads to a theoretical moment-strain

relationship for each test specimen.

Experimentally, the same bending moments were measured directly with

the aid of load cells connected to the hydraulic jack. These results led to an

experimental moment-strain relationship.

The theoretical and experimental moment-strain relationships for each of

the test specimens Bl, B 2 and B 3 are separately compared. It was found that in all

cases the correlations of the calculated and experimental values of M are good.

Typical moment-strain diagrams are shown in Fig. 4.4(1).

4.5 Summary

To help develop the prediction procedure for Vu, a moment-strain

relationship is established. The computer program, used for the calculation of the

moments from the reinforcement strains, is examined with the bending tests on the

determinant reinforced concrete beams.

The slab strips of the flat plate model structures all had similar overall

depths; their main differences were in the width of the slab strips and the steel

ratios. The overall depths of the three beam specimens tested were the same as

those of the slab strips, the widths and the steel ratios of the beams were also

similar to those of the slab strips.

A comparison of the predicted and measured bending moments of the

beam specimens indicate that the proposed moment-strain relationship is accurate

within the above range of test variables. Thus the computer program may be used

62

for the calculation of the slab strip moments from the reinforcement strain data.

(see Section 7.2.1).

63

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65

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CHAPTER 5

EXPERIMENTAL PROGRAMME AND GENERAL BEHAVIOUR OF THE FLAT PLATE MODELS

72

CHAPTER 5

EXPERIMENTAL PROGRAMME

AND GENERAL BEHAVIOUR OF THE

FLAT PLATE MODELS

In this chapter, the analysis, design and construction of five half-scale flat plate

models representing two adjacent edge- and corner-panels of a real structure are

discussed. Also represented are the behaviour of the model structures up to failure

including the mechanisms of failure, load-deflection characteristics, angles of twist

of the spandrels in relation to the load density of the slab, and finally the column

reactions.

It is worth mentioning that half-scale model tests of this type are a highly

labour intensive and expensive business. Before the present project, only two such

models have been tested successfully: those at the University of N e w South Wales.

5.1 The Half-Scale Models

The purpose of the experimental programme is to observe the behaviour of

the flat plate floors with spandrel beams of different depths, and to obtain essential

data for use in the establishment of the prediction procedure for the punching shear

strength, Vu. Also, the slab restraining effects on the spandrel strength (see

Chapter 3) call for the testing of models with proper boundary conditions, i.e. each

model should contain full-length spandrel(s) that are joined to the side face(s) of the

exterior columns.

Fig. 5.1(1) shows a half-scale flat plate floor, with spandrel beams in the

north-south direction. The overall depths of the slab and the spandrel are 100 m m

73

and 200 m m respectively. All columns have a height of 1500 m m and a cross

section of 200 x 200 mm. The floor slab consists of 16 square panels of 2700 mm

span lengths. Thus to include one edge- and one corner-column as well as the full

lengths of the spandrel beams in our test models, it should at least cover two

adjacent panels at the corner of the flat plate floor.

The shaded area of Fig. 5.1(1) includes two adjacent panels at the corner

of the real structure. The first test structure (Model Wl, Fig. 5.1(2)) was

constructed to represent panels "ABGF" and "BCHG" at the corner and edge

respectively. In this model, to simulate the restraining effects of the next panels of

the building, columns "F", "G" "H" and "C", as well as the boundary "CH", were

made stiffer. This aspect of the model design was carried out by taking full

consideration of the relative stiffnesses of the curtailed portions. Thus the model

would afford proper redistribution of moments following the yielding of the slab

reinforcement, and the application of load on the above region would provide the

same actions around the corner and edge columns as the whole slab was loaded.

In this study a total of five models were constructed in-situ and tested

under a uniformly distributed load up to failure. Four of them i.e. Models Wl to

W4 had spandrel beams of different depths; the remaining one (Model W5) was

without a spandrel or torsion strip.

In comparison with the models tested at the University of NSW, the

present model structures have the following advantages. The whiffle-tree loading

system works in tension therefore the instability problem is eliminated. The load

cells provided at the supports of all six column positions, allow the recording of the

column reactions at the edge- and corner-positions. Thus for the first time, the

failure mechanisms and other behavioural features of the slab-column connections

with spandrel beams at the edge- and corner-locations could be recorded up to

74

failure. Note that to accelerate the construction of the models, instead of concrete

columns, re-usable prefabricated steel sections (of equivalent stiffnesses) have been

bolted to the concrete slabs. The test programme also allowed for the measurement

of the column reactions and the slab reinforcement strains due to the slab self

weight.

5.2 Analysis and Design of the Model Structures

5.2.1 Analysis of forces using idealized frame method

Most codes of practice including AS3600-1988, and ACI 318-83,

recommend two methods for the analysis of the slab bending moments:

(i) the simplified method in which each floor of the building is analysed separately,

and

(ii) the idealized frame method in which the effects of the columns or the behavior

of the multistory structure is also taken into consideration.

In the idealized frame method, subject to some limitations, a multistory

three-dimensional structure with flat plate floors may be divided into two series of

approximately parallel idealized frames, running through the structure both in the

longitudinal and transverse directions. Each frame basically consists of the

columns, a column strip and two half middle strips.

The bending moments, shear forces and axial loads in each of the idealized

frames can be analysed using the traditional linear structural analysis method, and

the complete analysis of the frame should include all possible loading combinations.

However, the present study is limited to the case of the under-reinforced concrete

75

flat plates under a uniformly distributed vertical loading. Thus based on the

recommendations of AS3600-1988 a loading combination of 1.25 (live load) + 1.5

(dead load), with an ultimate load of 9 kPa has been adopted.

After analyzing the idealized frame for moments in the horizontal members

which have been treated as beams, the moments have to be distributed appropriately

to the column and half middle strips. However, based on the recommendation of

Regan (1981), instead of distributing the moments between the column and half-

middle strip(s), the following arrangements for the layout of the reinforcement

transferring moments to the edge- and corner-columns have been adopted.

For slabs with shallow spandrel beams or torsion strips, the width of the

idealized frame was divided into the affected and remaining widths. This is

schematically shown in Fig. 5.2(1). According to this procedure all the total

positive and negative bending moments of the idealized frame are assumed to be

resisted within a strip width of slab equal to the affected width; minimum tensile

reinforcement (see Clause 9.1.1 of AS3600-1988) is to be provided in the

remaining width.

A similar procedure has also been adopted for slabs with torsion strips

without closed ties. However when the spandrel provides near full bending

restraint for the connecting slab, all the bending moments have been assumed to be

distributed uniformly across the entire width of the idealized frame.

It is important to note that the above effective transfer width adopted for

the flexural moment transfer, has also been recognized to be appropriate by Moehle

(1988) and the ACI-ASCE Committee 352 (1988).

76

5.2.2 Design of the flat plates and the spandrels

Fig 5.1(2) shows the overall dimensions of Model Wl, with beams ABC,

and C H at its edge. The same reinforcement layout designed for the half-scale real

structure has been adopted for the flat plate model. The overall depth of the slab is

100 mm.

Note that in the remaining models (Models W2 to W5), the overall

dimensions of the slab, the slab depth and the slab reinforcement (except in the

vicinity of columns) were similar to those of Model W l . In these models beam

" C H " was omitted (see Fig. 5.2(2)). This was mainly because: (i) it has only

localized influence on column "C"; (ii) its effects on the slab-column connections at

"A" and "B"are negligible; and (iii) the assumptions that omissions of beam "CH" -

with the high stiffnesses provided by columns " C " and " H " - would not affect the

simulation of the real structure by the test models. Column " C " had a width twice

that of column "A". Therefore comparison of the results of the two comer columns

"C" and "A" also provides some information regarding the effects of column width

on the strength of slab-column connections.

The spandrel beams of Models Wl to W4 were designed with the

minimum practical transverse and longitudinal reinforcement, i.e. one 12 m m

diameter longitudinal bar at each corner, and 4 m m hard drawn wire stirrups at a

maximum spacing of 0.12ut as specified in Clause 8.3.8(b) of AS3600-1988 for an

isolated beam in torsion. The significance of these minimum reinforcements has

been discussed previously (see Section 3.4). The widths of the spandrel beams

were constant at 200 m m , and the main test variable was the ratio of the effective

depth of the spandrel beam to that of the slab, di/d.

77

To investigate the effects of the spandrel transverse reinforcement on the

punching shear strength of the slab-column connections, Model W 5 was

constructed without any transverse reinforcements (closed ties) in the torsion strip.

The design details of the spandrel beams of Model W l to W 4 together with those of

the torsion strip of Model W 5 are presented in Fig. 5.2(3).

5.2.3 Design of the steel columns (of equivalent stiffnesses)

For ease of construction instead of concrete columns, the concrete flat plate

slab was supported on six prefabricated steel sections (with equivalent stiffnesses).

The three force components at the pin supports at the bases of the test column were

measured by the use of specially designed load cells.

The columns included in these five models were extended downward to

the theoretical contraflexure point (approximately 1/2 of the column height below

the floor). Since the columns above the slab were omitted it was necessary to

double the column stiffness to resist twice the moments. This was achieved by

halving the column lengths. A typical prefabricated steel section simulating an in-

situ reinforced concrete column is shown in Fig. 5.2(4).

At each column position, the slab was confined by a prefabricated steel

column at the bottom and a steel plate on the top. They were held monolithically by

twelve 16 m m mild steel bolts in column " H " and eight 16 m m bolts in each of the

remaining columns (see Fig. 5.2(5)).

5.3 Materials

The ready-mixed concrete used to cast the test models was designed to

provide a cylinder strength of at least 20 M P a after 28 days. Concrete strengths at

78

the time of tests are listed in Table 5.3(1). Measured slumps ranged from 70 to 85

mm. Each of the strengths given is the average from tests of nine 150 x 300 mm

cylinders. Three of these cylinders were taken at the beginning, three at the middle,

and three near the end of the concrete pouring process.

Four types of steel reinforcement were used in the test structures: i) F 62

welded wire reinforcing fabrics, meeting the requirements of Australian Standard,

AS 1304-1984, for the bottom and top slab reinforcement; ii) 4, 5, 6 and 8 mm

diameter hard-drawn steel reinforcing wires meeting the requirements of AS 1303-

1984 for the required additional steel bars in the slab, as well as the spandrel ties;

iii) Y12 mm diameter deformed bars meeting the requirement of AS 1302-1982 for

the longitudinal reinforcement of the spandrels, and slab reinforcement in some

boundaries of the slab; iv) R16 mm diameter mild steel round bars, meeting the

requirement of AS 1302-1982 as the vertical studs of the test columns .

A typical load-strain variation for the above reinforcing bars is depicted in Fig.

5.3(1); and the rest are given in Appendix VI. Other reinforcement data are

presented in Table 5.3 (2).

5.4 Construction of the Models

5.4.1 Formwork

To fabricate the form work, first the steel columns were located on the top

of the steel pedestals (see Fig. 5.4(1) ) via five dummy load cells. Fig. 5.4 (2)

shows the adjustable support system that allowed for proper alignment of the test

columns both in the horizontal and vertical directions. Details of a typical support

system is shown in Fig. 5.4(3).

79

Formply was used to fabricate the formwork. The fabrication was in such

a way that it could be re-used. It was supported on Acrow extendable beams which

in turn were supported by two channel sections spanning in the north-south

direction (see Fig. 5.4(4)). Before bolting the steel channels to the test rig, with the

aid of Acrow props they were held in position and leveled within 0.5 m m accuracy.

When the formwork was in place, the levels were checked again, and adjustments

were made as necessary.

To resist the outward pressure of the wet concrete the slab formwork was

surrounded by angles and channels at all sides. Dowels with a diameter of 18 m m

were screwed to the formwork (from underneath) to provide 32 holes in the slab

which afforded the installation of the whiffle-tree loading system.

5.4.2 Reinforcement details

The slab reinforcement consisted of two layers of F 62 welded wire

fabrics, with a wire size of 6.3 m m . The pitch distances for both longitudinal and

cross-wires were 200 m m . Hard-drawn steel reinforcing wires were also used

where required. A typical arrangement of slab reinforcement is shown in

Fig.5.4(5).

According to Clauses 9.1.1 and 9.4.3 of AS3600-1988, a minimum steel

ratio of 1/ fsy is required for the strength of slab in bending and for crack control (or

shrinkage and temperature effects ). The welded wire fabrics (F62) used met this

requirement.

The F 62 meshes had an overall size of 2400 mm x 7000 mm. To avoid

bond slip of the top reinforcing bars of the slab in the main moment direction, they

were taken right up to the edge of the slab, then bent down and around to form the

80

bottom reinforcement. This formed a U-shape (see Fig. 5.4(6)). In the north-

south edges of the slab a similar procedure was also followed to anchor the top steel

bars of the slab. The amount of the steel meshes required for the whole slab is

three F 62 meshes per model. The 7000 mm length of the meshes after folding was

sufficient to cover the entire width of the slab in the main moment direction (or the

east-west direction). To keep the continuity of the reinforcement in the north-south

direction, adjacent meshes were welded to each other, at a section near the expected

contraflexure point (see Fig. 5.4(7)).

To avoid any loss of strength because of welding, a series of 6.3 mm

hard-drawn wires with different types of welded connections were tested. A

comparison of the variation of the tensile load versus deformation of these

specimens ( see Fig. 5.4(8)) indicates that 3 spot welds within a width of 120 mm

as shown in Fig. 5.4(7)e does not decrease the strength of the wire meshes at the

joints. Therefore F 62 meshes that met the above requirement would provide both a

safe and an economic reinforcement for the test structures.

As much of the steel as possible was tied in mats or cages before being

placed in the formwork. Special chairs were used to maintain the proper vertical

clearances and the steel was securely tied to the formwork so that it would not be

displaced during casting.

To reduce congestion of reinforcement specially at corner columns,

longitudinal reinforcing bars of the spandrel were bolted to the end plates which

were located at the northern and southern faces of columns A and C respectively.

A typical detail of the reinforcement which resulted from the design of the

half-scale flat plate structures are shown diagrammatically in Figs. 5.4(9) to

81

5.4(11), and in photograph in Fig. 5.4(12). For further details reference can be

made to Appendix V.

5.5 Casting and Curing

About 2.5 cubic metres of concrete were required to cast each model

and the nine 150 x 300 m m test cylinders. The ready-mixed concrete was pumped

to the slab through 100 m m flexible pipes. A n internal (poker) vibrator was used

to consolidate the concrete. The surface of the slab was finished immediately with a

wooden float. T w o to three hours after casting, the top surface of the slab was

smoothed with a steel trowel. There was no evidence of bleeding of the concrete in

any of the models. About eighteen hours after the concrete was placed, the slab

surface was covered with moist hessian and plastic sheets.

The test cylinders were cast at the same time. They were vibrated

internally and were cured in the same way as the flat plate model. Each model and

the test cylinders were stripped after approximately ten days. The casting and

testing dates of test Models W l to W 5 are shown in Table 5.3(1).

5.6 Instrumentation and Testing Procedure

The tests described in this thesis consisted essentially of applying a

uniformly distributed vertical load to half-scale flat plate structures, measuring loads

and reactions, and making certain measurements of the deformed structure.

This section is concerned with a brief description of how the load was ap

plied, how the load and reactions were measured, what measurements of the

structure were made, and what procedure was followed in carrying out a test.

82

5.6.1 Loading system

The flat plate models were cast-in-situ and were supported within a re

action frame. The reaction frame is shown diagrammatically in Fig. 5.6(1); as

depicted in Fig. 5.6(2) the reaction frame consists of six vertical steel sections

(pedestals), each supporting one of the six columns of the flat plate model. The

steel pedestals were tied to each other (both on the top and bottom) to resist the

lateral forces that would be induced during the test.

One 20-tonne double acting hydraulic ram was located at the centre of each

panel of the two-panel test structure. The load from each ram was distributed

through a whiffle-tree loading system to sixteen 100 mm square pads on the top of

each panel to simulate a uniform load. The downward (tension) load applied by the

hydraulic rams to the slab was resisted by the main reaction frame (or test rig). In

this system the rams' hinge-supports were bolted to the reaction frame; their other

ends were connected to the whiffle-tree loading system via a specially designed

electrical tension load cells. These load cells provided the means of measuring the

load density in each panel.

Due to the membrane effects in flat plate models (see Appendix I) a failure

load much higher than the design ultimate load was expected. Therefore the beams

and bolts of the whiffle-tree loading system as well as the main reaction frame were

designed for a load of approximately 3 times the design ultimate load. The plan

view and elevation of the loading system as well as some connection details are

shown in Fig. 5.6(3).

Synchronization of the rams was achieved by pumping through one single

hydraulic control system. Details of the hydraulic system are given in Fig. 5.6(4).

83

5.6.2 Testing procedure

To get the initial data (just before the application of the dead load by

stripping the slab) all the load cells and strain gauges were connected to the data

acquisition control system. Then the initial data were recorded onto a floppy disk

via a Hewlett Packard microcomputer (see Section 5.6.3). After removing the

formwork and hanging the whiffle-tree loading system the second set of load data (

due to dead load + weight of the loading system, approximately equal to 3.05 kN /

m^) were recorded. Also at this stage the zero deflection and rotation readings were

recorded.

For the third set of data onward the test loads were applied in increments

of approximately 5 percent of the expected failure load. To apply a load increment,

the hydraulic pressure in the loading rams was raised to a desired level. The load

was then held constant for a few minutes. During this time data for deflection ,

angles of twist of the spandrel, column reaction, and load density were recorded.

The new cracks were also identified.

The loading was continued up to failure. In all models immediately after

failure, there was a significant drop in load. At this stage all the recorded data were

transferred to floppy disks and the hydraulic system were disconnected. About

three hours were required to conduct each test.

5.6.3 Reaction measurements

As shown in Fig. 5.4(3) the steel columns simulating the in-situ concrete

columns were supported by pedestals via electrical compression load cells. With the

aid of these specially designed load cells, the three reaction components at the pin

84

supports (contraflexure points) at the bases of the six steel columns were measured.

Construction details for a typical load cell is illustrated in Fig. 5.6(5).

A Hewlett Packard 3054A data acquisition control system with a capacity

of 50 channels and a Hewlett Packard 9826 computer were used to record the

strains and other electrical signals. The first 20 channels were used for logging the

load cell readings; the remaining 30 channels were connected to strain gauges

attached to selected reinforcing bars of the slab.

To obtain the abovementioned electrical signals from the strain gauges and

load cells (via the data acquisition control system) and convert them to steel strains

and column reactions a computer program has been developed. The functions of

this program includes:

(i) printing the zero strain data;

(ii) instantaneous scanning of all the 50 channels, and printing the strain data at each

stage of loading;

(iii) reporting the faulty strain gauges by printing "wires disconnected" in front of

the gauge number;

(iv) printing magnitudes of shear force, bending moment, and transverse moment at

each column support for all stages of loading;

(v) printing the magnitudes of load in each ram as well as the load density for all

stages of loading;

85

(vi) drawing and printing the variations of shear force, bending moment, and

transverse moment versus load density at each column support (from the beginning

of the test) for each stage of loading (if required);

(vii) recording all the data on to a floppy disk operated by the computer.

It is worth mentioning that the time spent for the development of the

program was about three months of continuous work.

Fig. 5.6(6) shows the direction of the main slab reinforcement and the

bending moment for the flat plates. Variations of the column reactions of Models

Wl to W5 at the edge- and corner-positions are shown in Figs. 5.6(7) to 5.6(19).

The column reactions of the column positions Wl-C and W4-B are excluded simply

because they did not fail.

5.6.4 Strain measurement

Steel strains were measured at numerous locations in the test structures.

The purpose of measuring steel strains was two-fold. To determine the moments in

the slab and to determine the distribution of shear force along the critical perimeters

of the columns (see Section 2.5).

The strains were measured with electrical resistance strain gages (TML-

PLS-10-11) with a gauge factor between 2.06 to 2.11. A total of 30 gauges were

used on Model Wl, the number was increased to 40 for the remaining models.

Note that the extra 10 strain gauges provided in Models W2 to W5 were monitored

manually.

Fig. 5.6(20)a shows the locations of the strain gauges along the spandrel

beam AB. They were attached to the top steel bars of the slab and located along

86

measuring station No. 1 at a distance d/2 from the face of the spandrel, where d is

the effective depth of the slab. For each of the flat plate model structures (Wl to

W5), the readings of all the strain gauges were recorded up to failure. The strain

data corresponding to the last loading stage or the ultimate state is represented in

Fig. 5.6(20)b"l" . Similar experimental plots are also provided for the slab spans

BC, FG, and GH (see Appendix III). These strain profiles were then used as a

basis for the calculation of the slab bending moments.

The slab strain data may also be used as a basis for the definition of the

column critical perimeter. This is discussed in Appendix IV.

5.6.5 Measurement of deflections

In Model Wl the vertical deflections of the slab were measured at the

centre of each panel and the column line "BG". For the remaining models, to

provide a sound picture of the deflected shape of the test models, vertical

deflections of the slab were measured at 11 locations(see Fig. 5.6(21)a.

As shown in Fig. 5.6(2l)b, steel rulers were hung at those specified

locations with the aid of small hooks which in turn were glued to the slab. A level

was used to book the rulers, where the hanging rulers were functioning as the

leveling staffs with an accuracy of 0.5 mm. The level was located at a distance to

book all the rulers at each loading stage.

The deflection data for the centers of the north and south panels as well as

the column line BG (see Fig. 5.6(21)a) of Models Wl to W5 respectively are

t Note that Model W2 with a deep spandrel beam at its edges failed by the formation of a negative yield line at the face of the spandrel. At the time of failure the strain gauges, located along the measuring station 1 (see Fig. 5.6(2Q)a), were either disconnected or were indicating reinforcement strains more than the yield strain (i.e about 0.0044). Therefore in Fig. 5.6(20)b a constant strain of 0.005 was considered for this model.

87

presented in Figs. 5.6(22) to 5.6(26). It may be seen that, in all the models the

deflections at the center of the north panel, at the ultimate state, are higher than the

remaining measurement points. This may be attributed to the lower stiffnesses of

the slab-column connections of the north panel, and consequently the lower in-

plane forces (due to membrane effects).

A comparison of the slopes of the load-deflected curves of Models Wl to

W 5 indicates that (up to the ultimate state) the load gradients for the slabs with

spandrel beams are steep. Whereas for the slab with torsion strips, this slope

became flat near the ultimate state (see Fig. 5.6(26)). Thus in the absence of the

spandrel beams, the flat plate structures reached its maximum loading capacity first.

Further deformation then led to the failure of the system.

5.6.6 Measurement of the angle of twist of the spandrels

In this section the instrumentations used for the measurement of the angles

of twist of a typical spandrel beam in relation to its supporting column, Column

" B " of Fig. 5.2(2), are illustrated. The behaviour of the spandrels up to the

ultimate load are also discussed. The angles of twist of the spandrel beams of

Models W l to W 5 were measured with dial gauges of 0.001 in. graduation (see

Fig.5.6(27)). Fig 5.6(28) shows the locations of the dial gauges, installed at

Sections 1, 2, 3, and 4 along the spandrels. These gauges were located on the side

of the beam, two on each section, at a vertical spacing of 340 m m . The distance

between the sections is 450 m m , i.e. each set is located at one-sixth point. The

angle of twist of Sections 2, 3 and 4 of the spandrel in relation to Section 1 (located

at the center of the adjacent column) may then be calculated by means of the test

data obtained from the dial gauges.

88

The present models consisted of two comer columns "A" and " C " and one

edge column "B". Fig. 5.6(29) shows the variation of the angles of twist of the

spandrel beam of Models W l , W 2 , W 3 and W 5 in relation to the edge column "B".

It may be seen that by increasing the load density, the angle of twist also increases.

The maximum twist occurs between Sections 1 and 2, and as it gets closer to the

panel centreline (at a distance of 1350 m m ) the rate of twist decreases. A similar

variation has also been observed for comer location "A".

A comparison of the angles of twist of these models at the ultimate state

(see Fig. 5.6(29)) indicates that as the depth or the strength of the spandrels

decrease, the corresponding angle of twist increases.

The maximum angle of twist of Model W2 (with deep spandrels) is about

0.004 radian, whereas that of Model W 5 (without spandrel beam) is 0.049 radians,

which is almost 12 times that of Model W 2 .

The above discussion leads to the conclusion that the angle of twist of the

deep spandrel beams in relation to their adjacent columns are negligible. However a

larger angle of twist occurs in the absence of the spandrel.

5.7 Cracking and the Ultimate Load

The behaviour of all the models was the same. At the ultimate state they

all sustained loads far in excess of the design ultimate load.

In flat plate models with spandrel beams, at the ultimate state, the first

positive yield line occurred at midspan parallel to the spandrel. Further increase in

load led to the development of negative yield lines along the front faces of the edge-

and comer-columns.

89

Model W l reached a maximum load of 30.63t kPa. At this load both the

corner column "A" and the edge-column "B" failed suddenly and violently (see

Figs. 5.7(1) and 5.7(2)). In both column positions "A" and "B", spiralling skew

cracks occurred at the faces of the spandrel adjacent to the column. The cracks

were clearly the result of torsion in the spandrel.

Model W2, with the largest spandrel beam in the test series, carried a

maximum load of 28.91 kPa. Again, at the ultimate state individual negative yield

lines developed at the front faces of the edge- and corner-columns. Further

increases in load led to the joining up of these yield lines, at which instant a sudden

and violent failure occurred (see Fig. 5.7(3)). A postmortem examination of the

test model showed that the negative flexural reinforcement of the slab across the

face of the spandrel beams were fractured after necking of the bars. The same was

true for the bars across the positive yield line.

Model W3 had the smallest practicable spandrel beams in this test series.

The maximum load carried by this model was 24.69 kPa. This model had a similar

behaviour to that of Model Wl. In this model punching shear failure occurred at all

the column positions ("A", "B" and "C") simultaneously (see Figs. 5.7(4) to

5.7(6)).

The overall depth adopted for the spandrel beam of Model W4 was 50 mm

thinner than that of Model W2, and the load at which failure occurred was 28.95

kPa. Model W4 also had a behaviour similar to that of Models Wl and W3. In this

model punching shear failure occurred at the corner column positions "A" and "C"

simultaneously (see Figs. 5.7(7) and 5.7(8)). The behaviour of this model led to

t The slab self-weight is also included,

90

the conclusion that Model W 2 had the weakest deep spandrel beam for which the

mode of failure was flexural.

Model W5 had no closed ties in its torsion strips along the edges. The

failure was initiated by the formation of negative yield line segments along the front

face of the edge- and corner-columns. The maximum load carried by the slab was

19.01 kPa. In this model, the first punching shear failure occurred at column "B";

and further application of load led to the punching shear failure at column positions

"A"and"C".

In all the column positions, "A", "B" and "C", of Model W5, 45° cracks

occurred in the upper face of the spandrels adjacent to the columns. The cracks

were clearly the result of torsion in the torsion strip (see Figs. 5.7(9) to 5.7(11)).

Apart from Model W5, the edge column positions R-3A and R-4A of the flat plate

models tested by Rangan and Hall (1983), also had torsion strips. In all these

models the ultimate steel strains of the positive slab reinforcement in the main

moment direction (at midspan) were about 0.0022, whereas the corresponding yield

strains are on average 0.0044. Thus no positive yield line at mid-span preceded the

punching shear failure.

Typical crack pattern of the present model structures are depicted in Figs.

5.7(12) and 5.7(13). Further details may be obtained from a parallel study by Latip

(1988).

5.8 Concluding Remarks

The experimental work carried out herein has produced data on punching

shear failure at 13 edge and corner column positions namely, W2-A to W5-A,

Wl-B to W3-B, W5-B and W2-C to W5-C. The data thus obtained together with

91

the observed behaviour of the test models up to failure provide useful information

for the development of the proposed prediction procedure for the punching shear

strength, Vu. This procedure is discussed in the next chapter.

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Deflection, mm

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151

Table 5.3(2) Reinforcement properties

Model Wire Diameter Area

No. -6 Type strain xlO~° MPa MPa mm mrtv

W l

W2

W3

W4

W5

Common

in all

models

W5 F62

W6.3-1 W8-1

W4-1 F62 W6.3-2

W8-2

W4-2

F62

W6.3-3

W8-3

W4-2

F62

W6.3-3

W8-4

W4-2

F62

W6.3-3

W8-3

Y12

R16

4266

4380

4536

4264

4390

4380

4280

4700

4460

4380

4100

4400

4460

4380

4100

4450

4460

4380

4100

4400

4128

2260

492

570

550

480

499

570

487

563

520

570

489

506

520

570

489

500

520

570

489

506

431

262

210613

231745

209333

208926

204114

231745

210274

199540

201589

231745

226743

203500

201589

231745

226743

204496

201589

231745

226743

203500

200000

201000

4.99

6.30

6.30

8.00

3.98

6.30

6.26

7.91

3.98

6.30

6.26

7.91

3.98

6.30

6.26

7.93

3.98

6.30

6.26

7.91

12.00

16.00

19.56

31.17

31.17

50.06

12.44

31.17

30.78

49.14

12.44

31.17

30.78

49.14

12.44

31.17

30.98

49.39

12.44

31.19

30.76

49.1

113.10

201.06

CHAPTER 6

RESEARCH SCHEME FOR THE DEVELOPMENT OF THE PREDICTION PROCEDURE FOR Vu

152

CHAPTER 6

RESEARCH SCHEME

FOR THE DEVELOPMENT OF THE

PREDICTION PROCEDURE FOR Vu

This chapter serves as a key link between the first half of the thesis which

deals with the fundamentals and the experimental aspects of the thesis project, and

the second half which concentrates on the development and verification of the

prediction procedure for Vu. After a brief review of the contents of Chapters 1 to 5,

the development of the prediction procedure for the punching shear strength, Vu, is

schematically described with the aid of block diagrams. This provides a sound

background for the analytical aspects of the present study which are discussed in

Chapters 7 to 9.

6.1 General Remarks

The present investigation includes experimental and analytical studies of

reinforced concrete flat plates with spandrel beams. To better understand the

strength behaviour of the slab-column connections as well as to quantify such

behaviour, a well planned model test programme was carried out in conjunction

with a theoretical study. The main purpose is to develop a complete prediction

procedure for the punching shear strength, Vu.

Development of this procedure required a theoretical study on (i) the

distribution of the total bending moment and shear force at a slab-column

connection between the front and side segments of some critical perimeters within

the slab, and (ii) the interaction of torsion, shear and bending in the spandrel

beams. These studies are presented in Chapters 2 and 3 respectively. Note in (i)

153

that for the distribution of the shear force a moment-strain relationship has also to

be developed. This is discussed in Chapter 4.

The experimental study consists of tests on five under-reinforced half-scale

flat plate models, four with spandrel beams of different depths and steel ratios, and

one without any spandrel. Each structure consists of two adjacent panels at the

corner and edge of a building floor with carefully controlled boundary conditions.

The models were tested up to failure, under a uniformly distributed load. Details of

the model test programme together with the test results of Models W l to W 5

covering a total of 13 corner- and edge-column positions have been described in

Chapter 5. Note that in each of the models the edge- and corner-columns are

designed in such a way that they would fail simultaneously. All but two did fail at

the ultimate load. Thus the five models gave a total of 13 failure records at edge-

and corner-positions.

An outline of the proposed prediction procedure for the punching shear

strength, V u is given in Section 6.2. The development involves firstly, the

establishment of semi-empirical formulas for M i and Vi, based on the equations

presented in Chapter 2. This is followed by the derivation of semi-empirical

equations for \|/, based on the discussions in Chapter 3. These are respectively

presented in Sections 6.3.1 and 6.4.1. Section 6.4 also covers a discussion on the

accuracy of the prediction procedure.

6.2 Outline of the Research Scheme

Based on pilot studies carried out by the author an analytical procedure was

formulated for the prediction of Vu. In order to establish the undetermined internal

forces incorporated, a well planned test programme has been carried out.

154

The models were instrumented and tested in such a way as to provide data for the

quantification of the prediction procedure.

Fig. 6.2(1) shows schematically the proposed procedure for the prediction

of the punching shear strength, Vu. It may be seen that in addition to Vu there are

seven other undetermined forces, namely Mci, Mc2, Vi, Mi, V2, M2, and T2.

These forces which act in the vicinity of the edge- and corner-columns are

illustrated in Fig. 6.2(2).

The seven undetermined forces and moments may be classified into three

groups:

(i) To be measured experimentally and for which semi-empiricd formulas have to

be developed. These include Mi and Vi.

(ii) To be estimated using known structural analysis procedures. These include

Mci and Mc2-

(iii) To be determined in conjunction with those in groups (i) and (ii) using the three

equilibrium equations plus one interaction equation for torsion, shear and moment

acting on the spandrel. These include M2, V2, T2 .and Vu.

Note in (iii) that in order to establish the interaction equation it is necessary

to first determine the parameter co0, and the factors \\f and X. While co0 has been

found in the present study to be a constant some formulas have to be developed for

the estimation of \|/ and X.

In summary, in order to fully develop the prediction procedure for Vu,

semi-empirical formulas must first be established for Mi, Vi, \|f and X. Therefore

155

one of the main objectives of the ultimate load tests of the present series of half-

scale models (as detailed in Chapter 5) was to provide experimental data for setting

up these semi-empirical formulas. Measurements of the required experimental data

of the slab-column connections of these model structures are discussed in the

following section.

6.3 Determination of the Internal Forces within the Slab-Column-

Spandrel Connections

The block diagram of Fig. 6.3(1) shows the procedure for the

measurement of the eight undetermined internal forces Mci, Mc2, Vu, Mi, Vi, M2,

V2, and T2. It may be seen that the moments Mci and Mc2 and the force Vu can

direcdy be measured with the aid of horizontal and vertical load cells provided at the

column supports (see Section 5.6). Further, by the development of a new

technique for the measurement of Mi and Vi, all the eight parameters may then be

determined by means of statics. The procedure used for the measurement of Mi

and Vi is summarized in the following subsection.

6.3.1 Semi-empirical equations for Mi and Vi

Mi and Vi respectively are the slab bending moment and shear force at the

front segment of the critical perimeter (see Fig. 6.2(2)). These forces cannot be

measured directly. Only the basic quantities (i.e. the strains and the force

components) are measurable with the aid of electrical load cells and a data logger.

Therefore a procedure needs to be developed for their determination.

The proposed measurement procedure is summarized in the block diagram

of Fig. 6.3(2). According to the analytical model developed in Section 2.5, the flat

plate slab may be divided into a series of parallel slab strips.

156

To measure the slab strip moments, strain gauges were attached to selected

positive and negative slab reinforcements. An accurate measurement of these

moments from the strain data requires a carefully developed moment-strain

relationships.

As a result of a separate experimental and analytical investigation (see

Chapter 4), a relationship between the slab reinforcement strains and the bending

moments acting in a reinforced concrete slab up to the ultimate state has been

established. This relationship was used for the measurement of the slab strip

moments and consequently the determination of Mi.

Measurement of Vi is based both on the above "measured" moments and

the data obtained from the vertical load cells at the center of the column supports.

These data were then used in a computation procedure (see Section 2.5) for the

calculation of Vi.

The accuracy of the measurement procedure for Mi and Vi has been

examined through linear regression analysis, see Chapter 7. The semi-empirical

formulas developed for the determination of Mi and Vi are also presented in the

same Chapter.

6.4 Prediction of Vu

As discussed in Section 6.2, for the prediction of Vu, in addition to the

semi-empirical equations for Mi and Vi (see Section 6.3), the interaction of torsion,

shear and bending in the spandrel beams should also be determined.

157

6.4.1 Calibration of the proposed interaction equation

Based on the ultimate behaviour of the present model structures an

interaction equation for spandrel beam has been developed in Chapter 3, i.e. Eq.

3.4(15). This equation involves the parameter co0 and the factor \j/ that are to be

calibrated experimentally.

With the aid of the measurement scheme described in Fig. 6.3(1) the

internal forces of the slab-column connections of all the column positions have been

measured. These values are used for the determination of co0 and development of

semi-empirical equations for \j/. Details of the calibration procedure is illustrated in

Chapter 8.

The accuracy of the measured \\f values has been examined through

exponential regression analysis. The reliability of the proposed interaction equation

is also discussed in Chapter 8.

6.4.2 The prediction formulas

As discussed in Sections 6.3.1. and 6.4.1, the developed semi-empirical

equations for M i and Vi are presented in Chapter 7, and those for y in Chapter 8.

These equations together with the proposed interaction equation for spandrels (Eq.

3.4(15)) and the derived equilibrium equations (see Section 2.4) have been used for

the derivation of the prediction formulas for Vu. The formulas are presented in

Chapter 9.

Note that the prediction formulas of Chapter 9 are explicit and simple. They

may be used for the prediction of the punching shear strength of the slab-column

158

connections of flat plates, with or without spandrel beams, at the edge- and comer-

positions.

6.4.3 Comparison and discussion of results

In the light of the experimental results reported herein a comparative study

is carried out. This indicates that the present prediction procedure is more accurate

than that recommended in AS3600-1988. The latter procedure also suffers, at

times, the serious drawback of considerably overestimating the punching shear

strength.

159

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CHAPTER 7

DISTRIBUTION OF MOMENT AND SHEAR ALONG THE CRITICAL PERIMETER

164

CHAPTER 7

DISTRIBUTION OF MOMENT AND SHEAR

ALONG THE CRITICAL PERIMETER

7.1 General Remarks

Chapter 2 covers the theoretical background for the distribution of the total

moment, Mci, and the total shear, V u along some critical perimeters, at the edge-

and corner-column positions. The distribution is ready determinable provided

magnitudes of M x and Vi can be accurately predicted.

To develop semi-empirical formulas for the prediction of Mi and Vi, these

forces are to be measured first. The measurement procedure used has been

summarized in the block diagram of Fig. 6.3(2). According to this procedure the

slab m a y be divided into a series of parallel slab strips perpendicular to the

spandrel. The bending moment diagram along each slab strip may then be

determined with the aid of the slab reinforcement strain data. By establishing the

moment diagrams, its slope (i.e. the shear force) can also be calculated. These are

further discussed in Section 7.2.

Theoretically the measured values of Mi should be proportional to the

corresponding slab yield moments. Therefore by the calibration of the measured

M i values, some semi-empirical formulas can be developed. This is detailed in

Section 7.3.

The semi-empirical formulas developed for the prediction of Vi, are

presented in Section 7.4. In these equations Vi is expressed as a function of (i) the

locations and the strengths of the slab reinforcement, (ii) the clear span of the slab

165

in a direction perpendicular to the slab edge, and (iii) the stiffness of the spandrels

(at the side face(s) of the columns) and the slab.

7.2 Measurement of the Internal Forces at Slab-Column Connections

To measure the internal forces at the slab-column connections of the flat

plates, magnitudes of M i and Vi are to be measured first. As discussed in Section

2.5, the proposed measurement procedure is mainly based on the measured slab

strip moments.

7.2.1 Slab strip moments

Fig. 7.2(1) shows the size and location of the slab strips of Models Wl to

W 5 . It may be seen that the slab regions bounded by the panel center line(s)

adjacent to the edge- and corner-columns are divided into a series of parallel slab

strips.

Theoretically, for the determination of the moment diagram along the slab

strip i, at least three measured moments are required. Therefore along each slab

strip three measuring stations 1, m and 3 respectively at distances X = 0, X = X m

and X = X 3 were considered (see Fig. 7.2(1)). Note that X = 0 corresponds to a

distance equal to d/2 from the face of the spandrel beam, where d is the effective

depth of the top steel bars of the slab. The absolute values of the slab strip

moments of strip i at the measuring stations 1, m and 3 are respectively M n , M 3 i

and Mmi, where the subscript i stands for the measuring station (i.e. 1, m or 3).

To obtain a better picture of the variation of bending moments and shear

forces in the vicinity of columns, the width of the slab strips adjacent to the

166

columns were reduced (see Fig. 7.2(1)). This is to capture the very steep strain

gradients in the vicinity of the columns (see Fig. 5.6(20)b).

As a result of a separate experimental and analytical investigation (see

Chapter 4), a relationship between the slab reinforcement strains and the bending

moments acting in a reinforced concrete slab up to the ultimate state has been

established. This relationship was used for the measurement of the slab strip

moments.

Based on the data obtained from the strain gauges at the measuring stations

1, m and 3, and with the aid of the above moment-strain relationship, magnitudes of

Mn, M3i and M^ for all the slab strips of the Models Wl to W5 are calculated. A

computer program has been developed for the calculation of the slab strip moments.

The calculation results are presented in Appendix VII. These data are used for the

determination of the shear force Vi in Section 7.2.3.

7.2.2 Slab moments Mx and Mm

Fig. 7.2(1) shows the plan view of a typical flat plate floor at a corner

location. In this figure the slab strips E and C respectively located opposite the

edge- and corner-column positions are defined as the critical slab strips; the widths

of these strips (as discussed in Appendix IV) are equal to the widths of the columns

(i.e. C2).

Mi and Mm are respectively defined as the slab moments at the measuring

stations 1 and m of the above critical slab strips. Note that the location of the

measuring station 1 overlaps with that of the front segment of the critical perimeter

(i.e. at a distance d/2 from the column face).

167

Obviously, the measurement procedure for the determination of the slab

strip moments Mi and Mm of the critical slab strips E and C would be similar to

that used for the other slab strip moments (see Section 7.2.1). Magnitudes of Mi

and Mm are then measured for all the following positions:

(i) the A series corner column, i.e. column positions Wl-A to W5-A,

(ii) the C series corner column, i.e. column positions W2-C to W5-C,

(iii) the B series edge column, i.e. column positions Wl-B to W5-B , and

(iv) the edge columns R-3A and R-4A tested by Rangan and Hall (1983).

The measured values of Mi and Mm can then be calibrated against their

respective slab yield moments. This leads to the establishment of the semi-

empirical formulas for Mi and Mm, which are discussed in Section 7.3.

7.2.3 Shear force Vi

For each flat plate model structure the measured slab strip moments as

defined in Section 7.2.1., and the data obtained from the vertical load cells at the

center of the column support systems are used for the calculation of Vi. That is, the

substitution of these data in the calculation procedure developed in Section 2.5 led

to the determination of Vi.

The measured values of Vi are then calibrated in Section 7.4. This leads

to the determination of the semi-empirical formulas for Vi. The accuracy of the

proposed measurement system is also discussed in section 7.4.

168

7.3 Distribution of the Total Bending M o m e n t

Distribution of the total bending moment between the front and side faces

of the columns m a y be determined by the prediction of M i (see Section 6.3). In

addition to M i , the predicted values of M m are also required in the prediction

procedure for Vi. Therefore in this section both for the prediction of M i and M m

semi-empirical formulas are developed.

The measured Mi and Mm and the calculated values of their corresponding

slab yield moments are calibrated separately for the same column positions reported

in Section 7.2.2.

Note that in the calculation of the yield moment of the slab strips, each

strip is considered as an isolated beam. Then the corresponding yield moments are

calculated. The calculation results are listed in Appendix VII.

A simple regression analysis is then performed to draw the "best fit"

straight line through the scatter graph of the above test data. The results are

discussed in the following subsections.

7.3.1 Edge column positions

Fig. 7.3(1) shows the variation of the measured Mi values versus their

corresponding slab yield moments, M i y , for the edge column series reported in

Section 7.2.2. The semi-empirical formula of the "best fit" straight line through the

scattered data may be expressed as

Mi,edge = 0.83 M1> y i e l d+ 0.18 7.3(1)

169

This expression may then be simplified as

Mi,edge = 0.83M1>yield 7.3(2)

where Mledge and Mlyield are respectively the estimated and yield moments of the

slab at the front segment of the critical perimeter of the edge columns.

According to Fig. 7.3(1) the coefficient of determination (R2) for the

fitted curve is 0.997, indicating a reliable fit and confirming the accuracy of the

semi-empirical formula (Eq. 7.3(2)).

As discussed in Section 5.7, for all the models with spandrel beams

(Models Wl to W4), failure started by the formation of a positive yield line at mid-

span (parallel to the spandrels). Further increase in load led to the punching shear

failure of the edge- and/or corner-column positions. This indicates that at the

ultimate state, the positive slab reinforcement corresponding to the measuring

station m of the critical slab strips C and E also yielded (see Fig. 7.2(1)). Therefore

Mm would be equal to the slab yield moment.

In Section 5.7 it is also shown that in the flat plate models which had no

realistic spandrel beams, the punching shear failure occurs prior to the formation of

a positive yield line at mid-span.

Note that the spandrel beams of these models are either without any closed

ties (Model W5), or with a di/d ratio less than 1 (Models R-3A and R-4At), where

di and d are respectively the effective depths of the spandrel and the slab.

t Rangan and Hall (1983)

170

The measured M m values for the above model structures are presented in

Table 7.3(1), and the corresponding slab yield moment are reported in Appendix

VII. A comparison of these data indicates that in the absence of the spandrels, the

measured Mm values are 0.70 times their corresponding slab yield moments.

This discussion thus shows that for slabs with spandrel beams Mm is equal

to the corresponding slab yield moment at mid-span, whereas for slabs without

spandrels this value reduces to 70% of the slab yield moment.

7.3.2 Corner column positions

Fig. 7.3(2) shows, the variation of the measured Mi values versus their

corresponding slab yield moments (Mly) for the A and C series comer columns,

i.e. column positions Wl-A to W5-A, and W2-C to W5-C.

The semi-empirical formula of the "best fit" straight line through the

scattered data may be expressed as

Ml,corner = Ml,yield 7-3(3)

where Mlcomerand Mlyield are respectively the estimated and the yield moments of

the slab at the front segment of the critical perimeter of the comer columns.

According to Fig. 7.3(2) the coefficient of determination (R2) for the fitted

curve is 1, again indicating a reliable fit and confirming the accuracy of the semi-

empirical formula, i.e. Eq. 7.3(3).

For the prediction of Mm, again a discussion similar to that of the edge-

columns may be followed (see Section 7.3.1). That is, for slabs with a spandrel

171

beams M m is equal to the corresponding slab yield moment at mid-span, whereas

for slabs without spandrels this value reduces to 7 0 % of the slab yield moment.

7.3.3 Accuracy of results

Both for the edge- and corner-column positions, the ratio of the measures

to predicted M i and M m values are presented in Table 7.3(1), respectively in the

columns titled Mutest/ Mi,predicted, and Mm,test/ Mm,predicted. The ideal ratio is

unity, where the predicted value is equal to the corresponding measured one. It

may be seen that the semi-empirical equations are accurate and consistent in their

prediction with a mean test/predicted ratio of 1.02 and a standard deviation of 0.06

for the Mi,test/ Mi,preciicted ratio. The corresponding values for the ratio of Mm,test

/ Mm,predicted are respectively 1.00, and 0.02.

In summary, according to the present procedure the estimated Mi value for

the corner column position is equal to the corresponding slab yield moment along

the front segment of the critical perimeter. This value reduces to 0.83 times the slab

yield moment at the edge locations. Also for slabs with realistic spandrel beams

(di/d > 1), M m is equal to the corresponding slab yield moment, whereas for the

slabs with torsion strips (di/d < 1), with or without closed ties, M m reduces to 7 0 %

of the corresponding slab yield moment. Note that the slab yield moment (in all

cases) is measured over a width C2 of a slab, where C2 is equal to the column

width.

172

7.4 Distribution of the Total Shear Force

7.4.1 Theoretical background

As discussed in Section 6.3, with the prediction of Vlt the distribution of

the total shear force can also be determined. Vi may be predicted by the

development of some semi-empirical formulas. Theoretically the measured values

of Vi should be proportional to the slope of the bending moment diagrams of the

critical slab strips. These slopes, as per Section 2.5, may be expressed in terms of

Mb Mm and Lc, where Mi and Mm are respectively the critical slab strip moments

at the measuring stations 1 and m, and Lc is the clear span (see Fig. 7.4(1)).

The effects of the stiffnesses of the slab boundaries on the ultimate loading

capacity of the slab is discussed in Appendix I. According to this discussion, in the

flat plates with spandrel beams, the restraint provided by the spandrels and columns

against the horizontal displacements of the slab, causes a portion of the load to be

carried by an arch or dome action, which is able to utilize the strength of the

materials much more efficiently than the normal slab actions. And as a result the

loading capacity of the slab as well as the ultimate bending and shear capacities of

the slab-column connections significantly increase.

Therefore if the di/d ratio represents the relative stiffness of the spandrel

beam in relation to the slab, due to the dome action, Vi should increase as the di/d

ratio increases.

In summary, to determine the distribution of the total shear force along the

critical perimeter, some semi-empirical formulas for Vi should be developed. In

these formulas Vi may be expressed as a function of Mi, Mm, Lc and di/d. Note

173

that Mi and M m reflect the contribution of the size and location of the slab flexural

reinforcements as well as the concrete strength.

7.4.2 The formulas

Fig. 7.4(1) shows the plan view of a typical flat plate. The shaded area

represents the critical slab strips C and E, in which the variation of the bending

moment due to vertical loading is expressed as

M = AX2 + BX + C 7.4(1)

Assuming Vi as the slope of the moment diagram at X = 0, we have

Vi = B 7.4(2)

Substituting for B from Eq. 7.4(2) into Eq. 2.5(3) gives

V 1 • x m ( x*3- X m ) <

M ™ + M - > + X 3 ( X ^ X.) - M'> 7 4 ( 3 )

where Mi, M m and M 3 respectively are the absolute values of the moments at X =0,

X = Xm and X = X3 (see Fig. 7.4(1)).

In general M3=Mi, taking X3 and Xm as fractions of the clear span, Lc, measured

from face to face of supports. Eq. 7.4(3) may be rewritten as

Vl = y(Ml

T+ Mm) 7.4(4)

174

where y is a constant with respect to M i and M m-

According to Eq. 7.4(4), V1 is proportional to Mi, M m and Lc, where M i

and M m are in turn proportional to the strength and the location of the flexural bars

of the slab. Therefore Vi is a function of the size and location of the slab

reinforcement, the clear span Lc, and the compressive strength of the concrete.

In the following subsections, the influence of the effective depth ratio

(di/d) on the magnitude of the parameter y will be examined for the edge- and

comer-columns.

7.4.3 Edge column positions

Eq. 7.4(4) is a theoretical formula for the prediction of Vi in terms of Mi,

M m , L c and the parameter y. To determine this parameter Eq. 7.4(4) may be

rewritten as

Y =(LcVl,edge)

Tedge Mi + M m '" W

Theoretically yedge should be proportional to the effective depth ratio di/d

(see Section 7.4.1). Thus, to develop a semi-empirical formula for the estimation

of this parameter, the measured values of M i , M m , Vj>edge (i.e. Vi for edge

columns) and L c against the di/d ratio are plotted for the B series edge columns.

These include column positions W l - B to W 5 - B and those tested by Rangan and

Hall (1983), i.e. column positions R-3A and R-4A.

175

A simple regression analysis is then performed to draw the "best fit"

straight line through the scatter graph of the above test data (see Fig. 7.4(2)).

This gives:

Yedge = Lr(3.19+1.56-kj ? 4(6)

where

L c \0.85 L r = ( 2 3 7 7-4(7)

By substituting L,. from Eq. 7.4(7) into Eq. 7.4(6), and Yedge from Eq.

7.4(6) into Eq. 7.4(5), and solving for V^ge, we have:

V.-4,.- 0.75(2.04 + ^-±#-- 7.4(8)

According to Fig. 7.4(2) the coefficient of determination (R2) for the fitted

curve is 0.988, indicating a reliable fit and confirming the accuracy of Eq. 7.4(8).

In this equation Mi and Mm are in kN-m, Lc in m, and Vi>edge^ in kN.

7.4.4 Corner column positions

A procedure similar to that in Section 7.4.3 may be used for the

determination of the semi-empirical formula for Vi,corner (i-e- Vi f°r corner

columns). Therefore the magnitude of Vi,corner again may be expressed as a

* Note that V l e d g e refers to the total shear force acting along the front segment of the critical perimeter over the slab width C2 (see Fig. 7.4(1))

176

function of the measured slab moments Mi, M m , Lc and the effective depth ratio

(di/d). This leads to

Vi„ = 0.24(6.90 + ^^-^ 7.4(9)

In Eq. 7.4(9) again Mi and Mm are in kN-m, Lc in m, and Vi,corner in kN.

Note that the data used for the derivation of Eq. 7.4(9) are those from the results of

the model structures Wl-A to W5-A, and W2-C to W5-C.

7.4.5 Comparison of results

Both for the edge- and comer-column positions, the ratiosof the measured

to predicted values of Vi are presented in Table 7.4(1), in the column titled

Vi,test/Vi,Predicted- Again the ideal ratio is unity, for which the predicted value is

equal to the corresponding measured one. It may be seen that the semi-empirical

formulas are accurate and consistent in their prediction with a mean test/predicted

ratio of 0.99 and a standard deviation of 0.06 .

It is important to note that the developed semi-empirical formulas for the

determination of Vi (Eqs. 7.4(8) and 7.4(9)) lead to the determination of the actual

distribution of V u (i.e. is the total shear force at the column center) between various

sides of the critical perimeter.

In summary the proposed prediction formulas for the actual distribution of

V u along the critical perimeter incorporates the effects of the size and location of the

flexural reinforcement of the slab, the clear span, Lc, the compressive strength of

the concrete, and the restraint provided by the spandrel against the horizontal

displacement of the slab.

177

7.4.6 Discussion

Rangan and Hall (1983) presented some empirical formulas for the

estimation of Vi. Based on the concept of beam analogy, they divided the slab into

a series of parallel slab strips. Unit slab shears were then obtained through

calculations of the slopes of the unit moment curves which were derived from the

measured slab reinforcement strain data.

Later Rangan (1987) found that their proposed empirical formulas

underestimate the shear force Vi. Then he referred to the absence of reliable

information, and on that basis assumed that the distribution of the average shear

stress along the critical shear perimeter is uniform. This proposal, although without

a reliable basis, was adopted by the new Australian Standard AS3600-1988.

The presented semi-empirical formulas (see Sections 7.4.3 and 7.4.4)

cover directly the effects of the following significant variables which influence the

distribution of shear force over the critical perimeter:

(i) the size and location of the slab reinforcement,

(ii) compressive strength of the concrete,

(iii) clear span (Lc) in the main moment direction, and

(iv) the effective depth ratio (di/d), which is proportional to the induced

compressive membrane action in the slab.

178

According to the above discussion Rangan and Hall (1983) did not

consider the membrane effects in their analysis. This might be one of the reasons

for their underestimated Vi values.

It is worth mentioning that for the case of the slabs without spandrels,

Regan(1981)as well as Alexander and Simmonds(1987) both emphasized the

effects of the size and location of the slab steel bars on the distribution of the total

shear force Vu along the critical perimeter.

In summary, the semi-empirical equations developed herein for the

prediction of Vi leads to the determination of the shear force variation around the

critical perimeter. To be able to determine the distribution of the total bending

moment and shear force, as discussed in this chapter, means that part of the

prerequisites for developing the proposed prediction procedure for Vu are

completed.

179

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Yield Moments, kN-m

Fig. 7.3(1) Measured moments (M x) versus the corresponding yield moments of slab-edge positions

£ *

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Yield moments, kN-m

Fig. 7.3(2) Measured moments ( M i ) versus the corresponding yield moments of slab-corner positions

181

Corner colum

Critical Perimeter

Spandrel Beam

Critical Perimeter

is Edge column

F

Critical Slab strip c

d/2

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Measuring station 1

Critica Lc

d/2

slab strip E

Measuring station m (at midspan)

Fig. 7.4(1) Variation of m o m e n t along the critical slab strip

182

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Table 7.4(1) Measured

Model

No.

Wl-A

W2-A

W3-A

W4-A

W5-A

W2-C

W3-C

W4-C

W5-C

Wl-B

W2-B

W3-B

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10.17

13.99

12.56

9.27

12.93

14.38

14.16

10.91

24.71

10.34

15.24

14.81

15.57

23.87

16.27

Mean

Standard Deviation

and predicted values of V

l.pred v l.test

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10.5

11.46

14.52

12.19

10.25

12.53

13.64

15.56

11.88

25.51

9.29

14.71

14.61

15.94

24.98

16.07

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0.99

0.89

0.96

1.03

0.90

1.03

1.05

0.91

0.92

0.97

1.11

1.04

1.01

0.98

0.96

1.01

0.99

0.06

CHAPTER 8

PREDICTION FORMULAS FOR SPANDREL PARAMETERS

186

CHAPTER 8

PREDICTION FORMULAS FOR

SPANDREL PARAMETERS

8.1 Scope

To complete the prediction procedure for the punching shear strength Vu,

an interaction equation has been developed in Chapter 3 (Eq. 3.4(15)). This

equation contains the undetermined co0 and \j/ which are to be calibrated

experimentally.

The theoretical considerations for the determination of these parameters are

explained in Section 8.2. In Section 8.3.1 the procedure used for the determination

of a>o is discussed; also for each series of the slab-column connections located at the

comer A, corner C, and the edge positions, magnitudes of y are calculated.

The semi-empirical formulas developed for the estimation of y values for

each of the column series are presented in Section 8.3.2. Note that this section also

includes the effects of column width on the \j/ values. Section 8.3.3 includes a

discussion on the validity of the semi-empirical formulas and the reliability of the

proposed interaction equation. The assumptions adopted for the analysis of slab-

column connections with torsion strips without closed ties are discussed in Section

8.4. Finally, the specifications for deep spandrels are presented in Section 8.5.

8.2 Theoretical Consideration and Spandrel Parameters

Fig. 8.2(1) shows schematically the procedure used for the determination

of the parameters adopted in the interaction formula developed in Chapter 3 for the

187

spandrel beams (Eq. 3.4(15)). In this equation the restraining effects of the slab on

the rotation and elongation of the spandrel, respectively are incorporated in terms of

co0 and \|/. The parameter co0 and the factor \|/ are both undetermined and have to be

calibrated experimentally.

As shown in Fig. 8.2(1) the proposed interaction equation consists of five

undetermined quantities including the forces T2, V2, and M 2 , the parameter co0 and

the factor \|f. The measuring procedure described in Chapter 7 allows the

determination of V2, T2, and M 2 . Substitution of these values into Eq. 3.4(15)

leads to an equation in terms of co0 and \)/ for each column position.

As discussed in Section 3.2.1, the magnitude of \|/ for the slabs with deep

spandrel is equal to unity; according to the discussion of section 5.7 Model W 2 is

also a flat plate with deep spandrels. Therefore \|/ may be assumed equal to 1 for

the edge column position W 2 - B . The parameter co0 may now be determined by

substituting \\f = 1 and measured strength parameters of column position W 2 - B into

Eq. 3.4(15).

The overall slab depthsof the present model structures and those tested by

Rangan and Hall (1983) were constant. Therefore it is assumed that the slab

restraint on the rotation of the spandrel is also constant, and consequently co0 has

the same value for all the tested structures (see Section 3.2.2).

As depicted in Fig. 8.2(1), substitution of the measured co0 and the

measured forces and moments of each model structure into the interaction equation

gives the corresponding \|/ values.

188

The semi-empirical equations for the prediction of \j/ m a y then be obt*ained

through regression analysis of the measured values of \j/. That is, regression

analysis can be used for the curve fitting of these measured values in the scattered

graph of Y versus 8^ for each of the following sets of column positions:

(i) the series "A" comer-columns,

(ii) the series "C" corner-columns, and

(iii) the series "B" edge columns plus those tested by Rangan and Hall (1983).

Note that in each of the model structures the width of corner-column C

was twice that of corner-column A. Theoretically the measured \|/ values for

corner-column C are higher than those for corner-column A (see Section 3.2.3).

Therefore to show the effects of column width on the \|/ values by the column width

factor(k), this factor m a y be expressed as X = \\rc - \j/A, where \|/c and \|/A are

respectively the slab restraining factors for the series " C " and "A" comer columns.

Details of the measuring process used for the determination of GO0 and \j/\

as well as the derivation of some semi-empirical equations for \\f and X are

discussed in the following sections.

"I" see the definition of Section 3.4

189

8.3 Determination of the Spandrel Parameters

8.3.1 Measurements of co0 and y

Substituting y = 1 into Eq. 3.4(15) gives

Also by substitution of a and P respectively from Eqs. 3.4(6) and 3.4(9) into Eq.

8.3(1) and rearranging terms we have

4Alsflyu(l- j^j

where

A = A ^ + - ^ 8.3(3)

Vat

As discussed in Section 8.2 for Model W2-B, the magnitude of \|/ is equal

to 1. Substitution of the measured T2, V2, and M 2 of this model into Eq. 8.3(2)

leads to

co0 = 446 N/mm 8.3(4)

where co0 is the additional spandrel transverse strength developed by the restraining

effects of the slab.

190

Note that in the present analysis all the computations were done with the

aid of a computer. First, all the raw data of each column positions were stored in a

separate data file. Then for each stage of the analysis, a computer program has

been developed for the determination of the required parameters.

According to Section 8.2, by determining co0, the magnitudes of y can

easily be determined for each column position. This may be obtained by

substitution of the measured 0)o (= 446 N / m m ) , and the measured relevant forces

and moments of each column position into the interaction equation, Eq. 3.4(15).

To calculate y, Eq. 3.4(15) may be rewritten as

V|/ 'A+AiV

2A 2

8.3(5)

where

A1 = (A^4(A2f1^r 8'3<6>

and

A2 = 200,000(ccP)1/2 8-3(7)

in which A, a and =3 respectively are defined in the Eqs. 8.3(3), 3.4(6) and 3.4(9).

The measured values of y obtained through Eq. 8.3(5) are presented in Table

8.3(1).

Theoretically these measured values of y should be inversely proportional

to the corresponding spandrel strength parameters (8) of the spandrel beams; that is

191

the value of y increase as 8 decreases (see Section 3.3.1). Thus, to examine the

accuracy of this theory the measured y values are plotted separately against the

corresponding spandrel strength parameters (8) for each of the following three sets

of column positions:

(i) the series "A" comer columns, i.e. column positions Wl-A to W5-A,

(ii) the series "C" corner columns, i.e. column positions W2-C to W5-C, and

(iii) the series "B" edge columns, i.e. column positions Wl-B to W3-B, W5-B and

those of Rangan and Hall (1983), i.e. column positions R-3A and R-4A.

An exponential regression analysis is then made for each of the above test

series (see Figs. 8.3(1) and 8.3(2)). The coefficients of determination (R2) were

determined with the aid of a standard computer package*. The values of R 2

obtained indicate that they are adequate fits and the proposed interaction equation is

reliable.

It is worth mentioning that although the interaction equation was originally

developed for spandrel beams, it is also applicable to the case of torsion strips

without closed ties (see the measured y values of Model W 5 in Figs. 8.3(1) and

8.3(2)). However, further experimental work is required for a full investigation of

the slab-column connections of flat plates with torsion strips without closed ties.

The assumptions employed for the analysis of Model W 5 are discussed in Section

8.4.

"Cricket graph 1.2" developed for the Apple Macintosh by Cricket Inc.

192

8.3.2 Prediction formulas for y and X

The results of the exponential regression analysis of the measured values

of y are shown in Figs. 8.3(1) and 8.3(2) (see Section 8.3.1). It may be seen that

the values of R 2 obtained are close to 1, which confirm the adequacy of the fits.

The analysis leads to three semi-empirical formulas for the three series of column

positions. These equations may then be used for the estimation of the slab

restraining factor y.

Expressing the predicted y values for the series "A" and "C" corner

columns respectively by y A and y c and that for the edge series by yedge, the

standard regression analysis leads to the following semi-empirical formulas:

yc = 6.92 - 4.41 log (8) 8.3(8)

yA =5.21-3.14 log (8) 8.3(9)

Vedge = 3.24 - 1.64 log (8) 8.3(10)

A comparison between Eqs. 8.3(8) and 8.3(9) indicates that the slab

restraining factors for series " C " comer columns (yc) are larger than those of series

"A" (yA). Note that the series " C " corner columns have a width twice that of series

"A". The theoretical background for this observation has been presented in Section

3.2.3. According to this theory the difference in the slab restraining factors y c and

y A may be due to the effects of the column width (C2). As shown in Fig. 3.2(2)b

for an arbitrary spandrel beam attached to the corner columns A and C, the

corresponding slab restraining factors y c and y A may be related as

VC = A. + Y A 8.3(11)

193

Substituting y A and y c from Eqs. 8.3(8) and 8.3(9) into Eq. 8.3(11) and

solving for X, gives

X= 1.71 -1.27 log (8) 8.3(12)

Now incorporating the effects of the column width C2, the slab restraining factor

for comer columns may be expressed as

¥comer = ¥ + >-1^200 " 0 8.3(13)

where C2 is the column width in mm, X may be obtained from Eq. 8.3(12) and y is

equal to y A or

y = 5.21-3.14 log(8) 8.3(14)

in which the strength of the spandrel beam (8) may be obtained using Eq. 3.4(11).

8.3.3 Comparison and discussion of results

Both for the edge- and comer-column positions, the ratios of the measured

to predicted values of y are presented in Table 8.3(1) in the column titled ytest/

ypredicted- It may be seen that the semi-empirical formulas are accurate and

consistent in their predictions with a mean test to prediction ratio of 0.99 and a

standard deviation of 0.03. Further, the coefficients of determination (R^) of the

mathematical model used for the prediction of y and X are close to one (see Section

8.3.1). From these results it may be concluded that the proposed interaction

equation for spandrel beams (Eq. 3.4(15)) is reliable.

194

8.4 Slabs with Torsion Strips Without Closed Ties

Theoretically the restraining effects of the slab enhances (i) the spandrel

longitudinal strength, i.e. from Aisfiy, to yAlsfiy, and (ii) the spandrel transverse

strength i.e. from co, to co + co0. For flat plate column connections with torsion

strips without closed ties, a discussion similar to that of Section 3.3 may be used to

conclude that the restraining effects of the slab also enhances the longitudinal and

transverse strengths of these strips.

The characteristics of torsion strips without closed ties may also be

represented by the parameters a and P which are respectively the longitudinal and

transverse steel ratios (Eqs. 3.4(6) and 3.4(9)). These equations may be rewritten

as

a~ 200000 K }

Ut(cO + COQ)

P= 50000 8'4(2)

where in Eq. 8.4(2), due to the absence of the closed ties the spandrel transverse

strength co is equal to zero.

By the determination of a and p\ the spandrel interaction equation

developed in Chapter 3 (Eq. 3.4(15)) may also be applied to the case of the slab-

column connections with torsion strips without closed ties. However Eqs. 8.4(1)

and 8.4(2) indicate that the determination of a and p requires the areas, the

strengths and the locations of the longitudinal reinforcements of the torsion strips to

be defined first. This may be obtained by using the design recommendations of the

ACI-ASCE Committee 352 (1988).

195

According to the paper of A C I - A S C E Committee 352 (1988) when the

spandrel beams are absent, the slab edge should be reinforced to act as a spandrel

beam. The recommended slab edge reinforcement is intended to control cracking.

For the edge connections without closed ties, the bars running parallel to

the slab edge should be placed (where practicable) within the bars perpendicular to

the edge. The recommended reinforcement for the edge connections is depicted in

Fig. 8.4(1). It may be seen that the location of the bars remote from the slab edge

is designated by e, where

0.75Ci < e < Ci 8.4(3)

in which Q is the column width. The vertical and the horizontal distances between

the parallel bars are respectively designated by x and y (see Fig. 8.4(1)). Therefore

At and ut which are respectively the area and the perimeter of the rectangle defined

by x and y may now be calculated. Note that in this figure, Ais is the total cross-

sectional area of those parallel bars which are located at the comers of the rectangle

defined by x and y, and have sufficient anchorage lengths to develop the yield

strength. The same discussion is also applicable to the case of comer columns.

These recommendations are employed for the determination of the spandrel

strength parameter 8(= ccpdi/d) and the slab restraining factor y for the column

positions W5-A, W5-B and W5-C. The measured y values are shown in Figs.

8.3(1) and 8.3(2). The coefficients of determination for the fitted curves show that

the calculated values of y for Model W5 have a good correlation with the other test

data. This is an indication of the reliability of the employed procedure. Note that in

the calculation of (3, the magnitude of co0 is also 446 N/mm.

196

8.5 Slabs with Deep Spandrel Beams

Theoretically for the case of slabs with deep spandrel beams y = 1; and at

the ultimate state failure occurs by the formation of a negative yield line at the face

of the spandrel (see Section 3.2.1). To quantitatively define the slabs with deep

spandrels a series of five half-scale flat plates (i.e. Models W l to W 5 ) have been

tested up to failure. Note that the spandrels of all these models had the minimum

practical reinforcements as specified in Section 3.4.

According to the discussion of Section 5.7.1, Model W2 was considered

as a flat plate slab with deep spandrel beams of minimum strength. Thus it is

assumed that the interaction equation developed for shallow spandrels is also

applicable to Model W 2 . As per Section 3.2.1 in this case the slab restraining

factor, y, is unity.

To quantify the definition of the slabs with deep spandrel beams let us

calculate the strength parameter of the spandrel beams (8t) at the column position

W2-B. Substituting a and (3 from Eqs. 3.4(6) and 3.4(9) into Eq. 3.4(11) gives

8 = Alsflyut f^f^ + co0^ x 10-10 8.5(1)

For column position W2-B we have : A\s = 452.39 mm2; fiy = 431 MPa;

ut = 716 m m ; A w s = 12.44 m m2 ; fwy = 449 MPa; s = 90 m m ; di/d = 3.2; and co0 =

446 N/mm. Substituting these values into Eq. 8.5(1), we have

8 = 23 8.5(2)

' See the definitions of Section 3.4

197

Note that the spandrel beam of Model W 2 had i) the minimum practical

reinforcement as specified in Section 3.4, and ii) an effective depth ratio, di/d,

equal to 3.2, where di and d are respectively the effective depth of the spandrel and

the slab. Thus it may be concluded that for spandrels with minimum practical

reinforcement, a ratio of di/d > 3.2 would provide near full bending restraint for the

connecting slab, and they can be considered as deep spandrel beams.

In general, according to Eq. 8.5(2), the spandrel beams with a strength

parameter of 23 or higher can also be considered as a deep spandrel beam. As

noted above, the specified spandrel strength parameter of 23 for deep spandrels is

based on the test results obtained from the model structures with minimum practical

longitudinal and transverse reinforcement (see Section 3.4). To examine the

applicability of the specified parameter, 23, for other longitudinal and transverse

steel ratios, more model structures have to be tested.

It is worth mentioning that for the present model structures with minimum

spandrel steel ratios, due to the restraining effects of the slab, their ultimate

strengths are much higher than those of isolated beams. Thus it may not be

advisable to provide reinforcement more than the minimum specified amount in the

spandrels.

8.6 Summary

Expressing the slab restraint on the rotation of the spandrel beam by the

parameter C0o and on its elongation by the factor y it has been established that:

198

(i) the spandrel transverse strength, co is increased to co + co0, where co0 = 446

N/mm,

(ii) the slab restraining factor,y, decreases as the strength of the spandrel beam, 8,

increases;

(iii) for the case of comer columns, an increase in column widths, increases the

slab restraining factor,y, and consequently the punching shear strength of the slab-

column connections.

Note that the present study does not cover the effects of the overall depth

of the slab on the magnitude of co0, neither are the effects of the width of the edge

columns on the magnitude of y.

In Section 8.3.2 two semi-empirical formulas are developed for the

determination of the y values, i.e. Eqs. 8.3(10) and 8.3(13) respectively for the

edge- and corner-connections. The adequacy of the developed semi-empirical

formulas is discussed in Section 8.3.3. It was found that the mathematical basis for

the interaction equation (Eq. 3.4(15)) is reliable.

The interaction equation (Eq. 3.4(15)) may also be applied to the case of

slab-column connections with torsion strips without closed ties, provided that the

recommendations of Section 8.4 are followed. Of course, more test results are

required to justify the applicability of the proposed interaction equation for this type

of structures.

Finally it is established that in the spandrel beams with the minimum

practical reinforcements (as specified in Section 3.4), a ratio of di/d > 3.2 would

provide near full bending restraint for the connecting slab. This ratio is

199

corresponding to a spandrel strength parameter of 23. Thus the interaction equation

for spandrels (Eq. 3.4(15)) is applicable to flat-plate column connections with

spandrel beams with a strength parameter less than or equal to 23.

200

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Fig. 8.3(1) Relationship between y and the corresponding spandrel strength-corner locations

202

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Fig. 8.4(1) Typical details at discontinous edges of flat plates with torsion strips without closed ties(ACI-ASCE Committee 352 report,1988)

Table 8.3(1) Measured

Model

No.

Wl-A

W2-A

W3-A

W4-A

W5-A

W2-C

W3-C

W4-C

W5-C

Wl-B

W2-B

W3-B

W5-B

R-3A

R-4A

Ytest

kN

1.79

0.95

3.08

1.28

5.15

0.92

4.04

1.34

5.97

1.38

1.00

2.23

3.43

2.20

2.55

Mean

Standard Deviation

predicted values of \\f

Ypred

kN

1.79

0.95

3.00

1.32

5.19

0.95

3.92

1.35

6.04

1.49

1.00

2.16

3.40

2.21

2.62

Vtest

Vpred

1.00

1.00

1.03

0.97

0.99

0.97

1.03

0.99

0.99

0.93

1.00

1.03

1.01

1.00

0.97

0.99

0.03

CHAPTER 9

PREDICTION PROCEDURE FOR PUNCHING SHEAR STRENGTH Vu

206

CHAPTER 9

PREDICTION PROCEDURE FOR

PUNCHING SHEAR STRENGTH Vu

9.1 General Remarks

To the author's knowledge apart from the present work there are no test data

available for the failure of the corner connections for flat plates with spandrel

beams. Neither are there for the edge connections of the flat plates with spandrels

except that for Models R-3A and R-4A reported by Rangan and Hall (1983). In

these two models the overall depth of the spandrel was similar to that of the slab.

As a result of the limited amount of relevant test data, the available

procedures for the prediction of the punching shear strength, Vu, are rare for flat

plate models with spandrel beams. In 1987 Rangan published a prediction

procedure that was subsequently incorporated in the new Australian Standard A S

3600-1988.

Herein, the proposed procedure for the prediction of Vu is presented in

Section 9.2. This procedure is available for predicting the punching shear strength

of slab-column connections at the edge- and comer-positions. The formulas may

be applied to slab-column connections with shallow spandrel beams, with torsion

strips, or with torsion strips without closed ties. It is worth mentioning that the

formulas also cover fully the following important parameters and effects:

(i) the strength and location of the flexural reinforcement of the slab;

207

(ii) the restraining effects of the slab on the rotation and elongation of the spandrel

beams;

(iii) the concrete strength;

(iv) the overall geometry of the connection;

(v) the membrane effects on the distribution of the total shear force along some

critical perimeter.

In Section 9.3 the AS 3600-1988 approach based on Rangan's formulas is

examined and finally the accuracy of the two procedures are compared in Section

9.4.

9.2 The Proposed Procedure

9.2.1 Background

Based on the test results of five half-scale reinforced concrete flat plate

models an analytical procedure is developed for the prediction of the punching shear

strength, Vu. In this procedure semi-empirical formulas are established for the

determination of M i and Vi, respectively the bending moment and shear force

along the front segment of the critical perimeter. These equations are then used for

the calculation of all the forces and moments along the critical perimeter by means

of statics.

In the light of the experimental results it has been found that in slab-

column connections with shallow spandrels and/or torsion strips, failure was

initiated by the formation of a negative yield line across the front face of the

208

column. Further increase in load leads to the punching shear failure of the

connection under the combined effects of torsion, shear and bending. Based on the

test results of the punching shear strength values for a total of 15 column positions,

the restraining effects of the slab on the strength of the spandrel beams and torsion

strips have been analysed.

The effects of the slab restraint on the rotation of the spandrel may be

expressed in terms of the parameter co0 and that on the elongation of the spandrel by

the factor \|/. The parameter co0 and the factor \\f are then incorporated in an

interaction equation used for the prediction of the punching shear strength, Vu. The

explicit procedure given in this Chapter provides acceptable estimates of the

connection strength. It requires a reasonable amount of computational effort.

9.2.2 Assumptions and applicability of the procedure

The present procedure may be used for predicting the punching shear

strength of slab-column connections of flat plates at the edge- and comer-positions.

The assumptions used in this procedure may be classified into two groups:

(i) those taken in the analytical model which led to the determination of the semi-

empirical formulas for M i and Vi (see Chapter 2), and

(ii) those used for the derivation of the interaction equation for spandrel beams (Eq.

3.4(15)), which are discussed in Chapter 3.

The procedure accounts for the simultaneous effects of axial force, shear,

bending and torsion in the slab-column connection resulted from an externally

applied uniformly distributed vertical load on the slab.

209

Effects of creep, shrinkage, temperature, and foundation movement are

beyond the scope of the present work. Also the determination of the design forces,

i.e. the worst combination of action effects at the connection is not considered

herein.

The proposed procedure is applicable mainly to monolithic slab-column

connections of the flat plates with spandrel beams. However, to investigate the

effects of the closed ties on the magnitude of Vu, in Section 9.2.6 the applicability

of the proposed procedure is also discussed for the connections without closed ties.

Note that the present procedure can not be used for slab-column

connections with drop panels, prestressed reinforcement, shear head, or column

capitals. The concrete to be used is assumed to be of normal weight and with a

design compressive strength not greater than 40 MPa.

9.2.3 Corner connections

As discussed in Section 6.2 determination of Vu involves the simultaneous

solution of (i) the three equilibrium equations for the comer connections (Eqs.

2.4(1) to 2.4(3)), and (ii) the interaction equation for the spandrels, Eq. 3.4(15),

with the undetermined forces and moments of Vu, T2, V2 and M2.

Solving these equations for Vu gives

\/ k3k4ki - k4k2 + k3(Vi,corner) Qn/u V u " k3 -k4k5

y'ZUJ

where

210

kl . M O W p B - f fVcorner - k - f ^ 9.2(2)

k2 = k 6 M C i -dl

2+bl V , „ - Ml>con,er 9.2(3)

k3 = 9.2(4)

k4 = ( ^ ) " 2 9.2(5)

k5 - ^ l 9.2(6)

and

k6 = 1 - D j £ ° s 9.2(7)

Note that in Eqs. 9.2(1) to 9.2(7), M C i and Mc2 are respectively the total

unbalanced moments transferred to the column centre in the main and transverse

moment directions. These moments may be estimated using known structural

analysis procedures. The strength parameters V1>comer and M1 ) C o m e r as well as the

slab restraining factor \|/Corner may be obtained through the semi-empirical formulas

developed in Chapters 7 and 8 respectively.

Eqs. 7.3(3) and 7.4(9) are respectively the developed semi-empirical

formulas for the calculation of Mi,comer and Vi,corner- These equations may be

written as

Miômer = Miy 9.2(8)

.ind

211

Vl,corner = 0.24^6.90 + V^'corner + M m ) 9 2 ( 9 )

where Mi,conier and Miy are respectively the predicted and yield moments of the

slab along the front segment of the critical perimeter of the comer columns. In Eq.

9.2(9) the magnitude of Mm is proportional to Mmy (see Section 7.3.2), where Mmy

is the yield moment of the critical slab strip at midspan (see Fig. 7.2(1)). Note that

according to the discussion in Section 7.3.2, Mm for the slabs with realistic

spandrel beams (di/d > 1) may be expressed as

Mm=Mmy 9.2(10)

Also note that in Eq. 9.2(9), Vi,corner is the predicted shear force along the front

segment of the critical perimeter.

The semi-empirical equation for the slab restraining factor (Eq. 8.3(13))

may be rewritten as

where

\|/ = 5.21-3.14 log(8) 9.2(12)

and

21=1.71- 1.271og(6) 9.2(13)

212

Eqs. 9.2(11) to 9.2(13) show that the slab restraining factor \\rcomeT is a

function of the column width C2 (in mm), the column width factor X, and the

spandrel strength parameter (8). The definitions for a and P respectively the

longitudinal and transverse steel factors, as well as that for 5 have all been

presented in Section 3.4.

9.2.4 Edge connections

The prediction of Vu for the edge connections again involves the

simultaneous solution of (i) the three equilibrium equations for the edge connections

(Eqs. 2.4(4) to 2.4(6)), and (ii) the interaction equation for the spandrels, Eq.

3.4(15), in which the undetermined forces and moments are Vu, T2, V2, and M2.

Solving these equations gives.

V 2kik3k4-k4k2 + k3(V1,edge) oo/i/n V u~ k3-k4k5 9.2(14)

where

k. = 200,000f-^-r2rVedge-k-^C2) 9.2(15)

VYedgey v ^ J

k2 = k 6 M C i -—2~IVi,edge - M1 ? e d g e 9.2(16)

and k3 to k6 are respectively defined by Eqs. 9.2(4) to 9.2(7).

The semi-empirical formulas, Eqs. 7.3(2) and 7.4(8) may be used for the

calculation of Mi,edge and Vi,edge respectively. These equations may be rewritten

as

213

Mi,edge = 0.83 M i y 9.2(17)

and

V,^ - 0.75^2.04 + ^(Mired,.^) 9 2(i8)

where Mi,edge and Vi^dge .are respectively the bending moment and the shear force

along the front segment of the critical perimeter of the edge columns. As discussed

in Section 7.3.1 the magnitudes of M m for slabs with realistic spandrel beams

(di/d > 1) may also be defined by Eq. 9.2(10).

The semi-empirical formula for the slab restraining factor, Eq. 8.3(10),

may be rewritten as

Vedge = 3.24 - 1.64 log(8) 9.2(19)

Note that the effects of column width on the slab restraining factor for the edge

connections is beyond the scope of the present study.

9.2.5 Connections with torsion strips

The edge- and comer-column connections of flat plates with torsion strips,

may be defined as the connections for which the effective depth ratio (di/d) is less

than unity where di and d are respectively the effective depths of the torsion strip

and the slab. The determination of V u for connections with realistic spandrel beams

(for which di/d > 1) has already been discussed in Section 9.2.3 and 9.2.4.

214

Similar procedures may also be used for connections with torsion strips, except for

the calculation of M m which is described below.

In Section 7.3.3 it was concluded that for slabs with torsion strips, Mm is

equal to 7 0 % of the slab yield moment. This may be expressed as follows

Mm = 0.70 Mmy 9.2(20)

where Mm and Mmy are respectively the predicted and yield moments at midspan of

the critical strips (see Fig. 7.2(1)).

9.2.6 Connections with torsion strips without closed ties

The determination of Vu for the slab- column connections with torsion

strips has been discussed in Section 9.2.5. A similar procedure may also be used

for the connections with torsion strips without closed ties, provided the

recommendations in Section 8.4 are followed.

9.2.7 Connections with deep spandrel beams

Flat plate column connections with deep spandrel beams have been defined

in Section 8.5. For these connections the flexural strength is limited by the

development of a flexural yield line at the face of the spandrel beam, in which case

the spandrel beams do not reach their design strengths. In other words no yield

surface around the connection will develop. As a result,the so called punching shear

does not occur.

215

9.2.8 Accuracy

As discussed in Section 9.2.1, the prediction of the punching shear

strength V u involves the prediction of Mi, Vi and y first. The semi-empirical

formulas for the determination of M i and Vi and the slab restraining factor \|/ are

summarized in Sections 9.2.3 to 9.2.6. In addition to M i , Vi .and \\f, the

magnitudes of Mci and M C 2 respectively the total unbalanced moments transferred

to the column centre in the main and transverse moment directions should also be

substituted into the prediction formulas for Vu. These moments may be estimated

using known structural analysis procedures. However as discussed in Appendix I,

in these procedures the membrane effects are not incorporated. Therefore instead of

the estimated values, the measured magnitudes of Mci and M c 2 are substituted in to

the prediction formulas for Vu.

The proposed procedure for the prediction of Vu (Section 9.2.2 to 9.2.5) is

applied to the following test results:

(i) Series "A ", comer columns, i.e. column positions Wl-A to W5-A,

(ii) Series "C ", corner columns, i.e. column positions W2-C to W5-C,

(iii) Series "B", edge columns, i.e. column positions Wl-B to W3-B, and W5-B,

and

(iv) edge columns R-3A and R-4A tested by Rangan and Hall (1983).

The flow chart of the computer program used for the calculation of Vu is

shown in Fig. 9.2(1). The measured values of V u are then used as a basis for

comparing the predicted and test values. The ratios of the measured to predicted

216

values of V u for all the column positions are computed; they are also presented in

Table 9.4(1) in the column titled VU)test / Vu,predicted. It may be seen that the

correlations of the predicted and test values are good.

9.3 The AS 3600-1988 Procedure

9.3.1 Background

The new Australian Standard AS3600-1988 provides a prediction

procedure for the calculation of the punching shear strength of flat plate column

connections with spandrel beams. The procedure detailed in Clause 9.2 of the

Standard is based on Rangan's formulas (1987).

Fig. 9.3(1) illustrates the situation in the vicinity of an edge column.

Rangan (1987) assumes that the critical shear perimeter for failure is at a distance

d/2 away from the face of the column. In this figure M * and V * are the bending

moment and the shear force transferred to the column centre. The shear force is

transferred partly by Vi at the front face and the remainder by V2 at each side face.

The moment transfer occurs partly as the yield moment, M i , of the slab

reinforcement along the front segment of the critical shear perimeter, some due to

the eccentricity of the shear force Vi and the remainder as torsional moment T2 at

each side face.

Rangan (1987) in his paper declared that in the absence of reliable

information, it appears simple to assume that the shear force transferred at each face

is equal to (b0u)Vu, where b0 is the width of the face and u is the critical shear

perimeter. With this assumption and a known value of Mi, he calculated the forces

at each face of the critical section by means of statics. H e also assumed that a

punching shear failure is initiated either by the failure of the slab strips at the side

217

face in combined torsion and shear, or by the failure of the slab strip at the front

face (and the back face, if any) in shear.

It is important to note that Rangan, on the basis of his test results for the

punching shear failure at the edge column positions R3-A and R-4A, assumed that

the restraining effects of the slab enhances the torsional and shear capacities of the

spandrels by a constant factor of 4. On this basis he developed the failure criterion

for the spandrel beams.

9.3.2. Formulas

The relevant prediction formulas for the punching shear strength (Vu), as

recommended by A S 3600-1988 may be summarized as follows

(i) for connections without closed ties,

V " = V"°uM* « ( D 1.0 + 8V*ad

(ii) for connections with torsion strips with minimum quantity of closed ties, V u

shall be taken as

Vu,min - ujyj* 9.3(2) 1.0 + 2V*a2

(iii) for connections with spandrels with minimum quantity of closed ties, V u shall

be taken as

5 l U 0 D S

l ^ V n o ^ 1

* u,min— „\/\ * 9.3(3)

i-u + 2 V * a b w

218

(iv) for connections with spandrels (or torsion strips), containing more than the

minimum quantity of closed ties,

VS V u = V u , m i n ^ ( 7 T ^ - l 9.3(4)

where Vu,min is calculated using Eq. 9.3(2) or 9.3(3), as appropriate.

In no case shall V u be taken greater than

»u»max — * 'u>min'M v y.5(j)

where x and y are the shorter and longer dimensions respectively for the cross-

section of the torsion strip or spandrel beam. Note that the minimum area of closed

ties shall satisfy the following inequality :

Aws>0^yj_ 93(6)

In Eqs. 9.3(1) to 9.3(3), the ultimate shear strength of a slab with no moment

transfer is defined as

Vuo = ud(fcv) 9.3(7)

where

f„ = 0.11(1 + fyfil* 0 - 3 4 ^ 9-3(8)

Pi, 4 9-3(9)

219

a = ci + 2 9.3(10)

and

yi=y + DL + DT 9.3(11)

In the above equations Aws is the cross-sectional area of the bar from which the ties

are made; bi is the width of the spandrel; Q and C2 are side dimensions of column;

d is the effective depth of slab averaged around the critical shear perimeter, Di is the

overall depth of the spandrel; DL and DT respectively are the diameters of the

longitudinal and transverse reinforcement of the spandrel; Ds is the overall depth of

the slab; fc' is the compressive strength of concrete; fwy is the yield strength of the

transverse reinforcement; M* is the unbalanced moment; V* is the shear force

transferred to the column centre; finally Y is the longest overall dimensions of the

effective loaded area, and X is the overall dimension measured perpendicularly to

Y.

9.3.3 Accuracy

The accuracy of the AS 3600-1988 procedure is examined in the light of

the published data (Rangan and Hall, 1983) and those obtained from the present

series of model tests. Again the prediction of the punching shear strength, Vu,

requires the substitution of the estimated values of M* and V* into the prediction

formulas for Vu. The values of M* and V*may also be estimated using known

structural analysis procedures. However as discussed in Appendix I, in these

procedures the membrane effects are not incorporated. Therefore instead of the

estimated values, the measure values of M* and V* are substituted into the

prediction formulas for Vu.

220

For the prediction of Vu, based on the AS3600-1988 approach, another

computer program has been developed by the author. The flow chart of the

program used for the calculation of V u is given in Fig. 9.3(1). The measured values

of V u are then used as a basis for comparing the predicted and test values. Again

the ratios of the measured to predicted values of V u for all the column positions are

computed; they are also presented in Table 9.4(1) in the column titled

Vu,test/ Vu,Predicted. It may be seen that this procedure overestimates the punching

shear strength of flat plates with spandrel beams.

9.4 Comparison of the Procedures

For each of the two prediction procedures the ratios of the measured to

predicted values of V u for all the column positions are presented in Table 9.4(1) in

the respective columns titled VU)test/ VU)Predicted. It may be seen that the Australian

Standard prediction procedure overestimates the punching shear strength of flat

plates with spandrel beams. In particular, for the case of comer columns, the

predicted values can be as high as 2.6 times the model test strength (see column

position W 2 - C in Table 9.4(1)). O n the other hand, the prediction procedure

proposed by the author is accurate and consistent with a mean test/prediction ratio

of 0.99.

The improved accuracy by the present approach may be attributed to the

fact that it can fully cover the following important parameters and effects that were

left out by AS3600-1988. In the proposed approach it was established that

(i) the slab restraining factor \\f is a function of the strength of the spandrel beams

(whereas the Australian Standard method assumes \|/ = 4);

221

(ii) the distribution of shear force around the critical perimeter is a function of the

slab steel ratio, clear span of the slab and the size of the spandrel beams in relation

to the slab; whereas AS3600-1988 assumes a uniform distribution of average shear

stress along some critical perimeter;

(iii) the interaction equation for the spandrel beam includes the combined effects of

torsion, shear and moment (whereas the Standard ignores the interaction of bending

in the edge- and comer-columns);

(iv) flat plates with deep spandrels have a mode of failure very much different from

those with shallow beams and they are carefully defined in the proposed procedure

(whereas the Australian Standard method does not specify deep spandrels).

9.5 Summary

A simple procedure is presented for the prediction of the punching

shear strength of the slab-column connections of flat plates at the edge- and comer-

locations. The formulas may be applied to flat plates with shallow spandrel beams,

or with torsion strips without closed ties.

Apart from the present work the only available published prediction

procedure for Vu, covering the case of the flat plates with spandrel beams, is the

one developed by Rangan (1987). This procedure has been incorporated in the new

Australian Standard AS 3600-1988.

The accuracy of the AS 3600-1988 procedure is examined in the light of

the published data (Rangan and Hall, 1983) and those obtained from the present

series of model tests. This indicates that the present prediction procedure is more

accurate than that recommended in AS3600-1988. The latter procedure also

222

considerably overestimating the punching shear strength of the slab-column

connections with realistic spandrel beams.

R a w data -f

Predicted values of M j and

IV^ ( from Table 7.3(1) )

r a

P k3to

Eqs.

Calculation of

(Eq.3.4(6))

(Eq.3.4(9))

k respectively from

9.2(4) to 9.2(7)

YES

Calculation of

V1)Cor„er (EqA2(9))

Vcorner (Eq.9.2(ll))

k (Eq.9.2(2))

k (Eq.9.2(3)) 2

Calculation of

l.edge

V, edge

(Eq.9.2(18))

(Eq.9.2(19))

(Eq.9.2(15))

(Eq.9.2(16))

V =? u

(Eq.9.2(l))

(End^

V =? u

(Eq.9.2(14))

<S>

Fig. 9.2(1) Flowchart of the proposed procedure for the prediction of Vu

224

side face of critical section

/5£v front face of critical section

R +V

Fig. 9.3(1) Transfer of forces between slab and edge columns (Rangan, 1987)

Yes

u =q+2Cx + 2d

I

start

C l' C2' d> fc'> Aw s >

fwy,s, D l , D s , M * , V * ,

bx , x, y, D L , D T

Calculation of

f c v (Eq. 9.3(8))

a (Eq. 9.3(10))

^l (Eq. 9.3(11))

Calculation of V, u,min (Eq. 9.3(3))

K=0.2 y,/f '1 W!

No

1 u =C 1 + C 2 + d

T Calculation of \0

(Eq. 9.3(7))

No

Calculation of V„

Eq. 9.3(1)

Calculation of Vurnin

(Eq. 9.3(2))

(jjin-dT)

Cont....

V = ? (Eq. 9.3(4))

V =V u u.min

R=Dj / bx

If R>1 then R=l/R

I

R=Ds/a

If R>1 then R=l/R

T ]

Calculation of VU)max 0.5

V =3V . (R) u.max u.min v '

1 It vu > v u m a x

then Vu = V ax

I Calculation of

V„

End

Fig. 9.3(2) Flowchart of the AS3600-1988 procedure for the prediction of V

226

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CHAPTER 10

CONCLUSIONS

227

CHAPTER 10

CONCLUSIONS

As a part of a long-term study on the strength behaviour of reinforced concrete

flat plates with spandrel beams a series of five half-scale models has been tested.

With the aid of the experimental results, a prediction procedure is developed for the

punching shear strength at the corner- and edge-column positions. Based on the

present study, conclusions can be drawn in the following five areas:

(i) failure mechanisms,

(ii) moment and shear transfer between the slab and the columns,

(iii) derivation of an interaction equation for spandrel beams,

(iv) development of a prediction procedure for the determination of the punching

shear strength, Vu, and

(v) the accuracy of the proposed prediction procedure.

They are given in Sections 10.1, 10.2, 10.3, 10.4 and 10.5 respectively

with recommendations for further study enumerated in Section 10.6.

10.1 Failure Mechanisms

At the exterior slab-column connections of flat plates with spandrel beams

or torsion strips, depending on the strength of the spandrel or torsion strip at the

side faces of the column, failure could occur in one of two modes:

228

Mode 1- Development of a negative yield line across the front faces of the edge and

corner columns followed by the formation of inclined spiralling cracks in the

spandrel beam.

Mode 2- Flexural failure due to the formation of a negative yield line along the

spandrel-slab connection.

Note that in spandrel-slab connections, when the spandrel strength

parameter (8) is equal to 23 or more, failure would follow Mode 2. The spandrels

of these connections are called deep spandrel beams. For slabs with shallow

spandrel beams or torsion strips failure would be similar to Mode 1.

10.2 Moment and Shear Transfer Between Slab and Columns

To develop a prediction procedure for the punching shear strength, Vu, it

was necessary to determine the distribution of the total bending moment and the

total shear force transferred to the column center. Based on pilot studies carried

out by the author a procedure was formulated for the measurement of the individual

forces acting on a slab-column connections. The experimental data are then used

for setting up semi-empirical formulas for the prediction of M i and Vi. After

determining these strength parameters, all the forces and moments along the critical

perimeter of the column may be calculated by means of statics (see Section 6.3).

10.3 Proposed Interaction Equation for Spandrel Beams

To complete the prediction procedure for the punching shear strength, Vu,

an interaction equation for spandrel beams has been developed. In this equation the

restraining effects of the slab on the rotation and elongation of the spandrels

229

respectively are incorporated by means of co0 and y. The parameter co0 and the

factor \|/ have been calibrated experimentally.

It is worth mentioning that in the proposed procedure for the case of comer

columns, the effects of the column width on the \|/ values are expressed in terms of

the column width factor, X.

10.4 Prediction of Vu

A simple procedure is presented for the prediction of the punching shear

strength of slab-column connections of flat plates at the edge- and corner-locations.

The procedure involves

(i) determination of Mi and Vi with the aid of the proposed semi-empirical

formulas( Eqs. 9.2(8) and 9.2(9) for the comer column positions, and Eqs. 9.2(17)

and 9.2(18) for the edge column positions); and

(ii) determination of the slab restraining factor, V, as a function of the spandrel

strength parameter, 8, using the proposed semi-empirical formulas(Eq. 9.2(11) for

the corner column positions, and Eq. 9.2(19) for the edge column positions).

Then using (i) and (ii) together with the derived equilibrium equations, the value for

V u can readily be computed.

A short computer program has been developed for the calculation of Vu.

This program contains simple and explicit equations which are suitable for adoption

in design codes.

230

10.5 Versatility and Accuracy of the Proposed Procedure

The proposed procedure covers the effects of the following significant

variables which influence the punching shear strength of slab-column connections

of flat plates:

(i) the overall geometry of the connection,

(ii) the concrete strength,

(iii) the strength and location of the flexural reinforcement of the slab,

(iv) the restraining effects of slab on the rotation and elongation of spandrel beams;

(v) the enhanced strength of the slab-column connections due to membrane effects.

In the light of the experimental results reported herein a comparative study

is carried out. This indicates that the present prediction procedure is more accurate

than that recommended in AS3600-1988. The later procedure also suffers, at

times, the serious drawback of considerably overestimating the punching shear

strength.

10.6 Recommendations for Further Study

Further studies of the behaviour of slab-column connections of flat plates

should aim at determining the effects of the following parameters on the punching

shear strength:

(i) the width and the steel ratio of the spandrel beam;

231

(ii) the width of the edge columns;

(iii) the depth of the slab;

(iv) the loading patterns (including concentrated load, line load and uniformly

distributed load).

Part of the above recommendations have already been examined with the

aid of the test results of models M2, M3 and M4, constructed during 1989 with

grants provided by the Australian Research Council to Professor Y. C. Loo. It is

expected that another 3 reinforced concrete flat plate models will be constructed

during 1990. Also as a result of the progress made to date, a third year of research

is planned for 1991 to concentrate work on prestressed post-tensioned flat plates.

REFERENCES

232

REFERENCES

ACI-ASCE Committee 426, (1974), The Shear Strength of Reinforced

Concrete Member-Slabs. Proc, ASCE, Vol. 100, ST8, pp. 1543-

1591.

ACI Committee 318, (1983), Building Code Requirements for Reinforced

Concrete. ACI 318-83, American Concrete Institute, Detroit, Mich,

111 pp.

ACI-ASCE Committee 352, (1988), Recommendations for Design of

Slab-Column Connections in Monolithic Reinforced Concrete

Structures, (ACI 352. IR-88), ACI Structural Journal, V. 85, No. 6,

Nov-Dec, pp. 675-696.

Alexander, S. D. B. and Simmonds, S. H., (1987), Ultimate Strength of

Slab-Column Connections. ACI Structural Journal, May-June, PP.

255-261.

Bazant, Z. P. and Cao, Z., (1987), Size Effect in Punching Shear Failure

of Slabs. ACI Structural Journal , V. 84, No. 1, Jan.-Feb., pp. 44-

51.

British Standard Institution, (1972), The Structural Use of Concrete,

CP110: 1972, Part 1, Design, Materials and Workmanship.

Chen, C, (1986), Shear Strength of Reinforced Concrete Flat Plate. Jian

Zhu Jie Gou Xue Bao / Journal of Building Structures, Vol. 7, No. 1,

pp. 49-57.

233

Code for the Design of Concrete Structures for Buildings, (1984), (CSA

A23.3-M84), Canadian Standards Association, Rexdale, 281 pp.

Comite Europeen Du Beton, (1978), CEB-FIB Model Code for Concrete

Structures, Bulletin d'information, N125E, Paris.

Elfgern, L., (1972), Reinforced Concrete Beams Loaded in Combined

Torsion, Bending and Shear. Publication Tl:3, Division of Concrete

Structures, Chalmers University of Technology, Goteborg, Sweden.

Elfgern, L., Karlsson, I. and Lasberg, A., (1974), Torsion-Bending-

Shear Interaction for Concrete Beams. Journal of the Structural

Division, ASCE, vol. 100, ST8, pp. 1657-1676.

Falamaki, M. and Loo, Y.C., (1988), Ultimate load Test of a Half-Scale

Reinforced Concrete Flat Plate with Spandrel Beams.

Proceedings,l 1th Australasian Conference on Mechanics of Stractures

and Materials, University of Auckland, N.Z., August ,pp. 282-288 .

Falamaki, M. and Loo, Y.C., (1989), Failure Mechanisms of Reinforced

Concrete Flat Plates with Spandrel Beams. Second East Asia-Pacific

Conference in Stractural Engineering and Construction: Achievements,

Trends and Challenges, Proc. EASEC-2, Chiangmai , Thailand,

January, Vol. l,pp. 249-254.

Falamaki, M. and Loo, Y.C., (1990), Strength Tests of Half-Scale

Reinforced Concrete Flat Plate Models with Spandrel Beams. Invited

paper International Conference on Structural Engineering and

Computations, Beijing, 25-28 April, 17 pp.

Gilbert, S.G., Glass, C, (1987), Punching Failure of Reinforced

Concrete Flat Slabs at Edge Columns. The Structural Engineer,

Volume 65B, No. 1, March, pp. 16-21.

Gonzalez-Vidosa, F.; Kotsova, M.D. and Pavlovic, M.N., (1988),

Symmetrical Punching of Reinforced Concrete Slabs: an Analytical

Investigation Based on Nonlinear Finite Element Modeling, ACI

Structural Journal (American Concrete Institute) V. 85, N. 3, May-

June, pp. 241-250.

Hatcher, D. S., Sozen, M. A. and Siess, C. P., (1961), A Study of Tests

on a Flat Plate and a Flat Slab. Structural Research Series No. 217,

Civil Engineering Studies, University of Illinois, Urbana, Illinois,

143 PP.

Hawkins, N. M., (1974), Shear in Reinforced Concrete. SP-42, ACI,

Detroit, Vol. 2, pp 817-846.

Hsu, T.T.C., (1983), Torsion of Reinforced Concrete. Van Nostrand

Reinhold, 516 pp.

Jiang, Da-Hua; Shen, Jing-Hua, (1986), Strength of Concrete Slabs in

Punching Shear. Journal of Structural Engineering VI12 n 12 Dec,

pp. 2578-2591.

235

Lampert, P. and Thurlimann, B.,(1969), Torsion sversuche an stahlbeton

balken (Torsion-Bending Test on Reinforced Concrete Beams), Bericht

Nv.6506-2, Institute for Baustatik, Eldgen Ossische Technische

Hochschule, Zurich, Switzerland, 116 pp.

Latip, S., (1988), Deflection of Reinforced Concrete Flat Plate Structures.

B. Eng. Thesis, The University of Wollongong, N S W , Australia .

Long, A.E., (1975), A Two-Phase Approach to the Prediction of the

Punching Strength of Slabs. J. Am. Concr. Inst., 72, Feb., 2, pp. 37-

47.

Moe, J., (1961), Shearing Strength of Reinforced Concrete Slabs and

Footings under Concentrated Loads. Development Department

Bulletin No. D47, Portland Cement Association, Skokie, Illinois,

April, 130 pp.

Moehle, J. P., (1988), Strength of Slab-Column Edge Connections. ACI

Journal, V.85, No. 1, Jan.-Feb., pp. 89-98.

Moehle, J. P., Kreger, M.E., Leon, R., (1988), Background to

Recommendations for Design of Reinforced Concrete Slab-Column

Connections, ACI Stractural Journal, Nov.-Dec, pp. 636-644.

Nielsen, M.P., Andreasen, B.S. and Chen G., (1988), Dome Effect in

Reinforced Concrete Slabs. Proceedings of the 11th Australian

Conference on Mechanic of Structures and Materials, University of

Auckland, N e w Zealand, PP. 1-9.

236

Onsongo,W.M. and Collins, M.P., (1972), Longitudinally Restrained

Beams in Torsion. Pubn. No. 72-07, Dept. of Civil Engineering,

University of Toronto, 35 pp.

Rangan, B. V. and Hall, A. S., (1983), Forces in the Vicinity of Edge

Columns in Flat Plate Floors. Uni-Civ Report No. R-203, University

of N.S.W. Kensington, Vol. 1, 240 PP. and Vol. 2, 149 PP.

Rangan, B. V., (1987), Punching Shear Strength of Reinforced Concrete

Slabs. Transactions of the Institution of Engineers, Australia, Civil

Eng., Vol. CE29, No. 2, PP. 71-78.

Rankin, G.I.B., Long, A.E., (1987), Predicting the Punching Strength of

Conventional Slab-Column Specimens. Proc. Instn Civ Engrs, Part

1 82, April, pp. 327-346.

Rausch, E., (1929), Berechnung des Eiseenbetons ge gen Berdrehung

(Analysis of Torsion and Shear in Reinforced Concrete). Technische

Hochschule, Berlin, 53 pp., (in German).

Regan, P. E., (1981), Behaviour of Reinforced Concrete Flat Slabs.

CIRIA Report 89, Constructional Industry Research and Information

Association, London, 90 PP.

Solanki, H., Sabnis, G.M., (1987), Punching Shear Strength of Curved

Slabs. Indian Concrete Journal, V. 61, No. 7, July, pp. 191-193.

Standard Association of Australia, (1988), AS3600-1988, Concrete

Structures, Sydney, NSW.

237

Standard Association of Australia, (1982), AS1302-1982, Steel

Reinforcing Bars for Concrete, Sydney, N S W .

Standard Association of Australia, (1984), AS 1303-1984, Hard-drawn

Steel Reinforcing Wire for Concrete, Sydney, N S W .

Standard Association of Australia, (1984), AS1304-1984, Hard-drawn

Steel Wire Reinforcing Fabric for Concrete, Sydney, N S W .

Structural Use of Concrete : Part 1, (1985), Code of Practice for Design

and Construction, (BS 8110 : Part 1 : 1985), British Standards

Institution, London, 126 pp.

APPENDIX I

DESIGN ULTIMATE LOAD AND MEMBERANE EFFECTS

239

APPENDIX I

DESIGN ULTIMATE LOAD AND

MEMBERANE EFFECTS

Test Results

The design and the corresponding measured ultimate loads of the half-scale

flat plate Models W l to W 5 , together with the size and location of their spandrels

are presented in Table AI(1). It may be seen that the ratio of the measured to design

ultimate load, on average is equal to 2.93. The minimum ratio is 2.11, which refers

to Model W 5 , with no spandrel beams at its boundaries; the maximum ratio is equal

to 3.4, which corresponds to Model W l with spandrel beams on two sides.

This increase in strength may be atrributed to (i) the change in the

geometry of the structure at failure, and (ii) the size and location of the slab

reinforcement.

The change of geometry at ultimate sometimes has a considerable effect on

the load-carrying capacity. These effects are called membrane effects.

Compressive membrane effect or dome effects often predominate at small

deflections; tensile membrane effects predominate at larger deflections (Nielsen et

al. 1988).

As shown in Fig. AI(1), the spandrel beams and the columns restrain the

horizontal displacement of the slab. This restraining effects cause a portion of the

load to be carried by an arch or dome action, which is able to utilize the strength of

the materials much more efficiently than normal slab action. As a result the loading

240

carrying capacity of the slab as well as the resisting moments and shear forces of

the slab-column connections significantly increase.

According to Table AI(1) the ratio of the measured to the design ultimate

load of the flat plate models increases as the size or the stiffness of the slab

boundaries increases. Thus it may be concluded that by increasing the size of the

spandrel, the restraint to the horizontal displacement of the slab increases. This in

turn increases the load carrying capacity of the slab, and consequently the moment

and the shear capacities of the slab -column connections. Thus it is necessary to re

examine the normal design procedures on utilizing the load carrying capacities of

the materials involved in reinforced concrete flat plates.

The above discussion indicates that a portion of the applied load on the

slab is carried by an arch or dome action, and the remaining by the normal slab

action. Therefore, to determine the required steel area for the concrete section only

that portion of the load to be carried by the normal slab action, is to be considered.

Observations

In the analysis of flat plate floors for a design ultimate load P, known

structural analysis procedures m a y be used for the distribution of forces and

moments. Assuming M * and V* are respectively the unbalanced moment and the

shear force to be transferred to the column centre, the corresponding required steel

areas are then calculated for the ultimate limit state. The above discussion indicates

that due to the dome effects the ultimate loading carrying capacity of such a slab

may be expressed as

P u = Pd + P Al(l)

241

where Pd and P are respectively the load carrying capacities of the slab by the dome

action and by the normal slab action and Pu is the ultimate load carrying capacity of

the slab.

A similar discussion may also be applied to M* and V*, that is

Mu* = Md* + M* Al(2)

Vu* = Vd* + V* Al(3)

where Md* and Vd* are respectively the unbalanced moment and the shear force due

to the dome effects, and M u * and Vu* are respectively the ultimate unbalanced

moment and the shear force.

This discussion leads to the conclusion that by designing the slab-column

connection for M * and V*, due to the dome action the load carrying capacities of the

sections are respectively M u * and Vu*. Thus in the proposed procedure the

estimated values of M C i and MC21" (using known structural analysis procedures)

can not be used in the prediction formulas developed for the punching shear

strength Vu. Simply because the other components of these formulas have already

been affected by the dome action.

Note that the estimation of Md* and Vd* which are induced forces due to

the dome effect is beyond the scope of this thesis. Therefore in the prediction

formulas for Vu, instead of the estimated values of M C i and M c 2 , the

corresponding measured values are substituted.

t Mci and M C 2 are respectively the unbalanced moments at the column centre in the main and

transverse moment directions

242

I I 1 1 1 U 1 1 I 1

Fig. AI(1) D o m e effects in flat plates (Nielsen et al, 1988)

243

Vi

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APPENDIX II

INERACTION EQUATIONS AND PREDICTION FORMULAS FOR Vu

245

APPENDIX H

INTERACTION EQUATIONS AND

PREDICTION FORMULAS FOR Vu

As shown in (i) and (ii) below, the prediction formulas for Vu may be

expressed in terms of M x and Vi. Magnitudes of these moments and forces can be

obtained with the aid of the semi-emperical formulas developed in Chapter 7.

Therefore, if the errors of estimation for M i and Vi are respectively em and ey, the

prediction values for V u would also be proportional to em and ey. The relationship

between V u and the errors of estimation (em and ey) depends on the order of the

interaction formula used for the combined effects of torsion, shear and bending in

the spandrels.

The prediction formulas for Vu, derived on the basis of the second order

interaction equation, Eq. 3.3(6), are presented in part (i). According to these

formulas, the predicted values of V u are proportional to M i 2 and Vi2, and therefore

proportional to em2 and ev2. In other words V u is proportional to the square of the

errors of estimation of M i and Vi.

The prediction formulas for Vu, derived on the basis of the first order

interaction equation, Eq. 3.3(7), are presented in part (ii). Following a procedure

similar to that discussed for the second order interaction equation, it can be shown

that, V u is proportional to the first order of the errors of estimation of M i and Vi.

246

(T) Prediction of V3, on the hasis of Ea. 3.3(6)

Following a procedure similar to that of Section 3.4 for the derivation of

the interaction equation for spandrel beams in terms of co0 and \j/ gives

(^)2 + (a^)2=4ap(^^)xl0l° AII(1>

A simultaneous solution of Eq. AII(l) and the three equilibrium equations for the

edge column (Eqs. 2.4(4) to 2.4(6)), leads to the following prediction formula for

Vu

v -K7 + V K 72 - K 6 K 8 ATTf2.

V u " 4(K 22 K 3

2 + K42) {)

where

rDi -D,\ ,, di + bi. K1=Mcl-Mci[Êr&)-M1-^f^V1 AII(3)

AH(4)

AH(5)

AH(6)

K5 = 4 a p r ¥ - ^ x l 01 0 AHC7)

K6 = 4(K22K3

2 + K42) An(g)

K2

K 3 =

K4 =

ci - bi

~ 2

\2ut

_At

247

K 7 = 4(K1K2K32 + K42Vi) An(9)

K8 = 4Ki2K3

2 + 4K42Vi2 - 16K32K4

2K5 AII( 10)

According to Eqs. AII(2) to AII(IO) magnitude of Vu is proportional to Mx2 and

V-2

(ji) Prediction of V,. on the hasis of Ea. 3.3(7,

According to Section 3.4, the corresponding interaction equation for

spandrel beams, Eq. 3.4(15), may be rewritten as

T 2 V 2

Kj+K3" = K 9 AII(ll)

where k3 and k^ are respectively defined by Eqs. AII(5) and AII(6), and

K 9 = 2 0 0 0 0 0 ^ ] " 2 ( V - ^ A n ( 1 2 ,

Again a simultaneous solution of Eqs. A2(ll) and the three equilibrium equations

for the edge columns (Eqs. 2.4(4) to 2.4(6)), leads to the following prediction

formula for Vu:

V u _ / K 1 K 3 K 4 - K i K 3 + K4Vi K4-K2i3 * L AHC13)

248

where kl and k2 are respectively defined by Eqs. AII(3) and AH(4)

According to Eq. AE(13), for this case, magnitude of Vu is proportional to

M i and Vi.

Note that the prediction formulas developed in parts (i) and (ii) above, are

both for the case of edge columns. It can be easily shown that similar conclusions

can also be obtained for the case of comer columns.

APPENDIX III

STRAIN DATA OF FLAT PLATE MODELS AT

THE ULTIMATE STATE

250

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APPENDIX IV

THE CRITICAL PERIMETER

254

APPENDIX IV

THE CRITICAL PERIMETER

As depicted in Fig. AIH(l) to AEI(3) in slabs with both shallow and deep

spandrels the strain gradients near the columns are extremely steep. In other words

over a certain width of the slab the measured moments are almost equal to the yield

moments. Therefore determination of a proper slab width over which the measured

slab moments M i are close to the slab yield moments, would increase the accuracy

of the predicted values for Mi.

The present results (see Appendix IE) indicate that at the ultimate state, for

slabs with deep spandrels (Model W 2 ) the strain gradients are extremely steep all

along the spandrel. The strain gradients for slabs with shallow spandrels are also

steep in the vicinity of the columns; however magnitudes of the strains in the steel

bars which are located outside the column width C 2 are slightly less than the ones

within C2. Therefore the slab width over which (in both the shallow and deep

spandrels) the measured slab moments and the slab yield moments are very close,

can be assumed equal to C 2 (see Fig. AlV(l)a).

Thus according to the above discussion a better prediction of Mi would be

achieved if w e consider a width equal to C 2 for the front segment of the critical

perimeter. Regan (1981) also observed the same phenomena over the width C2, for

the case of flat plates without spandrels. However, AS3600-1988 suggests a larger

width for the front segment of the critical perimeter, which is not based on any

physical model with realistic spandrel beams.

The proposed critical perimeter and the critical shear perimeter suggested

by AS3600-1988 are compared in Fig. AIV(l).

255

In view of the above, the proposed critical perimeter is adopted as a basis

for the calculation of the slab strip moments.

256

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APPENDIX V

REINFORCEMENT DETAILS AT COLUMN POSITIONS

W2-A TO W5-A, W2-B TO W5-B

AND W2-C TO W5-C

258

W 6.3_T 8 B_2

F62-T&B ^ B ~ Y12-B

-Ve Reinf. .Side (F

2W 8 _ T_2

F62-T& B

+ Ve Reinf., Mid-span

•Ve Reinf..Side (Al

Fig. AV(1) Reinforcement details for column positions W2-A

and W2-F

259

W 6.3_T & EL3

T&B ^B~

-Ve Reinf. ,Side (F)

2W 8 - T- 3

F 6 2 -T&B

2 W 8 . L 3 W6.3_T & B_3

F62-T& B

+ Ve Reinf., Mid-spon

-Ve Reinf.,Side (A)


and W3-F

260

6 2 -T & B ^35"

-Ve Reinf. ,Side (F

2W 8 _ T_ t*

: 6 2 -T& B

1 W 8 _ T _ 4 F 6 2 - T & B

+ Ve Reinf., Mid-spQn

•Ve Reinf.,Side (A)


and W4-F

261

W 6.3.T& B.3

Ve Reinf. ,Side (F

2 W 8 _T_4

F62-T& B

+ Ve Reinf, Mid-span

•Ve Reinf.,Side (A)


and W5-F

262

F62-T&B

+ Ve Reinf.

F62-T & B

F62-T £ B

•Ve Reinf.

Side (B)

Fig. AV(5) Reinforcement details for column positions W2-B

d W2-G

263

F62-T&B

\ 2W8-T_3

.3W6.3..UB.

2W8-"L3

F 6 7. - T R B

Ve Reinf

M i ri -spa

3 W8 . T-3

2 W 6.3 - T_ 3

•Ve Rpinf.

Side (B)

F 6 2 - T & 8

200x200


and W3-G

264

F62-T&B

2W5.T-4 L '__

2Y12.T

2 W 8 . T. k

3W6.3 J 8 B

2V/8iT_u+

+ VeReinf.

Min -span'I

W6.3.T & B

•Ve Rpinf.

Side (B)

F62-T& B

3W6.T

Fig. AV(7) Reinforcement details for column positions W4-13

and W 4 - G

265

F62-T& B

W 8 _ T_ 4

2. B

--\r

• '

. <

•

U 100 • 1

83

01 76

4W8..T.>


and W5-G

266

«e-

•Ve Reinf.

Side (H)

F62-T&B

2Y12-T

3 W6.3-T& B-2

• Ve R e i n f

M idspan

F 6 2_T &B A

f Li

/

COO x 200

_Ve Reinf

Side ( C ) 1

? Y 12 _T

P±

^

84

4 267

300

26

Fig. AV(9) Reinforcement details for column positions W2-C

and W2-II

267

<£-

•Ve Reinf.

Side (H)

F6 2-T&B /

2Y12-T

3 W 6.3-T g B\-3

+ Ve Re i n f

Midspan

F 6 2_T &B

400x200

.Ve Reinf

Side ( C

3 W 0 _T- a

<

CNJ I

82

> •

**=>

24

m


and W3-H

268

+ V e Reinf

Midspan

F 6 2_T & B / /

/

•—1

COO x 200 1W 8 _T

.Ve Reinf

Side ( C

2 W 8 _ J.U

„

o *» 6 u )

0 u

"(NJ

8 2

o

a U-t

3:

ih 34

228

2 50

Fig. AV(ll) Reinforcement details for column positions W4-C

and W4-H

269

< &

-Ve Reinf.

Side (H)

F62-T&B X-

2 Y 1 2 - T

3 W6.3-T&B)-3

550x^00

W6.3 _T 8- B_3

*Ve Reinf

Midspan

F 6 2_T & B / f

V

/

— ... _

1

1 1

1 i 1

1

COOx 200 W6.3.T & B_3

.Ve Reinf TT"/ - ' "ut

Side ( C ) -=i

* * a—

-g » -TT ~D o

'-A

100

85

76

21


and W5-H

APPENDIX VI

LOAD-STRAIN DIAGRAMS FOR THE REINFORCING BAR

10.0

7.5-

2

a 5.0

Strain x 10 "6

Fig. AVI(l) Load-strain diagram for 5 m m hard drawn wires (type 1)

2

T3

18.0

15.0

12.0

9.0

6.0

3.0

0.0

Strain x 10 -6

Fig. AVI(2) Load-strain diagram for 6 m m hard drawn wires (type 1)

272

Strain x IO"6

Fig. AVI(3) Load-strain diagram for H m m hard drawn wires (type 1)

2 M xf cd

8

7

6

5

2

I

Q WLÎ ,4.1 1_JL

/

/

I J I I l-J 1—1 l_l I I ,1.1 I I , 1 , 1 1 I I 4—4 I I, I I, 4 I I I I I I 1 1 1 . ,1 I

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 m O i r t O i f l O m o m O i n

Strain x 10 "6

0 uO


273

•O cd

l — i i, i i 1 i i i > . I i i i i I j I i J_I_ i , i — i i i i i i i ' i i i • i i i i • i

O O O O O O O O O in o in o

H H N

0 0 in N

0 0 0 CO

Strain x

0 0 0 0 in o co *j 10'6

O O O O O O O O O O in o in o in <• m in * *

Fig. AVI(5) Load-strain diagram for 6.3 m m hard drawn wires (type 2)

2 -M

32 r

28

24

20

"S 16 cd

O 12

8

4

S /

/

/ /

y

/

'A i • • i. -I—.L 1_JL- JL_i_ 1 • • • • I — I — I — i l l

0 0 ft

0 0 0 H

0 0 LO H

0 0 0 N

0 0 ft

Strain x

0 0 0 CO

10 -6

0 0 ft CO

0 0 0 *'

' I • • * » I I I I I I

0 0 0 0 0 0 ft 0 ft <r m m


8 r

Z M

cd

O 3

0 L ..i. I I . I . I i i i i i i i i i ,i i J i > i i i . i i i i i » i 1 1 1

274

i i i > • i.

0 0 ft

0

% H

0 0 ft H

0 0 0 N

0 0 m N

0 0 0 CO

0 0 ft CO

0 0 0 *

0 0 ft rt

0 0 0 ft

0 0 in m

Strain x 10""

Fig. AVI(7) Load-strain diagram for 4 mm hard drawn wires (type 2)

Z

-a cd

o

_I_J_ _L i i I I i i i i — L

o o o o o o o o o o ft 0 ft 0 ft N CO CO <t v

Strain x 10 "6

• i • i

0 0 0 0 0 0 0 ft 0 ft ft UJ3

Fig. AVI(8) Load-strain diagram for 6.3 m m hard drawn wires (type 3)

275

Z M

32 r

28

24

20

TJ cd

i 16

12

8

4

UL^. I i i i i L - - i — i i i - J i i i — J L - i « -i -i i • » • i • i • * » «. i * .*- • • .i i . - • • • • i i . i i f *.

0 0 ft

0 0 0 H

0 0 ft H

s 0 N

0 0 ft

Strain x

0 0 0 CO

10 "6

0 0 ft CO

0 0 0 •f

0 0 ft <r

0 0 0 ft

0 0 in m


52.5

Z

-a cd

17.5

Strain x 10

Fig. AVI(IO) Load-strain diagram for 12 m m deformed bars (type 1)

276

32 r

Z M TJ cd

i • • ' • i • • i • ' • i i i * i—i i ,..>., I i — i — i — i — L . J i — i i I

0 0 ft

0 0 0 H

0 0 ft H

0 0 0 N

0 0 ft

Strain x

0 0 0 CO

10 •6

0 0 ft CO

0 0 0 *

0 0 ft V

0 0 0 ft

0 0 ft in

Fig. AVI(ll) Load-strain diagram for F62 meshes (type 1)

APPENDIX VII

MEASURED SLAB STRIP MOMENTS (AT THE ULTIMATE STATE)

AND THE CALCULATED YIELD MOMENTS OF THE CRITICAL SLAB

STRIPS

APPENDIX VH

MEASURED SLAB STRIP MOMENTS AT THE ULTIMATE STATE

AND THE CALCULATED YIELD MOMENTS OF THE CRITICAL SLAB STRIPS

The absolute values of the measured slab strip moments per strip width of Models

W l to W 5 (at the ultimate state) are presented in Table AVII(l) to AVII(4). As

discussed in Section 7.2.1, these moments are specified by M n , M m j , and M31,

where the subscript i stands for the strip numbers and subscripts 1, m, and 3

respectively refer to the measuring stations 1, m and 3 (see Fig. 7.2(1)).

279

Table AVn(l) Slab strip moments per strip width at the ultimate state

Column Mn Mmj M31 Mi2 Mr-^ M32

positions _____

Wl-A

W2-A

W3-A

W4-A

W5-A

W2-C

W3-C

W4-C

W5-C

Wl-B

W2-B

W3-B

W4-B

W5-B

R-3A

R-4A

kN-m

0.00

0.00

0.00

0.00

0.00

4.92

3.57

4.18

3.61

4.08

4.93

3.57

3.38

2.30

4.56

3.68

kN-m

0.00

0.00

0.00

0.00

0.00

4.99

4.97

4.95

3.62

4.99

4.99

4.96

4.95

3.62

4.56

3.68

kN-m

0.00

0.00

0.00

0.00

0.00

4.11

3.57

3.38

2.50

4.08

4.11

3.57

3.38

2.30

4.56

3.68

kN-m

0.00

0.00

0.00

0.00

0.00

2.03

1.45

1.65

1.47

2.27

2.75

1.98

2.18

2.00

2.63

2.08

kN-m

0.00

0.00

0.00

0.00

0.00

2.03

2.02

2.02

1.47

3.77

2.78

2.75

2.75

2.01

2.58

2.76

kN-m

0.00

0.00

0.00

0.00

0.00

1.67

1.45

1.38

0.93

2.26

2.28

1.98

1.88

2.01

2.63

2.40

280

Table AVH(2) Slab strip moments per strip width at the ultimate state

(contd.)

Column M13 M ^ M33 M u M m 4 M 3 4

positions _^____^_

Wl-A

W2-A

W3-A

W4-A

W5-A

W2-C

W3-C

W4-C

W5-C

Wl-B

W2-B

W3-B

W4-B

W5-B

R-3A

R-4A

kN-m

0.00

0.00

0.00

0.00

0.00

1.45

1.12

1.18

1.07

1.21

1.48

1.18

1.40

1.07

1.60

1.69

kN-m

0.00

0.00

0.00

0.00

0.00

1.48

1.47

1.47

1.07

1.48

1.48

1.47

1.47

1.07

1.38

1.37

kN-m

0.00

0.00

0.00

0.00

0.00

1.22

1.06

1.00

0.63

3.90

2.46

1.92

2.39

1.14

1.66

1.28

kN-m

2.72

2.76

5.94

3.47

4.41

2.95

5.14

4.71

4.34

4.57

1.47

5.90

3.34

4.10

11.56

7.04

kN-m

2.86

2.69

2.66

2.65

1.83

2.96

2.94

2.93

2.78

5.25

1.48

1.47

1.47

3.76

1.38

1.40

kN-m

2.01

2.27

3.27

5.18

1.19

8.29

2.21

4.05

1.36

3.86

4.01

2.00

4.31

1.76

11.56

5.92

281

Table AVII(3) Slab strip moments per strip width at the ultimate s

(contd.)

Column M15 M-^ M 3 5 M16 M m 6 M 3 6

positions

Wl-A

W2-A

W3-A

W4-A

W5-A

W2-C

W3-C

W4-C

W5-C

Wl-B

W2-B

W3-B

W4-B

W5-B

R-3A

R-4A

kN-m

2.72

1.50

1.06

1.15

1.18

0.00

0.00

0.00

0.00

1.22

1.46

1.17

1.43

1.07

2.00

1.77

kN-m

2.86

1.48

1.47

1.47

1.07

0.00

0.00

0.00

0.00

1.48

1.48

1.47

1.47

1.07

1.38

1.23

kN-m

3.56

3.28

1.96

2.89

1.10

0.00

0.00

0.00

0.00

3.43

2.13

1.40

2.06

1.14

2.00

1.58

kN-m

2.27

2.73

1.98

1.88

2.00

0.00

0.00

0.00

0.00

2.27

2.75

1.98

2.27

2.01

2.63

2.08

kN-m

2.77

2.77

2.76

2.75

2.01

0.00

0.00

0.00

0.00

2.77

2.88

2.76

2.75

2.01

2.59

2.40

kN-m

2.27

2.29

1.98

1.88

2.01

0.00

0.00

0.00

0.00

2.26

2.28

1.97

1.88

2.00

2.63

2.40

Table AVII(4) Slab strip moments per strip

width at the ultimate state (contd.)

Column

positions

Wl-A

W2-A

W3-A

W4-A

W5-A

W2-C

W3-C

W4-C

W5-C

Wl-B

W2-B

W3-B

W4-B

W5-B

R-3A

R-4A

Mn

kN-m

4.08

5.05

3.57

3.39

2.42

0.00

0.00

0.00

0.00

3.18

4.95

3.56

4.04

3.62

4.56

3.68

M m 7

kN-m

4.99

4.99

4.96

4.95

3.62

0.00

0.00

0.00

0.00

4.99

4.99

4.96

4.95

3.62

4.56

3.68

M 3 7

kN-m

4.08

4.11

3.56

3.38

2.30

0.00

0.00

0.00

0.00

4.08

4.11

3.56

3.38

2.48

4.56

3.68

283

Calculated Yield Moments of the

Critical Slab Strins

Table AVII(5) Yield moments of the critical slab strips per strip

width (contd.)

Column Location fc d As A's Moment

Position

Wl-A

Wl-A

W2-A

W2-A

W3-A

W3-A

W4-A

W4-A

W5-A

W5-A

W2-C

W2-C

W3-C

W3-C

W4-C

W4-C

W5-C

Column Face

Midspan

Column Face

Midspan

Column Face

Midspan

Column Face

Midspan

Column Face

Midspan

Column Face

Midspan

Column Face

Midspan

Column Face

Midspan

Column Face

MPa

34.60

34.60

35.13

35.13

26.30

26.30

23.48

23.48

26.81

26.81

35.13

35.13

26.30

26.30

23.48

23.48

26.81

mm

81.00

84.80

87.00

87.00

81.25

84.80

87.00

84.80

83.00

84.80

84.50

84.80

82.00

84.60

82.00

84.80

84.67

m m 2

62.34

62.34

61.95

91.95

160.23

61.95

80.51

61.95

111.34

61.95

62.34

62.34

123.90

62.34

111.73

62.34

93.12

nim2

62.34

62.34

61.95

91.95

160.23

61.95

80.51

31.17

111.34

61.95

62.34

62.34

123.90

62.34

111.73

62.34

93.12

kN-m

2.72

2.86

2.76

2.69

5.94

2.66

3.47

2.65

4.41

2.66

2.96

2.96

5.14

2.94

4.71

2.93

4.34

284

Table AVII(6) Yield moments of the critical slab strips per strip

width (contd.)

Column Location fc d As A's Moment

Position

Wl-B

Wl-B

W2-B

W2-B

W3-B

W3-B

W4-B

W4-B

W5-B

W5-B

R-3A

R-3A

R-4A

R-4A

Column Face

Midspan

Column Face

Midspan

Column Face

Midspan

Column Face

Midspan

Column Face

Midspan

Column Face

Midspan

Column Face

Midspan

MPa

34.60

34.60

35.13

35.13

26.30

26.30

23.48

23.48

26.81

26.81

44.00

44.00

28.00

28.00

mm

84.75

85.22

84.00

84.80

85.20

84.80

87.00

84.80

83.00

85.22

82.00

82.00

82.00

82.00

m m 2

131.69

131.69

31.17

31.17

178.59

31.17

98.78

31.17

129.85

129.85

408.55

50.32

220.03

53.43

m m 2

131.69

131.69

31.17

31.17

178.59

31.17

98.78

31.17

129.85

129.85

50.32

100.63

53.43

61.61

kN-m

5.22

5.25

1.47

1.48

6.91

1.47

3.99

1.47

5.07

5.70

13.89

1.91

7.92

2.00

285

PAPERS PUBLISHED BASED ON THIS THESIS

Falamaki, M. and Loo, Y.C., (1988), Ultimate load Test of a Half-Scale

Reinforced Concrete Flat Plate with Spandrel Beams. Proceedings, 11th

Australasian Conference on Mechanics of Structures and Materials,

University of Auckland, N.Z., August ,pp. 282-288 .

Falamaki, M. and Loo, Y.C., (1989), Failure Mechanisms of Reinforced

Concrete Flat Plates with Spandrel Beams. Second East Asia-Pacific

Conference in Structural Engineering and Construction: Achievements,

Trends and Challenges, Proc. EASEC-2, Chiangmai , Thailand, January,

Vol. 1, pp. 249-254.

Falamaki, M. and Loo, Y.C., (1990), Strength Tests of Half-Scale Reinforced

Concrete Flat Plate Models with Spandrel Beams. Invited paper

International Conference on Structural Engineering and Computations,

Beijing, 25-28 April, 17 pp.

Documents

1990 Punching shear strength of reinforced concrete flat