1994 Punching shear strength of reinforced and post ...

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1994

Punching shear strength of reinforced and post-tensioned concrete flat plates with spandrel beamsChoong-Luin ChiangUniversity of Wollongong

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Recommended CitationChiang, Choong-Luin, Punching shear strength of reinforced and post-tensioned concrete flat plates with spandrel beams, Doctor ofPhilosophy thesis, Department of Civil and Mining Engineering, University of Wollongong, 1994. http://ro.uow.edu.au/theses/1259

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PUNCHING SHEAR STRENGTH OF REINFORCED AND POST-TENSIONED 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

JT* 0%. F*

THE UNIVERSITY OF WOLLONGONG

by

C H O O N G - L U I N C H I A N G , PEng., B.E. (Honours Class I),

Grad IEAust

DEPARTMENT OF CIVIL AND MINING ENGINEERING

1994

"The world and all that is in

it belong to the Lord; the earth and all who live on

it are his. He built it on deep waters

beneath the earth and laid its foundations in the

ocean depths."

- Psalms 24.1-2

11

DECLARATION

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

such institution except where specifically indicated.

Choong-Luin Chiang

August 1994

iii

ACKNOWLEDGMENTS

The author deeply appreciates the opportunity provided by his thesis supervisor,

Associate Professor Yew-Chaye Loo, in the pursuit of this doctoral degree. The close

supervision, frank and fruitful discussions, and fatherly guidance given by Professor Loo

have been instrumental in honing the author's research skills over the course of this

study.

Sincere gratitudes are extended to the University of Wollongong for the scholarship.

The thesis research forms an integral part of an award of the Australian Research

Council funded project of Professor Loo. The author is grateful to the Council for

supporting the monumental task of this research project, over the past three-and-a-half

years.

The author also acknowledges the generous contributions in material and services

provided by the following firms in Wollongong and Sydney:

- Acrow Pty Ltd;

- Alliance Equipment Pty Ltd, Wollongong;

- Austress PSC, Sydney;

- Cleary Bros. Concrete Pumping;

- Nippy Crete Concrete; and

- VSL Prestressing, Sydney, in particular Mr. Jeff Lind (former Senior Engineer).

The technical staff of the Department of Civil and Mining Engineering, University of

Wollongong, in particular, Messrs. N. Gal, R. Webb, F. Hornung (who has since left),

iv

G. Caines, C. Allport and A. Grant are remembered for their invaluable assistance in the

laboratory work.

Acknowledgments, gestures of friendship and comradeship are also extended to special •

groups of current and ex-fellow students, sharing a common goal in model construction:

- J.R.A. Wagg and F. Stillone (Undergraduate class of '91);

- T.V. Chau, S.A. Clarkson, T. Mustaffa and D. Chaisumdet (Undergraduate class of

'92);

- H.T. Pengelly, K.F. Lei, H.M. Hii and J. O'Connors (Undergraduate class of '93); and

- A. LaManna, A. Wachtel, S. Merange and G. Canlis (Undergraduate class of '94).

Special acknowledgment is due to my wife, Sheila, for her constant support and

encouragements given, in sickness and in health, be it near or far, throughout the period

of this study. Her typing and proof-reading skills have been a God-send.

Finally, the author is indebted to all his other family members in Malaysia, USA and

Hong Kong for their understanding and encouragements, right from the very beginning.

v

ABSTRACT

In the design of reinforced or prestressed concrete flat plates with spandrel beams, one

of the most critical aspects that needs to be considered is the punching shear strength, Vu

of the slab-column-spandrel connections. Research on this behaviour has been carried

out in recent years by various engineering academics and professionals. One of the more

encompassing analytical approach in determining Vu , is the procedure developed by Loo

and Falamaki* for reinforced concrete slab-column-spandrel connections at the edge-

and corner-column positions. In the case of prestressed concrete flat plates, there is as

yet no reliable procedure for the determination of Vu , particularly for the above stated

slab-column-spandrel connections. Thus, the main objective of the present study is to

develop an analytical procedure for the prediction of Vu , for these types of structural

connections. This procedure is an extension of the analytical approach developed by

Loo and Falamaki* whereby the effect of prestress is taken into account and

incorporated into the appropriate equilibrium equations.

The development albeit an extension of a reliable analytical method requires accurate

test results from large-scale models with proper boundary conditions. Tests up to failure

were carried out on four prestressed post-tensioned concrete flat plate models, two with

spandrel beams of different widths and depths, while the other two have a torsion strip

and a free edge respectively. These half-scale models, each weighing approximately 5

tonnes, represent two adjacent panels at the corner of a typical flat plate structure and

* Loo, Y.C. and Falamaki, M. (1992), "Punching Shear Strength Analysis of Reinforced Concrete

Flat Plates With Spandrel Beams", ACI Structural Journal, V.89, No.4, July-August, 1992, pp.375-

383.

vi

Abstract

were tested under simulated uniformly distributed loads. For ease of construction, each

of the flat plate models was supported on six concrete column sections, which were cast

and rigidly connected onto prefabricated steel sections at the bottom instead of having

concrete columns. Specially designed load cells were used to measure the three reaction

components at the hinged base of each of the column supports. Strain gauges were also

attached to the reinforcing bars and prestressed tendons 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.

Besides the experimental work, three concurrent studies were also carried out. The first

was a comparative study of the various methods of punching shear strength analysis of

reinforced concrete flat plates with spandrel beams. It was found that the Wollongong

Approach developed by Loo and Falamaki [1992] was superior to alternative methods

including that recommended by the Australian Standard for Concrete Structures (AS

3600-1988). The second was a parametric study of the effect of spandrel beam on the

punching shear strength analysis, using the Wollongong Approach. From this study, a

range of parametric limitations were identified within which the Wollongong Approach

may be applied.

Finally, based on the experimental results, a theoretical study was undertaken which led

to the development of a prediction procedure for the punching shear strength, Vu of

post-tensioned concrete flat plates with spandrel beams, the details of which are

presented herein. As with the Wollongong Approach, the proposed procedure (referred

to in this thesis as the "Modified Wollongong Approach") is applicable to the analysis of

failures at the corner- and edge-column positions. In addition to the consideration of

parameters such as the overall geometry of the connection, the concrete strength and

vii

Abstract

the slab restraint on the spandrel, the procedure also takes into account the following

parameters:

(1) the size and location of the prestressed tendons,

(2) the magnitude of the applied prestress, and

(3) the profiles of the draped tendons.

In comparing the accuracy and reliability of the proposed analytical procedure with the

Australian Standard approach, the latter was found to be less reliable in that it

consistently overestimates the value of Vu , especially at the corner-column positions.

This is also consistent with the compared results of the comparative study on reinforced

concrete flat plates with spandrel beams.

vin

Table of Contents

TABLE OF CONTENTS

Title Page i Declaration iii Acknowledgments iv

Abstract vi

Table of Contents ix

List of Figures xv List of Tables xix Notation xxii

1. Introduction ...1 1.1 General Remarks.. 1

1.2 The Prediction of Vu 2

1.3 Aims of Research 3

1.4 Scope of Research and Experimental Work 4

1.5 Theoretical Background 8

1.5.1 Punching shear .8

1.5.2 Theoretical aspect 10

1.6 Overall Layout of Thesis 13

2. Literature Review 15 2.1 Background of Previous Work 15

2.2 Codes of Practice 18

2.3 Methods of Punching Shear Strength Analysis 19

2.3.1 Australian Standard A S 3600-1988 19

2.3.2 American Concrete Institute ACI318-89 19

2.3.3 British Standard BS 8110:1985 20

2.3.4 The Wollongong Approach 20

2.3.5 Others 21

2.4 Previous Experimental Work on Post-Tensioned Concrete Flat Plates 22

2.4.1 J.F.Brotchie and F.D.Beresford [ 1967] 22

2.4.2 N.H.Burns and R.Hemakom [1977] 22

2.4.3 Regan [1985] 23

2.4.4 D.A.Foutch, W.L.Gamble and H.Sunidja [1990] ....23

2.4.5 A.E.Long and D.J.Cleland [1993] 24

2.5 Comments on Previous Experimental Work 24

ix

Table of Contents

3. Performance of Analytical Methods for Punching Shear

Strength of Reinforced Concrete Flat Plates 25

3.1 General Remarks 25

3.2 Objectives and Scope 25

3.3 Methods of Analysis - A Review 27

3.3.1 Australian Standard Procedure [SAA, 1988] 27

3.3.2 The ACI Method [ACI, 1989] 29

3.3.3 The British Standard Method [BSI, 1985] 31

3.3.4 The Wollongong Approach [Loo and Falamaki, 1992] 33

3.4 Presentation and Comparison of Results 34

3.4.1 Model tests 34

3.4.2 Predicted results. 37

3.4.3 Comparison: Flat plates with torsion strips 39

3.4.4 Comparison of ACI and British Standard methods incorporating

reduction factors: Flat plates with torsion strips 41

3.4.5 Comparison: Flat plates with spandrel beams 44

3.5 Conclusions 46

4. Parametric Study of the Effect of Spandrel Beams on the

Punching Shear of Strength of Reinforced Concrete Flat

Plates 47

4.1 Introduction 47

4.2 Formulas for Spandrel Strength Parameter 47

4.3 Generating Models of Varying Features of Spandrel Beam 49

4.3.1 Varying the width of spandrel beam 49

4.3.2 Varying the depth of spandrel beam 54

4.3.3 Varying the area of longitudinal steel reinforcement in spandrel

beam 58

4.4 Summary of Parametric Study Results 62

5. Theoretical Approach to Incorporate Effect of Prestress in the

Analysis of Punching Shear 64

5.1 Effect of Prestress on Punching Shear 64

5.1.1 General 64

5.1.2 Current concrete codes of practice 67

5.1.3 Effect of prestress in the Wollongong Approach 70

5.1.4 Equilibrium equations 73

5.1.5 Spandrel beam parameters 75

X

Table of Contents

5.1.6 Interaction equation 78

5.1.7 Modifications to interaction equation 80

5.2 Evaluation of Ultimate Moment and Shear 81

5.3 Analysis of Punching Shear 83

5.3.1 Corner connections with spandrel beam 83

5.3.2 Edge connections with spandrel beam 84

5.3.3 Connections with torsion strip 85

5.4 Ultimate Flexural Strengths Mju and Mm for Post-tensioned Concrete

Sections 86

5.5 Losses of Prestress 88

5.5.1 Immediate losses 88

5.5.2 Time-dependent losses 91

5.5.3 Summary of prestress losses considered in the

calculation of Vu 92

6. Test Program - An Introduction to the Test Models 93

6.1 General 93

6.2 Details of Models 93

6.2.1 Dimensions 94

6.2.2 Support system 94

6.3 Design of Models 96

6.3.1 Design of prestressing and reinforcement 96

6.3.2 Prestressing and reinforcement details 102

6.3.3 Cable ducting and anchorage system 107

6.3.4 Concrete specifications 109

6.3.5 Design of spandrel beam 109

7. Test Program - Construction, Instrumentation and Testing of

Mod e l s 113


7.2 Construction of Models 114

7.2.1 Setting up of formwork 114

7.2.2 Cable ducts 118

7.2.3 Placement of ducts and reinforcement 122

7.2.4 Casting and curing of concrete and removal of formwork 127

7.2.5 Post-tensioning of prestressing tendons 129

7.2.6 Grouting 131

7.3 Instrumentation of Models 133

XI

Table of Contents

7.3.1 Strain gauges 134

7.3.2 Load cells 137

7.3.3 Whiffle-tree loading system 137

7.3.4 Deflection measurement 139

7.4 Testing of Models 141

7.4.1 During load application 141

7.4.2 After load application 143

7.4.3 Precautions prior to and during testing 144

7.4.4 Problems encountered during model testing 145

8. Presentation and Discussion of Test Results 146

8.1 General 146

8.2 Crack Patterns and Failure Mechanism 147

8.2.1 Model PI 147

8.2.2 Model P2 150

8.2.3 Model P3 153

8.2.4 Model P4 .'.157

8.3 Deflections 160

8.3.1 Model PI 160

8.3.2 Model P2 162

8.3.3 Model P3 163

8.3.4 Model P4 165

8.4 Punching Shear Strength and Unbalanced Moments 167

8.4.1 General 167

8.4.2 . Experimental results for shear, moment and torsion 168

8.4.3 Failure modes at the corner- and edge-column connections 172

8.4.4 Comparison of measured vertical column reactions and

total applied load 173

8.5 Observations 176

9. Determination of Flexural Strength of Critical Slab Strips 177


9.2 Flexural Behaviour of Uncracked and Cracked Sections 177

9.2.1 Uncracked section analysis 179

9.2.2 Cracked section analysis 180

9.3 Flexural Strength Theory 181

9.3.1 Formulas for the prestressed tendon strain components 181

9.3.2 Determination of Mu for a prestressed concrete section 182

xii

Table of Contents

9.4 Comparison of Measured and Calculated Values for Mj and Mm 188

10. Calibration of the Prediction Formulas for Spandrel

Parameter 191


10.2 Research Scheme for Determination of Semi-empirical Formulas

for \|/ 192

10.3 Measurement of the Slab Restraining Factor \\f 193

10.3.1 Derivation of equations to measure \|/ 193

10.3.2 Measured values of \j/ 197

10.4 Calibration of Semi-empirical Formulas for \[/ 199

10.5 Comparison Between Measured and Predicted Values of \|/ 204

11. Analysis of Punching Shear Strength 206


11.2 The Prediction Formulas for Vu 207

11.2.1 Corner-connections with spandrel beam 207

11.2.2 Edge-connections with spandrel beam 208

11.2.3 Connections with torsion strip or free edge 209

11.3 Computer Program for Vu 209

11.4 Comparison of Results 211

11.4.1 Predicted punching shear results by the proposed approach

(or the Modified Wollongong Approach) 212

11.4.2 Predicted punching shear results by codes of practice

and comparison 212

11.5 Comments on the Accuracy and Reliability of the Proposed Approach

(or the Modified Wollongong Approach) 216

11.5.1 Flat plates without spandrel beams (Models PI and P4) 216

11.5.2 Flat plates with spandrel beams (Models P2 and P3) 218

11.6 Summary 219

12. Conclusion 221


12.2 Performance and Limitations of the Wollongong Approach 222

12.3 Experimental and Theoretical Studies of Post-tensioned

Concrete Flat Plates 223

12.3.1 Deformation and Failure Mechanisms 223

12.3.2 Prediction of Vu by the proposed approach

xiii

Table of Contents

(Modified Wollongong Approach) 224

12.3.3 Accuracy of the proposed approach

(Modified Wollongong Approach) 225

12.4 Recommendations for Further Study 225

References 227

Appendix A Calibration of Prestressing Jack 234

Appendix B Deflection Measurements 237

Appendix C Plots of Experimental Moment, Shear and Torsion Results for

Post-tensioned Concrete Flat Plate Models 249

Appendix D Tabulations of Experimental Moment, Shear and Torsion Results for

Post-tensioned Concrete Flat Plate Models 254

Appendix E Flexural Strength Theory 259

Appendix F Strain Measurement Results 266

Appendix G Sample Calculations of Mj Using Measured Tendon Strain 272

Appendix H Sample Calculations for Punching Shear Strength Using the

Proposed Approach (Modified Wollongong Approach) 275

Appendix I Details of Spandrel Beams or Torsion Strips and

Critical Slab Strips 284

Appendix J Listings of Source Code for BASIC Computer Programs

to analyse Vu 286

Papers Published Based on This Thesis 307

xiv

List of Figures

LIST OF FIGURES

Fig. 1.1 Typical half-scale flat plate floor Fig. 1.2 Test models Fig. 1.3 Types of shear failures in a slab Fig. 1.4 Free-body diagrams for edge and corner connections Fig. 3.1 Relative strength of concrete cubes and cylinders Fig. 3.2 Flat plate model Wl with a typical spandrel beam section details

Fig. 3.3 Measured Vu versus predicted Vu - models with torsion strips Fig. 3.4 Measured Vu versus predicted Vu - models with spandrel beams Fig. 3.5 Measured Vu versus predicted Vu by A C I and British methods

incorporating reduction factors - models with torsion strips Fig. 4.1 Generated models with varying width of spandrel beam

Fig. 4.2 Graph of punching shear Vu versus spandrel strength parameter 5

of generated models with varying width bsp and shallow spandrel

beams


of generated models with varying width bsp and very deep

spandrel beams Fig. 4.4 Generated models with varying depth of spandrel beam


of generated corner-connection models with varying Dj


of generated edge-connection models with varying!)/

Fig. 4.7 Generated models with varying amount of longitudinal steel reinforcement in spandrel beam


of generated corner-connection models with varying A[s Fig. 4.9 Graph of punching shear Vu versus spandrel strength parameter 8

of generated edge-connection models with varying A/iV Fig. 5.1 Effect of prestress on principal stresses in the centroid of a slab

strip Fig. 5.2 Forces acting in the corner column region of a post-tensioned

concrete flat plate Fig. 5.3 Forces acting in the edge column region of a post-tensioned

concrete flat plate Fig. 5.4 Forces, strains and stresses in a prestressed concrete section

Fig. 6.1 Typical column support details Fig. 6.2 Design layout of cable ducts Fig. 6.3 Design layout of steel reinforcement Fig. 6.4 Profile of cable ducts along short span of flat plate model

xv

List of Figures

Fig. 6.5 Profile of cable ducts along long span of flat plate model Fig. 6.6 Stress versus strain plot for 5.05 m m diameter prestressed tendon Fig. 6.7 Stress versus strain plot for 10-rnm diameter steel reinforcing bar Fig. 6.8 Tendon and reinforcement details at column strips of post-

tensioned concrete flat plate models Fig. 6.9 Shape and section view of cable duct Fig. 6.10 Anchorage system Fig. 6.11 Stress versus strain plot for 6-mm diameter steel bar Fig. 6.12 Details of a typical spandrel beam (for Model P2) Fig. 7.1 General set-up of a flat plate model Fig. 7.2 Details of slab formwork and the supporting frame Fig. 7.3 Typical details of formwork and support system for spandrel

beam at the slab edge Fig. 7.4 Bending process of duct to final shape Fig. 7.5 Typical fabricated cable duct Fig. 7.6 Typical 12-mm bearing plate Fig. 7.7 P V C grout vent for cable ducts Fig. 7.8 Tendons protruding from the edge formwork Fig. 7.9 Top steel reinforcement over corner column A Fig. 7.10 Top steel reinforcement over edge column B Fig. 7.11 Spandrel beam reinforcement for Model P2 Fig. 7.12 Casting of concrete slab for flat plate model Fig. 7.13 Concrete test cylinders Fig. 7.14 Barrel and wedge used for anchorages

Fig. 7.15 Post-tensioning of tendons Fig. 7.16 Grouting of prestressed tendons Fig. 7.17 Hewlett-Packard 9826 computerised data acquisition system Fig. 7.18 Locations of strain gauges on steel reinforcement Fig. 7.19 Locations of strain gauges on tendons in cable ducts Fig. 7.20 Close-up view of load cells at the base of steel column Fig. 7.21 General view of whiffle-tree loading system Fig. 7.22 Dial gauges set up on top of slab to measure deflections Fig. 7.23 Set up of the Linear Variable Differential Transformer (LVDT) Fig. 7.24 Set up of synchronization control system for the two double

acting hydraulic rams Fig. 7.25 Post-mortem examination of test Model P3 after failure

Fig. 8.1 Crack pattern on top of Model PI Fig. 8.2 Crack pattern at soffit oi Model PI Fig. 8.3 Shear failure (Model PI - column A) Fig. 8.4 Crack pattern on top and sides of Model P2 Fig. 8.5 Crack pattern at soffit and sides of Model P2

Fig. 8.6 Flexural failure (Model P2) Fig. 8.7 Shear failure (Model P3 - column B) Fig. 8.8 Crack pattern on top and sides of Model P3

xvi

List of Figures

Fig. 8.9 Crack pattern at soffit and sides of Model P3 Fig. 8.10 Shear failure (Model P4 - column B) Fig. 8.11 Crack pattern on top and sides of Model P4

Fig. 8.12 Crack pattern at soffit and sides of Model P4 Fig. 8.13 Deflection measurement stations

Fig. 8.14a Load-deflection plots - Model PI (Locations 1, 5 and 6) Fig. 8.14b Load-deflection plots - Model PI (Locations 1, 3 and 4) Fig. 8.14c Load-deflection plots - Model PI (Locations 7, 8 and 9) Fig. 8.15a Load-deflection plots - Model P2 (Locations 1, 5 and 6) Fig. 8.15b Load-deflection plots - Model P2 (Locations 2, 3 and 4) Fig. 8.15c Load-deflection plots - Model P2 (Locations 7, 8 and 9) Fig. 8.16a Load-deflection plots - Model P3 (Locations 1, 5 and 6) Fig. 8.16b Load-deflection plots - Model P3 (Locations 2, 3 and 4) Fig. 8.16c Load-deflection plots - Model P3 (Locations 7, 8 and 9) Fig. 8.17a Load-deflection plots - Model P4 (Locations 1, 5 and 6)

Fig. 8.17b Load-deflection plots - Model P4 (Locations 2, 3 and 4) Fig. 8.17c Load-deflection plots - Model P4 (Locations 7, 8 and 9) Fig. 8.18 Experimental shear, moment and torsion recorded at corner-

connections A and C and at edge-connection B for Model P2 Fig. 9.1 Flow chart to determine the theoretical ultimate flexural strength

of prestressed concrete slab strips Fig. 9.2 Flow chart to determine the flexural strength of prestressed

concrete slab strips using strain measurements Fig. 9.3 Graph of measured versus calculated Mj and Mm results Fig. 10.1 Research scheme to establish the semi-empirical formulas to

determine \|/

Fig. 10.2 Graphs of y versus log10(78) for connections A and C , and \\f

versus logio(A-S) for connections B

Fig. 11.1 Measured versus predicted Vu for post-tensioned concrete flat plate models without spandrel beams (PI and P4)

Fig. 11.2 Measured versus predicted Vu for post-tensioned concrete flat plate models with spandrel beams (P2 and P3)

Fig. Cl.l Plot of moment transfer to columns A, B and C versus applied load up to failure (Model PI)

Fig. CI.2 Plot of shear strength around columns A, B and C versus applied

load (Model PI) Fig. CI.3 Plot of torsional moment at' columns A, B and C versus applied

load (Model PI) Fig. C2.1 Plot of moment transfer to columns A, B and C versus applied

load (Model P3) Fig. C2.2 Plot of shear strength at columns A, B and C versus applied load

(Model P3) Fig. C2.3 Plot of torsional moment at columns A, B and C versus applied

load (Model P3)

xvu

List of Figures

Fig. C3.1 Plot of moment transfer to columns A, B and C versus applied load (Model P4)

Fig. C3.2 Plot of shear strength at columns A, B and C versus applied load (Model P4)

Fig. C3.3 Plot of torsional moment at columns A, B and C versus applied load (Model P4)

Fig. E. 1 Typical strain and stress distribution of a prestressed concrete cross-section

Fig. E.2 Idealised stress block at ultimate moment conditions Fig. E.3 Strain distribution of prestressed tendon at three stages of

loading

xviii

List of Tables

LIST OF TABLES

Table 3.1 Table 3.2 Table 3.3 Table 3.4 Table 3.5 Table 3.6

Table 3.7

Table 3.8

Table 3.9

Table 3.10 Table 3.11 Table 3.12 Table 4.1

Table 4.2

Table 4.3

Table 4.4

Table 5.1 Table 6.1

Table 6.2

Table 7.1 Table 8.1 Table 8.2

Table 8.3

Table 8.4 Table 8.5 Table 8.6 Table 8.7

Comparison of punching shear results - models with torsion strips Comparison of punching shear results - models with spandrel beams Pearson correlation coefficients for models with torsion strips Deviations of predicted Vu for models with torsion strips Frequency distribution of predicted Vu for models with torsion strips Comparison of factored punching shear results by A C I and British

Standard methods - models with torsion strips Pearson correlation coefficients for models with torsion strips (after incorporating reduction factors) Deviations of predicted Vu for models with torsion strips (after incorporating reduction factors) Frequency distribution of predicted Vu for models with torsion strips (after incorporating reduction factors) Pearson correlation coefficients for models with spandrel beams Deviations of predicted Vu for models with spandrel beams Frequency distribution of predicted Vu for models with spandrel beams Typical generated data and results from parametric study of varying

width of spandrel beam in Model M2-A Typical generated data and results from parametric study of varying

depth of spandrel beam in Model M2-A Typical generated data and results from parametric study of varying longitudinal reinforcement in spandrel beam of Model M2-A

Summary of parametric study results of limits or ranges of 8 for reliable

punching shear strength analysis Summary of formulas for prestress losses used in the analysis of Vu Summary of tendon and reinforcement details in the critical column strips

and spandrel beam Yield strengths of steel reinforcement used for longitudinal bars and

closed ties in spandrel beam Laying sequence of cable ducts Concrete strengths of half-scale models Experimental shear, moment and torsion recorded at corner-connections

A and C and at edge-connection B for Model P2 Measured punching shear, unbalanced moments and applied loads at

failure for post-tensioned concrete flat plate models Comparison of vertical column reactions and applied loads for Model PI

Comparison of vertical column reactions and applied loads for Model P2 Comparison of vertical column reactions and applied loads for Model P3 Comparison of vertical column reactions and applied loads for Model P4

xix

List of Tables

Table 9.1 Strain measurements at ultimate for all models, at front face of columns and midspans of critical slab strips

Table 9.2 Measured and calculated values of Mj and Mm for all models

Table 10.1 Variables used to determine the slab restraining factor, \j/

Table 10.2 Results for the calculated variables and the slab restraining factor, \j/

Table 10.3 Calculated values of 8 , 7 , X , log^^S) and log^o(^8) for all post-

tensioned concrete flat plate models PI to P4

Table 10.4 Coefficients of determination ( R2) and correlation coefficients (R) of the

plots of \j/ versus log 10(78) and \j/ versus log10(X8)

Table 10.5 The measured and predicted values of \|/

Table 11.1 Punching shear strength predictions for post-tensioned concrete flat plate models PI to P4 using the proposed approach (Modified Wollongong

Approach) Table 11.2 Predicted punching shear strength results using the methods

recommended by A S 3600-1988, ACI318-89 andBS 8110:1985

Table 11.3 Deviations of predicted Vu for models without spandrel beams Table 11.4 Deviations of predicted Vu for models with spandrel beams Table B. 1 a Deflection measurements of Model PI (using rulers) Table B. 1 b Deflection measurements of Model PI (using LVDTs) Table B .2a Deflection measurements of Model P2 (using rulers) Table B.2b Deflection measurements of Model P2 (using LVDTs) Table B.3a Deflection measurements during first day of testing of Model P3 (using

rulers) Table B.3b Deflection measurements during second day of testing of Model P3

(using rulers) Table B.3c Deflection measurements during first day of testing of Model P3 (using

dial gauges) Table B.3d Deflection measurements during second day of testing of Model P3

(using dial gauges) Table B.3e Deflection measurements during first day of testing of Model P3 (using

L V D T s ) Table B.3f Deflection measurements during second day of testing of Model P3

(using L V D T s ) Table B.4a Deflection measurements during first day of testing of Model P4 (using

rulers) Table B.4b Deflection measurements during second day of testing of Model P4

(using rulers) Table B.4c Deflection measurements during first day of testing of Model P4 (using

dial gauges) Table B.4d Deflection measurements during second day of testing of Model P4

(using dial gauges) Table B.4e Deflection measurements during first day of testing of Model P4 (using

LVDTs)

xx

List of Tables

Table B.4f Deflection measurements during second day of testing of Model P4

(using LVDTs) Table D. 1 Shear, moment and torsion at columns A, B and C for Model PI Table D.2 Shear, moment and torsion at columns A, B and C for Model P2 Table D.3a Shear, moment and torsion at columns A, B and C for Model P3 - Day

One Table D.3b Shear, moment and torsion at columns A, B and C for Model P3 - Day

T w o Table D.4a Shear, moment and torsion at columns A, B and C for Model P4 - Day

One Table D.4b Shear, moment and torsion at columns A, B and C for Model P4 - Day

T w o Table F. 1 Strain measurements of reinforcing steel for Model PI Table F.2a Strain measurements of reinforcing steel for Model P2 Table F.2b Strain measurements of prestressed tendons for Model P2 Table F.3a Strain measurements of prestressed tendons for Model P3 - Day One Table F.3b Strain measurements of prestressed tendons for Model P3 - Day Two Table F.4a Strain measurements of prestressed tendons for Model P4 - Day One Table F.4b Strain measurements of prestressed tendons for Model P4 - Day Two Table F.4c Strain measurements of reinforcing steel for Model P4 - Day Two Table 1.1 Details of the spandrel beams or torsion strips Table 1.2 Column dimensions of post-tensioned concrete flat plate models

Table 1.3 Details of the critical slab strips Table 1.4 Details of prestressing in critical slab strips

xxi

Notation

NOTATION

a = the segment length of the critical shear perimeter parallel to the vector

direction of M

A = the area of the critical section

A' = the area of concrete under idealized rectangular compressive stress block AFL = anchorage friction losses

Ag = the gross area of the prestressed concrete section AL], AL2 = anchoring losses A/v = the total area of the longitudinal steel reinforcement in the spandrel beam

Ms.gen - the generated longitudinal steel area Ap = the area of prestressed tendons Api , AP2 = the area of the prestressed tendons in the main and transverse bending

directions Asc = the area of the compressive steel in the slab Ast = the area of tensile steel reinforcement in a cross-section Asti = the area of negative steel reinforcement in critical slab strip Astm = the area of positive steel reinforcement in critical slab strip At = the area of the rectangle defined by the centres of the four longitudinal

corner bars of the spandrel beam A't = area of the transformed cross-section Aws = the area of closed ties in the spandrel beam

b = the width of the critical slab strip

bSp = the width of spandrel beam bsp,gen = the generated width of the spandrel beam bw = the width of the spandrel beam

C = the resultant compressive force of magnitude

Cc = compressive force in the concrete above the neutral axis

Cs = a steel component of the resultant compressive force

d = the effective depth of the reinforced concrete slab dc = the effective depth of the compressive reinforcement

de = the effective depth of prestressed concrete slab de j = the depth of negative steel reinforcement in critical slab strip

dn = the depth to the neutral axis of a section dp = the depth of the prestressed tendons with respect to the extreme

compressive fibre dpi = the depth of the prestressed tendons in the main bending direction dp2 - the depth of the prestressed tendons in the transverse bending direction

xxii

Notation

ds = the depth of the tensile steel with respect to the extreme compressive fibre

dsc = the depth of the compressive steel with respect to the extreme compressive fibre

dsp = the effective depth of spandrel beam

dspi = the depth of tensile steel reinforcement in spandrel beam D, = the depth of spandrel beam

Di,gen = the generated depth of the spandrel beam Ds = the overall depth of the slab

e = the eccentricity of prestressing force from the centroid of the uncracked section

ej = the eccentricities of the prestressed tendons in the main bending direction e2 = the eccentricities of the prestressed tendons in the transverse bending

direction Ep = the modulus of elasticity of prestressed tendon Es = the modulus of elasticity of steel

fcf

h

J

kj ,k2 ,

k4,k5 , and ky

I

fc?>

**

lp, ls and lsc

Lpa

M,

M2

= the tensile stress of the concrete section

= the characteristic cube strength of the concrete Jcu

fey = the limiting concrete shear stress fly = the yield strength of the longitudinal steel reinforcement in the spandrel

beam /^y = yield strength of closed ties in the spandrel beam

= the moment of inertia of the gross prestressed concrete section

= the analogous polar moment of inertia

= the Jk-factors used for determining Vu (Modified Wollongong Approach)

= the lever arm between the internal compressive and tensile resultants = the lever arms between the concrete compressive force and the respective

forces at the prestressed and non-prestressed steel levels

= distance along the tendon from the jacking end

= the negative yield moment of the slab over the front segment of the

critical perimeter = the negative yield moment of the slab over the side segment of the critical

perimeter

xxiii

Notation

MCj = the total unbalanced moments transferred to the column centre in the main moment directions

M(22 = the total unbalanced moments transferred to the column centre in the transverse moment directions

Mcr - the cracking moment to be resisted by the section when the first crack is formed

Mm - the bending strength of the critical slab strip at the corresponding mid-

span M0 = decompression moment, that is the moment necessary to produce zero

stress in the concrete at the extreme tension fibre

MSp = pure bending for the spandrel beam Mu = ultimate moment M M V = pure bending in spandrel beam at ultimate

M = the design bending moment at the column

n = the modular ratio of steel reinforcement

n} = the number of prestressed tendons at the front face of the column in the

main bending direction n2 = the number of prestressed tendons at the side face of the column in the

transverse bending direction np = the modular ratio of prestressed tendons

pa = the force in the tendon at any point Lpa (in metres) from the jacking end

pe = the final or effective prestressing force Pel* Pe2 = tne prestressing forces being applied at the front faces of the column-slab

connections

Pe]x,Peiy = prestress components Pe2x ' P&y p. = the force in the tendon immediately after transfer p = the prestressing force imposed at the jack

P - the vertical component of the prestress usually acts in the opposite

direction to the load induced shear

s = the spacing of closed ties in spandrel beam

T = the resultant tensile force j 2 = the negative yield moment of the torsional moment over the side segment

of the critical perimeter

Tp = tensile forces in the prestressing tendon Ts = tensile forces in the reinforcing steel Tsp = pure torsion for the spandrel beam

T^ = P u r e torsion in spandrel beam at ultimate

xxiv

the critical shear perimeter

the perimeter of the rectangle defined by the centres of the four longitudinal comer bars of the spandrel beam

design shear force acting at the centre of the column the shear force over the front segment of the critical perimeter the shear force over the side segment of the critical perimeter the ultimate shear resistance of a concrete section which is uncracked the ultimate shear resistance when the section is cracked in flexure the vertical component of the effective prestress force at the section which is generally deemed to be negligible and may be ignored

pure shear for the spandrel beam the punching shear strength of reinforced and prestressed concrete flat

plates the measured punching shear strength of prestressed concrete flat plates the punching shear strength where there is no moment transfer to the

column pure shear in spandrel beam at ultimate

the horizontal dimensions for the rectangular perimeter of the rectangle defined by the centers of the four longitudinal comer bars of the spandrel

beam the vertical dimensions for the rectangular perimeter of the rectangle defined by the centers of the four longitudinal comer bars of the spandrel

beam the distance from the centroid of the critical section to the point where

VJTOZX acts

the depth of the centroidal axis of the uncracked transformed section the distance of the centroidal axis to the extreme tensile fibre

the product of the longitudinal steel ratio

the sum in radians of the absolute values of all successive angular

deviations of the tendon over the length Lpa the transverse steel ratio

the ratio of the long side to the short side of the loaded area

an angular deviation or wobble term which depends on the duct diameter

the spandrel strength parameter

the elastic concrete strain

the extreme fibre compressive strain at ultimate moment

strain of the non-prestressed tensile steel

strain measurement of the prestressed tendons

elastic strain in the prestressing tendon

xxv

Notation

^sc

<5cp

Gpu

'sc

'scu

'su

concrete strain at tendon level

the strain of tendon at ultimate load condition

g = the strain in the compressive steel reinforcement

g = the strain of non-prestressed tensile steel at ultimate

O y . 0 2 = residual proportion of prestress after deducting friction losses

0 L = the diameter of longitudinal steel bar

0 T = the diameter of closed tie

7 = the column width factor

yv = the portion of the unbalanced m o m e n t transferred to the column

7" = the ratio of the depth of the idealised rectangular compressive stress block to the depth of the neutral axis at the ultimate strength in bending

xjjf = the slab restraining factor

X = the slab thickness factor

jl = a friction curvature coefficient which depends on the type of duct

v c = a permissible shear stress carried by the concrete

* = the m a x i m u m shear stress at the critical section V 9p = an angle of inclination of the prestressing tendon

01 = an angle of inclination of the prestressing tendon in the main bending

direction

02 = an angle of inclination of the prestressing tendon in the transverse bending

direction

p = the percentage of tension steel of the slab in the direction perpendicular to

the edge cr = the normal compressive stress

Gj = principal tensile stress

<j, = the tensile stress of the section due to the applied prestress

the concrete stress at the tendon level

the ultimate stresses in the bonded tendons

cr = the stress in the compressive reinforcement

cr = the ultimate stresses in the compressive steel

cr = the ultimate stresses in the non-prestressed tensile

cr = ratio of the effective tendon prestress to the ultimate strength of tendons pe

1 = shear stress

coo = the additional transverse strength of spandrel beam due to restraining

effects of the slab tan 00 = friction loss per unit length in tendon

£,,£.- = elastic prestress losses

xxvi

Chapter I : Introduction

CHAPTER 1

INTRODUCTION

1.1 General Remarks

Flat plates are slab structures without drop panels and column capitals at the column and

slab connections. From an aesthetic and economic point of view, the flat plate structure

has the edge over other forms of slab systems due to the significant savings in

construction costs and an aesthetically pleasing sight from the slab soffit. In addition to

these, the elimination of beams and girders reduces the clear spacing of multi-storey

buildings thus creating additional floor space for a given building height. For these

reasons, flat plates are widely used for multi-storey structures such as office buildings

and car-parks. The structural capacity of the slab-column connection can be enhanced

by the introduction of spandrel beams spanning the free edges of the slab.

Besides being normally reinforced, flat plates may also be prestressed. Post-tensioned

concrete floors form a majority of all prestressed concrete construction and are

economically competitive with reinforced concrete slab structures in the popular medium

to long-span construction range.

Prestressing overcomes many of the disadvantages associated with reinforced concrete

slabs. Deflection, which is usually a major design consideration, is better controlled in

post-tensioned slabs. B y a careful control of cambers by prestressing, "deflection" may

be reduced substantially or eliminated altogether. Besides controlling the deflection,

prestress also inhibits cracking and may be used to produce near crack-free floor

systems. Prestress has also been regarded as one of the options to improve the punching

shear capacity of concrete slabs.

1


Punching shear failure is referred to the local shear failure which would occur around the

loaded area or the column support. It is one of the most critical considerations when

determining the thickness of flat plates at the column-slab intersection.

This study is focused on the reliability of the various methods of predicting the punching

shear strength, Vu , of reinforced concrete flat plates. In addition, it emphasises on the

development of a reliable prediction method for Vu of post-tensioned concrete flat plates

with or without spandrel beams.

1.2 The Prediction of Vu

The analysis of the punching shear strength, Vu , of the slab-column-spandrel

connections of reinforced and post-tensioned concrete flat plates, at the edge- and

comer-column positions, has been closely studied by engineering academics and

professionals in recent years. In so far as the development of a reliable method of

predicting Vu for a slab-column-spandrel connection is concerned, several analytical

methods have been developed and proposed for reinforced concrete flat plates.

Unfortunately, most of these methods do not encompass the effects of prestress in the

determination of Vu for post-tensioned concrete flat plates. Those which do seem not to

work satisfactorily. Therefore a detailed study is required to be undertaken so as to

develop a reliable analytical method for post-tensioned concrete flat plates. A general

analytical method for Vu , is said to be reliable if it can predict both the punching shear

strength of the connections as well as the accompanying failure mechanisms.

The problem associated with predicting the punching shear strength Vu , may be

summarised by posing the following questions:

• How would the strength of the spandrel beams affect the magnitude of Vul

2


• H o w would the characteristic compressive strength of concrete affect the

failure mechanism and the punching shear strength at the edge- and comer-

column positions?

• What are the effects of the spandrel strength on the mechanisms of failure?

• Would the most suitable critical shear perimeter be the same as for reinforced

concrete flat plates?

• How to incorporate the effect of prestress into the governing equilibrium

equations?

• Would the interaction of torsion, bending and shear in the spandrel beam in the

vicinity of the connection be similar for both reinforced and post-tensioned

concrete flat plates?

1.3 Aims of Research

The purpose of this research is to study the structural behaviour of post-tensioned

concrete flat plates with or without spandrel beam, with regard to their punching shear

strength capabilities. This is done by testing half-scale post-tensioned concrete flat plate

models to failure under uniformly distributed loads. From the results of these laboratory

experiments, essential data may be obtained for establishing an accurate prediction

procedure for the punching shear strength of post-tensioned flat plates.

An analytical procedure is to be developed for the evaluation of the punching shear

strength of corner and edge column-slab connections of post-tensioned concrete flat

plates with spandrel beams or torsion strips. The proposed procedure is an extension to

3


an earlier analytical approach developed by Loo and Falamaki [1992] of which the scope

of research was focused on reinforced concrete flat plates with spandrel beams or

torsion strips. This method is also referred to as the "Wollongong Approach".

In addition, a separate comparative study is also made on the accuracy and reliability of

the Wollongong Approach in predicting the punching shear strength of reinforced

concrete flat plates. In this study, the Wollongong Approach is compared to three

national concrete codes of practice, namely AS 3600-1988 [SAA, 1988], ACI318-89

[ACI, 1989] andBS 8110:1985 [BSI, 1985].

Based on the Wollongong Approach, a parametric study is made on how the spandrel

beam in terms of width, depth and amount of longitudinal steel reinforcement, affects the

spandrel strength parameter. This is used in turn to study the effects of the spandrel

beam on the Wollongong Approach in determining the punching shear strength of

reinforced concrete flat plates.

From the collected experimental data on post-tensioned concrete flat plates, a study is

also carried out to incorporate the effect of prestressing into the above analytical

procedure to evaluate the punching shear strength of post-tensioned concrete flat plates

with or without spandrel beams.

1.4 Scope of Research and Experimental Work

The research is based mainly on the analytical procedure for punching shear strength of

reinforced concrete flat plates detailed in the Wollongong Approach [Loo and Falamaki,

1992]. This is aided by experimental work conducted on post-tensioned concrete flat

plate models with spandrel beams.

4


The flat plate models are constructed to a half-scale representation of a typical flat plate

floor structure. The continuity of each slab model is closely simulated by having two

panels supported by six columns. The focus of this study is on the comer and edge

connections of the models which were of post-tensioned concrete construction with

bonded tendons.

The models are closely studied in their failure mechanisms in punching shear at the

column-slab and column-slab-beam connections. The deflections at every loading stage

up to failure and the crack patterns of the models are also studied and discussed.

The plan view of the half-scale flat plate floor system is presented in Fig. 1.1 whereas the

experimental flat plate models are illustrated in Fig. 1.2 . In Fig. 1.2 are shown the

loading points which simulate the uniformly distributed loads and the locations where

vertical deflections are measured.

5

Chapter 1: Introduction

rt -e-• • • •

• • • •

• • • •

• • • i "

• F B G - H • • r

.A _B ,

2700 -««« m>-

2700 -* m~-

,c

m

rI

1

2700 - ^ tm-

.D

2701 -mt

1

•

•

Jj

F,

3 % * - •

r O in so CN

O in VO CN

O ' in VO CN

o l

m vo CN

in vo CN

1'

o3 I

Cuj

O • i—i ut—>

o CD

00

Fig. 1.1 - Typical half-scale flat plate floor

6


Section a-a

.80

80 J t160

.80 U22JT-20

Models Pi and P4

Model P2

r

OO <N

U 2

Column

400 x550

7

_ 400 x400

-H250H* T Model P3

6

NQ

2

Column H H

200 x 400 -

5

3

•400x 300

o m

1 o uO

B|-400x 300

H H |

• EL

Spandrel beam $ or

torsion strip

o

725. 1325 1325

3525

200 x -300 •

o o r-

im

Whiffle-tree loading arrangement at the bottom of the slab

^ II ' II

t

Legend

4 Locations for deflection • measurement

gj Loading points for testing

CS Column strips

Note: All measurements are in milimetres

Downward direction of applied loadings

Fig. 1.2-Test models

7


1.5 Theoretical Background

1.5.1 Punching shear

In the design of slabs and other similar stmctures such as footings, the shear strength

often determines the required thickness of the member, especially in the vicinity of a

concentrated load or a column support [Gilbert and Mickleborough, 1990].

Consider a column-slab connection shown in Fig. 1.3 . Shear failure may occur on one

of two critical sections. The slab may act essentially as a wide beam and shear failure

may occur across the entire width of the member, as illustrated in Fig. 1.3a . This is

beam-type shear (or one-way shear) failure and the shear strength calculations are

similar to that of a beam. The critical section for this type of shear failure is usually

assumed to be located at a distance equal to the effective depth of the slab, from the face

of the column or concentrated load. Beam-type shear is often critical for footings but

rarely causes concern in the design of floor slabs.

An alternative type of shear failure may occur in the vicinity of a concentrated load or

column. This is illustrated in Fig. 1.3b . The slab fails in a local area around the column,

which forms a truncated cone or pyramid caused by the critical diagonal tension crack

around the column [Park and Gamble, 1982]. This is known as punching shear failure

(or two-way shear failure). The mechanism of punching shear failure may be described

as follows.

The first crack to form in the slab when subjected to applied load, is a roughly circular

tangential crack around the perimeter of the column due to negative bending moments

in the radial direction. This is followed by radial cracks (due to negative bending

moments in the tangential direction) which extend from the perimeter of the column (see

8

Chapter 1 : Introduction

Fig. 1.3b). These diagonal tension cracks tend to originate near the mid-depth of the

slab, and are therefore more similar to web-shear cracks than flexure-shear cracks. The

stiffness of the slab surrounding the cracked region tends to preserve the shear transfer

to the column by aggregate interlock at higher loads. Punching shear failure may occur

eventually, when the applied load is high enough to overcome the slab stiffness in the

vicinity of the cracked region.

u.uuiu.un

Assumed critical section t

z

(a) Beam-type shear failure

Assumed critical section

PWWi t

Section

Actual failure mode

Radial cracks

Tangential cracks

(b) Punching shear failure

Fig. 1.3 - Types of shear failures in a slab


The critical section assumed in some concrete codes [SAA, 1988 and ACI, 1989], and

the cracks that occur in the concrete in the actual failure mode, are shown in Fig. 1.3b .

The critical section for punching shear is usually taken to be geometrically similar to the

loaded area and located at a distance half the effective depth away from the face of the

loaded area [SAA, 1988 and ACI, 1989], although the distance may be specified as 1.5

times the effective depth [BSI, 1985]. The critical section (or failure interface) is

assumed to be perpendicular to the plane of the slab.

1.5.2 Theoretical aspect

The theoretical aspect of this research is aimed at developing a prediction procedure for

the punching shear in post-tensioned concrete flat plate. Following the work by Loo and

Falamaki [1992], there are three groups of undetermined forces at the column-slab-

spandrel connection in a flat plate. These forces are shown acting on free-body

diagrams in Fig. 1.4 and are briefly described below.

(a) Mj and V2

These are internal actions to be measured experimentally and for which semi-empirical

formulas have to be developed. They are the negative yield moment of the slab and the

shear force respectively, over the front segment of the critical perimeter.

(b) MCi and MC2

These are actions to be computed using known structural analysis procedures. They are

the total unbalanced moments transferred to the column centre in the main and

transverse moment directions respectively.

10


Spandrel beam

Column

Critical perimeter

Spandrel beam

Column

Critical perimeter

(a) Corner connection (b) Edge connection

(c) Typical section

Fig. 1.4 - Free body diagrams for edge and corner connections

3 0009 03132123 0

11


(c) M2,V2,T2, and Vu

These are actions to be determined in conjunction with (a) and (b) using the three

equilibrium equations plus an interaction equation for torsion, shear and moment acting

on the spandrel or torsion strip [Loo and Falamaki, 1992].

Note that M2 , V2 and T2 are the negative yield moment of the slab, shear force and

torsional moment respectively over the side segment of the critical perimeter. Finally, Vu

is the punching shear strength of the slab.

In order to establish the interaction equation in (c), the following parameters, ©0 and \\i

have been determined previously [Loo and Falamaki, 1992]. While co0 is the additional

transverse strength of spandrel beam (if any) due to restraining effects of the slab, \\i is

the slab restraining factor.

In order to fully develop the prediction procedure for the punching shear strength Vu ,

semi-empirical formulas have to be established for M;, V}, and \\i .

The main objective of the ultimate load tests of a series of nine half-scale reinforced

concrete flat plate models undertaken by Falamaki and Loo [1992] was to provide

experimental data for setting up these semi-empirical formulas. This was similarly

carried out for post-tensioned concrete flat plate models, in which the semi-empirical

formulas for \i7 was re-calibrated to take the effect of prestress into consideration in the

determination of Vu .

Further discussion is undertaken in Chapter 5 on how to incorporate the effects of

prestress into the punching shear prediction approach introduced by Loo and Falamaki

[1992],

12


1.6 Overall Layout of Thesis

Following the introduction presented in this chapter, Chapter 2 gives a review of current

and past literature relevant to the research on punching shear, Vu , of reinforced as well

as post-tensioned concrete flat plates with or without spandrel beams. Based on

published results, a comparative study is made in Chapter 3 on the performance and

reliability of the current methods of analysing Vu . The methods considered are the

Australian Standard AS 3600-1988 [SAA,1988], the American Concrete Institute

ACI318-89 [ACI,1989], the British Standard BS 8110:1985 [BSI, 1985] and the

Wollongong Approach [Loo and Falamaki, 1992].

A parametric study is carried out in Chapter 4 on the effects of varying some variables of

the spandrel beam, on the spandrel strength parameter, 8 . This is undertaken as a step

towards determining the effects on the punching shear, Vu (in the Wollongong

Approach) for reinforced concrete flat plates. From this study, the limits of the

parameter, 8 , are defined for the applicability of the prediction procedure for analysing

Vu-

In Chapter 5, an overview is made as to how the effects of prestress are included in the

analysis of Vu of post-tensioned concrete flat plates by the major codes of practice

mentioned above. A theoretical approach that incorporates the effects of prestress into

the Wollongong Approach is also introduced. The effects are taken into consideration in

the three basic equilibrium equations and the interaction equation for the forces acting at

the column-slab-spandrel connections. In this chapter, the semi-empirical equations

from the Wollongong Approach are also described and discussed with emphasis on

applicability to post-tensioned concrete flat plate models.

13


The overall experimental approach and the construction procedure for the post-

tensioned concrete flat plate models are described fully in Chapter 6 and Chapter 7

respectively. The test results obtained for all the post-tensioned concrete flat plate

models are presented in Chapter 8. These include crack propagation, the deflections and

the measured punching shear strengths.

The determination of the ultimate flexural strengths (based on the measured strains of

steel and tendons) of the critical slab strips [Falamaki and Loo, 1992] of post-tensioned

concrete flat plates are shown in Chapter 9. In this chapter, a comparison is also made

between the measured and calculated moments at ultimate for the critical slab strips.

From this comparison, the reliability of using the measured moments in analysing Vu , is

determined.

The semi-empirical formulas to determine the slab restraining factor, \j/ for the prediction

of Vu , in post-tensioned concrete flat plates are calibrated in Chapter 10, using the

method employed by Falamaki [1990]. Using these calibrated formulas and the available

test data, the punching shear strength, Vu for both comer- and edge-column

connections are determined in Chapter 11. All the predicted Vu results using the

proposed approach (or the Modified Wollongong Approach) for post-tensioned

concrete flat plates are then compared with the experimental results to determine the

accuracy of the proposed prediction procedure. In this chapter, comparisons are also

made on the reliability of the analytical methods recommended by the Australian,

American and British concrete codes for the punching shear strength of post-tensioned

concrete flat plates.

Finally, the conclusions on the outcome of the theoretical and experimental aspects of

the study of the punching shear strength of flat plates are presented in Chapter 12

together with recommendations for further study.

14

Chapter 2 : Literature Review

CHAPTER 2

LITERATURE REVIEW

2.1 Background of Previous Work

The earliest code recommendation on flat slabs was an empirical design method included

in the 1920 ACI Code. Some designers viewed this empirical approach with scepticism

as it may not have been based on valid experimental results, although some revisions of

this method have been included in subsequent codes in many countries including the

previous British Code CP110:1972.

In order to overcome the limitations on the applicability of such empirical methods, an

alternative approach to slab design was developed and introduced in the Californian

Code of 1933. With some minor modifications, it became the now widely known

equivalent frame method which was incorporated into the ACI codes from 1941 to

1963.

A major reconsideration of slab design was undertaken in America in the late fifties and

early sixties. A series of small-scale slabs, each comprising nine 1.5 m x 1.5 m square

panels, was tested at the University of Illinois [Sozen and Siess, 1963]. It included a flat

slab without column capitals or drop panels i.e. a flat plate, but with beams at all free

edges and two flat slabs with capitals and drop panels. In addition, a larger scale flat

slab without capitals or drop panels was also tested by the Portland Cement Association

[Guralnick and LaFraugh, 1963].

These experimental studies involved careful and thorough instrumentation, which

together with complementary analytical studies, gave rise to the improved equivalent

15


frame method of design used in the following and current A C I code [ACI, 1989]. This

approach unifies the design of all two-way slab systems and in respect of flat slabs,

introduces an allowance for the relative flexibility of the slab-column connections.

This research was a significant step towards an improved slab design approach although

since then further tests on multi-panel slabs have been undertaken with no significant

advances in the design methods.

Throughout the period of the development of flat slabs, there has been a secondary type

of experimental work concerned with local conditions at the slab-column junctions and

aimed largely at establishing expressions for resistance to punching shear. Thirty-nine

reinforced concrete slab models of 1.8m x 1.8m size were tested at the University of

Illinois by Elstner and Hognestad [1956], from which 34 models failed in punching

shear. The test specimens have generally been intended to represent regions of slabs

around individual columns bounded by lines of radial contraflexure. The reduction of

the area of slab to be modelled has enabled the scale of these specimens to be at times

much larger than that of most multi-panel work.

Originally, attention was focused on internal columns loaded symmetrically, but more

recent studies have included internal columns (subjected to unsymmetrical loading), edge

columns and comer columns [Zaghlool et al, 1970]. The scope of work has also

extended to prestressed slabs [Brotchie and Beresford, 1967, Bums and Hemakom,

1977, Regan, 1985, and Foutch, Gamble and Sunidja, 1990] and lightweight slabs

[Beresford and Blakey, 1963]. Tests of these types are the basis for the code of practice

equations for punching shear resistance. These equations have been amended and

extended as more data became available. As an example of this process, AS 3600-1988

[SAA, 1988] is the first Australian code to include provisions for spandrel beams in flat

plates.

16


Until very recently, there have been two separate strands of development: tests of multi-

panel structures and tests of isolated slab-column connections.

The former have the disadvantage of requiring a very large amount of work in order to

obtain essentially a single test result in terms of ultimate load behaviour. Testing

techniques are unavoidably complicated if distributions of moments in the slabs and

reactions and moments in the columns are to be determined with any degree of accuracy.

The specimens are either relatively large or built to such a small scale that there must be

doubts as to the quantitative significance of the results obtained.

The latter type of specimen has the drawback of having arbitrary boundary conditions

and may not represent the behaviour of a similar region in a real structure, where lines of

contraflexure can shift as the loading varies. In all cases where unbalanced moments are

transmitted from the slab to a column, there is a possibility that the specimen may fail in

flexure when the moment resistance is fully utilised. On the other hand, in a real flat-

plate, the moment might remain constant as the shear increased with further loading.

Due to these limitations inherent in past testings, since then there have been several

research projects using fairly large scale specimens (of size ranging from 3m x 1.8m to

3m x 3m), which may be termed "partial structures". Slabs supported by four corner

columns have been tested by Zaghlool et al [1970] and Ingvarsson [1974]. Kinnunen

[1971] and Long et al [1978] have used specimens supported by two opposite edge

columns or by one edge and one internal column. Models of internal regions extending

from a column to lines of maximum span moments have also been tested by Vanderbilt

[1972] and Long and Masterson [1974].

A series of laboratory tests were conducted by Rangan and Hall [1983 a] and Rangan

[1987] on half-scale models of edge panels of a reinforced concrete flat plate floor.

17


These were the basis for the punching shear design provisions contained in the

Australian Standard for Concrete Stmctures, AS 3600-1988 [SAA, 1988] for reinforced

as well as prestressed concrete flat plates.

2.2 Codes of Practice

There are existing codes which cover the methods of analysis of the punching shear in

reinforced as well as post-tensioned concrete flat plate structures. Among them are the

Australian Standard AS 3600-1988 Concrete Structures [SAA, 1988] and the American

Concrete Institute ACI 318-89 [ACI, 1989].

The punching shear strength analysis and design of reinforced and prestressed concrete

flat plates with spandrel beams are not included in the American Concrete Institute code

of practice [ACI, 1989]. The current Australian Standard AS 3600-1988 [SAA, 1988]

provides a method for computing the punching shear strength of both reinforced and

prestressed concrete flat plates with torsion strips and spandrel beams. The method for

prestressed concrete flat plate is a simplistic extension of the procedure developed for

reinforced concrete structures. This was done by including the nominal prestressed

concrete stress in the relevant equations. The accuracy and reliability of these equations

have not been verified although the basic procedure for reinforced concrete flat plates

was shown to be at times grossly inadequate [Loo and Falamaki, 1992].

Other national code of practice for concrete stmctures also reviewed herein is the British

Standard BS 8110:1985 [BSI, 1985]. The recommendations for the method of

determining the shear strength of prestressed beams are also applicable to prestressed

slabs in the British Standard method, whereby consideration is given to uncracked or

cracked sections in flexure. An analytical method recently developed by Loo and

18


Falamaki [1992] for predicting the punching shear of reinforced concrete flat plates is

also discussed along with other relevant literature on this topic.

Experimental work on post-tensioned concrete flat plates undertaken by researchers

which have been published in Australia, United Kingdom and North America such as

ACI Stmctural Journal are reviewed in Section 2.4 . The models that were tested and

the reported findings of their stmctural behaviour at failure loads are also discussed in

Section 2.5 .

2.3 Methods of Punching Shear Strength Analysis

2.3.1 Australian Standard AS 3600-1988 [SAA, 1988]

The provisions for punching shear in the Australian Code have been developed from the

results of a series of laboratory tests conducted by Rangan and Hall [1983a] on half-

scale reinforced concrete edge column-slab specimens.

The extension of Rangan and Hall's proposals to cover prestressed concrete slabs has

been described as logical and simple [Gilbert and Mickleborough, 1990]. The

procedures which are based on a simple model of the slab-column connection will be

discussed in more details in Chapter 3 (Section 3.3.1) for reinforced concrete slabs, and

in Chapter 5 (Section 5.1.2a) for prestressed concrete flat plates.

2.3.2 American Concrete Institute ACI318-89 [ACI, 1989]

For prestressed concrete slabs under combined bending and shear, the ACI code

recommends an analytical method which is a modified form of the procedures for

19


reinforced concrete slabs. These procedures are based on the proposals of Di Stasio

and Van Buren [I960].

Gilbert and Mickleborough [1990] stated that although the ACI approach provides

reasonable accuracy with test results [Hawkins, 1974], it is essentially a linear working

stress design method and has fallen from favour over the past 20 years. Such

approaches are described as irrational, too cumbersome and difficult to use, particularly

in the case of edge and comer columns both with or without spandrel beams.

However, it is noted that the ACI method does not specify any provisions for the

presence of spandrel beams in reinforced and prestressed concrete slabs.

2.3.3 British Standard BS8110 1985 [BSI, 1985]

In the current British code, the effect of prestressing on punching shear is not clearly

specified for flat plates although it mentioned that the approach is similar to method of

determining shear resistance for prestressed concrete beams.

2.3.4 The Wollongong Approach [Loo and Falamaki, 1992]

The scope of the Wollongong Approach did not include the effect of prestress in the

determination of the punching shear strength of flat plates. One of the aims of this study

is to incorporate the prestressing effect into this approach.

20


2.3.5 Others

The European CEB-FIP Model Code [1978] has provision for determining the punching

shear at a distance of half the effective depth from the face of the loading area. This is

consistent with the Australian and the ACI Codes.

However, the formula recommended can only be used for circular columns with

diameter not exceeding 3.5 times the effective depth. It may not be applicable for

rectangular columns with perimeter exceeding 11 times the effective depth and with a

ratio of length to breadth which exceeds 2.0. It considers the prestress as a permanent

compressive force whereby the structure is then treated as a reinforced section subject to

an applied axial force to enhance its shear resistance.

Some other researchers have developed formulas for calculating the punching shear of

flat plates. Most of them are based on the existing codes of practice. For example,

Regan [1981] has developed an expression which is similar to the British Standard but

with the critical perimeter located 1.25 times the effective depth away from the column

faces. In his approach, the influence of prestress on the punching shear strength is

accounted for by adding the decompression load to the punching resistance of a

geometrically similar slab without prestress. The decompression load is the load required

to annul the effect of the prestress, i.e. to give a zero stress at the extreme fibres at the

top of the slab [Regan, 1985].

Chana [1991] suggested that the simplest method of introducing the effect of prestress

on the punching shear resistance is through the use of an effective reinforcement ratio.

The actual ratio of prestressed reinforcement is converted to an equivalent ratio of

ordinary steel reinforcement in the calculation for punching shear strength. This is

similar to the original proposal by the Concrete Society [1979], whose approach has

21


been described as "optimistic" as it considers that the applied prestress can still achieve

its ultimate state prior to the punching shear failure of the slab.

2.4 Previous Experimental Work on Post-Tensioned Concrete Flat Plates

2.4.1 J.F. Brotchie and F.D. Beresford [1967]

Brotchie and Beresford [1967] tested a large scale ( 8 m x 14 m) flat plate model with

15 columns (5x3 arrangement) at 3 m spacing. The slab was prestressed by draped

unbonded tendons without having any non-stressed steel reinforcement.

The model was tested to failure under uniformly distributed load and sustained a

flexural failure. The shear capacity of the model was sufficient to prevent a punching

shear failure.

2.4.2 N.H. Burns and R. Hemakom [1977]

Bums and Hemakom tested a one-third scale flat plate model having 16 columns (4x4

arrangement). It was also prestressed by unbonded tendons. The physical behaviour of

the model was observed up to the point of collapse.

Punching shear failure occurred only at the interior columns whereas the exterior

columns failed by developing positive and negative yield lines near the supports and

midspan respectively. Extensive cracks were recorded around the interior columns but

not around the exterior columns.

22


The authors concluded that the shear strength of the slab was affected by the concrete

compressive stress (P/A), the amount of bonded reinforcement and the tendon

arrangement.

2.4.3 P.E. Regan [1985]

Regan [1985] carried out and reported fifteen tests of bonded post-tensioned concrete

slabs of 180 mm to 225 mm thick, and of 3 m x 1.5 m size. These models were

prestressed only in one direction and normally reinforced in the transverse direction.

Concentrated upward loads were applied through steel plates located at the centre of the

models, while reactions were provided by four tie bars close to the short edges. All but

three of the tests were reported to have failed in punching shear, and that the applied

prestress have contributed to the punching shear strength of the models.

2.4.4 D.A. Foutch, W.L. Gamble and H. Sunidja [1990]

Foutch et al [1990] conducted tests on four two-third scale single-column flat plate

models which were prestressed with unbonded tendons.

Two of their models failed in flexural mode while the other two failed in shear. They

concluded that these tests support the use of prediction formulas for shear stress which

includes a contribution from the effect of prestress for an edge column-slab connection.

The current ACI code only allows the effect of prestress to be taken into account for

interior columns.

23


2.4.5 A.E. Long and D.J. Cleland [1993]

Long and Cleland [1993] tested five 14-scale model slabs that closely simulated an

exterior panel for the purpose of studying the failure characteristics of edge panels in

post-tensioned unbonded slabs. All the models are of identical size and supported by

two edge columns but with varying levels and distribution of prestress.

All the models were reported to have failed in a punching shear mode and it was found

that higher prestress levels increase the ultimate punching shear capacity.

2.5 Comments on Previous Experimental Work

The models tested by all the researchers mentioned in Section 2.4 have one common

feature: they were post-tensioned by unbonded tendons, as allowed by the ACI318-89

[1989]. The Australian Standard AS3600-i988 does not recommend unbonded tendons

in post-tensioning. All unbonded tendons have to be subsequently grouted after full

prestressing of the tendons.

Another common oversight by the previous experiments on post-tensioned concrete flat

plates is the effect of having spandrel beams along the edge of the slab on the punching

shear strength of the column-slab-spandrel connections. Further, punching shear failure

behaviour at comer- and edge-column positions was not the main subject of these earlier

studies.

The present thesis research covers both these topics which were neglected by previous

researchers.

24

Chapter 3 : Performance of Analytical Methods

CHAPTER 3

PERFORMANCE OF ANALYTICAL METHODS FOR PUNCHING

SHEAR STRENGTH OF REINFORCED CONCRETE

FLAT PLATES

3.1 General Remarks

The punching shear strength df concrete flat plates is one of the topics of intensive

research in recent years by various concrete stmctures researchers. This chapter

reviews four current methods of analysing the punching shear strength at the corner-

and edge-column positions of reinforced concrete flat plates. They include those

recommended in the Australian Standard AS3600-1988 [SAA, 1988], the American

Concrete Instimte ACI318-89 [ACI, 1989] and the British Standard on Concrete

Practices (BS8110) [BSI, 1985] as well as the approach developed at the University of

Wollongong, Australia [Loo and Falamaki, 1992]. Based on half-scale model test

results, a comparative study of these four analysis methods is made with regard to their

Umitation, accuracy and reliability. It is found that the Wollongong Approach in general

gives the best performance in predicting the punching shear strength of flat plates with

torsion strips and those with spandrel beams. The Australian Standard procedure

performs just as satisfactorily for flat plates with torsion strips but tends to be unsafe for

those with spandrel beams. Both the A C I and the British methods are applicable only to

flat plates with torsion strips; they also tend to give unsafe predictions for the punching

shear strength.

3.2 Objectives and Scope

Various major national codes of practice for concrete stmctures have provisions for the

25


analysis of punching shear strength of concrete flat plates. However, there appears to be

no published material available on the relative accuracy and reliability of these methods

of analysis. The relative merits or otherwise of the various methods would be useful for

the design engineers, especially those who practise internationally.

A brief review is given herein of the analytical method recommended in

the Australian Standard AS3600-1988: Concrete Stmctures [Standard

Association of Australian, S A A 1988];

• the American Concrete Institute, Building Code Requirements for Reinforced

Concrete Stmctures A C I 318-89 and Commentary - A C I 318R-89, [ACI,

1989];

the British Standard B S 8110: Part 1, "Code of Practice for Design and

Construction" , [British Standards Institute, BSI 1985].

In addition, an analytical method recently developed by Loo and Falamaki [1992] (the

Wollongong Approach) is also discussed.

Based on- the half-scale model test results published by Rangan and Hall [1983a],

Rangan [1990b] and Falamaki and Loo [1992], the relative accuracy and reliability of all

four methods of analysis are investigated. The comparison is carried out by plotting the

experimental punching shear strength values against the predicted values due to each of

the four methods, where applicable. Pearson correlation coefficients are also obtained to

illustrate the strength of the relationship between each prediction method and the

experimental values.

Other standard statistical indicators such as maximum, average and standard deviations

as well as frequency distribution are also employed to determine the relative

26


performance of the four analytical methods in question. The comparative study provides

considerable insight into the relative merits or otherwise of the four methods of analysis.

3.3 Methods of Analysis - A Review

Of the four methods included in this study, the Australian Standard procedure and the

Wollongong Approach cover reinforced concrete flat plates with spandrel beams or

torsion strips whereas the A C I and the British methods are limited to cases with free

edges only (i.e. torsion strips without closed ties). These methods are briefly described

in the following sections.

3.3.1 Australian Standard Procedure [SAA, 1988]

The Australian Standard procedure is mainly the work of Rangan [1990a]. It is based

heavily on the experimental study conducted by Rangan and Hall [1983a] on half-scale

models of the edge panels of flat plate floors. According to Rangan [1987], the

equations developed are deemed to consider the following cases :

(a) a slab in the vicinity of an interior column, without spandrel beams and closed

ties;

(b) a slab in the vicinity of an edge column, with a spandrel beam on the side

provided with minimum closed ties;

(c) a heavily loaded slab in the vicinity of an edge column, with a spandrel beam as

in part (b) above but having more than the minimum closed ties;

(d) a slab in the vicinity of an edge column, without spandrel beam and closed ties,

i.e. a free edge.

The analysis procedure for an edge-column connection is assumed valid for a corner-

27


column connection.

When the moment to be transferred is zero, the equation for punching shear only

incorporates four parameters, namely the critical perimeter, the effective depth of slab,

the column dimensions and the compressive strength of concrete. The critical shear

perimeter is defined as the (failure) section half the effective depth away from the faces

of the column.

The punching shear strength where there is no moment transfer is given as:

Vuo = udfcv ...(3-1)

where u is the critical shear perimeter, d is the effective depth of the slab and fcv is the

Umiting concrete shear stress for which relevant formulas are given in the Australian

Standard.

When there is moment transfer to the column, in addition to the above-mentioned

parameters, the equation also considers the total unbalanced moment, and the shear

force to be transferred to the column support.

In the presence of spandrel beam and closed ties, other parameters taken into account

include the dimensions and yield strength of closed ties, as well as the area and perimeter

defined by the longitudinal bars at the four comers of the closed ties.

For a typical slab with moment transfer to the column, and having a spandrel beam with

the minimum requirement of closed ties, the punching shear strength is given as:

28


1-2 [ ] Vm Vu = —* -C

3"2) M u

[ 7 + — ^ ] 2 y*abw

where (DiJDs) is the ratio of the depths of the beam and the slab; M and V are

respectively the design bending moment and shear force at the column; a is the segment

length of the critical shear perimeter parallel to the vector direction of My and bw is the

width of the spandrel beam.

This approach has been derived based on two assumptions [Rangan, 1987] :

(a) the distribution of the shear stress is taken to be uniform along the critical

perimeter;

(b) for both the edge-column and comer-column positions, a constant slab

restraining factor of 4 is adopted for the slab-column connections with spandrel

beams or torsion strips with closed ties.

Some shortcomings of these assumptions have been identified and discussed by Loo and

Falamaki [1992]. For further details of this method, the reader is referred to the

Australian Standard [SAA, 1988].

3.3.2 The ACI Method [ACI, 1989]

The method recommended in the ACI code for punching shear design under combined

shear and moment is mainly based on the work of Di Stasio and Van Buren [I960]. In

this method, the moment is assumed to be transferred to the column by flexure of the

slab and by torsion of the slab edge strip or torsion strip [Cope and Clark, 1984]. Also,

29


the moment is transferred along a critical section around the column, similar to the

Australian Standard procedure, over which the punching shear failure is assumed to

occur.

In the presence of an unbalanced moment, consideration is given to the effect of torsion

by applying an analogous polar moment of inertia of the critical section, and a calculated

fraction of the net moment to be transferred to the column by this torsion. The

maximum shear stress at the critical section is expressed as:

* = r JvMly (3_3) A J

where A is the area of the critical section, / is the analogous polar moment of inertia,

Y v is the portion of the unbalanced moment transferred to the column and y is the

distance from the centroid of the critical section to the point where Vmax acts.

In addition, the ACI method stipulates that the maximum shear stress vL* is not to

exceed a permissible shear stress, vc carried by the concrete which is:

vc = ( 0.166 + ^-)ifc (< 0.33 VZ) -O-4)

where Pc is the ratio of the long side to the short side of the loaded area.

Note that in this approach, the effects of the torsion strips with or without closed ties

are taken into consideration but those of the spandrel beams cannot be included.

Furthermore, according to ACI-ASCE Committee 426 [1977], the predicted results are

safe where the ratio of the long and short sides of a rectangular column or loaded area

30


is higher than 2.0 .

In this simplistic approach, the main parameters are the compressive strength of concrete

and the dimensions of the critical perimeter. It does not take other factors into account

such as the slab restraining effect and the transverse strength of the torsion strip or the

spandrel beam.

3.3.3 The British Standard Method [BSI, 1985]

The general approach in determining the punching shear strength recommended in

BS8110:1985 is similar to the previous British Concrete Code CP 110:1972 [BSI,

1972]. The values for the design nominal shear stresses in CP110 are based on the

recommendations of the Institution of Stmctural Engineers Shear Study Group [1969].

As in the Australian Standard, it also assumes that the distribution of shear around the

column is uniform, although this is not the case in general [Loo and Falamaki, 1992].

According to this method, the punching shear stress is given as

^c=C^Tu^f- .-(3-5)

where p is the percentage of tension steel of the slab in the direction perpendicular to the

edge and/CM is the characteristic cube strength of concrete.

The relationship between the compressive cube strength and compressive cylinder

strength is presented in Fig. 3.1. The cylinder strengths are between 76% to 96% of the

corresponding cube strengths [Neville, 1978].

31


It is important to note that the compressive cylinder strength results obtained for the

reinforced concrete models tested in Wollongong and the University of New South

Wales are to be converted to their respective compressive cube strengths, before

applying the British Standard method in determining their punching shear strengths.

In contrast to the Australian and the ACI codes, BS8110 takes the shear failure

perimeter to be a rectangle with its sides at 1.5 times the average effective depth from

the column faces. The parameters included in calculating the punching shear are the

steel ratio of slab, effective depth and cube root of the compressive strength of concrete

(where the concrete strength must exceed 25 MPa). Further allowance is made in the

form of a reduction factor of 1.25 to be applied to the calculation in determining the

punching shear strength.

r % OS

% * « — " ^ OO <3 l>

m V

J - >

5» O

> ca m 4J rl PH

a o u

60

SO

40

30

20

10

-

-

-

/

V

c

y /

i

/

f

a/

f

1 1

/

1

0 10 20 30 40 50 60

Compressive cylinder strength (MPa)

Fig. 3.1 - Relative strength of concrete cubes and cylinders

As in the ACI method, the effects of spandrel beam on the punching shear strength

cannot be accounted for in the BS8110 procedure. There are other limitations :

32


(a) compressive strength of concrete not to exceed 40 M P a ;

(b) effective depth of slab not to exceed 400 m m ;

(c) main steel ratio of the slab not to exceed 3%.

Similar to the ACI approach, the British method is also considered simplistic in that

factors such as the restraining factor of the slab and the transverse strength of the torsion

strip which are not taken into consideration.

3.3.4 The Wollongong Approach [Loo and Falamaki, 1992]

The Wollongong Approach is the result of a comprehensive research into the punching

shear strength of reinforced concrete flat plates. Applicable to flat plates with spandrel

beams or torsion strips, the approach comprises explicit formulas through which the

punching shear strength at the comer- and edge-column connections can be determined.

A detailed presentation of the method has been made elsewhere [Loo and Falamaki,

1992] and it is not repeated herein. However, it may be noted that the Wollongong

Approach takes full account of the following important parameters and effects that were

neglected or left out in the other three methods :

(a) The slab restraining factor is not a constant but a function of the strength of the

spandrel beam.

(b) The distribution of shear force around the critical perimeter is not uniform but a

function of the slab reinforcement ratio, clear span of the slab, and the size of the

spandrel beam in relation to the slab.

(c) The interaction of torsion, shear and moment in the spandrel beam.

(d) Flat plates with deep spandrels have a bending mode of failure very much

different from those with shallow beams and they are carefully defined.

33


3.4 Presentation and Comparison of Results

3.4.1 Model tests

As part of a comprehensive study of the punching shear strength of concrete flat plates,

a total of nine half-scale reinforced concrete models were constmcted and tested to

failure over a period of three years (from 1987 to 1990). Test program of such a scale is

not believed to have been attempted elsewhere previously. Full details of this

experimental work and its results have been reported by Falamaki and Loo [1992]. For

ease of reference, the plan dimensions of the models and a typical cross section of the

slab-column-beam connection are presented in Fig. 3.2a and 3.2b respectively.

-O

Column

Column

F G _

X

A B

H

C

i

\

i

y

725

2650

.

3

l

•««£

<

2700

540C

X

Imm

2700 mm -• >

"••>

(a) Plan

3375

Fig. 3.2 - Flat plate model Wl with a typical spandrel beam section details

(continued next page)

34


Longitudinal reinforcement Slab reinforcement

Slab

Closed ties

(nominal reinforcement)

Spandrel beam

(b) Section x-x

Fig. 3.2 - Flat plate model Wl with a typical spandrel beam section details

(continued from previous page)

Table 3.1 - Comparison of punching shear results - models with

torsion strips

Colum

n

W5-A

M5-A

W5-B

M5-B

W5-C

M5-C

R3-A

R4-A

R90-A

R90-B

R90-C

_ * Type

C/NS

C/NS

E/NS

E/NS

C/NS

C/NS

E/NS

E/NS

E/NS

E/NS

E/NS

Mtuasured

\

(kN)

32.59

34.42

71.24

87.81

37.30

38.36

88.76

75.42

120.20

108.20

21.90

A S 3600-1988

Predicted

(kN)

37.01

40.70

53.04

49.79

43.66

42.82

87.23

85.03

114.95

148.35

24.12

Mean:

MeasmssJ Predicted

0.88

0.85

1.34

1.76

0.85

0.89

1.02

0.89

1.05

0.73

0.91

1.015

ACI318-89

Predicted v«

(kN)

57.50

69.95

91.18

144.23

80.22

90.23

130.17

103.84

108.02

136.38

58.58

Measured Predicted

0.57

0.49

0.78

0.61

0.46

0.43

0.68

0.73

1.11

0.79

0.37

0.638

BS8110:1985

Predicted

(kN)

41.20

64.32

80.05

173.66

53.27

77.49

166.60

127.85

130.88

138.43

59.22

Measured Predicted

0.79

0.54

0.89

0.51

0.70

0.50

0.53

0.59

0.92

0.78

0.37

0.647

Wollongong

Predicted V u

(kN)

35.21

29.37

74.62

94.21

35.07

53.91

94.90

83.73

76.60

144.24

23.00

Measured Predicted

0.93

1.17

0.95

0.93

1.06

0.71

0.94

0.90

1.57

0.75

0.95

0.987

C-corner column; E-edge column; NS-without spandrel beam.

35


Table 3.2 - Comparison of punching shear results - models with

spandrel beams

Column

R90-D

Wl-A

W2-A

W3-A

W4-A

Wl-B

W2-B

W3-B

W2-C

W3-C

W4-C

M2-A

M3-A

M4-A

M2-B

M3-B

M4-B

M3-C

M4-C

Type**

E/S C/S

C/S

C/S

C/S

E/S

E/S E/S C/S C/S

C/S

C/S C/S

C/S

E/S

E/S

E/S C/S

C/S

Measured

(kN)

36.20 0

50.15

48.08

43.38

47.07

117.63

120.36

93.57

45.17

44.33

46.32

53.90

25.70

58.97

123.22

76.50

130.24

24.30

60.09

AS 3600-1988

Predicted

Vu (kN)

51.71

119.07

120.94

70.42

96.12

146.92

150.05

94.99

113.54

73.38

82.93

82.90

127.31

114.77

116.24

214.32

137.82

131.89

102.75

Mean:

Measured

Predicted

0.70

0.42

0.40

0.62

0.49

0.80

0.80

0.98

0.40

0.60

0.56

0.65

0.20

0.51

1.06

0.36

0.95

0.18

0.58

0.593

Wollongong

Predicted

Vu (kN)

45.89

55.05

58.23

42.91

55.85

148.53

105.87

115.76

44.44

38.36

49.55

35.36

10.59

46.64

98.15

78.07

97.18

32.90

58.58

Measured

Predicted

0.79

0.91

0.83

1.01

0.84

0.79

1.14

0.81

1.02

1.16

0.93

1.52

2.42

1.26

1.26

0.98

1.34

0.74

1.03

1.027

The test to failure of these nine models, each weighing about five tonnes, provides a

total of 24 punching shear strength results, including 16 for comer-column and 8 for

edge-column positions. Also included in the present study are the punching shear

strength results from two similar models tested by Rangan and Hall [1983a] and from

four single edge-column specimens tested by Rangan [1990b]. These make a total of 16

results for comer-column connections and 14 for edge-column connections.

These test results are tabulated in column 3 of Tables 3.1 and 3.2 . Note that the

** S-with spandrel beam. 0 This specimen did not fail in punching; the reported result is the maximum value.

36


Wollongong results are those prefixed with the letter W or M and those of Rangan and

associates, with an R. Note that the four results R90-A, B, C and D are for single

column specimens. Out of these 30 results, 11 are for flat plates with torsion strips

while the rest are for those with spandrel beams of various sizes.

3.4.2 Predicted results

The four methods of analysis reviewed herein, namely the Australian Standard

procedure, the ACI method, the British Standard recommendations and the Wollongong

Approach, are used to compute the punching shear strength of the slab-column and slab-

column-beam connections of the models. Remember that the British and the ACI

methods are applicable to flat plates with torsion strips only.

Table 3.1 presents the predicted Vu values by all the four methods for the models with

torsion strips; Table 3.2 contains the predicted results by the Australian Standard

procedure and the Wollongong Approach for the models with spandrel beams. Also

included in the tables are the ratios of the measured and predicted results.

In Fig. 3.3, the measured Vu are plotted against the predicted values for the models with

torsion strips. The corresponding correlation plots for the models with spandrel beams

are given in Fig. 3.4 . Superimposed on these figures are the 30-percentile lines. In

these plots, a correlation point falling below the 45° equality line represents an

overestimate or unsafe prediction. Further, a point falling below the -30% line means

that the analysis method has overestimated the strength by more than 30%. The -30%

line may be taken as a safety criterion to reflect the Australian Standard stipulation of a

shear capacity reduction factor of 0.7 .

37


200

+30% Equality (underestimate) ,

-30% a / (overestimate)

Legend A S 3600-1988 ACI318-89 B S 8110:1985 Wollongong

0 50 100 150 200

Predicted V u (kN)

Fig. 3.3 - Measured Vu versus predicted Vu - models with torsion strips

250

200

55

> T3

P ca cd

150

lOO -

50

O

+30% (underestimate)

Equality

-30% (overestimate)

- A,

Legend A S 3600-1988

A Wollongong

1 ' 1 ' I n

O 50 100 150 200 250

Predicted Vu (kN)

Fig. 3.4 - Measured VM versus predicted Vu - models with spandrel beams

38


The Pearson correlation coefficients [Nie, Hull and Bent, 1975] for the correlation plots

in Fig. 3.3 and 3.4 are summarised in Tables 3.3 and 3.4 respectively. Note that the

closer the correlation coefficient is to unity, the stronger is the relationship between the

measured and the predicted results.

3.4.3 Comparison: Flat plates with torsion strips

In Table 3.1 for models with torsion strips, it can be seen that with a mean ratio of the

measured and predicted Vu very close to unity, either of the two Australian methods is

superior to the ACI and British methods. The ACI method, on average, overestimates

the punching shear strength by 56.7% * while the British approach overestimates by

54.6% *.

It is apparent in Fig. 3.3 that the predictions by the Wollongong Approach and the

Australian Standard procedure has one correlation point each above the +30% line.

While the former has two predictions below the -30% line, the latter has only one below

the same -30% line. Contrary to these, "the ACI procedure has 8 out of 11 correlation

points falling below the -30% line and the British method has 8 out of 11. The

frequency distributions of these points are presented in Table 3.5 .

A survey of the Pearson correlation coefficients in Table 3.3 leads to the conclusion that

the strength of the relationship between each of the four prediction methods is almost as

high as the other three. However, a reading of the standard deviation values in Table

3.4 shows that both the Australian methods deviate far less than the ACI or the British

l Percentage overestimate for ACI method = ( - 1 ) x 100% = 56.7%

0.638 1 Percentage overestimate for British approach = ( - 1 ) x 100% = 54.6%

0.647

39


method. Further, the m a x i m u m and average deviations of both Australian methods are

much smaller than the other two methods.

With a comparatively smaller deviation in addition to a safe and consistent distribution in

their Vu predictions, both the Australian methods are by far superior to those

recommended by the ACI and the British Standard for flat plates with torsion strips.

Table 3.3 - Pearson correlation coefficients for models with torsion strips

Methods of analysis

A S 3600 -1988

ACI318-89

BS 8110:1985

Wollongong

Correlation coefficient

0.868

0.840

0.836

0.860

+ Table 3.4 - Deviations of predicted Vu for models with torsion strips

Methods of analysis

AS 3600 -1988

ACI318-89

BS 8110:1985

Wollongong

Maximum

deviation

- 0.43

+ 1.67

+ 1.70

+ 0.41

Average

deviation

+ 0.04

+ 0.72

+ 0.67

+ 0.05

Standard

deviation

18.18

36.69

44.14

18.19

+ Maximum deviation = the maximum value of (Predicted Vu - Measured V„)/Measured Vu Average deviation = the average value of (Predicted Vu - Measured VJ/Measured Vu

Standard deviation = lit (Predicted Vu - Measured VUY

40


Table 3.5 - Frequency distribution of predicted Vu for models

with torsion strips

Methods of

analysis

AS 3600 -1988

ACI318-89

BS 8110:1985

Wollongong

Over-estimates

30 - 60 %

1

8

8 2

0 - 30 %

6 2

3

6

Under-estimates

0-30% 3

1

-

2

30 - 60 %

1

-

-

1

3.4.4 Comparison of A C I and British Standard methods incorporating reduction

factors: Flat plates with torsion strips

The mean ratios of measured and predicted values of Vu for the ACI and British

Standard methods may be applied as reduction factors to bring their respective

predictions of Vu closer to the measured Vu values. For example, the predicted Vu

values by the A C I procedure are to be multiplied by a factor 0.638, whereas the

corresponding predictions by the British Standard method should be multiplied by a

factor of 0.647 . The factored punching shear results for both the A CI and British

Standard methods are shown in Table 3.6 .

With a mean ratio of the measured and predicted Vu very close to unity, both the ACI

and British methods have improved considerably in their prediction of Vu when the

appropriate reducing factors are introduced.

41


Table 3.6 - Comparison of factored punching shear results by ACI and

British Standard methods - models with torsion strips

Column

W5-A

M5-A

W5-B M5-B

W5-C

M5-C R3-A

R4-A

R90-A R90-B

R90-C

Type*

C/NS

C/NS

E/NS

E/NS

C/NS

C/NS E/NS

E/NS E/NS

E/NS E/NS

Measured

V„ (kN)

32.59 34.42 71.24

87.81

37.30

38.36 88.76 75.42

120.20

108.20 21.90

ACI318-89

Predicted

V„ (kN)

36.69 44.63

58.17 92.02

51.18

57.57 83.05

66.25 68.92

87.01 37.37

Mean:

Measured

Pr edicted

0.89 0.77

1.22

0.95 0.73

0.67 1.07

1.14 1.74

1.24

0.59 1.001

BS 8110:1985

Predicted

V„ (kN)

27.17 41.62

51.79 112.36 34.47 50.14

107.79 82.72

84.68 89.56 38.32

Measured

Pr edicted

1.20 0.83

1.38

0.78 1.08

0.77 0.82 0.92 1.42

1.21 0.57 0.997

This is clearly illustrated in Fig. 3.5 wherein the resulting predictions by the A C I

procedure has 7 correlation points within the - 3 0 % to + 3 0 % envelope whereas the

British Standard method has 10 correlation points within the same envelope. While the

former has three predictions below the - 3 0 % line, the latter has only one below the same

- 3 0 % line.

A reading of Table 3.7 shows that the strength of the relationship between the A C I and

British methods with the introduction of the reduction factors remains unchanged with

correlation coefficients of 0.840 and 0.836 respectively. The smaller deviations and

narrower frequency distributions of these points (Table 3.8 and 3.9 respectively) indicate

that these modified results obtained are safer and more reliable.

42


+30% (underestimate)

150

125 -

55

to

CO CO

5S

100

/ */k. O^ (overestimate)

50 -

Equality

3 0 %

Legend A ACI318-89 n BS 8110:1985

I ' I ' I ' I ' I '

O 25 50 75 100 125 150

Predicted V u (kN)

Fig. 3.5 - Measured Vu versus predicted Vu by A C I and British methods

incorporating reduction factors - models with torsion strips

By incorporating the reduction factors of 0.638 and 0.647 respectively to the ACI

procedure and the British Standard method, their predictions of Vu are more

satisfactory. This means that both the A C I and the British Standard methods may be

confidently used to predict Vu provided the appropriate reduction factors are applied.

Table 3.7 - Pearson correlation coefficients for models with torsion strips

(after incorporating reduction factors)

Metihods of analysis

ACI318-89

B S 8110:1985


0.840

0.836

43


Table 3.8 - Deviations of predicted Vu for models with torsion strips

(after incorporating reduction factors)

Methods of analysis

ACI318-89

BS 8110:1985

Maximum

deviation

+ 0.71

+ 0.75

Average

deviation

+ 0.10

+ 0.08

Standard deviation

19.79

17.85

Table 3.9 - Frequency distribution of predicted Vu for models with torsion

strips (after incorporating reduction factors)

Methods of

analysis

ACI318-89

BS 8110:1985

Over-estimates

30 - 60 %

3

1

0 - 30 %

3

5

Under-estimates

0 - 30 %

4

5

30 - 60 %

1

-

3.4.5 Comparison: Flat plates with spandrel beams

For the Wollongong Approach, the mean ratio of the measured and predicted Vu given

in Table 3.2 is 1.027 . This is far more satisfactory than the Australian Standard

procedure which has a mean ratio of 0.593 (indicating that, on average, the procedure

overestimates the punching shear strength of flat plates with spandrel beams by 68.6%).

The superiority of the Wollongong Approach is confirmed in Fig. 3.4 where all the

correlation points except two, lie within the 30 percent lines; the exceptions are

conservative predictions. For the Australian Standard procedure, out of 19 correlation

points, 14 are below the - 3 0 % line. This underscores the unsafe tendency of its

predictions.

44


Table 3.10 - Pearson correlation coefficients for models with

spandrel beams

Methods of analysis

AS 3600 - 1988

Wollongong


0.434

0.891

It is obvious in Table 3.10 that the Wollongong Approach correlates well with the test

results. The Pearson correlation coefficient of 0.891 is much higher than that for the

Australian Standard procedure (0.434). From the additional statistical comparisons

shown in Tables 3.11 and 3.12, it can be further concluded that the Wollongong

Approach, with a much smaller deviation and safer distribution in its predictions, is more

accurate and reliable than the Australian Standard procedure in analysing Vu for flat

plates with spandrel beams.

++ Table 3.11 - Deviations of predicted Vu for models with spandrel beams

Methods of analysis

AS 3600 -1988

Wollongong

Maximum

deviation

+ 4.43

-0.59

Average

deviation

+ 1.10

- 0.01

Standard

deviation

60.18

15.43

Table 3.12 - Frequency distribution of predicted Vu for models

with spandrel beams

Methods of

analysis

AS 3600-1988

Wollongong

Over-estimates

30 - 60 %

14

-

0 - 30 %

4

9

Under-estimates

0 - 30 %

1

8

30 - 60 %

-

2

45


Falamaki and Loo [1991] have observed that the Australian Standard procedure was

developed without adequate support of model test results. In particular, none of the test

results quoted by Rangan [1987] was for models with spandrel beams. This may be the

main reason for the poor performance by the Australian Standard procedure.

3.5 Conclusions

For the punching shear strength analysis of flat plates at the corner- and edge-column

positions, the Wollongong Approach is found to be accurate and reliable. The

performance is consistent for flat plates with spandrel beams and with torsion strips.

The Australian Standard procedure on the other hand gives accurate and reliable

predictions for flat plates with torsion strips but at times gravely over-estimates the

strength of flat plates with spandrel beams.

Both the ACI and the British Standard methods are limited to flat plates with torsion

strips without closed ties. Further, they tend to give unsafe predictions. On average, the

ACI prediction is 1.57 "times the measured strength and the corresponding value for the

British Standard method is 1.55 times. On the other hand, by applying the reduction

factors of 0.638 and 0.647 respectively to the ACI and the British Standard methods, the

resulting predictions of Vu are more accurate and satisfactory. This means that both the

ACI and the British methods may be confidently used to predict Vu for flat plates with

torsion strips, provided that the appropriate reduction factors are applied.

46

Chapter 4 : Parametric Study of Spandrel Beams

CHAPTER 4

PARAMETRIC STUDY OF THE EFFECT OF SPANDREL BEAMS

ON THE PUNCHING SHEAR STRENGTH OF REINFORCED

CONCRETE FLAT PLATES

4.1 Introduction

This chapter describes a parametric study of the Wollongong Approach [Loo and

Falamaki, 1992] in analysing the punching shear, Vu , of reinforced concrete flat plates

with spandrel beams. This study is focused on how the spandrel strength parameter may

affect the analysis of Vu , using the Wollongong Approach. The investigation is

undertaken by varying the spandrel beam dimensions and the total area of steel

reinforcement in the spandrel beam, keeping other variables constant, such as overall

depth of slab, column size, and'compressive strength of concrete.

It is shown that, in general, the punching shear increases with deeper and wider spandrel

beam but only up to a certain point, beyond which the formulations are not applicable.

The same also applies to the amount of longitudinal steel reinforcement in the spandrel

beam whereby limitations exist, within which the Wollongong Approach formulas may

be applied.

4.2 Formulas for Spandrel Strength Parameter [Falamaki and Loo, 1992]

The spandrel strength parameter as represented by 5 is the product of the longitudinal

steel ratio, a , the transverse steel ratio (3 , and the ratio of dspld where dsp and d are

respectively the effective depths of the spandrel beam and the slab of the reinforced

concrete flat plates. Thus,

47


d o = a p - f ...(4-1)

a

where,

a = ^!lA_ .(4.2) 200000 v J

= C/f[co + co0]

50000

The variables /4/^ and fly are the total area and yield strength of the longitudinal steel

reinforcement in the spandrel beam. Ut refers to the perimeter of the rectangle defined

'by the centers of the four longitudinal corner bars of the spandrel beam (i.e. Ut = 2

[x+y]), where x and y are the horizontal and vertical dimensions respectively for the

rectangular perimeter.

While the transverse spandrel strength parameter is defined as:

co = ^LsL ...(4-4)

its additional transverse strength co0 was provided by the slab restraint for the spandrel

rotation and has been established empirically by Loo and Falamaki [1992] to be a

constant value, or

to0= 446 N / m m •••(4-5)

Note that Aws and fwy are the area and yield strength of closed ties in the spandrel

beam. The derivation of the Eqs. (4-2) and (4-3) for a and P respectively, as well as the

48


definitions for © and co0 are fully explained elsewhere [Falamaki, 1990] and are

elaborated further in Section 5.1.5.

4.3 Generating Models of Varying Features of Spandrel Beam

The generation of models with different characteristics of spandrel beam is undertaken

with the aid of a simple BASIC computer program, specifically written to run, save and

retrieve the various data input. It is also able to provide data output in ASCII format so

as to facilitate data import to other software uses such as LOTUS 123, Cricket Graph

and Microsoft Word for Windows.

Three types of model generation were undertaken, namely, by varying the spandrel beam

width and its corresponding depth as well as varying the amount of longitudinal steel

reinforcement in the spandrel beam. All these variations have the effect of influencing

the spandrel strength parameter 6 , which is then used to study the effect on the

determination of punching shear Vu by the Wollongong Approach.

4.3.1 Varying the width of spandrel beam

The width of spandrel beam (denoted as bsp) may be related to the parameter 5 in Eq.(4-

1) through the variable x , the horizontal dimension of Ut in Eq.(4-3), which in turn is

directly related to the parameter P . Since 5 and P are directly proportional to each

other, the parameter 8 may be used to relate the varying width bsp to the punching

shear, Vu .

The width of the spandrel beam for ten generated models are set by generating a range

of data for bsp as shown

49


bsp.gen = bsp - 10° + i20 ...(4-6)

where bsp,ge„ is the generated width of the spandrel beam, bsp is the spandrel width based

on a given model and / refers to the sequence from 1 to 10 for the ten generated

models. Fig. 4.1 shows ten generated models of varying width of spandrel beam.

All the data for the generated models remains unchanged except for the variable bSp

which in turn will generate a range of the parameter 5 to be used to determine the

punching shear, Vu of the generated models. For example, typical generated data and

results for models based on ModelM2-A are as shown in Table 4.1 .

Longitudinal reinforcement

Spandrel beam Closed ties

Varying width

Column

Fig. 4.1 - Generated models with varying width of spandrel beam

50


Table 4.1 - Typical generated data and results from parametric study of

varying width of spandrel beam in Model M2-A

Generated model (/)

1

2

3

4

5

6

7

8

9

bSp (mm)

140.0

160.0

180.0

200.0

220.0

240.0

260.0

280.0

300.0

a

1.0790

1.0790

1.0790

1.0790

1.0790

1.0790

1.0790

1.0790

1.0790

P 3.0097

3.4589

3.9081

4.3574

4.8066

5.2558.

5.7050

6.1542

6.6034

5

3.4846

4.0047

4.5248

5.0448

5.5649

6.0850

6.6051

7.1252

7.6453

r«(N) -70966

17581

31731

35356

35711

34741

33169

31313

29330

From Table 4.1, it can be seen that for the first generated model with small spandrel

width (i.e. bSp = 140 m m ) , the Wollongong Approach produces a negative Vu . From

the second to fifth model, Vu increases with bsp , after which Vu steadily decreases.

This is clearly illustrated in Fig. 4.2a with a plot of Vu versus 5 .

Three categories of the parametric study of the varying width have been identified which

are dependent on the depth of the spandrel beam or the ratio dsplde . Xi dspld < 1.2 , the

generated models have shallow spandrel beams; if 1.2 < dSpld < 2.5, they are deemed to

be with intermediate depth of spandrel beams; where dspld > 2.5, they are considered to

be with very deep spandrel beams.

Fig. 4.2a shows a plot for punching shear Vu versus spandrel strength parameter 5 for

generated corner-connection models with varying width bSp and shallow spandrel beams

(dspld < 1.2) using selected models M2-A , W3-A , and W3-C. The corresponding plot

for edge-connection models with shallow beams is presented in Fig, 4.2b using models

W3-B and M2-B .

51


V u (N)

100000

50000 -

-500O0

Ocn.M2-A Oen.W3-A Ocn.W3-C

-100000 r—•—| —i j r

2.0 4.0 6.0 8.0

Spandrel strength parameter 5

(a) Corner-connection models

Vu (N) 200000

100000

-100000 -

-200000

-300000

-400000

Oon.M2-B -JII— Gcn.W3-B

1 I ' I ' I ' I ' I ' 3.0 4.0 5.0 6.0 7.0 8.0 9.0


(b) Edge-connection models

Fig 4.2 - Graph of punching shear Vu versus spandrel strength parameter 5

of generated models with varying width bsp and shallow spandrel beams

52


It may be seen from Fig. 4.2 that the lower limit range of 8 for realistic analysis of Vu in

corner-connection models with shallow spandrel beams is between 3.6 and 3.9.

Similarly, the lower limit range is nearly identical (at between 3.4 and 4.0) for edge-

connection models with shallow spandrel beams.

Fig. 4.3a shows a plot for punching shear Vu versus spandrel strength parameter 5 for

generated corner-connection models with very deep spandrel beams (dspld > 2.5) using

selected models M3-A , W2-A , and W2-C. The corresponding plot for edge-connection

models with very deep beams is presented in Fig. 4.3b using models W2-B and M3-B .

Vu (N) lOOOOO -o— Oen.M3-A

-+— Oen.W2-A -a— Oen.W2-C

50000

-50000 -

-100000

10.0 20.0 30.0 40.0


(a) Corner-connection models


of generated models with varying width bsp and very deep spandrel beams


53


V u (N)

200000

150000

100000

50000 -

0 -

-50000

-lOOOOO

-150000 -

-200000

-o— Oen.M3-B -*— Oen.W2-B

18.0

i — ] i | i

lO.O 20.0 30.0 40.0


(b) Edge-connection models


of generated models with varying width bsp and very deep spandrel beams

(continued from next page)

Fig. 4.3 shows that the range of 5 for realistic analysis of Vu in corner-connection models

with very deep spandrel beams is between 20.0 and 27.0 . On the other hand, only a

lower limit is identified (at 5 = 18.0) for edge-connection models with very deep

spandrel beams.

For generated corner-connection models with intermediate spandrel depth (not

graphically shown here), the range of 5 for realistic analysis of Vu is between 10.0 and

15.0 while the lower limit of 5 for the corresponding edge-connection models is 9.5 .

4.3.2 Varying the depth of spandrel beam

It is obvious in Eq.(4-1) that the depth of spandrel beam (denoted as D,) may be directly

related to the parameter 5 by the effective depth of the spandrel beam, dsp. The depth

54


D, may also be related to 5 through the variable^ , the vertical dimension of Ut, which

is a variable for determining the parameter P . From these two direct relationships, the

parameter 8 may also be used to relate the varying depth Dy to the punching shear Vu .

The depth of the spandrel beam for ten generated models are set by generating a range

of data for D, as shown

Dl,eu= Ds + (/- l)Dj •••(4-7)

where DJgen is the generated depth of the spandrel beam, Ds is the overall depth of the

slab, D; is the original spandrel depth based on a given model and / refers to the

sequence from 1 to 10 for the ten generated models. The generated models with varying

depth of spandrel beam is presented graphically in Fig. 4.4 .

Varying depth

D,6 Di,5 D,i4 D|3 D,,7

Longitudinal reinforcement

Closed ties

D u

D 1.9 D, 1,10'

Column

Fig. 4.4 - Generated models with varying depth of spandrel beam

55


All the data for the generated models remains unchanged except for the variable D,

which in turn will generate a range of the parameter 5 to be used to determine the

punching shear, Vu , of the generated models. For example, typical generated data and

results for models based on ModelM2-A are as shown in Table 4.2 .

From observation of Table 4.2, it can be seen that for the first two generated models,

one with a torsion strip (i.e. D, = 100 m m ) and the other with a shallow spandrel depth

(i.e. D1 = 103.6 m m ) , the Wollongong Approach produces a negative Vu . From the

third model onwards, Vu increases numerically with D} . This is clearly illustrated in Fig.

4.5 with a plot of Vu versus 5 .

The graphical relationship between the spandrel strength parameter, 8, and the calculated

punching shear results of the generated models shown in Fig. 4.5 is based on selected

corner-connection models M2-A , M3-A , Wl-A , W2-A , M3-C and W2-C .


varying depth of spandrel beam in Model M2-A

Generated

model (/)

1

2

3

4

5

6

7

8

9

10

Dj (mm)

100.0

103.6

107.2

110.8

114,4

118,0

121.6

125.2

128.8

132.4

a

1.0790

1.0790

1.0790

1.0790

1.0790

1.0790

1.0790

1.0790

1.0790

1.0790

P 3.9531

4.0339

4.1148

4.1956

4.2765

4.3574

4.4382

4.5191

4.5999

4.6808

8

3.7004

3.9549

4.2166

4.4855

4.7616

5.0448

5.3353

5.6328

5.9376

6.2495

VU(N)

-16470

-3986

7279

17502

26823

35356

43194

50412

57075

67306

56


Vu (N) 150000

10OOOO -

50000

-5OOO0

-1OOOO0

-150000 -

-200000

-*— Oen.M2-A — — Qon.M3-A -O-Oon.Wl-A -»— Oen.W2-A -x—- Oen.M3-C

Ocn.W2-C

35.0

O.O 10.O 20.0 30.0 40.0



of generated corner-connection models with varying Dj

It can been seen that the Wollongong Approach has limitations, beyond which the

analysis of Vu would not be feasible. From Fig. 4.5, it can be observed that the lower

limit for realistic analysis of Vu ranges from 8 = 3.0 to 8 = 9.0 whereas the upper

limit ranges from 8 = 25.0 to 8 = 35.0 .

A similar graphical representation for generated edge-connection models based on

selected edge-connection models M2-B , M3-B , Wl-B and W2-B is shown in Fig. 4.6 .

In this case it can be observed that the lower limit for realistic analysis of Vu ranges

from 8 = 4.0 to 8 = 11.0 whereas the upper limit is at 8 = 29.0 .

57


-*— <3en.M2-B -*— Oen.M3-B -O- Gen.Wl-B -A— Gen.W2-B

V u (N)

300000

200O00 -

100000

O

-100000

-20OO00

-300000

-400000

O.O lO.O 20.0 30.0 40.0



of generated edge-connection models with varying Dj

4.3.3 Varying the area oflongitudinal steel reinforcement in spandrel beam

As the variable for longitudinal steel reinforcement (denoted as A!s) is also directly

related to the spandrel strength parameter, 8 in Eq.(4-1), the parametric study of the

effect of A/s on the punching shear, Vu , may be undertaken by relating the parameter,

8 to Vu for the generated models.

The area of the longitudinal steel in the spandrel beam for say, eight generated models

are set by generating a range of data for Als as shown:

, Als Ah, gen = 2Ah - ' -j

...(4-8)

58


where Aisgen is the generated longitudinal steel area, A/s is the original longitudinal steel

area based on a given model and /' refers to the sequence from 1 to 8 for the eight

generated models with varying A[s (see Fig. 4.7).

Longitudinal reinforcement Ajs

Spandrel beam

• Original longitudinal reinforcement

O Additional longitudinal reinforcement

Fig. 4.7 - Generated models with varying amount of longitudinal steel

reinforcement in spandrel beam

This means that, all the data for the generated models remains unchanged except for the

variable Als which in turn will generate a range of the parameter 8 to be used to

determine the punching shear, Vu , of the generated models. For example, typical

generated data and results for models based on Model M2-A are as shown in Table 4.3 .

59



varying longitudinal reinforcement in spandrel beam of Model M2-A

Generated model (/)

1

2

3

4

5

6

7

Als (mm2)

791.68

678.59

565.49

452.39

339.29

226.20

113.10

a

1.8882

1.6184

1.3487

1.0790

0.8092

0.5395

0.2697

P 4.3574

4.3574

4.3574

4.3574

4.3574

4.3574

4.3574

8

8.8285

7.5673

6.3061

5.0448

3.7836

2.5224

1.2612

P«(N)

79142

69427

55371

35356

6276

-39076

-125269

From observation of Table 4.3, it can be seen that for the first five generated models, the

calculated Vu reduces numerically with a reducing A/s. From the sixth model onwards,

the Wollongong Approach produces a negative Vu . This is clearly illustrated in Fig. 4.8

with a plot of Vu versus 8 .

The graphical representation shown in Fig. 4.8 is the relationship between the spandrel

strength parameter, 8 , and the calculated punching shear results of the generated models

based on selected corner-connection models M2-A , M3-A , Wl-A , W2-A , M3-C and

W2-C . It can be seen that the Wollongong Approach has limitations beyond which the

analysis of Vu would not be feasible. From Fig. 4.8, it can be observed that the lower

limit for realistic analysis of Vu ranges from 8 = 4.0 to 8 = 6.0 whereas the upper

limit ranges from 8 = 25.0 to 8 = 34.0 .

60


Vu (N) 100000

-100000

-200000

Oon.M2-A Qen.M3-A Gen.Wl-A Oen.W2-A Gen.M3-C Oon.W2-C

O.O 10.0 20.0 30.0 40.0



of generated comer-connection models with varying A[s

Vu (N) 250000

200000

150000 -

100000 -

50000

-50000 -

•100000

*— Gon.M2-B •— Oen.M3-B

Oen.Wl-B W2-B

0.0 lO.O 20.0 30.0 40.0



of generated edge-connection models with varying A\s

61


A similar graphical representation for generated edge-connection models based on

selected edge-connection models M2-B , M3-B , Wl-B and W2-B is shown in Fig. 4.9 .

In this case it can be observed that the lower limit for realistic analysis of Vu ranges

from 8 = 3.0 to 8 = 8.0 whereas the upper limit is at 8 = 28.0 .

4.4 Summary of Parametric Study Results

From the limited parametric study of the Wollongong Approach in analysing Vu of

reinforced concrete flat plates with spandrel beams, it has been shown that, in general,

the punching shear increases with deeper and wider spandrel beam but only within some

boundary limits beyond which the analyses are not feasible. It should be noted that the

parametric study was undertaken, based not only on one actual model tested, but on

several models by varying the features of their respective spandrel beams. Therefore,

one would not expect the upper or lower limits of 8 to be perfectly a certain value for

the group of models considered. A range of upper and lower limits of 8 were found for

the several models considered, so as to be used as a guide within which the analysis of

Vu may be reliable. The same also applies to the amount of longitudinal steel

reinforcement in the spandrel beam.

All these variations have the effect of influencing the spandrel strength parameter 8

directly, which is conveniently used to study its relationship with the punching shear

strength Vu of the generated models. The results obtained from the parametric study are

summarised in Table 4.4 . It is noted that any generated models beyond the limits of 8

will not produce any realistic or reliable punching shear results.

62


Table 4.4 - Summary of parametric study results of limits or ranges of 8 for

reliable punching shear strength analysis

T y p e of parametric study of the

spandrel b e a m

(1) Varying the width bsp of

spandrel beam

(a) Shallow beams

dSpld <\.2

(b) Intermediate

depth of beams

1.2 < dspld < 2.5

(c) Very deep beams

dspld > 2.5

(2) Varying the depth D} of spandrel beam

(3) Varying the area of longitudinal

reinforcement Ais of spandrel beam

Corner-connections of generated models

Lower limit or

range of 5

3.6 to 3.9

10.0 to 15.0

20.0

3.0 to 9.0

4.0 to 6.0

Upper limit or

range of 8

-

-

27.0

25.0 to 35.0

25.0 to 34.0

Edge-connections of generated models

Lower

limit or

range of 8

3.4 to 4.0

9.5

18.0

4.0 to 11.0

3.0 to 8.0

Upper limit or

range of 8

-

-

-

29.0

28.0

63

Chapter 5 : Theoretical Approach

CHAPTER 5

THEORETICAL APPROACH TO INCORPORATE EFFECT OF

PRESTRESS IN THE ANALYSIS OF PUNCHING SHEAR

5.1 Effect of Prestress on Punching Shear

5.1.1 General

One of the fundamental reasons of introducing prestress into structural concrete

elements is to reduce the amount of cracks associated with a structure under severe

loading conditions. W h e n punching shear is likely to occur, the first sign is the

appearance of a roughly circular tangential crack around the perimeter of the column

[Park and Gamble, 1982]. This is followed by radial cracks which extend from the

perimeter of the column. These grow and are accompanied by more circumferential

cracks in the form of a quarter- or half-truncated cone around the loaded area or the

column support [Falamaki and Loo, 1992].

The effect of prestress on the formation and direction of inclined shear cracks can be

seen by examining the stresses acting on a small element located at the centroidal axis of

an uncracked beam or slab strip as shown in Fig. 5.1 [Gilbert and Mickleborough,

1990].

By using a simple Mohr's circle construction, the principal stresses and their directions

are readily determined. W h e n the principal tensile stress at reaches the tensile strength

of concrete, cracking occurs and the cracks form in the direction perpendicular to the

direction of o t ,

64


W h e n the prestress is zero, a{ is equal to the shear stress T and acts at 45° to the strip

axis, as shown in Fig. 5.1a . If diagonal cracking occurs, it will be perpendicular to the

principal tensile stress, i.e. at 45° to the strip axis. When the prestress is not zero, the

normal compressive stress cr (=P/A) reduces the principal tension o"j , as illustrated in

Fig. 5.1b . The angle between the principal stress direction and the strip axis increases,

resulting in a less inclined crack if cracking occurs.

Therefore, prestress may improve the effectiveness of any transverse reinforcement that

may be used to enhanced the shear strength of the structural element. As the prestress

caused the inclination of the crack to be smaller, a larger number of vertical stirrups or

ties are crossed by the crack and consequently a larger tensile force can be carried across

the crack.

If the prestressing tendon is inclined at an angle dp , the vertical component of the

prestress Pv (-P sin Qp = P Qp) usually acts in the opposite direction to the load which

induces shear. The force Pv may therefore be included as a significant part of the shear

strength of the cross-section.

65

Chapter 5: Theoretical Approach

~3Z 7 7

j ^

At A: o =

J W

^5= SSSJ

"PT

- — and A

x = lb

(a) IfP = 0: 11 Shear

- 1 A , (90-ev) = 45°

(b) IfP>0

Shear

Normal

v/(90^ev)>45°

o.

Fig. 5.1 - Effect of prestress on principal stresses in the centroid

of a slab strip

66


5.1.2 Current concrete codes of practice

(a) Australian Standard AS 3600 -1988 [SAA, 1988]

The effect of prestress is taken into consideration in the Australian Standard in the

following manner. In the case of no bending moment transfer to the column support in

the direction considered, the equation for calculating the ultimate shear strength of the

slab is a modification of Eq.(3-1).

Vuo= ude[fcv+ 0.3 Gcp] ...(5-1)

where the additional term, acp is the average intensity of effective prestress in the

concrete in each direction. The effect of prestress on the shear strength of the slab is

taken 0.3 times the effective prestress, on the basis of recommendation by the ACI318-

89 [ACI, 1989] as shown in Eq.(5-2). As identified in Chapter 3, u is the length of the

critical perimeter whereas de is the effective depth of the prestressed slab.

When there is moment transfer to the column, the procedures are the same as those for

reinforced concrete flat plates. The only difference is the additional prestress parameter

Gc in Eq.(5.1) . This is applied to the appropriate equations for various cases such as

Eq.(3.2) for a typical slab having a spandrel beam with the minimum requirement of

closed ties. The associated equations may be found elsewhere [SAA, 1988].

(b) American Concrete Institute ACI318-89 [ACI, 1989]

According to the ACI code, for slabs prestressed in two directions, in which no portion

of the column cross-section is closer to a discontinuous edge than four times the slab

67


thickness, the shear strength is calculated based on the principal tensile stress approach,

that is:

Vuc = t 0.29 J7c + 0.3 acp ] ude + Vp ...(5-2)

where Gcp is as described in Eq.(5-1) and Vp is the vertical component of the effective

prestress force at the section, which is generally deemed to be negligible and may be

ignored.

Based on research carried out by ACI-ASCE Committee 423 [1983] and Burns and

Hemakom [1977], the ACI method considers 0.3 times the effective prestress in the

analysis of Vu . This recommendation was adopted by the Australian Standard AS 3600-

1988 [SAA, 1988], as shown in Eq.(5.1).

Other limitations include that f'c should not be greater than 35 MPa and acp in each

direction should not be less than 1 MPa or greater than 3.5 MPa.

Where some portions of the column cross-section is close to a discontinuous edge by

within four times the slab thickness and in the presence of an unbalanced moment, the

calculation of Vu is the same as for a normally reinforced concrete slab section (by

applying Eq.3.3 as described in Chapter 3). Apparently, the ACI method considers that

the effect of prestress is negligible when such conditions exist.

(c) British Standard BS 8110 : 1985 [BSI, 1985]

The recommendations for the method of determining the shear strength of beams are

also applicable to slabs in the British Standard method, whereby consideration is given

to uncracked or cracked sections in flexure, as indicated by Kong and Evans [1987].

68


The ultimate shear resistance of a concrete section which is uncracked in flexure is given

by:

Vco = 0.67 bDs -Jo? +0.8 acpat ...(5-3)

where b is the width and Ds is the overall depth of the slab section, acp is the concrete

compressive stress due to prestress and at is the principal tensile stress determined

from

a, = 0.24 JZ -(5-4)

Note that f' refers to the characteristic cube strength of the concrete in this case. On

the other hand, the ultimate shear resistance when the section is cracked in flexure may

be calculated as:

Vcr = (1 - 0.55 f-) vc bde + Mo-Tf > 0J V/« H -(5-5)

J p v

where,

^SL - ratio of the effective tendon prestress to the ultimate strength of

tendons;

v = design shear stress as from Eq.(3-5) in which the total tension steel

area includes the prestressing tendons;

dp = effective depth to the centroid of the tendons;

= decompression moment, i.e. the moment necessary to produce zero

stress in the concrete at the extreme tension fibre;

e

M,

69


V and Mv = design shear force and design bending moment at the section under

consideration.

For a cracked section analysis, in which case M* is larger than M0 , the ultimate shear

resistance is to be taken as the lesser of Eq.(5-3) and Eq.(5-5).

5.1.3 Effect of prestress in the Wollongong Approach [Loo and Falamaki, 1992]

In order to fully understand the forces acting on the column-slab connection in a post-

tensioned flat plate, two free-body diagrams based on the Wollongong Approach are

shown in Figs. 5.2 and 5.3 for forces acting at a corner- and edge-column positions

respectively.

Some parameters shown have already been briefly described in Chapter 1. They include

Mj, Vj, M2 \ V2 , T2, MC1, MC2 and Vu.

Mj and Vj are the negative yield moment of the slab and the shear force respectively

over the front segment of the critical perimeter, whereas M2 , V2 and T2 are the

negative yield moment of the slab, shear force and torsional moment respectively over

the side segment of the critical perimeter.

It should be noted that the critical perimeter as defined by Loo and Falamaki [1992] for

the direct transfer of the slab bending moment and shear force to the column is also

defined in Figs. 5.2 and 5.3 . The front segment of the perimeter is located 0.5de from

the front face of the column, where de is the effective slab depth. The side segments of

the critical perimeter coincide with the column side face(s).

70


MQI and MQ2 ar& the total unbalanced moments transferred to the column centre in the

main and transverse moment directions respectively and finally, Vu is the punching shear

strength of the slab.

Additional parameters on the effect of prestress shown in Figs. 5.2 and 5.3 are the

prestressing force applied, the overall area of prestressed tendons and their

eccentricities.

Pe] and Pe2 are the effective prestressing forces acting at the front and side faces of the

column-slab connections respectively. The subscripts x and y denote the horizontal and

vertical components of these applied prestressing forces. Apj and AP2 are the area of

the prestressed tendons while ej and e2 represent the eccentricities of the prestressed

tendons in the main and transverse directions respectively. The term nj refers to the

number of prestressed tendons at the front face of the column in the main bending

direction. Similarly, n2 is the number of prestressed tendons at the side face of the

column in the transverse bending direction.

Other variables shown include the dimensions of the column (C/ and C2), the width and

depth of the spandrel beam (bsp and Df), the overall and effective depths of the slab (Ds

and de), the height of the column Lj and the horizontal reaction component at the base

of the column.

The procedure undertaken to incorporate the effect of prestress into the prediction

formulas developed by Loo and Falamaki [1992] is elaborated in the following sections.

71


prestressing tendons area Ar

Prestressing force Pe2 (per n 2 wires)

y

A

Prestressing force Pel (per nj wires)

prestressing tendons area A p 2

Prestressing tendon

L -I Dc

Slab

Spandrel beam

Column View 'x'

Ci " bsp/2

Fig. 5.2 - Forces acting in the corner column region of a

post-tensioned concrete flat plate

72


de/2

v Prestressing u force P

(per n wires)

Prestressing force P e 2

(per n 2 wires)

Prestressing force Pel

(per nj wires)

prestressing tendons area A p i

prestressing tendons area A, P2

Fig. 5.3 - Forces acting in the edge column region of a

post-tensioned concrete flat plate

5.1.4 Equilibrium equations

By referring to Fig. 5.2, for a corner connection, the three equilibrium equations may be

derived as shown.

By considering the equilibrium of vertical forces at the ultimate state, i.e. I*Fy = 0,

Vu = Vj + V2 - niPely ...(5-6)

B y enforcing the equilibrium of moments, w e have for IM0 = 0 ,

73


MC1 = T2 + M1+ [VJ - ni Pely] [Cj - (tsp ' de)] - nj Pelx e, + MT ...(5-7)

and for, ET 0 = 0 ,

MC2 M2

h [Ll - (^ ' D'>]

rhPeix -(5-8)

" MC2= U iDi-Ds)-

n2Pe2*Ll -(5"9)

2L;

In Eq.(5-7),

= ^ [ / f r - P , ] + E Q - * g ] (5_10) r 2L; 2

for position O not coinciding with 0;, and

= M C / [ D 7 - Z ) 5 ] ...(5-11) r 2Lj

for position (9 coinciding with Oh i.e. C/ = bsp

Similarly, for the edge connection as shown in Fig. 5.3,

Vu= V1 + 2V2 -niPely -(5-12)

74


MC1 = 2T2 + Mj + [V7 - n, Pely] [Cj - (b*p-de)] . nj Pe}xej + MT ...(5-13)

and

"«" i^}Si 2L7

Before discussing the interaction equation involving torsion, shear and bending in the

spandrel beam, the spandrel beam parameters are explained first.

5.1.5 Spandrel beam parameters

The formulation for the slab restraining factor \|/ has been discussed in detail in previous

work by Loo and Falamaki [1992] for reinforced concrete flat plates. It is dependent on

the strength parameter of the spandrel, the longitudinal reinforcement factor and the

transverse reinforcement factor which are represented as 8, a and p respectively.

The equation for 8 has been identified in Chapter 4 and its derivation is briefly discussed

here. The factor 8 is expressed as:

8 = o d A -(5-15)

in which, the effective depths of spandrel beam and slab, dsp and de respectively, are

determined using Eq.(5-34), except that the term Ap2 is replaced by Api for de . The

terms for a and (3 are derived from the ratio of the longitudinal and transverse spandrel

75


strength to their respective minimum strength requirement as governed by the Australian

Standard AS 3600-1988 [SAA, 1988] for concrete structures. They are:

a = r AlJl\ ...(5-16)

and,

P = ut [ffl + <°ol -(5-17)

S min

The terms for Ais and fiy are the total area and yield strength of the longitudinal steel

reinforcement, and Aws and fwy are the area and yield strength of closed ties in the

spandrel beam. The term Ut is defined in Section 4.2 .

Loo and Falamaki [ 1992 ] simplified the equations for a and |3 to:

a = ^hy ...(5-18) 200000

and

= Ut[a> + coQ] ...(5-19)

50000

The transverse spandrel strength parameter is defined as:

0) = Ksfwy ...(5-20)

76


while its additional transverse strength co0 is due to slab restraint for the spandrel

rotation and is related to the concrete cracking strength. It has been established

empirically by Loo and Falamaki [1992] to be a constant value, or

ta0= 446 N/mm ...(5-21)

Eq.(5-21) is regarded to be generally applicable, since any slack would be taken up by

the formula for evaluating the slab restraining factor, \\t [Loo and Falamaki, 1992].

From these expressions, the semi-empirical formulas to determine v|/ for both edge and

corner column positions were established for reinforced concrete flat plate models. The

application of the same semi-empirical formulas for post-tensioned concrete flat plate

models was not feasible, as shown by Lei [1993]. The Wollongong Approach gave an

over-estimated punching shear result for the corner column position whereas a negative

value was given for the edge column position. The reason for the inapplicability of the

Wollongong Approach for the post-tensioned concrete flat plate models was due to the

semi-empirical formulas for \\J, which do not take the effect of prestress into account, as

mentioned earlier in Section 1.5.2 (c). Therefore, this necessitates the testing of post-

tensioned concrete flat plate models to obtain data for establishing new sets of semi-

empirical formulas to determine the slab restraining factor, \|/. These will be elaborated

further in Chapters 7 and 10 respectively.

For information, the semi-empirical formulas for the slab restraining factor v|/ applied to

reinforced concrete flat plates are re-produced below. For edge columns,

75 3

yedge = 2.65 - 1.71 log10 [(—) 8] -.(5-22) .5

For corner columns where C2 > Cj,

11


^corner = 10.70 - 4.45 log1Q [T8] ...(5-23)

and for C 2 < Ch

ycomer = 5.04 - 3.12 log10 [yb] ...(5-24)

In the above expressions, the column width factor

T-[f]J|

5.1.6 Interaction equation

The interaction equation which takes into account the effects of torsion, shear and

bending in the spandrel beam is based on the theory originated by Elgren, Karlsson and

Losberg [1974]. This interaction formula, which described the failure of an isolated

beam with the compression zone on top, may be derived from:

*&_ + [-3_]2 _ £ _ -JL. + [--*-]' —- -?«-=/ ...(5-26)

Astflydsp 2At Awf^ Astfiy 2dsp A-wsfwy At fly

where Ast is the area of tensile steel reinforcement and At is the area of the rectangle

defined by the centres of the four longitudinal corner bars of the spandrel beam.

A f After substituting Ab = 2Ast and co = ^ into the Eq.(5-26), the following

s

expression is obtained.

78


2M2 + tJL/ I J ^ + [A.J J. J ^ = . _ Asflydsp "2A^ ® ^//y ~ 2 < V W %/fy

In cases of pure bending, pure torsion and pure shear, the ultimate beam moment,

torsion and shear are respectively,

Mus = | Asfiydsp -(5-28)

Tus = 2A( | M £ = V 2 4 C 0 S -(5-29) 1 2Ut \Ut®

v--*J^-w-$i

This leads to an interaction equation which may be expressed as:

M2_ + i^L/ + [ii/ = 1 ...(5-31) M^ Tus Vus

In order to be more conservative, the above interaction equation may be linearised as

[Elgrenetal, 1974]:

Ml_ + 1l_ + Yl_ = j ...(5-32) M T V iVJ-us xus rus

79


5.1.7 Modifications to the interaction equation

In order to take into account of the established spandrel parameters in Section 5.1.5,

modifications have to be made to the interaction equation obtained from Section 5.1.6.

Furthermore, the simplistic expression for pure bending will need to be modified to

include other relevant factors. Thus,

Mm = [A„fly + Ap2f„) [1 - 0.6 ^ ± M j s l ] d s p ...(5-33) ®sp®spJc

where the effective depth of the spandrel beam,

dsp = A-stflyd-s + A.p2Jpy

dp (5-34)

Astfly + \lfpy

By substituting (co + C0o) for ® and ^Msfly for AiJiy, we can derive the expressions

for pure bending, torsion and shear respectively for the spandrel beam as:

Msp = VM„, -(5-35)

T„ = 200000^- , p ^ -(5-36) sp Ut V 2

Vsp = 200000 J ^ - V o W -(5-37)

Incorporating these relationship, the interaction equation becomes:

80


M2_ + Il_ + Yl_ = , ...(5.38) M T V irisp *sp Ysp

or in its complete form

Ml + ^ = ^ = + $- = 1 -(5-39)

VM™ 200000±J<®L 2 0 0 0 0 0 ^ ^

i.e. — S — + - 7 ^ = = [¥ - - ^ ] 200000 i^. ...(5-40)

V2C/, J 217,

5.2 Evaluation of Ultimate M o m e n t and Shear

The procedure for evaluating the ultimate moment and shear acting at the front face of

the column and over the front segment of the critical perimeter, i.e. Mj and Vi

respectively is through the use of semi-empirical formulas developed by Loo and

Falamaki [1992]. These semi-empirical formulas are used to analyse the post-tensioned

flat plate models. The analysed Vu results can be compared with the experimental results

to test the suitability of utilising the same semi-empirical formulas or whether some

modifications may be necessary to ensure a closer prediction of the shear strength of the

models. This is carried out in Section 11.3 .

For corner connections,

Mi = Mr ...(5-41) lYl1, corner lvllu v

81


and

Mr + M Vitamer = '3-66 + 5.04 ^ ^ L » ...(5-42) u+

for dsp/de < 1.2 and for ^i/tie > 1.2 ,

Mr + M ^corner = '0-50 + 4.70 Mlfiorner " ...(5-43)

Similarly for edge connections,

Mledge = M]u -(5-44)

The corresponding ultimate shear is:

Vledge = 9.35 + 3.49 ¥lf& -(5-45) 4

for Jjp/cie < 1.2 and for d^de > 1.2 ,

* W = 2.84 + 5.62 Ml'edgeT + Mm -(5-46) 4

In the above equations, M\u and M m refer to the bending strengths of the critical slab

strip at the front segment of the critical perimeter and at the corresponding mid-span

respectively. The analysis of Af/„ and Mm is briefly discussed in Section 5.4 .

82


5.3 Analysis of Punching Shear

In order to analyse the punching shear strength of the post-tensioned concrete flat plate,

we need to consider the three equilibrium equations developed in Section 5.1.4 and the

interaction equation (5-40) derived in Section 5.1.7 . With these four equations, we can

proceed to solve simultaneously for the four unknown quantities, namely, Vu , T2 , V2

and M2 •

5.3.1 Corner connections with spandrel beam

Firstly, we consider the analysis formulas for corner connections with spandrel beam.

By condensing the equilibrium equations (5-6), (5-7) and (5-9) and the interaction

equation (5-40), we have for the punching shear strength

y = kfe^ ' ^4 + ^7 (5-47) k3 - k4k5

where,

k, = 200000 [ i w - W s - ^ * , ) ] U L ...(5-48) Mus V Vcorner

k2 = k6Mcl - MliCorner - kylCj - (bsp'de)] + nj Pelx ej ...(5-49)

42Ut k3 = -^r ...(5-50)

83


K = J ^ -(5-5D

k5 = 9LL^!L ...(5-52)

k6 = 1 - [Dl ' D*] ...(5-53) 6 2L}

and,

ky = V1>corner - n2 Pely -(5-54)

Note that in Eqs.(5-48) and (5-49), MC1 and MC2 are the unbalanced moments in the

main'and transverse moment directions to be transferred to the centre of column,

respectively.

5.3.2 Edge connections with spandrel beam

In the case of edge connections, a similar procedure may be followed to obtain the

explicit equation for the punching shear strength. Condensing Eqs.(5-12), (5-13), (5-14)

and (5-40) gives:

v 2krk3k4 - k2k4 + kjky ...(5-55) u ' k3 - k4k5

where,

84


k, = 200000 [yedge - L \ J-2&- ...(5.56) Lus Wedge

iK'de)-'.edge

k2 = k6Mcl - Mledge - kyiCj - * e ] + nj Pelx ej . ...(5-57)

and,

*7 = VI>edge - nj Pely ...(5-58)

The other quantities are as those defined for the corner connections.

5.3.3 Connections with torsion strip

As generally defined, where dsp is not greater than de, the spandrel is referred to as a

torsion strip (with or without closed ties).

Equations for Vu derived in Sections 5.3.1 and 5.3.2 are applicable to flat plates with

spandrel beams, but they may be similarly applied to connections with torsion strips by

replacing Eq.(5-41) for a corner connection [Loo and Falamaki, 1992] with,

_ r 400 ] " M ...(5-59) Ml toner ~ lbsp + C2

lu

Similarly, the equations for Vu are also applicable to an edge connection with torsion

strips [Loo and Falamaki, 1992] except that Eq.(5-44) is replaced by

85


Mltedge = i2b + c 1 M]u ...(5-60)

5.4 Ultimate Flexural Strengths Mju and Mm for Post-Tensioned Concrete

Sections

At the overload stage, the behaviour of a prestressed concrete flexural member is similar

to that of a reinforced concrete member such as a beam. The procedure used to

determine the moment capacity of a reinforced concrete section may be applied to a

prestressed concrete section, with the same initial assumptions associated with the

concrete compressive stress block.

Beyond the cracking load, the internal forces in a typical cracked section are as shown in

Fig. 5.4 . They consist of tensile forces Ts and Tp in the reinforcing steel and

prestressing tendon respectively, and a compressive force Cc in the concrete above the

neutral axis. If compressive reinforcement is also present, the compressive resultant Csc

acts in conjunction with Cc . Note that the section is at equilibrium at all load stages up

to the ultimate state when Mu prevails.

In adopting the concept of the critical strip widths [Loo and Falamaki, 1992], the

analysis of a flat plate takes the critical strip width to be equal to the width of the

corresponding column support. By definition, the bending moments of the critical slab

strip at the front segment of the critical perimeter and at the corresponding mid-span, are

Miu and Mm respectively.

The determination of these bending moments is described and elaborated in detail in

Chapter 9.

86


o— — |

4°"

H H

t t ^ U

\

<

z

^ Q

-df

-O T?

-C? T?

-O T3CIO

1)

1) Uu

o C o U

cfl

c l-u U-J

Cfl

O

Cut

Cfl

U O o P-4

Fig. 5.4 - Forces, strains and stresses in a prestressed concrete section

87


5.5 Losses of Prestress

The determination of prestress losses is based on the provisions given in the Australian

Standard AS 3600-1988 [SAA, 1988] and outlined in the Concrete Design Handbook

(in accordance to AS 3600-1988) published by Cement and Concrete Association of

Australia [1991]. This has also been discussed by Gilbert and Mickleborough [1992].

Losses may be classified as the immediate or short-term prestress losses and the time-

dependent or long-term losses of prestress.

The difference between the prestressing force imposed at the jack, Pjt and the force in

the tendon immediately after transfer, Pit is termed as the immediate loss. The gradual

loss of prestress that takes place with time is called the time-dependent loss. The final or

effective prestressing force, Pe remains after all losses.

5.5.1 Immediate losses

The main immediate losses are caused by:

• elastic deformation of the concrete as the prestress is transferred;

• friction along the draped tendon in a post-tensioned member;

• friction in the prestressing jack;

• friction in the anchorage; and

• anchorage slip.

(a) Loss due to elastic deformation

For a post-tensioned slab with more than one tendon and where the tendons are stressed

sequentially, the elastic deformation losses vary from one tendon to the other. They are

at a maximum in the tendon stressed first and a minimum (zero) in the tendon stressed

88


last. These losses are usually small and, for practical purposes, the average elastic

shortening loss may be obtained as:

1 E P p = 2 ltGci -( 61)

where Ep and Ec are the moduli of elasticity of prestressed tendon and concrete

respectively while <5cl is the concrete compressive stress at the tendon level immediately

after transfer. GCI may be calculated as [Warner and Faulkes, 1988]:

C- = "f + f T 1 ] 6 -(5~62) Ag *g

The terms Ag and lg are the gross area and moment of inertia of the prestressed section,

and e is the eccentricity of prestressing force from the centroid of the uncracked

section.

(b) Loss due to friction along the tendon

For post-tensioned members, friction losses occur along the tendon during the stressing

operation. Friction between the tendon and the duct results in a gradual reduction in

prestress with distance Lpa along the tendon from the jacking end.

This loss depends on the total angular change of the tendon, the distance from the

jacking end and the size and type of the sheathing containing the tendons.

89


The resultant prestress after loss due to friction along the tendon may be obtained by an

equation recommended by major concrete codes [ACI, 1989, BSI, 1985 and SAA,

1988].

Pa = p.^-a-tot + tpLpJ ...(5-63)

where,

Pa is the force in the tendon at any point Lpa (in metres) from the jacking end

P, is the force in the tendon at the jacking end

p. is a friction curvature coefficient which depends on the type of duct. For

galvanized steel duct, \i ~ 0.2.

atot is the sum in radians of the absolute values of all successive angular

deviations of the tendon over the length Lpa.

Pp is an angular deviation or wobble term which depends on the duct diameter.

For flat metal ducts: 0.016 < p> < 0.024.

(c) Loss due to friction in the jack

This is dependent on the type of jack being used. In this study, we may assume that the

loss is negligible.

(d) Loss due to friction in the anchorage

This loss (which is minimal) may be estimated as two percent of the jacking force

[Cement and Concrete Association of Australia, 1991].

AP = 0.02xPj ...(5-64)

90


(e) Loss during anchoring

In post-tensioned members, some slip or draw-in occurs on transfer of prestressing force

from the jack to the anchorage. The amount of slip depends on the type of anchorage

used. For wedge-type anchorages, the maximum slip A / may be taken as 6 mm

[Cement and Concrete Association of Australia, 1991]. The loss in prestress due to

anchoring may be obtained from:

AP = 2 Ztctnti) ...(5-65)

where,

Z = V^7^ tan co

tan co = friction loss per unit length

A/ = anchorage slip + draw-in

= 6 mm for current commercially available systems.

5.5.2 Time-dependent losses

Since the interval between casting and testing of the post-tensioned flat plate models

was relatively short, i.e. not more than 40 days, it is assumed that the time-dependent

losses in prestress was quite minor. Thus, this gradual losses are not taken into

consideration in the calculation for the final effective prestress Pe in the tendons prior to

the ultimate load tests of the post-tensioned flat plate models.

In general, the time-dependent losses in prestress are caused by the gradual shortening

of the concrete at the tendon level due to creep and shrinkage, and by relaxation of the

tendon itself.

91


5.5.3 Summary of prestress losses considered in the calculation of Vu

The formulas for prestress losses, to be used in the analysis of punching shear strength,

Vu , of post-tensioned concrete flat plate models in Chapter 11, are summarised and

presented in Table 5.1 .

Table 5.1 - Summary of formulas for prestress losses used in the

analysis of Vu

Immediate losses due to :

(a) Elastic deformation

(b) Friction along the tendon

(c) Friction in the anchorage

(d) Anchorage slip

Formulas used:

1 EP

P 2 Ec ct

p - p -MV-tot + Vp^pa]

AP = 0.02 x Pj

A P = 2 Z tan a*

92

Chapter 6 : Test Program - An Introduction

CHAPTER 6

TEST PROGRAM - AN INTRODUCTION TO

THE TEST MODELS

6.1 General

As part of a comprehensive study of the behaviour of concrete flat-plate structures, four

half-scale post-tensioned concrete models have been tested to failure under simulated

uniformly distributed load. These supplement similar tests on nine reinforced concrete

models [Falamaki and Loo, 1992]. Representing two continuous panels of a building

floor, each model contained six columns, including three corner and edge column-slab

connections with or without spandrel beam.

This chapter describes the overall layout of the flat plate models in terms of size and the

support system used. It also details the design of the models, with emphasis on the

layout of prestressing tendons and steel reinforcement, type of anchorage system used,

properties of materials used for tendons, ductings, steel and concrete, and finally the

design of the spandrel beam.

6.2 Details of Models

The details of the models in terms of dimensions, fabrication and instrumentation are

similar to the ultimate load tests conducted by Falamaki and Loo [1992] on reinforced

concrete flat plate models. The general layout of the test models for post-tensioned

concrete flat plates were illustrated in Figs. 1.1 and 1.2 (Chapter 1).

93

Chapter 6 : Test Program -An Introduction

6.2.1 Dimensions

In Fig. 1.1, the shaded area of the half-scale flat plate represents two adjacent panels at

the corner of a typical building floor. The test models shown in Fig. 1.2 were

constructed to represent these corner panels (denoted as ABGF and BCHG). Four

models were constructed and tested to failure. They are identified as models PI , P2 ,

P3 and P4 . Models PI and P4 were built without spandrel beam; Model PI a free edge

while Model P4 featured a torsion strip along the edge. On the other hand, Model P2

was constructed with a 300 mm by 160 mm spandrel beam, and Model P3 incorporated

a 250 mm by 120 mm spandrel beam.

Apart from the spandrel beam and torsion strip, all the models are identical in all other

aspects including having the same slab thickness of 80 mm. The overall dimensions of

the models including the cross sections of the slab models are given in Fig. 1.2 . These

dimensions are similar' to those adopted for the reinforced concrete models [Falamaki

and Loo, 1992].

6.2.2 Support system

In all the models, the restraining effects of the adjacent panels of the building (which

were curtailed) were simulated by enlarging the cross sections of columns F, G , H and

C in order to provide more stiffness in the redistributing of moments following the

yielding of the slab reinforcement. The sizes of the various column supports, which are

common in all the models are shown in Fig. 1.2

The models were tested in a self-contained steel space frame resting on the floor of the

laboratory. The columns of the model were extended downward to a theoretical

contraflexure point (at approximately half the column height below the floor). Since the

94


columns above the slab were omitted, it was necessary to double the stiffness of each

column by halving the column height so as to resist twice the moments. For ease of

construction each of the flat plate model was cast onto six prefabricated steel column

sections (with equivalent stiffnesses to the cast-in-place concrete columns).

Top steel plate (bolted onto flat plate)

Fig. 6.1 - Typical column support details

The height of the columns in each model varies, depending on the depth of the spandrel

beam at the edge. For example, in Model PI, the height of the column from the centre

of the slab to the centre of support is 410 mm whereas it is 430 mm for Model P2.

Details of a typical support system is shown in Fig. 6.1 . The adjustable support system

95


allowed for proper alignment of the columns both in the horizontal and vertical

directions. At all the support positions, the slab with the truncated concrete columns

were held monolithically to the reusable steel column sections by twelve 16 mm mild

steel bolts in column H and eight 16 mm bolts in each of the remaining columns. The

arrangement shown in Fig. 6.1 ensured a monolithic connection between the spandrel

beam (if existed), the column and the slab. This reusable column support system has

worked successfully in the tests of nine half-scale reinforced concrete models [Falamaki

and Loo, 1993].

6.3 Design of Models

The design of the models are divided into four main components, namely the design

layout of prestressing tendons and steel reinforcement in the slab, the design

specifications of prestress and reinforcement, other material requirements for cable ducts

and concrete strengths and finally, the design of a reinforced concrete spandrel beam for

Model P2 and Model P3 .

6.3.1 Design of prestressing and reinforcement

The typical model slab was designed with the collaboration of a Senior Engineer of VSL

Prestressing, Sydney in accordance with the Australian Concrete Standard AS 3600-

1988 [SAA, 1988]. The design layout of cable ducts and steel reinforcement, which are

identical in all the models are shown in Figs. 6.2 and 6.3 respectively. The design

superimposed service load for all the models was 8 kPa.

96


10-mm perimeter bars (lapped 300 m m )

1-tendon carrying

duct

Legend

(4 ) cable duct number

yv location of anchorage / 3 2 \ as measured from slab

soffit, in m m PvTi soffit to the bottom '—' of ducting, in m m © ®

150 150©© © © ©

© © Note: All measurements are in milimetres

Fig. 6.2 - Design layout of cable ducts

97


In Fig. 6.2, all the cable ducts spanning across both directions contained two

prestressing tendons except for the ducts running along the edges of the slab which

accommodates only one prestressing tendon. The size of the prestressing tendons used

was 5.05 mm diameter. The numerical values along the length of the cable ducts in both

directions indicate the profile of the ducts along the respective direction, that is the

required height of the duct measured from the soffit of the slab to the underside of the

duct.

The corresponding values at both ends of the cable ducts along the edges of the slab are

the heights of the duct measured from the soffit of the slab to the centreline of the duct.

Fig. 6.2 also shows that the cable ducts are spaced closer within the column strips at 250

mm and are spaced wider in between the column strips from 350 mm to 450 mm.

Besides the prestressing tendon layout, Fig. 6.2 also shows the layout of 10 mm

diameter reinforcing bars along the edges of the slab as perimeter bars, one on top of the

other. They are essentially required to strengthen the concrete edges during the post-

tensioning process.

As shown in Fig. 6.3, all the positive reinforcement bars are located over all the column

supports in both the longitudinal as well as the transverse directions. More

reinforcement are required for larger size columns such as column G and H . Negative

reinforcement are provided at the mid-span (bottom) of column strips along the main

bending direction.

All these additional steel bars are provided to increase the flexural strength at the column

supports as well as the critical column strips. Figs. 6.4 and 6.5 show the profiles of the

cable ducts along both directions in a cross-sectional view.

98


4 No. 10-mm bars x 900 m m long (top)

2 No. 10-mm bars @ 250 (bottom) x 2000 m m long


W/,'

m F - m %

7\

2 No. 10-mm bars (top)


450




*

il/


s .='


/-


o

2 No. 10-mm bars x 900 m m long (top) 450

Fig. 6.3 - Design layout of steel reinforcement

99


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Fig. 6.4 - Profile of cable ducts along short span of flat plate models

100


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Fig. 6.5 - Profile of cable ducts along long span of flat plate models

101


6.3.2 Prestressing and reinforcement details

The 5.05 mm diameter post-tensioning tendon has a mean tensile strength of 1500 MPa.

A typical stress-strain plot is shown in Fig. 6.6 . Each of the tendons in the model was

stressed to 75% of the tensile strength. This required a jacking force of 23.03 kN, 25%

of which was applied 24 hours after casting the models.

Stress (MPa) 2000 -j—

1600 -

1200 - /

BOO - /

400 - /

O -f , -| 1 1 '

O 10 20 30

Strain (x O.OOl)

Fig. 6.6 - Stress versus strain plot for 5.05 mm diameter prestressed tendon

In the design, the following allowances were made for the post-tensioning tendons:

• friction coefficient, \L = 0.20;

wobble coefficient, (3P = 0.015 radians/m.

A wedge draw-in of 3 mm was also allowed.

102


The value of 0.20 used for the friction coefficient, ji, in the design was as recommended

by the concrete codes [ACI, 1989, BSI, 1985 and SAA, 1988] for galvanized steel duct.

This was described in Part (b) of Section 5.5.1 .

On the other hand, the specified wobble coefficient, Pp of 0.015 was slightly less than

the recommended minimum of 0.016 . This indicated that the flat metal ducting used

would caused less frictional loss because each ducting carried only a maximum of two

prestressing tendons in the design layout for all the models (see Section 6.3.1).

A wedge draw-in or slip of 3 mm was specified, which was half of the maximum slip

allowed of 6 mm as described in Part (e) of Section 5.5.1 . This was due to the half-

scale size used for all the models tested.

The additional non-stressed steel reinforcement provided for the models were all of 10

mm diameter high tensile steel with an ultimate tensile strength of at least 450 MPa. A

typical stress-strain plot for the reinforcement is depicted in Fig. 6.7 . The minimum

concrete cover to reinforcement was specified at 15 mm.

103


Stress (MPa)

600

0 20 40 60 80 Strain (x 0.001)

Fig. 6.7 - Stress versus strain plot for 1 0 - m m diameter steel

reinforcing bar

The details of the tendons and steel reinforcement (in. the critical column strips in the

direction perpendicular to the slab edge, and spandrel reinforcement) are summarised in

Table 6.1 . It should be read in conjunction with Fig. 1.2 which outlines the various

widths of the column and middle strips of the models [SAA, 1988]. Fig. 6.8 reflects the

details as summarised in Table 6.1 .

104


Table 6.1 - Summary of tendon and reinforcement details in the critical

column strips and spandrel beam

Model .

PI

P2

P3

P4

Column

strip

CS1 and

CS3

CS2

CS1 and

CS3

CS2

CS1 and

CS3

CS2

CS1 and

CS3

CS2

Slab steel

Top

10 mm @ 60 mm + 2

L 10 mm + 2

@ 50 mm

Bottom

10 mm @ 250 mn + 2 - straight bars

10 mm @ 250 mr + 3 - straight bars

Prestressing tendon

3 No. ducts @

250 m m - 5.05 mi tendons + 5

3 No. ducts @ 250 m m - 5.05 mi

tendons + 10

As in Model PI

As in Model PI

As in Model PI

As in Model PI

As in Model PI

As in Model PI

Spandrel steel

None

2-10 m m at top and bottom with

6 m m closed ties

@ 300 mm spacings for a

300 m m x 160 m m deep spandrel beam

As in Model P2 but for a 250 m m x

120 m m deep

spandrel beam

2-10 m m at top and bottom

with 6 m m closed ties @ 250 m m

spacings for a 300

m m wide torsion strip

105


Corner Front face of column columns A & C \ / T°Psteel

Bottom steel

Duct with 1 tendon

Ducts with 2 tendons each

Fig. 6.8 - Tendon and reinforcement details at column strips of

post-tensioned concrete flat plate models

106


6.3.3 Cable ducting and anchorage system

The cable ducts used were made of 0.4 mm thick galvanized steel. Each duct consisted

of a 45 mm width strip placed on top of a 25 mm x 15 mm square corrugated strip with

two 10 mm wide edges. The two components of the ducts were held together with

water-proofing adhesive tapes. This duct size was used in order to accommodate at

least two 5.05 mm diameter prestressing tendons. The details of the cable ducts are

depicted in Fig. 6.9 .

10 mm 25 mm 10 mm \m%\ •j-^ • m*fm •{

/

sealed with waterproof P V C tape

Fig. 6.9 - Shape and section view of cable duct

15 mm

107


Twelve-mm thick steel bearing plates for the anchorage points were fabricated with two

6-mm diameter holes (at 35 mm apart) which were required to provide access for the

two prestressing tendons. Only one 6-mm diameter hole is needed for anchorage points

that span parallel and nearest to the edge of the slab since the ducts at these locations

were to accommodate only one tendon each. Fig. 6.10 shows details of the anchorage

system.

Grout vent r- 2 - 10 m m <j) perimeter bars (lap 400 m m )

0.5mm relief all round

Si 5 m m (j) prestressing tendon

Spacer 12 m m thick-bearing plate

(a) Section along cable

50 mm 100 mm

Barrels and wedges

- «)

6 m m fl) hole

1-tendon bearing plate

-M

6 m m (J) hole

2-tendon bearing plate

(b) Section a-a

Fig. 6.10 - Anchorage system

108

diopter 6 : Test Program - An Introduction

6.3.4 Concrete specifications

The ready-mixed concrete used to cast the model was designed to provide a

compressive strength of 22 MPa at transfer of applied post-tensioning and a

characteristic strength of 32 MPa at 28 days, although the 28-day strength for Model P4

was lower than expected at 27.0 MPa. Details of concrete strength for the models are

given in Table 8.3 in Chapter 8. The size of aggregates used in the concrete mixes for

all the models was specified as 10 mm.

Each of the concrete strengths given is the average value taken from the compression

tests of four 100 x 200 mm cylinders for Model PI but 150 x 300 mm cylinders were

used for Model P2, P3 and P4 . For Model PI, two batches of these cylinders were

taken near the edge of the model, i.e. strips AF and CH while the other two near the

centre of each panel. In Model P2 and subsequent models P3 and P4, only three

batches were taken, one from each of the column strips AF, BG and CH. A minimum of

three sets of test cylinders were set aside for testing for each model.

Typical measured slumps ranged from 80 to 100 mm for Model PI and 100 to 120 mm

for models JP2 , P3 and P4 .

6.3.5 Design of spandrel beam

A reinforced concrete spandrel beam was designed and constructed for Model P2 and

ModelP3 The design was in accordance with the Australian Standard AS 3600-1988

[SAA, 1988], The concrete specifications were the same as those specified for the slab

design.

109


For a spandrel beam of 300 m m wide and 160 m m deep or twice the slab thickness (in

Model P2), the design required, the use of eight 6-mm diameter ties at 300 mm spacings,

with four 10-mm longitudinal bars at the four corners of the spandrel beam. The design

for a spandrel beam of 250 mm wide and 120 mm deep (for Model P3) was identical to

that for Model P2 , as the closed tie requirement for both models was already at a

maximum for the given design load of 8 kPa. Therefore, the design of a spandrel beam

for Model P3 was essentially the same as in Model P2 except for a shallower spandrel

beam for Model P3 .

The 10-mm longitudinal bars are similar to those used for the slab steel reinforcement, of

which the yield strengths are shown in Table 6.2 . It also includes the yield strengths of

the 6-mm closed ties used in both spandrel beams (in Model P2 and P3) and in the

torsion strip (in Model P4). Each of the yield strength values shown was the average of

tensile strength test results for at least three steel bars of a given type and size.

A typical stress-strain plot for the 6-mm diameter steel used as closed ties is shown in

Fig.6.11 while details of the spandrel beam are depicted in Fig. 6.12 .

110


Table 6.2 - Yield strengths of steel reinforcement used for

longitudinal bars and closed ties in spandrel beam

Model

PI

P2

P3

P4

Yield strength of 10-mm

steel bars (MPa)

512

538

578

555

Yield strength of

6-mm steel wire (MPa)

-

450

489

541

Stress (MPa)

600

500

400 -

300 -

200 -1

100 -

O •' i | i | i | i |

0 10 20 30 40 50 60

Strain (x 0.001)

Fig. 6.11 - Stress versus strain plot for 6-mm diameter steel bar

111


10-mm bars

160 m m spandrel beam depth

15 mm cover all round

80 m m slab thickness

6-mm ties at 300 m m spacings

Fig. 6.12 - Details of a typical spandrel beam (for Model P2)

111

Chapter 7 : Test Program - Construction, Instrumentation and Testing

CHAPTER 7

TEST PROGRAM - CONSTRUCTION, INSTRUMENTATION

AND TESTING OF MODELS

7.1 General Remarks

This chapter details the construction, instrumentation, and testing of the post-tensioned

concrete flat plate models. Each of the models was cast-in-place and the test up to

failure was carried out in a self-contained space reaction frame. The test set-up as

shown in Fig. 7.1 is identical to that for the reinforced concrete models tested previously

[Falamaki and Loo, 1992].

Fig. 7.1 - General set-up of a flat plate model

113

Chapter 7: Test Program - Construction, Instrumentation and Testing

Instrumentation in the form of load cells, strain gauges, and linear variable differential

transformers (LVDTs) were used to measure the forces and displacement data of the

models up to the failure stage. A whiffle-tree loading system was used to simulate a

uniformly distributed load being applied on all the test models.

7.2 Construction of Models

The construction of the models commenced with the setting up of the simulated column

support system and the formwork for the slab and spandrel beam (if any). Fabrication of

the cable ducts and the cutting and bending of the steel reinforcement were undertaken

at about the same time. These were followed immediately by the placement of cable

ducts (with the prestressing tendons) and steel reinforcement onto the formwork of the

slab.

After casting and curing of the slab, post-tensioning was applied onto the prestressing

tendons in two stages, the first to be applied 24 hours after casting and the second after

the concrete strength has achieved the required strength for full transfer of prestress.

Grouting was necessary to ensure that the prestresed tendons in the ducts were

sufficiently bonded to the concrete slab.

7.2.1 Setting up of formwork

Before the formwork was set up, the column sections were supported into position on

top of the supporting steel pedestal by five dummy load cells. This adjustable support

system allowed for the proper alignment of the columns both in the horizontal and

vertical directions. A typical column support system is as shown in Fig. 6.1 .

114


Standard marine formply was used to fabricate the slab and edge formwork. The

advantage of this material was that it could be re-used for the subsequent models. The

formwork was supported on extendable steel beams which in turn were supported by

two channel sections spanning in the north-south direction.

Fig. 7.2 shows details of the support system for the slab formwork for each of the

models while Fig. 7.3 presents the formwork and support system used for the spandrel

beam construction in models P2 and P3 . The base formwork for the spandrel beam is

propped up by adjustable steel props.

Before bolting the steel channels to the test frame, they were held in position with the

aid of adjustable steel props and levelled to an accuracy of 0.5 mm. The sheets of

formply used were cut into various sizes, the largest being about a quarter of the area of

a single panel of the model. Once they were put together into their final positions, any

gaps in between were sealed with silicon sealant.

The edge formwork to the slab was constructed around the slab perimeter, and fixed in

position by steel angles and bolts so as to resist the expected outward pressure of the

fresh concrete during casting.

Twelve 16-mm mild steel threaded rods were slotted through the bolt holes of the steel

column H and eight 16-mm bolts at the remaining steel columns. They extended above

the top of the slab and were securely fastened with nuts onto a steel plate at the top and

the prefabricated steel section at the base (see Fig. 6.1).

115


Edge form Plastic tubing form

Steel plate form welded onto steel column

100 m m deep extendable steel beam

Steel bracing

Support frame (bolted to pedestal)

Steel column

Dummy load cells

Pedestal

Fig. 7.2 - Details of slab formwork and the supporting frame

116


Top of slab 300 m m

Spandrel beam

Channel screwed onto spandrel formwork and welded onto T-Plate

10 m m steel T-pIate welded on to pedestal to support spandrel formwork

Pedestal

Fig. 7.3 - Typical details of formwork and support

system for spandrel beam at the slab edge

117


P V C tubings of 27-mm internal diameter were slotted over these rods in order to

provide spaces allowing for concrete movements during prestressing. The tubing was

evenly spaced around the rods by having nuts at the top and bottom ends of the rods.

These tubings were removed once the concrete has sufficiently hardened. Steel plates

were welded around the sides of the columns as the required form for the truncated

concrete column at these locations.

7.2.2 Cable ducts

Galvanized steel was used to fabricate the cable ducts because it can be easily cut and

bent to the required shape and it was flexible enough to be compressed transversely in its

final form to fit into the thin slab. The galvanized steel sheet used was of 1.2 m by 2.4 m

and of 0.4 mm thick. Two different strip sizes, i.e. 45-mm and 75-mm wide were cut

from each sheet so as to facilitate the bending process to fabricate the cross-sectional

shape as shown in Fig. 6.9 . The strip for the top was joined to the bent strip section by

using waterproof PVC adhesive tape. A typical bending operation of the duct to its final

square corrugated shape is shown in Fig. 7.4 .

Once every length of the ducting (2.4 m) was fabricated, the ductings were riveted end-

to-end (where required) so as to make up the necessary spans of the slab in both

directions. All the joints were sealed with masking or PVC adhesive tape. This also

applied to any gaps along the length of the ducts. A typical fabricated cable duct is

shown in Fig. 7.5 .

118


© 75 m m

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119


Fig. 7.5 - Typical fabricated cable duct

To facilitate the positioning of the wires through the two-hole anchorages, the end of the

ducts were cut and splayed. The ducts for the one-hole anchorage did not require such

treatment. The 12-mm bearing plates for the anchorages (see Fig. 7.6) were fabricated

with the required 6-mm diameter holes and a welded knob on the inner face. The knob

enabled the duct to slot easily onto the plate to the required position.

Following this, holes were cut near the end of the ducts to insert the plastic grout vents

(see Fig. 7.7), which were secured to the ducting with adhesive tape. These were

required for grouting the tendons.

120


Fig. 7.6 - Typical 12-mm bearing plate

Fig. 7.7 - P V C grout vent for cable ducts

121


Finally, the ducts were bent to their required multiple curve profiles (see Figs. 6.4 and

6.5). The ducts were bent to the required profiles after having being placed onto chairs

of selected heights on the formwork and then tied securely down by wires through the

slab formwork.

7.2.3 Placement of ducts and reinforcement

The layout of the cable ducts was marked out on the top surface of the formwork with

masking tapes. Then holes were drilled through the edge formwork at the appropriate

points in order to fix the bearing plates as well as to allow the prestressing tendons to

extend through the anchorages and edge formwork. The 12-mm bearing plates at the

anchorages were then screwed onto the inside of the edge formwork. It was important

to ensure that the bearing plates were flushed with the edge formwork and perpendicular

to the base formwork. This was to ensure that the prestressing was properly distributed

when applied.

Spacer chairs of various sizes were glued onto the formwork for the ducts to sit on so as

to provide for the required profiles of the cable ducts. Since the ducts were relatively

stiffened by its shape, at various points thin wires were necessary to tie the ducts down

to satisfy the concrete cover requirement as well as to prevent them from being

displaced during casting.

The laying sequence of the ducting was worked out in a way as described in Table 7.1 .

This was necessary in view of the stiffness of the ducts which would prevent vertical

clearance of ducts at overlapping points. The laying sequence was found to be effective

in overcoming this difficulty.

122


Table 7.1 - Laying sequence of cable ducts

Laying sequence

1

2-

3

•4

5

6

7

8

9

10

11

12

13

14

15

Duct number*

22

23

21

10

11

9

30

12

14

13

29

15

24

20

25

Laying sequence

16

17

18

19

20

21

22'

23

24

25

26

27

28

29

30

Duct number*

19

1

2

8

28

16

18

26

3

7

17

27

6

5

4

Once each duct was set into place, the tendons were pushed into the duct by slotting

them through one edge formwork and out from the other end. The tendons extended

about 300 mm out from both edges of the formwork to allow for the ease of post-

tensioning at a later stage (Fig. 7.8).

The bearing plates were completely wrapped to the ducting with the use of masking tape

and silicon sealant. Then rubber tubes of 16-mm diameter were fitted over the vent of

the grout plates for the grouting of tendons as well as to prevent ingress of concrete

during casting.

* The numbering of cable ducts is in accordance with Fig. 6.2

123


Fig. 7.8 - Tendons protruding from the edge formwork

The 10-mm diameter steel reinforcing bars were cut and bent where necessary to the

required shape. These steel reinforcing bars are then placed into their final positions

near the columns, at mid-spans and at the perimeter of the slab. At the column regions,

the steel reinforcing bars were secured by tying them to the cable ducts.

The positive reinforcement bars near the mid-spans of column strips were set onto chairs

to maintain the proper vertical clearances and were secured by tying to adjacent ducts.

The perimeter bars were overlapped accordingly and set as close to the edge as possible.

Typical details are shown in Figs. 7.9 and 7.10 .

124


Fig. 7.9 - Top steel reinforcement over corner column A

Fig. 7.10 - Top steel reinforcement over edge column B

125


The typical layout for the 10-mm longitudinal bars and 6-mm closed ties in the spandrel

beam is shown in Fig. 11 .

Fig. 7.11- Spandrel beam reinforcement for Model P2

126


1.2A Casting and curing of concrete and removal of formwork

lust before casting, the top of the prefabricated steel section and inner sides of the steel

formwork were lubricated with grease. The PVC tubings for the column bolts were also

lubricated with grease. The former was to allow for some movement in the concrete

under prestress whereas the latter was to ensure easier removal of tubings after concrete

has hardened. Rubber tubes over the grout plates were sealed up with masking tapes to

prevent entry of concrete during casting.

Fig. 7.12 - Casting of concrete slab for flat plate model

The total volume of ready-mixed concrete required for the test model and the twelve test

cylinders was about 2.0 cubic metres. The ready-mixed concrete was pumped to the

slab through a 100-mm diameter hose. An internal (poker) vibrator was used to

127


consolidate the concrete. Extra precautions were taken to prevent damages to the strain

gauges attached on the reinforcement during the casting and vibrating processes (Fig.

7.12).

The surface of the slab was finished immediately with a wooden float. Two to three

hours after casting, the top surface of the slab was made smoother with a steel trowel.

There was slight bleeding of the concrete as the mix was deliberately made wet in order

to be more workable around the cable ducts.

About 24 hours after casting, the concrete was cured by covering the top surface with

moist hessian and plastic sheets. At the same time the edge formwork of the model was

stripped to facilitate the prestressing process.

Fig. 7.13 - Concrete test cylinders

128


The test cylinders were cast at the same time and cured in the same manner as the slab

(Fig. 7.13). They were vibrated internally and were cured in the same way as the flat

plate model. The cylinders were stripped after approximately eight days. Following

this, the plastic tubing forms were removed from around the column rods. Finally, the

strain gauges were checked to identify those which were not in working order.

7.2.5 Post-tensioning of prestressing tendons

According to the design specifications, 25% of the total jacking force was applied

approximately 24 hours after the slab has been cast. Note that this requirement could

not be carried out for Model PI due to a developed fault in the jacking equipment.

Nevertheless, the omission of this procedure would not cause severe effect due to the

half-scale size of the- model (in consultation with a Senior Engineer of VSL

Prestressing).

In the calibration of the hydraulic jack to be used in applying the required prestressing in

the model, it was determined that for a total jacking force of 23.03 kN per tendon, the

equivalent pressure from the jack was 2800 psi (see Appendix A). Therefore, 25% of

this was 700 psi. The tensioning was carried out only at the live end for each

prestressing tendon.

Before the prestressing can proceed, the barrels and wedges for the anchorages were

slotted in tightly against the exposed 12 mm plate along the sides of the slab. The steel

barrels and wedges used for anchoring the ends of the tendons are shown in Fig. 7.14 .

Unfortunately in the case of Model PI, during the process of applying the 25%

prestressing, the jacking equipment developed a fault. It was then decided that the 25%

129


prestressing has to be omitted for Model PI. A full prestressing was then applied 8 days

after casting when the fault in the prestressing jack has been rectified. The problem did

not occur in the other models, in which the recommended 25% prestressing was applied

as scheduled in each case.

Fig. 7.14 - Barrel and wedge used for anchorages

The prestressing was carried out in the required sequence so as to ensure an evenly

distributed tensioning in the wires during this process. The prestressing order was in

accordance with the cable duct numbers (see Fig. 6.2) as follows :

Initial order 1, 3, 5, 7, 9, 11, 13;

then 14 to 30;

and followed by 2, 4, 6, 8, 10, 12.

130


Fig. 7.15 - Post-tensioning of tendons

Fig. 7.15 shows the post-tensioning operation being carried out. It was important to

note that after applying the full prestressing, a barrier made of timber plank and steel

plate was erected around the perimeter of the slab as a precautionary or safety measure

against any prestressing failure before and during testing.

7.2.6 Grouting

The tendons of the model were grouted the following week after full prestressing.

Grouting was necessary to ensure that the tendons in the ducts are sufficiently bonded to

the concrete part of the slab.

131


air was The grout was pumped in through the grout vent at one end of the duct and

allowed to seep out from the vent at the other end (see Fig. 7.16). A force of about 3

k N was used to pump the grout into the ducts.

Fig. 7.16 - Grouting of prestressed tendons

Wherever a duct was blocked by concrete seepage during casting, the grout was

pumped in from the other end as well to ensure complete grouting of the tendons. O n

completion of grouting the tendons in each duct, the rubber tubes were sealed off by

tying them with wires. At the same time, the gaps around the bolts in the columns were

grouted to provide monolithic joint to each of the column-slab connections.

132


7.3 Instrumentation of Models

Electrical resistance strain gauges were attached to selected reinforcing bars and

prestressing tendons at strategic locations of the models so as to record the strains

during the testing stage.

Fig. 7.17 - Hewlett-Packard 9826 computerised data acquisition system

133

Chapter 7: Test Program - Construction, Instrumentation and Testi ng

Further instrumentation was undertaken by the placement of three load cells at the base

of each column support. These were used to measure the three reaction components at

each of the pin supports.

The uniformly distributed loading to be applied during testing was simulated by the

installation of the whiffle-tree loading system at the bottom of the slab. This system

allowed for hydraulic rams at the base to apply the required (tension) loading which

were also measured by load cells.

All the strains and load cell readings were logged using a Hewlett-Packard 3054A data

acquisition control system via a Hewlett-Packard 9826 computer (Fig. 7.17). The

vertical deflections were also measured at critical locations of the slab (see Section

7.3.4).

7.3.1 Strain gauges

The steel strains were measured at numerous locations in the test structure with

electrical resistance strain gauges (TML-PL-10-11) having a gauge factor between 2.06

to 2.11 and a length of 10 m m . A total of 18 such gauges were mounted on the 10-mm

steel reinforcing bars in Model PI. These strain gauges which were placed at the critical

perimeter (i.e. at half the effective depth d/2 from the front face of the columns) and at

the mid-spans of the critical column strips for Model PL The strain values recorded

were used to determine the ultimate moments at these locations (see Section 9.3.3).

For subsequent models, in addition to sixteen strain gauges on steel reinforcing bars

(Fig. 7.18), more strain gauges were used so as to compensate for any failed or faulty

ones at later load stages. A further 27 gauges of type Nl l-FA-5-120-11 (with gauge

134


factor of approximately 2.10 and gauge length of 5 m m ) , were m o u n t e d on the tendons

on specific locations as shown in Fig. 7.19 .

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Fig. 7.18 - Locations of strain gauges on steel reinforcement

135


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136


Before the strain gauges were glued onto the steel bars or tendon, the selected locations

have to be polished smooth and shiny by sandpaper. Once the gauges have been glued

onto the surface of the steel, waterproof sealant was applied over them to protect from

grease and moisture to ensure the effectiveness of the gauges.

The end-connectors of all the gauges were soldered onto connecting electrical wires,

which in turn were to be soldered onto cables connecting to the portable strain meters or

the computerised data acquisition system. It was very important to check the wirings

and connections for these strain gauges to make sure that they were all in good working

condition prior to the casting of the slab.

7.3.2 Load cells

Three specially designed electrical load cells were provided at the pin support

(contraflexure points) at the base of each of the columns to allow for the recording of

the three reaction components of the columns. The steel sections simulating the in-situ

concrete columns were each supported by a steel pedestal via these electrical load cells.

These load cells which are shown in Fig. 7.20 were placed before the slab formwork was

stripped.

7.3.3 Whiffle-tree loading system

After the slab formwork was set up, PVC dowels with a diameter of 16 mm were

screwed to the formwork (from underneath). These were to provide for the 32 points in

the slab which facilitate the installation of the whiffle-tree loading system at the base of

the slab. This arrangement is shown in Fig. 1.2 . A total of 32 100-mm square pads

were set on top of the slab to secure the supporting steel rods for the loading system.

137


Fig. 7.20 - Close-up view of load cells at the base of steel column

Fig. 7.21 - General view of whiffle-tree loading system

138


The whiffle-tree loading system was essentially a series of criss-crossing steel beams in

an inverted pyramid set-up of ascending beam sizes from top to bottom. Fig. 7.21

shows the general set-up of the whiffle-tree loading system.

Two 20-tonne double-acting hydraulic rams which were attached to the whiffle-tree

system at the base of each slab panel, were used to apply the downward pulling loading.

To measure the load density, each ram was attached to an electrical load cell.

Synchronization of the rams was achieved by pumping through one single hydraulic

control system.

7.3.4 Deflection measurement

The slab deflections under applied load were measured at nine locations as shown in Fig.

1.2 . Rulers were hung on the soffit of the slab at the nine specified locations for

deflection measurement for Model PL In subsequent models, dial gauges were used in

addition to the hung rulers at similar locations (Fig. 7.22). These dial gauges were set

up on top of the slab.

Linear Variable Differential Transformers (LVDTs) were also installed on top of the slab

at centre of each panel for finer deflection measurement (Fig. 7.23). These LVDTs were

initially calibrated (before testing of models) in such a way that the voltage readings may

be directly converted to provide the equivalent deflection measurement in millimetre.

139


Fig. 7.22 - Dial gauges set up on top of slab to measure deflections

Fig. 7.23 - Set up of the Linear Variable Differential Transformer (LVDT)

140


The L V D T s were calibrated by connecting them to 10 volts D C power supply. Then, a

voltmeter was connected to the DC power supply to get the individual DC output for

each LVDT. During the calibration, the displacement point of each LVDT was

displaced over a micrometre rule to obtain 10 displacement values ranging from zero to

50 mm. These measured values were calibrated and recorded against the corresponding

voltage readings obtained from the voltmeter. These set of readings were then plotted in

a linear relationship of voltage against displacement in millimetre.

The voltage readings for the LVDTs were recorded at each loading stage during testing

of the models. The obtained readings, which were in voltages, were then converted to

millimetre displacements by using the voltage-millimetre plots.

7.4 Testing of Models

7.4.1 During load application

The loading system was connected up onto a 20-tonne double acting hydraulic ram at

the base of the slab which has a synchronized control system for the rams (Fig. 7.24).

The Hewlett-Packard computer was run to record the initial set of data as the zero strain

readings prior to commencement of testing.

141


Fig. 7.24 - Set up of synchronization control system for the two

double acting hydraulic rams

initial deflection readings were taken with a level for the hung rulers (accuracy to 0.5

mm) and dial gauges (accuracy to 0.05 mm). The initial deflection readings for the

LVDTs were also recorded. This was carried out by obtaining the calibrated voltage

reading corresponding to zero displacement reading in each LVDT. By adjusting the

LVDTs' vertical displacement points up or down, the calibrated voltage readings were

obtained and recorded as the initial zero deflection readings before the testing

commenced.

For the subsequent sets of load data, the test loads were applied at an increment of

approximately 5% of the expected failure load:

142


Design load = 8 kPa.

i.e. load increment = 0.05 x 8 = 0.4 kPa

which is equivalent to: D-

0.4 kPa x ( 5.7 m x 3.53 m slab ) = 8 kN, say 10 kN.

This was done by raising the hydraulic pressure in the loading rams to the required level.

At every load stage, the applied load was held constant so that data for column

reactions, load densities, strain measurements and deflection readings were recorded.

The Hewlett-Packard computer was run to check the level of load density applied on the

model. Once the load density in the printout matched the required load level at that

stage, the computer was run to record the required set of column reactions and strain

data. At the same time, the deflection readings were taken at all the selected points.

Observations and markings were also made of any cracks formed as a result of the

applied load increment.

Generally, the first cracks occurred at around load stage 6 ( i.e. 60 kN ) for each of the

model. As the applied load neared the 100 kN mark, the subsequent load increment was

reduced from every 10 kN to every 5 kN until the model finally failed.

7.4.2 After load application

Load increment was continued up to slab failure whereby all recorded data were printed

out as hardcopy. The hydraulic system was then disconnected at the end of the testing.

A post-mortem examination of the test model was conducted whereby the pattern, size,

143


location and type of cracks before and after failure were identified and recorded (Fig.

7.25).

Fig. 7.25 - Post-mortem examination of test Model P3 after failure

7.4.3 Precautions prior to and during testing

After the full prestressing of the tendons, a barrier of timber planks and steel plate was

erected around the perimeter of the slab as a safety measure against any prestressing

failure before and during testing. This has to remain in place throughout the test of each

model.

For safety reasons, only authorised persons (with hard hats) were allowed to step on top

of and below the slab in between load stages, to mark the crack patterns. The hydraulic

144

Chapter 7: Test Program - Construction, Instrumentation ami Testing

pressure for the loading rams in between load stages was maintained steady (by an

experienced laboratory technician) while marking of cracks and deflection readings were

taken.

7.4.4 Problems encountered during model testing

There were insufficient number of nuts being screwed and tightened on the bottom

threaded ends of steel rods (for the whiffle-tree loading system). Consequently, this

resulted in the collapse of the loading system during the testing of Model P3 near the

ultimate load stage, requiring a resumption of the test the following day.

The loading system also failed during the testing of Model P4 , but this time due to the

wom-off end of a threaded steel rod (for the whiffle-tree loading system). This also

resulted in the test being carried out over two days.

These setbacks did not affect the test results recorded over a two-day period, as can be

seen in the deflection, stress and strain measurements in Appendices B, D and F. The

models exhibited sufficient repeatability to enable a repeated load to be applied from the

beginning until they reached the ultimate (beyond the last applied load on the first day of

testing).

145

Chapter 8: Presentation and Discussion of Test Results

CHAPTER 8

PRESENTATION AND DISCUSSION

OF TEST RESULTS

8.1 General

This chapter presents the test results obtained for slab deflections, unbalanced slab

moments to be transferred to the columns, as well as punching shear strength and the

crack propagation patterns of the models. In light of the test results, the ultimate

punching shear behaviour of post-tensioned concrete flat plates with or without spandrel

beams is also discussed.

The laboratory study has produced experimental data not hitherto available on the

behaviour of post-tensioned concrete flat plates with bonded tendons. These include the

load deflection plots at nine locations in each of the two-panel six-column slabs, the slab

crack patterns, the failure loads, the punching shear results and the ultimate unbalanced

slab moments.

This information provides an insight into the complicated punching shear failure

behaviour of the slab-column connections at the corner- and edge-column positions.

The data will be useful in an attempt to extend the punching shear analysis of concrete

flat plates with spandrel beams [Loo and Falamaki, 1992] to include the effects of

prestressing.

146


8.2 Crack Patterns and Failure Mechanism

8.2.1 Model PI

The crack patterns recorded at the top surface and the soffit of Model PI are shown in

Figs. 8.1 and 8.2 respectively. It may be seen that the crack patterns around the corner

columns A and C and the edge column B are characteristics of punching shear crack

failure. This is particularly obvious from the pattern on the top surface (Fig. 8.1).

The failure at each of the three columns is identified as the shear failure mode [Falamaki

and Loo, 1992] for which the failure zone takes the form of a truncated cone: for the

edge column it was a half cone; for the corner columns it was a quarter cone. The top

view of the failure cone for the corner column A may be seen in Fig. 8.3 . Note also that

most of the positive moment cracks were developed at the failure load which together

with the punching shear cracks formed the collapse mechanism.

The numbers marked along the cracks indicate the load stages during which the Viarious

cracks first occurred; and the letter F signifies the cracks recorded after failure.

147

Chapter 8 : Presentation and Discussion of Test Results

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148


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149


Fig. 8.3 - Shear failure (Model PI - column A )

8.2.2 Model P2

Of Model P2, the crack patterns are recorded in Figs. 8.4 and 8.5 respectively for the

top and bottom surfaces of the model. It is clear that positive yield lines were developed

at the soffit along the midspans in both the longitudinal and transverse directions.

These were accompanied by a distinct negative yield line developed along the joint

between the slab and the spandrel beam (Fig. 8.6). Negative moment cracks also

appeared around columns F, G and H as well as along the line joining the two central

columns (B and G). These enabled the formation of the distinctly flexural collapse

mechanism in each of the two slab panels.

150


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151


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152


Fig. 8.6 - Flexural failure (Model P2)

8.2.3 Model P3

In Model P3, the first positive yield line occurred at the soffit of each slab panel midway

between the columns and parallel to the spandrel beam. Further increases in load led to

the meeting of these yield lines. The formation of cracks at the midspan of the slab

soffit, extended gradually to the edges in the transverse directions. The increase in load

also cause crack formations at the vertical sides of the spandrel beam and slab. These

153


cracks were clearly the result of torsion in the spandrel. In addition, cracks were also

observed at the corner of the columns and then across the front face of columns,

particularly the corner-columns A, C and the edge-column B .

Model P3 finally failed by punching shear at the corner- and edge-column positions,

after the formation of a typical large yield line crack across the positive flexural

reinforcement at the midspan of the slab. Fig. 8.7 shows a typical punching shear failure

at the edge-column B of the model. The shear failure is clearly identified by the crack

pattern on the top of the slab as shown in Fig. 8.8 . The crack pattern of the slab soffit

is illustrated in Fig. 8.9 .

Fig. 8.7 - Shear failure {Model P3 - column B)

154


16 P^7 L

10 -15 mm crack width


155


Fig. 8.9 - Crack pattern at soffit and sides of Model P3

156


8.2.4 Model P4

The crack pattern of Model P4 is very similar to Model PI, as both were constructed

without spandrel beams although Model P4 had a torsion strip at the edge spanning from

column A to column C. The punching shear failure occurred first at the edge-connection

B (see Fig. 8.10), followed closely by similar failures at columns A and C as illustrated in

the crack patterns at the top of the slab in Fig. 8.11.

As in the failure mechanism of Model P3, the first positive yield line in Model P4

occurred at the soffit of each slab panel midway between the columns and parallel to the

torsion strip. As the applied load increases these yield lines joined together, and cracks

at the midspan of the slab soffit also extend gradually to the edges in the transverse

directions (see Fig. 8.12).

Fig. 8.10 - Shear failure {Model P4 - column B)

157


2 V ^ - F ,18 J,

*\

19

s

[ Punching shear A Fi crack

Spalling of I Punching concrete surface v crack

shear

Fhinching shear

crack 22! ^ 2


158


[u~ 0)

xn <*-*

1 3 3 F

23

t Liu

241

R 24

18

XL

"^T 19 ^W 20

Fig. 8.12 - Crack pattern at soffit and sides of Model P4

159


The first sign of punching shear failure occurred in the vicinity of the edge column B, in

which the crack width at the front face and corners of the column increased significandy

at failure. This was immediately followed by similar failures at the comer columns A and

C.

8.3 Deflections

The slab deflections under applied load were measured at nine locations as defined in

Fig. 8.13 . The deflection results for models PI, P2, P3 and P4 are detailed in Tables

B.l, B.2, B.3 and B.4 (Appendix B); the load-deflection plots up to failure load for

Model PI are presented in Figs. 8.14 . The corresponding load-deflection plots for

models P2, P3 and P4 are given in Figs. 8.15, 8.16 and 8.17 .

8.3.1 Model PI

For Model PI, the maximum deflection was measured at location 3 which was the centre

of panel ABGF with a deflection reading of 46.7 mm at the ultimate load. This was

followed by deflection readings of 45.3 mm and 42.0 mm at location 8 and location 5

respectively.

F| O

1 3 e o

_ _4 A ] - O

0. 5 e

B

7 ©

8 o

9 e

H

6 e

c

4 O Locations for deflection measurement

Fig. 8.13 - Deflection measurement stations

160


0 10 20 PO 40 50

Deflection (mm)

Fig. 8.14a - Load-deflection plots-

Model PI (Locations 1, 5 and 6)

O 10 20 30 40 50

Deflection (mm)

Fig. 8.14b - Load-deflection plots-

Model PI (Locations 2, 3 and 4)

0 10 20 30 40 50

Deflection (mm)

Fig. 8.14c - Load-deflection plots - Model PI (Locations 7, 8 and 9)

161


A survey of Fig. 8.14 indicates that the deflection is linear up to an applied load of 10.0

kPa for all locations. This is just above the design superimposed (service) load of 8 kPa.

8.3.2 Model P2

Similarly, the applied load versus deflection curves for Model P2 are shown in Fig.

8.15. In this case, the deflection is elastic until the applied load approached 10.5 kPa.

In the model, the maximum deflections occurred at location 8 (66.7 mm) and location 3

(66.3 m m ) , the centre of panels BCHG and ABGF respectively. Due to the stiffening

effect of the spandrel beam, the deflections at the edges of the two slab panels were

small all the way up to the failure load stage. This is evident in the plots for locations 4

and 9 in Figs. 8.15b and 8.15c .

A\i

PH

lied

ON

H *—r— 1 1

*^y

-o-l

-*-5

i

o

i

_

0 10 20 30 40 50

Deflection (mm)

Fig. 8.15a - Load-deflection plots

Model P2 (Locations 1, 5 and 6)

~ i — | — i — | — I 1 1 1 r

0 10 20 30 40 50 60 70

Deflection (mm)

Fig. 8.15b - Load-deflection plots


162


0 10 20 30 40 50 60 70

Deflection (mm)

Fig. 8.15c - Load-deflection plots - Model P2 (Locations 7, 8 and 9)

8.3.3 Model P3

For Model P3, the applied load versus deflection curves shown in Fig. 8.16 exhibit

similar pattern to that of Model P2. The six superimposed curves shown in Fig. 8.16 are

the load-deflection plots of deflection readings taken during the first and second day of

testing of the model. This was due to the failure of the loading system on the first day of

testing. As with the first two models, the deflection is elastic until the applied load

approached 10.0 kPa mark. As expected, the maximum deflections (combined first and

second day readings) occurred at location 3 (93.1 m m ) and location 8 (72.5 m m ) , the

centre of panels ABGF and BCHG respectively.

163


Again, the stiffening effect of the spandrel beam caused the relatively small deflections at

the edges of the two slab panels up to the failure load stage, as indicated in the plots for

locations 4 and 9 in Figs. 8.16b and 8.16c.

cd

PH

M T3

a o u

PH

<

LIU

ed 2U PH M T3 O

V

PH 1 Q

O H <

o -c • i

JrlWil /I2J7

1

-•

-B-2 (Day 1) -*-•* (Day 1) —0-4 (Day 1) —*-2 (Day 2) -n-3 (Day 2) -*-4 (Day 2)

i i

10 20 30 40

Deflection (mm)

10 20 30 40 50

Deflection (mm)





164


J U

la 2 0

PH

« 15 _o

Applied

I-L

i i

0 -c

frm

lip

« I

/

v/

1

•

-a-7 (Day 1) -•-8 (Day 1) -H5-9 (Day 1) —•—7 (Day 2) - 0 8 (Day 2) —*—9 (Day 2)

i i

0 10 20 30 40 50

Deflection ( m m )


8.3.4 Model P4

The applied load versus deflection curves for Model P4 are shown in Fig. 8.17. Again,

due to the failure of the loading system on the first day, the testing was resumed the next

day. This resulted in the superimposition of six curves for each load-deflection plot in

Fig. 8.17 . It can be seen that the deflection is elastic until the applied load has just

passed an applied load of 11.0 kPa. The maximum deflections (combined first and

second day readings) are at location 3 (132.0 m m ) and location 8 (85.0 m m ) , the centre

of panels ABGF and BCHG respectively.

165


cd

PH

•*3 ca o

20

15

10

•a 4H "P< PH

<

•

- ITT

-I

T ' 1

-B-

—o—

•

I (Day 1) 5 (Day 1) 5 (Day 1)

1 (Day 2) 5 (Day 2) 6 (Day 2)

1 r

20 40 60 80

Deflection (mm)

40 80 120 160 Deflection (mm)





20

30 60 Deflection (mm)

90


166


It should be noted that the deflection measurements were recorded over two days during

the testing of the slabs Model P3 and Model P4 due to the sudden failure of the whiffle

tree loading system at an applied load of 21.0 kPa and 12.7 kPa, respectively. The

models were re-loaded the next day and all measurements were re-taken from the

beginning up to failure. The deflection results shown for Model P3 and Model P4 are

the measurements taken during the two days of testing of each model up to failure.

8.4 Punching Shear Strength and Unbalanced Moments

8.4.1 General

The construction and test dates for the four models are given in Table 8.1 which also

shows the average concrete cylinder strength test results.

. The punching shear strength, Vu , is defined as the ultimate column reaction at the

contraflexure point. The vertical load cell installed at the bottom of each test column

(see Fig. 6.1) provided the records for this reaction. Note that the three exterior

connections A, B and C (see Fig. 1.2) of each model were designed to fail at the same

ultimate load. For all practical purposes, this objective was achieved for all the models.

Table 8.1 - Concrete strengths of half-scale models

Model

PI

P2

P3

P4

Date of casting

01-10-91

08-07-92

28-09-93

05-05-94

Date of testing

06-11-91

09-09-92

26/27-10-93

09/10-6-94

Average cylinder strength, M P a

35.7

33.9

47.8

27.0

167


8.4.2 Experimental results for shear, moment and torsion

The measured values of the unbalanced moments in the main and transverse bending

directions to be transferred to the column centre are termed as Mci and MQ2

respectively. The values of MQI and Mc2 are important parameters in the punching

shear strength analysis of concrete flat plates [Loo and Falamaki, 1992]. These moment

values were obtained as the products of the relevant horizontal column reactions and the

vertical distance between the column contraflexure point and the mid-depth plane of the

slab (Lj in Fig. 5.2). The horizontal reactions were measured using the corresponding

load cells located horizontally, as shown in Fig. 6.1 .

The measured stresses (for shear, moment and torsion) recorded at the three exterior

connections of Model P2 , under uniformly distributed load up to failure, are presented

in Table 8.2 . The recorded moment and torsion results refer to the measured

unbalanced moment transfer to the columns, MCj and MC2 respectively whereas the

recorded shear was the vertical reaction measured at the column until the ultimate value

of Vu , was reached.

Some plots of the experimental stresses (namely, moment, shear and torsion) against the

applied load up to failure, as recorded at the three exterior connections for Model P2 are

presented in Fig. 8.18 .

168


Table 8.2 - Experimental shear, m o m e n t and torsion recorded at corner-

connections A and C and at edge-connection B for Model P2

Applied

load

(kPa)

0

1.37

3.28

4.37

5.69

6.96

9.86

11.27

12.48

13.21

14.69

15.86

16.07

17.23

18.44

19.61

Shear (kN) around column

A

0

3.01

4.23

6.11

7.86

9.91

14.41

17.73

19.74

21.18

24.07

26.08

26.64

28.52

30.23

32.02

B

0

10.20

13.79

18.34

23.82

28.89

38.53

43.58

47.96

50.92.

56.57

60.73

60.95

65.89

71.01

76.33

C

0

3.36

4.64

6.10

8.32

10.00

14.91

17.65

19.86

20.66

23.27

25.00

25.48

27.39

29.16

32.97

Moment (kN-m) at column

A

0

0.52

0.75

1.20

1.64

2.29

4.49

6.56

7.70

8.68

10.17

11.17

11.57

11.88

11.95

12.01

B

0

2.44

3.41

4.73

6.32

7.90

10.35

12.47

14.62

19.54

23.84

24.75

25.61

26.55

27.51

28.50

C

0

1.52

2.04

2.66

3.50

4.24

5.30

6.04

6.77

7.25

7.94

8.67

8.80

9.69

10.47

11.17

Torsion (kN-m) at column

A

0

1.15

1.68

2.34

3.04

3.79

4.85

5.45

5.68

5.87

6.15

6.31

6.59

7.06

7.12

7.35

B

0

0.04

0.05

0.04

0.05

0.07

0.14

0.14

0.10

0.15

0.09

0.14

0.17

0.27

0.32

0.28

c 0

0.05

0.30

0.62

1.14

1.57

3.34

4.59

5.46

5.92

7.49

8.11

8.26

8.71

9.19

9.51

It may be observed from Table 8.2 and Fig. 8.18 that the recorded moments and shears

at the edge-connection B were comparatively higher than both corner-connections A and

C due to the higher proportion of the applied load being taken up by connection B.

Conversely, the recorded torsion was the least at edge-connection B with the net

transverse unbalanced moments at or near zero due to the near symmetry of the two-

panel flat plate models. These trends in the stresses are generally observed in the other

models as well. Similar plots of stresses versus applied load for models PI, P3 and P4

169


are presented in Appendix C. The corresponding tabulated results for these plots are

shown in Appendix D.

30 - Column A • Column B • Column C

T 0 5 10 15 20

Applied load (kPa)

(a) Plot of moment transfer to columns

A, B and C versus applied load up

to failure

O 5 IO 15 20

Applied load (kPa)

(b) Plot of shear strength at columns A,

B and C versus applied load up fo

failure

Fig. 8.18 - Experimental shear, moment and torsion recorded at corner-



170


10 -°— Column A -*— Column B -*— Column C

O 5 IO 15 20

Applied load (kPa)

(c) Plot of torsional moment at columns A, B and C versus

applied load up to failure

Fig. 8.18 - Experimental shear, moment and torsion recorded at corner-


(continued from previous page)

As a result of the ultimate load tests of four post-tensioned concrete flat plate models,

twelve punching shear strength results were available from the four models tested.

These are presented in Table 8.3 together with the measured values of Mci and Mc2

and the measured failure loads.

171


Table 8.3 - Measured punching shear, unbalanced moments and applied

loads at failure for post-tensioned concrete flat plate models

Connection

PI-A

Pl-B

Pl-C

P2-A

P2-B

P2-C

P3-A

P3-B

P3-C

P4-A

P4-B

P4-C

Measured

vu,m

26.93

68.75

27.72

32.02

76.33

32.97

30.97

73.35

31.78

27.64

66.43

28.22

Measured

MC1,

kN-m

7.39

15.25

8.70

12.01

28.50

11.17

12.25

33.30

13.54

7.34

16.42

9.55

Measured

MC2,

kN-m

7.26

0.24

6.14

7.35

0.28

9.51

7.29

0.28

8.23

6.31

0.13

5.10

Measured

failure load,

kPa

16.18

19.61

22.42

16.64

8.4.3 Failure modes at the corner- and edge-column connections

Models PI and P4 yielded smaller punching shear strength values for .all corner- and

edge-column connections, compared to that recorded for models P2 and P3 . This is

mainly due to the additional stiffening effect provided by the presence of spandrel beams

along the edge in models P2 and P3 .

* All numbers that end with A or C are corner connections; those that end with B are edge connections.

Unlike models P2 and P3, models PI and P4 did not have spandrel beams at the edge.

172


Model P2 underwent significant tendency towards a flexural failure mechanism (as

shown in Section 8.2) prior to the failure stage. At this stage, punching shear cracks

have developed well in advance particularly around the regions of the interior columns

as well as the coiner- and edge-column connections A , B and C . Therefore, when the

model finally failed, the corner- and edge-columns also failed almost simultaneously in

punching shear as a response to a sudden shift or re-distribution of the applied loading to

the exterior column supports. For all intended purposes, the comer- and edge-column

connections in Model P2 have failed in punching shear, albeit indirectly.

The failure load in Model P3 is slightly higher than that for Model P2, although Model

P3 has a smaller spandrel beam (of 250 mm wide by 120 mm deep) compared to that for

Model P2 (of 300 mm x 160 mm). The reason is that the compressive strength of

concrete for Model P3 is much higher at 47.8 MPa (see Table 8.1). Nevertheless,

Model P3 also failed in punching shear at the comer- and edge-column connections,

with punching shear results which are slightly less than that of Model P2 (see Table 8.3).

This is due to the shallower and narrower spandrel beam in Model P3 which has a lesser

stiffening effect in enhancing the punching shear resistance at these exterior column

connections.

8.4.4 Comparison of measured vertical column reactions and total applied load

As a check for the accuracy of the loading system used, the total applied load are

compared with the measured vertical column reactions in Tables 8.4, 8.5, 8.6 and 8.7 for

all the test models. It may be seen that the error is less than two percent in all cases.

This confirms the good accuracy of the loading and recording system.

173


Table 8.4 - Comparison of vertical column reactions and applied loads for

Model PI

Connection

Pl-A

Pl-B

Pl-C

Pl-F

Pl-G

Pl-H

ZVM =

V M,kN

26.93

68.75

27.72

33.55

71.76

30.48

259.19

Panel

North

South

ZP =

Applied load at failure, P, kN

130.35

131.09

261.44

Table 8.5 - Comparison of vertical column reactions and applied loads for.

Model P2

Connection

P2-A

P2-B

P2-C

P2-F

P2-G

P2-H

zvM =

V u,kN

32.02

76.33

32.97

32.44

72.28

34.25

280.29

Panel

North

South

•LP =

Applied load at failure, P, kN

139.58

141.51

281.09



Model P3

Connection

P3-A

P3-B

P3-C

P3-F

P3-G

P3-H

ZV H =

V M , k N

30.97

73.35

31.78

38.71

84.46

36.45

295.72

Panel

North

South

ZP =

Applied load at

failure, P, k N

148.32

149.46

297.78


Model P4

Connection

P4-A

P4-B

P4-C

P4-F

P4-G

P4-H

ZV„ =

Vu,kN

27.64

66.43

28.22

29.59

66.70

25.39

243.97

Panel

North

South

EP =

Applied load at

failure, P, k N

119.38

120.40

239.78

175


8.5 Observations

The failure mechanisms of the flat plate models are found to be greatly influenced by the

presence of the spandrel beam which enhanced the punching shear strength at the edge-

and comer-column positions. Spandrel beams of smaller width and depth possess less

stiffening effect to improve the punching shear resistance at the exterior column

positions.

The stiffening effect of the spandrel beam also contributes to smaller deflections at the

edges of the two slab panels up to the failure load stage.

176

Chapter 9: Determination of Ultimate Flexural Strength

CHAPTER 9

DETERMINATION OF ULTIMATE FLEXURAL STRENGTH

OF CRITICAL SLAB STRIPS

9.1 General Remarks

The critical slab strips (as briefly described in Section 5.4) are referred to slab strips of

width equal to the width of the corresponding column support, and span in the main

bending direction perpendicular to the spandrel beam.

As input to the analysis of punching shear Vu , the ultimate flexural strengths of the slab

strips are required to be calculated for the post-tensioned concrete flat plate models.

Based on a rational and well-established flexural strength theory, the strain-moment

relationship of a prestressed concrete section as described in Section 5.4 is used to

determine the ultimate moments Mj and Mm , at the front face of the critical perimeter

and the mid-span of the critical strip respectively. Procedures for determining the

moments are outlined for both uncracked and cracked sections of the slab strips.

9.2 Flexural Behaviour of Uncracked and Cracked Sections

In order to determine whether the section is uncracked or cracked, the applied and

cracking moments, M0 and Mcr , respectively are calculated. M0 is calculated by using

the direct design method of analysis [SAA, 1988]. The moment to be resisted by the

section, when the first crack is formed, is defined as the cracking moment, Mcr . It can

be determined from the expression:

177

Chapter 9 : Determination of Ultimate Flexural Strength

Kr = U'cf + °bp] (9.1}

where abp is the maximum compressive fibre stress of the section due to the applied

prestress and is given by:

Pe ?ee

°bp A; + 1Ty' -(9-2)

in which Pe and e are the effective prestressing force and the eccentricity of the

prestressed tendon respectively;^ is the distance of the centroidal axis to the extreme

tensile fibre; fcf is the tensile stress of the concrete section. According to AS 3600-

1988 [SAA, 1988],

fcf = 0.6 yjfc (9_3)

Further, It and A t are the moment of inertia and area of the transformed cross-section

respectively where,

2

It=~f- + bDs[yc- ^ - } 2 + (np-l)Ap[yc-dp}2

2 , /„ rs A r r -, 2 + in-I)An\yc-dsY + {n-l)Asc[yc-dsc}2 ...(9-4)

and

4 = bD, + (np-l)Ap + {n-l)Ast + {n-l)Asc ...(9-5)

The various terms used in Eqs.(9-4) and (9-5) are described as follows:

178


n = the modular ratio of steel reinforcement (i.e. E/Ec);

np = the modular ratio of prestressed tendons (i.e. Ep/Ec);

Es , Ep and Ec = the moduli of elasticity of reinforcing bars and the tendon, and

the concrete modulus, respectively;

Ast, Ap and Asc = the area of tensile steel reinforcement, prestressed tendons, and

the compressive steel reinforcement (if any), respectively;

b and Ds = the width and depth of the section, respectively;

ds, dp and dsc = the depth of the tensile steel, prestressed tendons and the

compressive steel, with respect to the extreme compressive

fibre, respectively; and

yc = the depth of the centroidal axis of the uncracked transformed

section, which is,

r n 2

(n-1) [Ast ds + Asc dsc] + (n -1) [A d ] + —±-

(n-1) [Ast + Asc] + (np-l)Ap + bDs

...(9-6)

If M0 exceeds Mcr, then the section is cracked, and the analytical method as outlined in

Section 9.2.2 is used. Otherwise, an uncracked section analysis is utilised as described

below.

9.2.1 Uncracked section analysis

The analysis of an uncracked section involves the calculation of the moment resisted by

the transformed section. That is,

M = abp -^ -(9-7) yt

179


in which the various terms are as defined earlier.

The moment of the uncracked section may also be determined experimentally through

the use of strain measurements of the prestressed tendons, Zp. The strain is then applied

to Eq.(9-2) to determine abp , [Gilbert and Mickleborough, 1990] or

„ - £ A (*, - D Ap S A {np -l)Ape

This experimentally determined value can then be used in Eq.(9-7) to obtain the

required moment M.

9.2.2 Cracked section analysis

When the tensile stress produced by the external moment M0 overcomes the

compression caused by the prestress, and the extreme fibre stress reaches the tensile

strength concrete, cracking occurs, i.e. M0 has reached or exceeded Mcr . The

determination of the flexural strength of the critical slab strip section up to its ultimate

state is based on the method used for the ultimate flexural strength of a typical beam

section as described by Gilbert and Mickleborough [1990].

This flexural strength theory which is to be applied to a prestressed concrete section is

discussed in Section 9.3; whereas the application of the method is elaborated in

Appendix G. In applying this theory, the non-yielding or yielding of tensile

reinforcement and the prestressed tendon are taken into account. This is done by

considering the stresses exerted by them at various load stages up to failure.

180


9.3 Flexural Strength Theory

The flexural strength theory is a reasonable and well-established theory used to

determine the ultimate strength of a cross-section in bending, Mu [Gilbert and

Mickleborough, 1990]. It considers the strength of both the concrete and the steel in

the compressive and tensile parts of the cross-section. This theory assumes a linear

variation of strain over the cross-section, the absence of tensile stress and an idealised

stress-strain relationship of both concrete and steel. The detailed explanations of these

assumptions and the adoption of an idealised compressive stress block for concrete, as

well as the prestressed tendon strain components are described in Appendix E.

9.3.1 Formulas for the prestressed tendon strain components

The ultimate flexural strength of a section may be calculated accurately, by obtaining an

accurate estimate of the final strain in the prestressed tendons and the steel

reinforcement. For a bonded tendon, the ultimate tensile strain in the prestressed

tendon, Epu, is usually considered to be the sum of several sub-components.

With reference to Fig. E.3 (Appendix E), at loading stage la, the elastic concrete strain,

Ece may be determined from:

PI p2

z" = t [ Y + i ] •(9-9)

The corresponding elastic strain in the prestressing tendon, Epe (denoted as stage lb in

Fig. E.3) is:

181


eDe = -&- ...(9-10) p A F

ApZp

At load stage 3, the concrete strain at tendon level may be expressed in terms of the

extreme compressive fibre strain, Ecu ( = 0.003) and the depth to the neutral axis at

failure, dn,as:

d„ - d„

e„r = eCM [ ~JLTJL 1 -(9-H)

dn

This concrete strain at tendon level, Ept may be measured experimentally by using the

recorded strain readings at failure from strain gauges attached on the prestressed

tendons during testing.

Therefore, the strain of tendon at ultimate load condition may be obtained as:

Zpu = Zpe + e « + Zpt ...(9-12)

9.3.2 Determination of Mu for a prestressed concrete section

In order to determine theoretically, the moment at ultimate Mu , using Eq.(E-l) in

Appendix E, the depth to the neutral axis dn and the corresponding final stress in tendon

0"pM must first be ascertained.

An iterative trial and error procedure is used to determine dn . About three iterations are

normally needed to obtain a good estimate of dn and hence Mu . The steps required to

carry out these calculations are as follows:

182


(X) By referring to a doubly reinforced section as shown in Fig. E.2 (Appendix E),

the resultant compressive force consists of a steel component Cs (= <JSCASC) and

a concrete component Cc (= 0.85fcby"dn). The strain in the compressive

reinforcement is thus,

(d„ - d.r) e * = eCM , -(9-13)

dn

ff Esc < Esy (= fsylEs), then asc = ESCES , otherwise osc = fsy , where asc is the

strain in the compressive reinforcement.

(2) The resultant tensile force comprises of a prestressed component Tp (= GpuAp)

and a non-prestressed component Ts (= GstAst)- The strain of non-prestressed

tensile steel at ultimate is

e* = eCM (ds ' dn) -(9-14)

dn

If Est < Esy, then Gst = EstEs, otherwise ast =fsy.

(3) To calculate the depth to the neutral axis at ultimate dn , a trial and error

procedure is utilised. Successive values of dn are selected and tried until the

value which satisfies the horizontal equilibrium equation is determined.

TP + TS=CC+CS ...(9-15)

Assuming that all non-prestressed steel are at yield, then from Eq.(9-15),

Cc = Tp + Ts- Cs

i.e. 0.S5 f'cby"dn = apu Ap +fsy (Ast - Asc)

183


resulting in,

0.85f'ci'bdn -fsv(Ast - Asc) <Jpu = — " sy st — ...(9-16)

AP

(A) The selected value for dn is also used to determine Epu from Eq.(9-12). When

the values of o>« and Epu satisfy the stress-strain relationship of the prestressing

tendon, the correct value of dn is found.

(5) Check that Esc > Esy and Est > ESy to ensure that the non-prestressed steel has

indeed yielded. Otherwise, GSC = ESCES and / or Gst = EstEs before applying

Eq.(9-16). More iteration for dn may be needed if this is the case.

(6) The ultimate moment may be determined by using Eq.(E-6) in Appendix E or

alternatively by taking moments about the extreme compressive fibre, i.e. at Ecu

level.

Mu = Cc(±y"dn) 4- Csdc - Tpdp - Tsds ...(9-17)

where dc, dp and ds are the depth of the compressive steel, prestressing tendon

and the non-prestressed tensile steel respectively.

The overall procedure for determining the theoretical ultimate moment may be

summarised as shown in the flow chart in Fig. 9.1 .

184


<<-

C sta" 3

Compute: Mcr Eq.(9-1) Mg (From structural analysis)

Input data: b, D, ds,dp,fc, Pe ,e etc.

No Uncracked Section /Analysis

Compute: E« Eq.(9-9) Epe Eq.(9-10)

Yes

Cracked Section j\nalysis

Compute:

M Eq.(9-7)

C St°P 3

Loop: -1

i = D to 2

Select:

d„ = i

Compute:

Eq.(9-ll)

Epo and apU

are valid

Compute:

EpU Eq.(9-12)

EpU and o-pu are not valid

Check Yes ' A b s ( 0pu. Opuplot) N ^ No

- Minimum

Compute:

°"pu.piot (From stress-straiii curve plot for tendon)

Compute:

cTpU Eq.(9-16)

Yes Check

No V f s and f

\

sc *• ' s y /

> /

1

Compute:

M u Eq.(9-17)

(^ End ^)

Fig. 9.1 - Flow chart to determine the theoretical ultimate flexural strength

of prestressed concrete slab strips

185


The resisting moment up to the ultimate or failure stage may be determined

experimentally by applying the strain measurements of the steel reinforcement and the

prestressed tendons to the appropriate equations described earlier. The method of

determining the resisting moment of the critical slab strips is similar to that shown in Fig.

9.1, except for the input of measured strain values for steel and tendons.

(Similar to Fig. 9.1^

prior to the loorj/

± y Loop: -1

i = D to 2

Yes

lg — ^sy ks

Yes

°"p ~ e py E p

Select:

d„ = i

No

f, = E S E S

No

°"p = eP^P

Compute:

M Eq.(9-17)

d„ is valid

C E"d 3

Yes

d„ is not valid

No

Compute:

"p.plot (From stress-strain curve plot for tendon)

ft

Compute:

Eq.(9-12)

Fig. 9.2 - Flow chart to determine the flexural strength of prestressed

concrete slab strips using strain measurements

186


These strain values are then used in Eqs.(9-12) and (9-16) to determine the depth of the

neutral axis dn which is finally included in Eq.(9-17) to calculate the required resisting

moment. The overall approach is shown in Fig. 9.2 . The strain measurements of the

prestressed tendons to be applied in this approach are presented in Table 9.1.

Table 9.1 - Strain measurements at ultimate for all models, at front face of

columns and midspans of critical slab strips

Connection

Pl-A*

Pl-B*

Pl-C*

P2-A

P2-B

P2-C

P3-A

P3-B

P3-C

P4-A

P4-B

P4-C

Strain measurements at ultimate (used to calculate for Mj and Mm)

Gauge at front face of column

(forM;)

0.002050

0.001009

0.002023

" 0.002404

0.002197

0.002049

0.002681

0.002157

0.003065

0.000512

0.000815

0.000209

Gauge at midspan of critical

slab strip ( for Mm)

0.000933

0.001204

0.001031

0.002558

0.002018

0.002035 #

0.002422

0.003119

0.002751

0.000519

0.004772

0.000678

Note: All strain measurements are of prestressed tendons, unless otherwise stated.

The measured strains are read off gauges on steel reinforcement, as strain gauges were not placed on tendons in Model PI .

# This is measured strain of steel reinforcement, as the strain gauges on tendons were faulty at this location for Model P2-C .

187


The strain measurements recorded up to failure of the steel reinforcing bars and

prestressed tendons for all the models are presented in Appendix F.

9.4 Comparison of Measured and Calculated Values for M; and Mm

The measured values for the ultimate moments at the front face of column, M; , and at

the midspan of the critical slab strip, Mm , are those determined using the experimentally

recorded strains of the prestressed tendons. On the other hand, the calculated values for

Mj and Mm , are computed by assuming that the prestressed tendons are yielding when

the model fails. A typical sample calculation of Mj using the measured tendon strain

for Model P2-B at the front face of the column support is presented in Appendix G.

The measured and calculated values of Mj and Mm , for all the models are presented

in Table 9.2 . It can be observed that 5 out of the 12 measured values for Mj are

determined with strain measurements of the prestressed tendons at yield. As a result,

these five measured values of Mj coincide with the calculated Mj values. On the

other hand, 6 out of the 12 measured values for Mm coincide with the calculated Mm

values.

The measured values for M1 and Mm correlate reasonably well with the corresponding

calculated values, as shown in Fig. 9.3 . The only exception is the correlation point for

Model P3-B, in which the measured Mj is less than the corresponding calculated value

by 15.25 % .

This is due to the exceedingly high concrete strength (of 47.8 MPa) being used in Model

P3 , which resulted in the tendon strain to be well below its yielding point when the

critical strip section (at the front face of column B) failed in punching shear (which

188


happened well before it had reached its ultimate flexural capacity). The failure

mechanism of Model P3-B has been documented earlier in Chapter 8.

From the comparison of the measured and calculated flexural strength values of the

critical slab strips at failure, it can be deduced that the correlation of the measured to the

calculated results is reasonably good. Therefore, the measured results for M1 and Mm

may be confidently used for the prediction of Vu by employing the Modified Wollongong

Approach in Chapter 11.

Table 9.2 - Measured and calculated values of Mj and Mm for all models

Connection

Pl-A*

Pl-B*

Pl-C*

P2-A

P2-B

P2-C

P3-A

P3-B

P3-C

P4-A

P4-B

P4-C

Measured

Mi, kN-m

7.047

10.370

7.529

7.292

10.790

7.932

7.131

9.005

7.615

6.974

10.713

7.773

Mm, kN-m

9.079

- 11.098

9.749

9.709

11.731

10.509 #

9.325

11.243

9.840

9.636

11.625

10.029

Calculated

Mj, kN-m

7.114

10.540

7.625

7.581

10.790

8.152

7.131

10.626

7.615

6.974

10.713

8.002

Mm, kN-m

9.226

11.098

9.749

10.007

11.731

10.590

9.346

11.243

9.840

9.729

11.625

10.422

The measured results were based on strains measurement of steel reinforcement, as strain gauges were not placed on tendons in Model PI .

# This measured result was based on strain of steel reinforcement, as the strain gauges on tendons were faulty at this location for Model P2-C.

189


12

B I

55

T3 U

ca u

s

-a

IO

8

8 IO

10.626

12 Mil and. M m , calculated

(kN-m)

Fig. 9.3 - Graph of measured versus calculated Mj and Mm results

190

Chapter 10 : Calibration of the Prediction Formulas

CHAPTER 10

CALIBRATION OF THE PREDICTION FORMULAS FOR

SPANDREL PARAMETER

10.1 General Remarks

In this chapter, the formulas for predicting the slab restraining factor on the spandrel

beam, \y , are calibrated. These can be used in Chapter 11 to compute the punching

shear strength values, Vu , of post-tensioned concrete flat plates using the Modified

Wollongong Approach developed in this thesis (in Chapter 5). It should be reiterated

that the Wollongong Approach in its original form [Loo and Falamaki, 1992] is

applicable only to reinforced concrete flat plates.

It has been stated in Section 5.1.5 that the semi-empirical equations established for

reinforced concrete flat plates could not be applied to post-tensioned concrete flat plates

[Lei, 1993]. Therefore, new semi-empirical formulas to determine the slab restraining

factor \}/ , for post-tensioned concrete flat plates have to be developed. Such

development can be done with the aid of the ultimate load test data on post-tensioned

concrete flat plate models.

The method of formulating the semi-empirical equations (5-22), (5-23) and (5-24) to

determine \|/ for reinforced concrete systems [Falamaki, 1990] is employed herein to

calibrate a new set of semi-empirical equations for the post-tensioned concrete flat

plates. The accuracy of the resulting semi-empirical equations is determined by

comparing the predicted values of \|/ with the corresponding measured values.

191

Chapter 10: Calibration of the Prediction Formulas

10.2 Research Scheme for Determination of Semi-empirical Formulas for \\f

The research scheme to establish the new semi-empirical formulas for \|/ is shown in Fig.

10.1 . The relevant equations in the Modified Wollongong Approach are set up in

Section 10.3.1 to provide an analytical guide for the determination of \|/ in Section

10.3.2 with the aid of test data obtained from the ultimate load tests of the half-scale

post-tensioned models (Chapters 8 and 9).

Section 10.3.1

Set up general equations as

guides for the measurement

of \|/

Section 10.4

Use measured \|/

to calibrate semi-empirical formulas

for \|/

(r

-)

• *

Obtain test data

Ji-kL

Section 10.3.2

Obtain measured

values of tj/

Section 10.5

Use test data to test

accuracy of semi-

empirical formulas for V|/

Fig. 10.1 - Research scheme to establish the semi-empirical formulas to

determine \|/

192


The measured values of \|/ are used as a basis for the calibration of the semi-empirical

formulas for \\t. This is done via a linear regression analysis of the measured \|/ in

Section 10.4. The linear regression analysis is used for the curve fitting of these

measured values of \]/ in the scatter graphs of \|/ versus 8 or the spandrel strength

parameter (see Section 4.2 for definition). This is carried out for each of the following

sets of column positions, which are governed by the column dimensions [Falamaki,

1990]:

• the series "A" corner columns ( Q > C2);

• the series "C" corner columns (C; < C2); and

• the series "B" edge columns (C; < C2).

From the linear regression analysis, the estimates of the regression coefficients are

obtained for the resulting semi-empirical formulas. In Section 10.5, the semi-empirical

formulas are used to predict the values of \|/. These predicted \\f values are compared

with the measured ones to determine the accuracy of the resulting semi-empirical

formulas.

10.3 Measurement of the Slab Restraining Factor \|/

10.3.1 Derivation of equations to measure \J/

The measurement of the slab restraining factor, \|/ , involves the use of the three

equilibrium equations (5-6), (5-7) and (5-9) for corner connections or equations (5-12),

(5-13) and (5-14) for edge connections plus the interaction equation (5-40).

193


In order to clarify the measuring procedure, the interaction equation (5-40) from Section

5.1.7 is re-produced below:

T2 V2 _ _ rw M2 n 0/wwl (ap + - p i — = [\|/ *-] 200000 l-^- ...(10-1) r Af i K p _ ' Mus ~ V V

Eq.(lO-l) is simplified by letting:

2 , ^2 A = AT + T / -(10-2)

and

A2 = 200000Va|3 ...(10-3)

Substituting Eqs.(10.2) and (10.3) into Eq.(lO.l) yields

A = [¥ - - ^ ] *2 - -(10-4)

^™ vV

In order to obtain an expression for \|/, Eq.(10-4) is re-arranged to take the form:

V A 2 - ^ A - ^ A 2 = 0 -(10-5) M„ us

This is a general quadratic equation of Jy, the solution of which gives:

194

Chapter 10: Calibration of the Prediction Formulas

a AK < M2 - -A) ± JAZ - 4A2( - -f- A2)

vv = T: -(10-6) 2 A 2

By squaring both sides of Eq.(10-6), the factor \|/ is determined as:

v2 A,K s2 M2 A + A4 + 4(A2y

v = [ L ^" ] 2 -dO-7)

zA2

By imposing \|/ = ymeasured > tne measured factor, and letting

A, = JA2 + 4(A2)

2 j ± -(10-8) us

Eq.(10-7) is simplified further, or

T measured L _ . J

It can be observed from Eq.(10-9) in conjunction with Eqs.(10-2), (10-3) and (10-8)

that all the variables, except M2 , V2 and T2 , can be determined using the following

equations as given in Chapters 5:

Eq.(5-18)for a;

Eq.(5-19) for (3;

• Eq.(5-33) for Mm\ and

195


Eq.(5-34) for dsp;

Note that in Eq.(10.2), At is the area of the rectangle defined by the centres of the four

longitudinal corner bars of the spandrel beam (i.e. At = xy), and Ut is the perimeter of

the rectangle defined by the centres of the four longitudinal comer bars of the spandrel

beam (i.e. Ut = 2x + 2y).

The equations for M2 , V2 and T2 as given in Eqs.(lO-lO), (10-11) and (10-12) are

derived from the three equilibrium equations (5-9), (5-6) and (5-7) respectively, for

comer-column connections. Similarly, Eqs.(10-13), (10-14) and (10-15) for M2 , V2 and

T2 are based on the equilibrium equations (Eqs.5-14, 5-12 and 5-13) for edge-column

connections.

Thus, for comer-column connections,

M2= lMc2+n2Pe2xL1]U-^i^1^ -dO-10)

V-, = V - V, + n,P, ...(10-11) v2 yu.test yl T "-llely

and,

T2 = MCI - M, - [V, - n, PeIy) [C, - y ^ I + «, Ku », • MT -(10-12)

In Eq.(10-12), MT may be calculated using Eq.(5-10).

196


For edge-column connections,

M2 = MC2 [1 - iD'2LD')] ...(10-13)

V2 = \ [ VUJest - V, + niPely ] ...(10-14)

and,

T2 = ^-[MC1-M1- [V, - n} Pely] [C, - ( ^ ~ ^ ] + n, Pe!x e, - MT } ...(10-15)

2 -u-*

Note that the expressions for M2 , V2 and T2 as shown from Eq.(lO-lO) to Eq.(10-15)

require the input of recorded test data for Ma , MC2 and Vutest (see Table 8.3 in

Section 8.4.2) as well as the measured values of M1 (see Table 9.2 in Section 9.4).

10.3.2 Measured values of \j/

Based on the test data recorded in Table 8.3 for MC1 , MC2 , Vujest and the calculated

M1 in Table 9.2, M2 , V2 and T2 may be determined via Eqs.(lO-lO), (10-11), (10-12),

(10-13), (10-14) and (10-15).

The values determine for M2 , V2 , T2 , a , P , Mus, dsp , At and Ut in Section 10.3.1

are then applied to Eq.(10-9) in order to obtain the measured values for the slab

restraining factor, \|/measured • The values obtained for At, Ut , dsp , a , p and Mm are

197


presented in Table 10.1 whereas those of M2 , V2 , T2 , A , A} , A2 and the measured

results of \|/ are given in Table 10.2 .

Table 10.1 - Variables used to determine the slab restraining factor \j/

Model

Pl-A

P2-A

P3-A

P4-A

Pl-B

P2-B P3-B

P4-B

Pl-C P2-C

P3-C

P4-C

dsp (mm)

56.2

112.5

83.5

57.7

58.3

113.8 84.7

59.0

56.2 132.1

92.2

57.7

At (mm2)

9920 26784

17976 9920

9920 26784

17976 9920

9920 26784

17976

9920

Ut (mm)

576 712

596

576

576. 712

596 576

576 712

596 576

a

0.40212

0.84509 0.90792

0.87179

0.40212

0.84509 0.90792 0.87179

0.40212 0.84509

0.90792 0.87179

P 5.13792

6.95489 5.86560 5.84267

5.13792

6.95489

5.86560 5.84267

5.13792

6.95489 5.86560 5.84267

Mus (kNm)

3.6433 12.061

9.3109 5.7255

3.7906 12.208 9.4582

5.8728

3.6433 14.285 10.357 5.7255

Table 10.2 - Results for the calculated variables and

the slab restraining factor \\f

Model

Pl-A

P2-A

P3-A

P4-A

Pl-B

P2-B

P3-B

P4-B

Pl-C

P2-C

P3-C

P4-C

M2

(kNm)

15.535

14.555

15.659

15.386

0.2400

0.2540

0.2676

0.1300

14.410 16.474

16.531

14.170 1

y2

(kN)

-3.0885

-1.3986 -1.3441

-3.7455

25.296

9.9884

11.252

23.928

-4.7596

-4.6990

-3.8199

-5.8798

T2

(kNm)

-4.6013

-2.3340

-2.7488

-4.8295

2.7604

2.5474

5.2154

3.1989

-3.6111 -4.4933 -2.6884

-3.2659

A

(xlO3)

-391.82

-92.720

-133.97 -413.31

339.11

131.09

286.76

368.42

-318.08

-184.35

-139.79 -294.44

(x 103)

1250.3

1069.1 1204.6

1536.5

368.68

191.70

326.09 392.14

1186.9

1057.6

1174.6 1450.4

A2

(xlO3)

287.48 484.87

461.54

451.38

287.48 484.87

461.54

451.38

287.48

484.87 461.54

451.38

Ttneasured

2.22913

1.01380 1.34517 1.54804

1.51546

0.11080

0.44078 0.70977

2.28324

0.81085

1.25660

1.63961

198


10.4 Calibration of Semi-empirical Formulas for \J/

The procedure used to calibrate a new set of semi-empirical equations for the slab

restraining factors, \|/ for post-tensioned concrete flat plates is similar to the method

employed by Loo and Falamaki [1992] for reinforced concrete systems. The semi-

empirical equations given in Eqs.(5-22), (5-23) and (5-24) for reinforced concrete flat

plates can be presented in the following general forms.

For edge columns,

Vt«, = A -BlogI0 [&S] -(10-16)

and for comer columns,

3

V — = A-B,w£)J£8] -C"»7> Cj U£>,

s

where A and B are coefficients that were ascertained from regression analysis of the

plots of the measured values of \|/ versus the corresponding logarithmic expression in

terms of the spandrel strength parameter, 5 (Eq.5-15).

It was established by Falamaki [1990] that the measured values of \|/ (i.e. ^measured)

should be inversely proportional to the corresponding spandrel strength parameters, 5 .

199


Therefore, the general forms for \|/ as shown in Eqs.(10-16) and (10-17) are also

adopted herein to determine \}/ for post-tensioned concrete flat plates.

Besides the slab restraining factor, x\f, the calibration of the semi-empirical formulas also

involves the spandrel strength parameter, 8 (Eq.5-15) and the column dimension factor,

C 3 175 7, for comer-column connections where according to Eq.(5-25), y = [—] — .

ci VDs

For the edge-column connections, the slab thickness factor,

X = [£] .-(10-18)

With the introduction of 7 and X, we have from Eq.(10.16),

NW = A -Blogl0[Xb] -(10-19)

for edge columns, and from Eq.(10.17),

Corner = *-B logw [tf] -d0"20)

for comer columns.

The calculated values for 5 , 7 , A, and the corresponding logarithmic expressions of (78)

for comer connections, and (X5) for edge connections are presented in Table 10.3 .

200


With the results in Table 10.2 and Table 10.3, the measured \\f values (from Table 10.2)

are plotted against log10Cy8) and log10(X5) from Table 10.3, for the comer and edge

connections respectively. These plots are presented in Fig. 10.2 . The linear regression

analysis is used for the curve fitting of these measured values of \j/ in the scatter graph of

y for each of the three column positions A, B and C (Section 10.2).

Table 10.3 - Calculated values of 8 , 7 , X , log10(75) and log10(A,5) for all

post-tensioned concrete flat plate models PI to P4

Model

Pl-A

P2-A

P3-A P4-A

Pl-B

P2-B

P3-B

P4-B

Pl-C

P2-C

P3-C P4-C

5

2.0633

11.725 7.8504

5.2036

2.1941

12.146 8.1492

5.4432

2.0633

13.758 8.6684

5.2036

7

0.286888 0.286888

0.286888

0.286888

-

2.295102

2.295102 2.295102

2.295102

X

-

0.823975 0.823975 0.823975

0.823975

-

Iog10 (7§)

-0.22773 0.52682

0.35260 0.17402

-

0.67536 1.49936 1.29874

1.07711

log10 OS)

-

0.25717 1.00033 0.82703

0.65177

-

A linear regression analysis is carried out for each of the plots given in Fig. 10.2 . The

"curve-fitting" process has been carried out with the aid of a standard computer

package* to create graphs and to determine the coefficients of determination (R2) . The

values of correlation coefficients, R were obtained (i.e. VR2) as an indication of the

adequacy of the curve-fittings.

"CA-Cricket Graph v. 1.3.1" developed for IBM and IBM-compatibles by Computer Associates and

Microsoft Corp.

201


\p - 1.86-1.58 log (y5)

3.0

2.0

W

1.0 -

0.0

R"2 - 0.9957

-<—\—'—i—'—r-^ -0.25 O.OO 0.25 0.50 0.75

log (y6)

(a) Corner connections A

•K - 1.98-1.88 log (X5)

V

1.5 J

l.O -

0.5 -

o.o

R"2

Pl-B

P4-B "

1 1 '

= 0.9975

\ P3-B

\ P2-B

1 ' 1 ' 1 0.25 0.50 0.75 1.00 1.25

log (A.5)

(b) Edge connections B

v»V = 3.50-1.76log (y5)

3.0

2.0 -

V

1.0

0.0 -|—i—|—i—|—i—| i I r

O.SO 0.75 l.OO 1.25 1.50 1.75

log(yo)

(c) Corner connections C

Fig. 10.2 - Graphs of y versus log 10(78) for connections A and C and

\j/ versus logio(M>) for connections B

202


Note that the closer the correlation coefficient is to unity, the better is the adequacy of

the fitted curve for the plots [Walpole and Myers, 1985]. The curve-fittings are shown

in Fig. 10.2a, 10.2b and 10.2c for connections A, B and C respectively, and the R values

are tabulated in Table 10.4 .

Table 10.4 - Coefficients of determination (R2) and correlation coefficients

(R) of the plots of \|/ versus logio (76) and \|/ versus logio (kd)

Connection

A B C

Position

Comer Edge Comer

R2

(from Fig. 10.2)

0.9957 0.9975 0.9946

R

0.9978 0.9987 0.9973

As shown in Table 10.4, with Rvalues of 0.9978, 0.9787 and 0.9973 for connections A,

B and C respectively, the linear curve-fitting for each of the plots is considered

adequate. The mathematical expressions for the fitted lines are taken to be the semi-

empirical formulas for \|/.

The semi-empirical formula to be applied for edge-column (connections B) is:

Vedge = 1-98 - 1-88 log10 [Xb] .(10-21)

whereas for comer-column (connections Q where C2>Ci,

Vcorner = 3.50 - 1.76 logw [7§] .(10-22)

203


and for C 2 ^ C j , in connections A ,

Vcorner = lM ' l"58 l°SW M ...(10-23)

10.5 Comparison Between Measured and Predicted Values of y

The predicted values of \j/ are determined using the semi-empirical equations (10-21),

(10-22) and (10-23). These are presented and compared with the measured \j/ values in

Table 10.5 together with the ratio of measured to predicted values of \)/.

The punching shear strengths, Vu , at all the connections are analysed using the semi-

empirical formulas developed in Section 10.4 . The results of these analyses are

presented in Chapter 11. The relationship between the analysed Vu and the ratio " J f '

¥ predicted

is studied.

It was found that, when the ratio measured is closed to unity, the semi-empirical T predicted

formulas give an accurate prediction of Vu .

Also, when the ratio ^measaKi > 1.0 , the semi-empirical formulas produce a T predicted

conservative prediction of Vu .

204


As shown in Table 10.5, with a mean value of 1.008 and a standard deviation of 0.047

for the ratio of measured to predicted values of V|/ , the resulting semi-empirical

equations give reasonably accurate predictions of \|/.

Table 10.5 - T h e measured and predicted values of \|/

Model

Pl-A

P2-A

P3-A P4-A

Pl-B P2-B P3-B

P4-B

Pl-C

P2-C P3-C

P4-C

T measured

2.22913

1.01380

1.34517 1.54804

1.51546

0.11080

0.44078 0.70977

2.28324

0.81085

1.25660

1.63961

T predicted

2.21981 1.02762

1.30289 1.58506

1.49651

0.09938 0.42519 0.75468

2.31137 0.86113 1.21422

1.60429

Mean

Standard Deviation

i measured

T predicted

1.004

0.987 1.032 0.977

1.013 1.115 1.037

0.940

0.988 0.942

1.035 1.022

1.008

. 0.047

205

Chapter 11 : Analysis of Punching Shear Strength

CHAPTER 11

ANALYSIS OF PUNCHING SHEAR STRENGTH


The punching shear strength, Vu , of the post-tensioned concrete flat plate models are

analysed herein using the Modified Wollongong Approach developed in Chapter 5 and

calibrated in Chapter 10. The Wollongong Approach [Loo and Falamaki, 1992] for

reinforced concrete flat plates was used as a basis for the development of the Modified

Wollongong Approach. This was achieved by incorporating the effects of prestress into

the Wollongong Approach.

The equations required for the analysis of Vu by the Modified Wollongong Approach are

summarised for ease of reference. The development and use of a computer program to

analyse the punching shear strength is also discussed. The computer program

encompasses the analytical methods recommended by A S 3600-1988 [SAA, 1988],

ACI318-89 [ACI, 1989], B S 8110:1985 [BSI, 1985] and the Modified Wollongong

Approach. T w o sets of sample calculations for Vu at the comer- and edge-column

connections using the Modified Wollongong Approach are presented in detail.

The predicted punching shear strength results for the eight comer- and four edge-

column connections of the post-tensioned concrete flat plate models, using the four

analytical methods are presented and compared with the experimental results. From

these comparisons, it is found that the Modified Wollongong Approach gives a more

accurate and reliable prediction of Vu than the other three methods.

206


11.2 The Prediction Formulas for Vu

11.2.1 Corner connections with spandrel beam

For a corner column-slab-spandrel connection,

kfak4 - k2k4 + k3ky

k3-k4k5

where,

kAMn + noP^xLr)^ [ a|3 n 1 n. k} = 200000 [ycorner - 6K C2 2 " ] J—*- -dl-2)

lv*us V T corner

k2 = k6Mcl - M 7 , c o r w - [ Q - - ^ ] + n, Pei;c ej ...(11-3)

jk, = - A - -(H-4) V2<7,

A,2u7.

5 2

••(11-5)

ci - bsP ...(11-6)

207


Wl - Ds\ T-i

h-l- M ^ ...(U-7)

and,

k7 = V1>corner- njPely ...(11-8)

11.2.2 Edge connections with spandrel beam

For an edge column-slab-spandrel connection,

v = 2k}k3k4 - k2k4 + k3hj ...(11-9)

" k3 - k4k5

where,

*, = 200000 lVedge - *£*] J-p_ ...d.-.O) mus V T edge

k2 = k6Ma - M1Mge - hflC, - L-^-l + „, PeIX e, ...(H-l D

and,

u T/ P ...(11-12) K7 ~ Vledge ' nlUly

208


The other quantities are as those defined for the comer connections.

11.2.3 Connections with torsion strip or free edge

Equations for Vu presented in Section 11.2.1 are applicable to connections with torsion

strips except that Eq.(5-41) for a comer connection [Loo and Falamaki, 1992], is

replaced by

400 °25

M ^ - l - j ^ Miu -dl-13)

Similarly, for edge connections with torsion strips [Loo and Falamaki, 1992], the

equations given in Section 11.2.2 may be utilised to analyse Vu by replacing Eq.(5-44)

with

"* • i»?U" *>- -(1M4)

11.3 Computer Program for Vu

The computation for Vu is aided by the use of a BASIC computer program to run all the

required simple and explicit equations contained in the proposed prediction procedure

(referred to in this thesis as the Modified Wollongong Approach). The program is quite

versatile in usage, featuring user-friendly data input system as well as facilities for easy

storage and retrieval of data.

209


Besides developing a computer program for the Modified Wollongong Approach, other

analytical methods such as the Australian Standard procedure, the ACI method and the

British Standard method are also programmed in a similar fashion. This enable a faster

and easier analysis of all the tested post-tensioned concrete models as well as to mn

computation on reinforced concrete models tested previously by Loo and Falamaki

[1992]. The listings of the source code for the four computer programs (for the

Modified Wollongong Approach, the Australian Standard procedure, the ACI method

and the British Standard method) are shown in Appendix J.

Two sets of typical sample calculations using the Modified Wollongong Approach, are

carried out with the aid of the program for models Pl-A and P3-B . These are

presented in Appendix H. The data input for the analysis of Vu for other post-

tensioned concrete models are presented in Appendix I.

Other features of the computer program for the Modified Wollongong Approach are:

• computation of the measured slab restraining factor, ti/measured (shown in Section

10.4); and

• analysis of the ultimate moments, M; and Mm .

Note also that the parametric study undertaken in Chapter 4 was made easier by writing

a simple interface routine to handle numerous punching shear analysis runs and data

management work.

210


11.4 Comparison of Results

The predicted results by the Modified Wollongong Approach are presented, in

conjunction with those by the methods recommended in AS 3600-1988 [SAA, 1988],

ACI318-89 [ACI, 1989] andBS 8110:1985 [BSI, 1985].

Table 11.1 presents the predicted punching shear strength results by the Modified

Wollongong Approach. Also included are the ratios of the predicted and measured Vu ,

for the eight comer- and four edge-column connections of the post-tensioned concrete

flat plate models. The parallel results by the other three methods are shown in Table

11.2.

The relative accuracy and reliability of all four prediction methods are determined by

comparison with the measured values of Vu . The comparison is carried out by plotting

the experimental punching shear strength values against the predicted values from each

of the four prediction methods, where applicable. These are referred to as the

correlation plots. Standard statistical results (i.e. maximum, average and standard

deviations) are used to determine the relative performance of the four prediction

methods.

As in Section 3.4.2 (Chapter 3), superimposed on these figures are the 30-percentile

lines and the 45° equality line. Any point falling below the -30% line means that the

analysis method has overestimated the strength by more than 30%. Also, the -30% line

may be taken as a safety criterion to reflect the Australian Standard requirement of a

shear capacity reduction factor of 0.7 .

211


11.4.1 Predicted punching shear results by the proposed approach (or the

Modified Wollongong Approach)

Table 11.1 presents the punching shear results for all the four post-tensioned concrete

flat plate models predicted using the Modified Wollongong Approach, herein referred to

as the proposed approach.

It can be observed in Table 11.1 that, with an overall mean ratio of predicted to

measured Vu of 0.987, the proposed approach gives reasonably accurate predictions of

the punching shear of all the models. The predictions are slightly more conservative for

models with spandrel beams (i.e. models P2 and P3) with a mean ratio of predicted to

measured Vu of 0.953 . The corresponding value is 1.021 for models without spandrel

beams (i.e. models PI and P4).

11.4.2 Predicted punching shear results by codes of practice and comparison

The punching shear strength results for the four flat plate models are compared with the

predicted values due to the Australian Standard procedure [AS 3600-1988] and the

methods recommended by the American Concrete Institute [ACI318-89] and the British

Standard Institute [BS 8110:1985]. These methods have been described earlier in

Section 5.1.2.

212


Table 11.1 - Punching shear strength predictions for post-tensioned concrete

flat plate models PI to P4 using the proposed approach (Modified

Wollongong Approach)

Connection

Pl-A

Pl-B

Pl-C

P4-A

P4-B

P4-C

Measured Vu

(kN)

26.93

68.75

27.72

27.64

66.43

28.22

P2-A

P2-B

P2-C

P3-A

P3-B

P3-C

32.02

76.33

32.97

30.97

73.35

31.78

Predicted Vu

(kN)

26.35

67.71

29.32

31.70

71.95

24.68

Mean ratio =

34.05

70.54

42.63

23.58

68.83

23.46

Mean ratio =

Overall mean =

Ratio

Predicted / Measured

0.978

0.985

1.058

1.147

1.083

0.875

1.021

1.063

0.924

1.293

0.761

0.938

0.738

0.953

0.987

All numbers that end with A or C are corner connections; those that end with B are edge

connections. Unlike models P2 and P3 , models PI and P4 do not have spandrel beams.

213


The outcomes of these codes of practice predictions are presented in Table 11.2 . It can

be seen that for the six slab-column connections of models PI and P4 (which are

without spandrel beams), both the Australian and the American code methods give a

mean ratio of the predicted and measured results of 1.619 and 1.892 respectively. This

means that the Australian Standard procedure gives a better prediction than the ACI

method, which overestimate Vu by as much as 2.46 times that of the corresponding

measured result (for comer connection Pl-C). On the other hand, the Australian

Standard procedure overestimates the strength by just below 2.15 times at the most (for

comer connection P4-C). Note that the Australian Standard results for edge

connections are generally good for models PI and P4 , at a predicted to measured Vu

ratio of 0.954 and 1.289 respectively.

A survey of Table 11.2 indicates that the British method with a mean ratio of the

predicted and measured results of 0.726, gives a conservative prediction overall for

models PI and P4 . The maximum underestimate is for edge connection B where the

prediction is about 60% of the measured value.

In the case of models P2 and P3 which are flat plates with spandrel beams for which the

ACI method and the British method are not applicable, only the Australian Standard

procedure is used for the comparison. This is shown in the lower portion of Table 11.2 .

The mean ratio of the predicted and measured results is 2.364 for models P2 and P3 .

These are more unsafe compared to the predictions for models PI and P4 (which are

without spandrel beams).

214


Table 11.2 - Predicted punching shear strength results using the methods

recommended by A S 3600 -1988, ACI318-89 and B S 8110:1985

Connection

*

Pl-A

Pl-B

Pl-C

P4-A

P4-B

P4-C

P2-A

P2-B

P2-C

P3-A

P3-B

P3-C

Measured

Vu,kN

26.93

68.75

27.72

27.64

66.43

28.22

32.02

76.33.

32.97

30.97

73.35

31.78

Overal

AS 3600-1988

Predicted

V„,kN

42.75

65.56

45.63

57.81

85.62

60.56

Mean =

96.09

127.20

116.85

73.03

84.37

78.36

Mean =

[ mean =

Predicted

Measured

1.587

0.954

1.646

2.092

1.289

2.146

1.619

3.001

1.666

3.544

2.358

1.150

2.466

2.364

1.992

ACI318-89*

Predicted

41.84

101.35

68.12

54.90

99.79

67.07

Mean =

Predicted

Measured

1.554

1.474

2.457

1.986

1.502

2.377

1.892

For Model P2, the

mean of n Predicted

Measured

= 2.737

For Model P3, the

mean of n Predicted

Measured

= 1.991

BS 8110:1985*

Predicted Vu,kN

18.88

42.82

26.31

17.47

39.38

24.22

Mean =

Predicted

Measured

0.701

0.623

0.949

0.632

0.593

0.858

0.726

-

Note that the predictions by the Australian Standard procedure are less safe for Model

P2 with a deeper spandrel beam (of 160 mm) than those for Model P3 with a shallower

spandrel beam (of 120 mm). This is shown in the fifth column at the lower portion of

* All numbers that end with A or C are corner connections; those that end with B are edge

connections. Unlike models P2 and P3 , models PI and P4 do not have spandrel beams.

* ACI318-89 and BS 8110:1985 do not cover the punching shear analysis of flat plates with spandrel

beams.

215


Table 11.2 . The mean ratio of the predicted and measured results for Model P2 is

2.737, which is higher than that for Model P3 at 1.991. Also, the higher overestimates

are for the comer connections P2-A and P2-C , with the predicted to measured Vu ratio

of 3.00 and 3.54 respectively.

11.5 Comments on the Accuracy and Reliability of the Proposed Approach (or

the Modified Wollongong Approach)

11.5.1 Flat plates without spandrel beams (Models PI and P4)

It is obvious in Fig. 11.1 that the predictions by the proposed approach has all its

correlation points well within the +30%- and -30%-percentile limits. In contrast, the

Australian Standard procedure has 2 and the British method has 3 (out of 6 points), in

those limits.

The four remaining correlation points for the Australian Standard procedure are

overestimates by more than 30% (i.e. below the -30% line); the three remaining points

for the British method are above the +30% line.

Contrary to these, the ACI procedure has all its correlation points falling below the

-30% line. This means that all its results are overestimates and by more than 30%.

A survey of the maximum, average and standard deviations in Table 11.3 leads to the

conclusion that the strengths of the prediction for all four methods (from the highest to

lowest) are in the following order : the proposed approach, British method, Australian

Standard procedure and ACI method.

216


* AS 3600-1988 ° ACI318-89 H BS 8110:1985

Proposed approach

i ' \~—*—r O 30 60 90 120

Predicted Vu (kN)

Fig. 11.1 - Measured versus predicted Vu for post-tensioned concrete flat

plate models without spandrel beams (PI and P4)

+ Table 11.3 - Deviations of predicted Vu for models without spandrel beams

Prediction methods

A S 3600-1988

ACI318-89

BS 8110:1985

Proposed approach

Maximum deviation

+ 1.15

+ 1.46

-0.41

+ 0.15

Average deviation

+ 0.62

+ 0.89

-0.27

+ 0.02

Standard deviation

22.01

32.36

16.28

3.25

+ Maximum deviation = the maximum value of (Predicted Vu - Measured VJ/Measured Vu Average deviation = the average value of (Predicted Vu - Measured V„)/Measured Vu

Standard deviation = [Z (Predicted V M - Measured vuY

N

217


11.5.2 Flat plates with spandrel beams (Models P2 and P3)

It may be observed in Fig. 11.2 that all 6 correlation points of the proposed approach fall

within the 30% percentile limits, whereas only one correlation point by the Australian

Standard procedure is bounded by the 30% line limits. The other correlation points are

all overestimates by more than 30% (i.e. below the -30% line).

It is obvious in Table 11.4 that the predictions by the proposed approach deviate far less

from the measured values than that by the Australian Standard procedure. This means

that the proposed approach produces a more accurate and reliable prediction of Vu for

flat plate models with spandrel beams than the Australian Standard procedure.

- AS 3600-1988 » Proposed approach

Equality

30% /(overestimate)

- " — i — • — i — • r O 30 60 90 120 150

Predicted V u (kN)

Fig. 11.2 - Measured versus predicted Vu for post-tensioned concrete flat

plate models with spandrel beams (P2 and P3)

218


Table 11.4 - Deviations of predicted Vu for models

with spandrel beams

Prediction methods

A S 3600-1988

Proposed approach

Maximum deviation

+ 2.54

+ 0.14

Average

deviation

+ 1.36

+ 0.03

Standard deviation

54.45

2.94

11.6 S u m m a r y

For both types of post-tensioned concrete flat plate models (i.e. with and without

spandrel beams), the overall performance of the proposed approach is compared to the

Australian Standard procedure [SAA, 1988]. It is found that their relative performance

is similar to the findings for reinforced concrete flat plate models (see Chapter 3), in

which the Wollongong Approach [Loo and Falamaki, 1992] is compared to the

Australian Standard procedure.

It has been suggested in Section 3.4.5 that the Australian Standard procedure was

developed without adequate support of model test results particularly for those with

spandrel beams. This was believed to be the reason for its poor performance [Falamaki

and Loo, 1991 and Loo and Chiang, 1993]. It appears that the lack of reliability of the

Australian Standard procedure also extends to post-tensioned concrete flat plates.

The performance of the ACI method [ACI, 1989] for models without spandrel beams is

also poor (with overestimates as high as 2.46 times the measured values). Most likely,

this is due to the omission of the effect of prestress on the punching shear strength of

219


exterior column connections (see Section 5.1.2b). O n the other hand, the British

method [BSI, 1985] may not be as accurate as the proposed approach, but its

predictions are conservative (with underestimated results as low as 59.3% of the

measured value).

220

Chapter 12 : Conclusion

CHAPTER 12

CONCLUSION


In Chapter 3, a comparative study is described of the relative performance of the

Wollongong Approach [Loo and Falamaki, 1992] in predicting the punching shear

strength of reinforced concrete flat plates. In this study, the Wollongong Approach is

compared with three national concrete codes of practice, namely A S 3600-1988 [SAA,

1988], ACI318-89 [ACI, 1989] and B S 8110:1985 [BSI, 1985]. Besides the relative

accuracy and reliability of the Wollongong Approach, some limitations of this approach

are also identified regarding the effects of varying spandrel beam features on the

punching shear strength of reinforced concrete flat plates. The conclusions drawn are

given in Section 12.2 .

As part of a long-term research effort on the structural behaviour and failure mechanisms

of post-tensioned concrete flat plates with spandrel beams, a series of four half-scale

models were constructed and tested to failure. With the aid of the experimental results,

the prediction procedure for reinforced concrete flat plates by Loo and Falamaki [1992] is

modified to analyse the punching shear strength at the comer- and edge-column positions

of post-tensioned concrete flat plates. This is referred to as the proposed approach (or the

Modified Wollongong Approach). Based on these related studies, conclusions can be

drawn in the following four areas:

(a) deformation and failure mechanisms of post-tensioned concrete flat plate models;

221


(b) development of the proposed approach, for the prediction of Vu of post-tensioned

concrete flat plates with and without spandrel beams; and

(c) the accuracy of the proposed approach.

These are elaborated in Sections 12.3.1, 12.3.2 and 12.3.3 respectively. Some

recommendations for further study are given in Section 12.4 .

12.2 Performance and Limitations of the Wollongong Approach [Loo and

Falamaki, 1992]

The Wollongong Approach in general gives the best performance in predicting the

punching shear strength of flat plates with torsion strips and those with spandrel beams.

The Australian Standard procedure performs just as satisfactorily for flat plates with

torsion strips but tends to be unsafe for those with spandrel beams (as it at times grossly

over-estimates the punching shear strength). The main reason for the inferior

performance by the Australian Standard provision is the inadequate support of model

test results, particularly for flat plates with spandrel beams.

Both the ACI and the British methods are applicable only to flat plates with torsion

strips, in which they also tend to give unsafe predictions of the punching shear strength.

However, by applying appropriate reduction factors of 0.638 and 0.647 respectively to

the ACI and British methods respectively, the accuracy of the predictions of Vu

improves dramatically (Section 3.4.4).

The parametric study of the Wollongong Approach in analysing Vu , of reinforced

concrete flat plates with spandrel beams shows that, the punching shear strength

222

Chapter 12: Conclusion

increases with deeper and wider spandrel beam. This is true only within some limits of

the spandrel strength parameter, 8 , beyond which the formulations are not feasible. The

same also applies to the amount of longitudinal steel reinforcement in the spandrel beam.

The lower and upper limits of 8 , in which the Wollongong Approach may be applied to

determine Vu , of reinforced concrete flat plates with spandrel beams are summarised

and presented in Table 4.4 (Chapter 4).

12.3 Experimental and Theoretical Studies of Post-tensioned Concrete Flat

Plates

12.3.1 Deformation and Failure Mechanisms

The patterns of deformation (crack propagation and deflection) and failure mechanisms

at mid-panels and at exterior slab-column connections of post-tensioned concrete flat

plate models with spandrel beams are similar to the reinforced concrete systems.

Depending on the stiffening effect offered by the spandrel beam or torsion strip along the

slab edge, the post-tensioned concrete flat plate models would fail in either shear or

flexural failure modes.

Both models without spandrel beam and the one with shallow spandrel (i.e. models PI,

P4 and P3 respectively) followed shear failure mode. O n the other hand, the model with

deep spandrel (i.e. Model P2) failed in a flexural mode. Note that when Model P2

finally failed, the corner- and edge-columns also failed almost simultaneously in

punching shear as a response to a sudden shift or re-distribution of the applied loading to

the exterior column supports, as described in (Section 8.4). It has also been shown that

223


the spandrel beam offered considerable stiffening effect in reducing the magnitude of

deflections, particularly along the edges of the slab.

12.3.2 Prediction of Vu by the proposed approach (Modified Wollongong

Approach)

j\n analytical procedure is developed in this thesis for the prediction of the punching

shear strengths of post-tensioned concrete flat plates at the edge- and comer-column

connections. This procedure, referred to as the proposed approach, involves the

following work.

(a) Adoption of the basic theoretical aspects of the Wollongong Approach as

developed by Loo and Falamaki [1992] (Section 5.1.3).

(b) Incorporation of the effect of prestress into the three basic equilibrium equations

plus an interaction equation to account for shear, moment and torsion in the

spandrel (Sections 5.1.4 and 5.1.7 respectively).

(c) " Re-calibration of the semi-empirical formulas for the slab restraining factor, \J/ ,

to be used as an integral part of the prediction formulas for Vu of post-tensioned

concrete flat plates (Section 10.4).

This process is aided by developing a BASIC computer program to calculate all the

required simple and explicit equations contained in the proposed prediction procedure.

The program is versatile in usage, featuring user-friendly routines for data input, storage

and retrieval.

224


12.3.3 Accuracy of the proposed approach (Modified Wollongong Approach)

It was shown in Tables 11.1 and 11.2 (Chapter 11) that the proposed approach

developed in this thesis gives a more accurate prediction than those of A S 3600-1988

[SAA, 1988].

For models without spandrel beams, the proposed approach produced the best

prediction results compared with those of A S 3600-1988 [SAA, 1988], ACI318-89

[ACI, 1989] and B S 8110:1985 [BSI, 1985], as shown in Tables 11.1 and 11.2 . The

proposed approach also performed better in predicting Vu for models with spandrel

beams than the Australian Standard procedure (Tables 11.1 and 11.2).

The Australian Standard procedure gave a slightly better prediction results for models

without spandrel beam than for models with spandrel beams. Nevertheless, it

overestimates the punching shear strength considerably. This is particularly the case at

comer-column connections (Table 11.2). It has been suggested that the Australian

Standard procedure was developed without adequate support of model test results

particularly for those with spandrel beams. This is believed to be the reason for its poor

performance.

12.4 Recommendations for Further Study

Further studies on the structural behaviour of post-tensioned concrete flat plates may be

focused on two aspects: analytical studies of proposed prediction procedure;

experimental studies of other types of model construction.

225


Based on the proposed approach, additional parametric studies may be carried out on

factors that may influence the analysis of Vu . These include the distribution of prestress

and other features of the slab and spandrel beam.

To supplement the model tests being carried out to date, other studies may cover:

(a) band beam slabs; and

(b) effects of repeated loads on post-tensioned concrete flat plates.

226

References

REFERENCES

American Concrete Institute (ACI 1989), Building Code Requirements for Reinforced

Concrete (ACI 318-89) and Commentary - ACI318R-89, Detroit, 353 p.

ACI-ASCE Committee 423 (1983), "Recommendations for Concrete Members

Prestressed With Unbonded Tendons (ACI423.3R-83)", Concrete International :

Design and Construction, V.5, No.7, July 1983, pp.61-76.

ASCE-ACI Committee 426 (1977), "Suggested Revisions to Shear Provisions for

Building Codes"', ACI Structural Journal, V.74, September 1977, pp.458-469.

Beresford, F.D. and Blakey, F.A. (1963), "Experimental Lightweight Flat Plate

Structure, Part VII - A Test to Destruction", Constructional Review, V.36, No.6,

June 1963, pp. 18-26.

British Standards Institution (BSI 1985), BS8110:1985 Structural Use of Concrete,

Part 1, Code of Practice for Design and Construction, London.

Brotchie, J.F. and Beresford, F.D. (1967), "Experimental Study of a Prestressed

Concrete Flat Plate Structure", Civ. Engg. Trans. IEAust, V.CE9, No.2, October

1967, pp.276-282.

Burns, H. and Hemakom, R. (1977), " Test of Scale Model Post-Tensioned Flat Plate",

Proc. ASCE, V.103, ST6, June 1977, pp.1237-1255.

CEB-FIP Model Code (1978), International System of Unified Standard Codes of

Practice for Structures, Comite Euro-International du Be ton, Paris, 348 pp.

227

References

Cement and Concrete Association of Australia (1991), Concrete Design Handbook (in

accordance withAS3600), 2nd Ed., Sydney, 1991, 237 p.

Chana, P.S. (1991), "Punching Shear in Concrete Slabs", The Structural Engineer,

Vol.69, No.15, 6 August 1991, pp.282-285.

Chiang, CL. and Loo, Y.C. (1993), "Punching Shear Tests of Half-Scale Post-

Tensioned Concrete Flat Plate Models with Spandrel Beams," Proc. 13th

Australasian Conference on Mechanics of Structures and Materials (ACMSM

13), Wollongong, July 1993, V.l, pp. 185-192.

Clarkson, S.A. (1992), Deflection of a Post-tensioned Concrete Flat Plate With

Spandrel Beams, B.Eng. thesis, University of Wollongong, New South Wales,

1992, 100 p.

Concrete Society (1979), Flat Slabs in Post-tensioned Concrete with Particular

Regard to the Use of Unbonded Tendons: Design Recommendations, London,

Technical Report 17, July 1979.

Cope, RJ. and Clark, L.A. (1984), Concrete Slabs Analysis and Design, Elsevier

Applied Science, London, 1984, 502 p.

Di Stasio, J. and Van Buren, M.P. (1960), "Transfer of Bending Moment Between Hat

Plate Floor and Column", ACI Structural Journal, V.57, September 1960,

pp.299-314.

228

References

Elgren, L., Karlsson, I., Losberg A. (1974), "Torsion-Bending-Shear Interaction for

Concrete Beams", Journal of the Structural Division, ASCE, Vol.100, N0.ST8,

August 1974, pp. 1657-1676.

Elstner, R.C. and Hognestad, E. (1956), "Shearing Strength of Reinforced Concrete

Slabs", ACI Structural Journal, V.28, No.l, July 1956, pp.29-58.

Falamaki, M. (1990), Punching Shear Strength of Reinforced Concrete Flat Plates

With Spandrel Beams, PhD thesis, University of Wollongong, New South Wales,

1990, 285 p.

Falamaki, M. and Loo, Y.C. (1991), Discussion 87-S15 , "Punching Shear Design in the

New Australian Standard for Concrete Structures", by Rangan, B.V. (1990) , ACI

Structural Journal, V.88, No.l, January-February 1991, pp.115-116.

Falamaki, M. and Loo, Y.C. (1992), "Punching Shear Tests of Half-Scale Reinforced

Concrete Flat Plate Models With Spandrel Beams", ACI Structural Journal,

V.89, No.3, May-June 1992, pp.263-271.

Falamaki, M. and Loo, Y.C. (1993), Authors' Closure (Discussion 89-S25) , "Punching

Shear Tests of Half-Scale Reinforced Concrete Hat Plate Models With Spandrel

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No.2, March-April 1993, p.219.

Foutch, D.A., Gamble, W.L. and Sunidja, H. (1990), "Tests of Post-Tensioned Concrete

Slab-Edge Column Connections", ACI Structural Journal, V.87, No.2, March-

April 1990, pp. 167-179.

229

References

Gilbert, RJ. and Mickleborough, N.C. (1990), Design of Prestressed Concrete, Unwin

Hyman Ltd., London, 1990, 504 p.

Guralnick, S.A. and LaFraugh, R.W. (1963), "Laboratory Study of a 45-foot Square

Hat Plate Structure", ACI Structural Journal, V.60, No.9, September, 1963,

pp. 1107-1184.

Hawkins, N.M. (1974), "Shear Strength of Slabs with Moment Transferred to

Columns", Shear in Reinforced Concrete, SP-42, Vol.2, American Concrete

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Hii, H.M. (1993), Serviceability of a Prestressed Concrete Flat Plate, B.Eng. thesis,

University of Wollongong, New South Wales, 1993, 169 p.

Ingvarsson, H., (1974), Experimentellt studium av betongplattor understodda av horn

pelare (Experimental Study of Concrete Slabs Supported By Corner Columns),

Meddelande 111, Institutionen for Byggnadsstatik, Kungliga Tekniska Hogskolan,

Stockholm, 1974.

Institution of Structural Engineers (1969), Shear Strength of Reinforced Concrete

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Kinnunen, S., (1971), Forsok med betongplattor understodda av pelare vid fh kant

(Tests on Concrete Slabs Supported by Columns at Free Edges), Rapport R2,

Statens Institut for Byggnadsstatik, Stockholm, 1971.

Kong, F.K. and Evans, R.H. (1987), Reinforced and Prestressed Concrete, 3rd Ed.,

Van Nostrand Reinhold (International), London, 1987, 508 p.

230

References

Lei, K.F. (1993), Punching Shear Strength Behaviour of Post-tensioned Concrete Flat

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Wales, 1993, 144 p.

Lin, T.Y. and Burns, N.H. (1981), Design of Prestressed Concrete Structures, 3rd Ed.,

John Wiley & Sons, New York, 1981, 646 p.

Long, A.E. and Masterson, D.M. (1974), "Improved Experimental Procedure for

Determining the Punching Shear Strength of Reinforced Concrete Hat Slab

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1974, pp.921-935.

Long, A.E., Kirk, D.W. and Cleland, DJ. (1978), "Moment Transfer and the Ultimate

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Long, A.E. and Cleland, D.J. (1993), "Post-Tensioned Concrete Hat Slabs at Edge

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Loo, Y.C. and Chiang CL. (1993), "Methods of Punching Shear Strength Analysis of

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July-August, 1992, pp.375-383.

231

References

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Sciences, McGraw-Hill Book Co., New York, 1975, 675 p.

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232

References

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Longman Cheshire, Melbourne, Australia, 1989, 553 p.

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233

Appendix A

APPENDIX A

CALIBRATION OF PRESTRESSING JACK

Conversion factor : 1 mm = 0.039 in.

I N = 0.22431b.

1 kPa = 145 psi.

(IMPa = 0.145 psi)

A.l Actual Stress on Prestressing tendon

5.05 m m diameter (or 0.197") of prestressing tendon

giving a cross-sectional area, Ap

n [ 0.197 ]2 _ Mne. . A = — — = 0.0305 sq.in.

P 4

The required prestressing force to be applied on each tendon was 23.03 kN. This was

equivalent to,

23.03 x 103 N x — = 5164.7 lbs. 9.81

Therefore, the applied prestress on each tendon was,

234

Appendix A

5164.7

0.0305 = 169335.4 psi

A.2 Stress Exerted by Jack

Since we require a prestress force of 23.03 kN to be applied on each tendon, the stress to

be exerted by the jack in order to achieve the required force may be determined by the

following manner.

Jack cylinder

32 m m (1.26")

Cross-sectional area, A

. n [ 1.972 - 1.262 ] 1fic . A = = 1.8 sq.in

For a stress application of 2000 psi, the required force needed to be exerted on the jack

cylinder was,

2000x1.8 = 3600lbs. (or 16.4kN)

and for an applied stress of 3000 psi, the required force was,

235

Appendix A

3000x1.8 = 5400lbs. (or24.5 kN)

In order to get an applied force of 5164.7 lbs. or 23.03 kN on each tendon, the applied

stress by the jack was,

2000 + [ 23-03 ' 16A x 1000 psi ] = 2819 psi 24.5 - 16.4 F F

which is the calibrated jacking force to be applied during the post-tensioning process.

236

Appendix B

APPENDIX B

DEFLECTION MEASUREMENTS

B.l Deflection Measurements of Model PI

Table B.la -Deflection measurements of Model PI (using rulers)

Applied load

(kPa)

0

4.46

5.93

7.33

8.69

10.18

11.59

13.18

14.89

16.18

Deflection measurement (mm) using rulers for both panels at location

1 0

0.5

1.0

1.5

2.0

3.5

5.0

7.0

10.5

37.0

^2 0

0.1

0.3

0.4

0.5

0.9

1.5

3.3

5.0

8.0

. 4 0

0.2

0.5

0.7

1.0

2.0

3.0

5.0

8.0

34.0

5

0 0.7

1.5

2.0

2.5

4.2

6.0

9.0

13.0

42.0

6 0

0.2

0.5

0.8

1.0

2.0

3.0

5.5

8.0

13.0

7 0

0.1

0.2

0.3

0.5

0.9

1.5

2.5

4.0

7.0

9 0

0.2

0.4

0.4

0.5

1.5

2.5

3.2

4.0

34.5

Table B.lb -Deflection measurements of Model PI (using LVDTs)

Applied load (kPa) 0

4.46

5.93

7.33

8.69

10.18

11.59

13.18

14.89

16.18

Deflection measurement (mm) using L V D T

Centre of panel ABGF (at location 3)

0

0.55

1.10

1.64

2.88

5.20

7.67

11.10

24.66

46.68

Centre of panel BCHG (at location 8)

0

0.41

0.96

1.51

2.47

3.97

5.89

8.90

15.07

45.25

237

Appendix B

B.2 Deflection Measurements of Model P2

Table B.2a -Deflection measurements of Model P2 (using rulers)

Applied load

(kPa)

0

1.37

3.28

4.37

5.69

6.96

9.86

11.27

12.48

13.21

14.69

15.86

16.17

17.23

18.44

19.61


1

0

0.6

2.4

2.8

3.4

4.1

5.2

8.0

12.5

15.4

19.2

22.8

25.8

31.6

37.9

46.1

2

0

0.1

0.5

0.7

1.0

1.2

1.7

2.2

3.3

4.5

5.3

6.9

7.4

8.2

10.4

13.4

4

0

0.1

0.3

0.5

0.6

0.8

1.2

2.7

3.4

3.5

3.6

4.3

5.3

5.7

6.4

7.7

5

0

0.1

0.5

1.1

1.8

2.8

3.3

9.0

13.3 '

16.5

19.6

24.5

29.8

33.7

36.5

41.9

6

0

0.3

1.1

1.4

2.0

2.7

3.9

7.3

12.0

14.2

17.6

20.1

28.1

36.0

42.1

49.3

7

0

0.2

0.4

0.5

0.8

0.9

1.2

2.4

3.2

3.7

4.2

5.0

5.6

5.8

9.7

13.4

9

0

0.1

0.3

0.3

0.4

0.5

1.3

3.0

3.8

4.3

4.7

5.1

6.0

7.1

8.2

11.9

238

Table B.2b -Deflection measurements of Model P2 (using LVDTs)

Applied load

(kPa)

0

1.37

3.28

4.37

5.69

6.96

9.86

11.27

12.48

13.21

14.69

15.86

16.17

17.23

18.44

19.61


Centre of panel ABGF (at location 3)

0

0.67

1.80

2.53

3.33

4.50

6.21

11.25

17.05

20.81

25.25

33.57

42.44

44.41

51.84

66.33

Centre of panel BCHG (at location 8)

0

1.14

1.74

3.43

3.62

4.62

6.28

11.72

17.75

22.05

25.98

34.61

43.18

45.28

52.85

66.68

Appendix B


Table B.3a - Deflection measurements during first day of testing

of Model P3 (using rulers)

Applied load

(kPa)

0

1.40

2.80

4.20

5.60

7.00

8.40

9.80

11.20

12.60

14.00

14.68

15.40

17.61

18.17

19.57

20.27


1

0

0

1.0

1.0

1.5

3.0

3.5

4.5

5.5

7.0

8.0

9.5

11.0

14.0

15.0

18.5

22.0

2

0

0

1.0

1.0

1.5

2.0

2.5

3.0

4.0

6.0

6.5

8.0

9.0

11.0

11.5

13.5

15.0

3

0

0

1.0

1.0

2.5

4.0

5.5

7.0

10.0

14.0

16.0

19.5

23.0

30.0

32.0

40.0

47.0

4

0

0

0

1.0

1.0

1.0

1.5

2.0

3.0

6.0

6.0

7.5

9.0

11.5

12.5

15.0

17.0

5

0

0

1.0

2.0

3.0

4.0

5.0

6.5

8.0

9.0

10.0

12.0

13.5

16.0

18.5

23.0

28.0

6

0

0

0

0

1.0

2.0

3.0

3.5

4.0

5.0

6.5

7.5

8.5

10.5

11.5

13.5

15.5

7

0

0

0

0.5

1.0

1.0

1.0

1.0

2.0

2.5

3.0

3.0

3.5

4.0

5.0

5.5

6.5

8

0

2.0

2.0

3.0

3.0

4.0

5.5

6.5

8.0

10.0

11.0

13.0

15.0

19.0

21.0

26.0

32.0

9

0

0

1.0

1.0

1.5

2.0

2.0

7.0

7.0

8.5

9.5

10.0

11.0

12.0

13.0

14.5

16.0

240

Appendix B

Table B.3b - Deflection measurements during second day of testing of Model P3 (using rulers)

Applied load

(kPa)

0

7.97

14.0

16.77

18.73

20.27

21.52

22.36

Deflection measurement (mm) using mlers for both panels at location

1

0

2.00

8.00

13.00

15.00

17.50

20.00

26.50

2

0

0.50

3.50

4.00

5.50

6.50

7.50

9.50

3

0

4.00

15.00

22.00

25.00

30.00

35.00

45.00

4

0

1.50

5.50

10.00

11.00

12.50

17.00

25.00

5

0

3.00

10.00

14.50

17.00

20.00

23.50

34.00

6

0

1.50

5.50

7.00

8.00

9.50

11.00

24.50

7

0

0.50

3.00

3.50

4.00

4.50

5.50

8.00

8

0

3.50

10.50

15.50

18.00

21.00

25.00

48.00

9

0

1.00

3.00

9.00

10.50

13.00

14.50

17.00

Table B.3c - Deflection measurements during first day of testing

of Model P3 (using dial gauges)

Applied load

(kPa)

0

1.40

2.80

4.20

5.60

7.00

8.40

9.80

11.20

12.60

14.00

14.68

15.40

17.61

18.17

19.57

20.27

Deflection measurement (mm) using dial gauges for both panels at location

1

0

0.25

0.64

1.13

2.44

2.53

3.53

4.34

5.56

6.92

7.89

9.12

10.80

13.09

14.34

17.94

22.09

2

0

0.13

0.22

0.45

0.76

1.15

1.76

2.17

3.15

4.76

5.73

7.11

8.41

10.09

10.75

12.67

14.06

4

0

0.16

0.39

0.70

0.97

1.36

1.91

2.49

3.76

4.72

6.97

8.34

9.78

12.43

13.50

16.58

18.98

5

0

0.32

0.93

1.72

2.60

3.52

4.84

5.79

7.46

8.59

9.80

11.39

13.10

16.60

17.86

20.82

22.41

6

0

0.24

0.63

1.11

1.66

2.16

3.15

3.43

4.35

5.46

6.27

8.49

9.51

11.59

12.38

14.05

16.75

7

0

0.80

1.14

1.33

1.50

1.68

1.94

2.15

2.51

3.09

3.41

3.99

4.48

5.05

5.50

6.47

7.40

9

0

0

0.12

0.27

0.41

0.62

0.93

1.24

2.01

2.58

3.99

4.82

5.68

7.14

7.78

9.51

11.31

Appendix B

Table B.3d - Deflection measurements during second day of testing


Applied load

(kPa)

0

7.97

14.0

16.77

18.73

20.27

21.52

22.36


1

0

2.90

8.18

11.38

12.17

15.53

18.53

2

0

1.36

4.25

6.01

6.67

7.92

8.84

-

4

0

2.09

6.17

8.67

9.82

11.39

12.66

28.50

5

0

5.00

12.02

16.15

18.12

21.13

25.14

-

6

0

3.52

7.47

9.84

10.89

12.52

14.52

-

7

0

1.77

3.50

4.47

4.89

5.52

6.21

-

9

0

1.05

3.70

4.86

5.79

6.72

8.45

19.01

Table B.3e - Deflection measurements during first day of testing

of Model P3 (using LVDTs)

Applied load

(kPa)

0

1.40

2.80

4.20

5.60

7.00

8.40

9.80

11.20

12.60

14.00

14.68

15.40

17.61

18.17

19.57

20.27


Centre of panel ABFG (at location 3)

0

0.30

0.70

1.90

3.10

4.50

5.90

7.40

10.10

12.90

16.70

18.90

22.90

27.90

32.40

40.10

46.30

Centre of panel BCGH (at location 8)

0

0.30

0.70

1.80

2.70

3.30

4.70

5.80

6.90

8.50

10.30

11.90

14.40

16.80

20.30

25.10

29.70

242

Appendix B

Table B.3f - Deflection measurements during second day of testing

of Model P3 (using L V D T s )

Applied load

(kPa)

0

7.97

14.0

16.77

18.73

20.27

21.52

22.36


Centre of panel ABFG (at location 3)

0

6.20

17.50

24.50

27.70

32.90

51.40

93.10

Centre of panel BCGH

(at location 8) 0

7.10

15.40

20.00

22.50

25.70

33.70

72.50


Table B.4a - Deflection measurements during first day of testing


Applied load

(kPa) 0

1.37

2.75

4.43

5.71

6.89

8.75

9.72

11.47

12.77

12.70


1 0

0

0.9

1.1

2.0

2.9

4.0

5.0

6.9

10.0

4.0

2 0

0

0.9

1.0

1.0

2.0

2.5

2.5

3.1

6.0

3.0

3 0

0.5

1.0

2.0

3.0

4.5

6.0

7.0

10.0

16.0

7.0

4 0

0

1.0

1.0

1.5

2.0

2.5

3.0

4.0

7.0

5.0

5 0

0.2

0.5

1.0

1.4

2.0

2.5

3.0

4.1

5.5

2.2

6 0

0

0

1.0

1.5

3.0

3.0

3.0

4.0

5.0

5.0

7 0

0

0

0

0.5

1.0

1.5

1.5

2.0

3.5

7.0

8 0

0

0.5

1.5

2.0

4.0

5.0

6.0

8.5

12.0

4.5

9 0

0

0

0

0.5

1.0

1.5

1.5

2.0

3.5

1.0

243

Appendix B

Table B.4b - Deflection measurements during second day of testing


Apphed

load

(kPa)

0

0.67

1.49

2.11

2.92

3.38

4.4

5.76

7.06

8.39

9.37

11.03

12.92

14.33

15.64

16.04

16.64


1

0

0

0.5

0.5

1.3

1.4

2.0

2.7

3.5

4.5

5.5

6.5

7.8

10.0

12.5

15.5

66.5

2

0

0

0.2

0.2

0.2

0.2

1.2

1.2

1.7

2.2

3.2

3.2

4.2

5.2

7.2

8.2

14.7

3

0

0.1

1.0

1.0

2.0

2.0

3.5

5.0

6.0

7.5

9.5

11.0

13.0

17.0

22.0

26.0

131.0

4

0

0

0.4

0.9

0.9

0.9

1.4

1.9

2.7

2.9

3.9

4.7

5.9

6.9

8.9

9.9

36.9

5

0

0

0.1

0.4

0.6

0.6

1.0

1.5

2.0

2.5

3.0

3.5

4.0

5.5

7.0

8.0

45.0

6

0

0

1.0

1.0

1.5

2.0

2.0

3.0

3.1

4.0-

5.0

5.0

6.0

8.0

11.0

13.0

69.5

7

0

0

0.5

0.9

1.0

1.0

1.2

2.0

2.1

3.0

3.0

4.0

4.5

6.0

7.0

8.5

23.0

8

0

0.2

1.0

1.0

2.0

2.0

3.0

4.0

5.0

6.0

7.5

9.0

11.0

14.5

19.5

23.0

87.0

9

0

0

0.5

0.5

0.8

0.9

1.0

1.5

2.0

2.0

3.0

3.0

4.0

5.0

6.0

7.0

36.0

244

Appendix B

Table B.4c - Deflection measurements during first day of testing


Applied load

(kPa)

0

1.37

2.75

4.43

5.71

6.89

8.75

9.72

11.47

12.77

12.70


1

0

0.45

0.97

0.97

1.38

1.68

2.88

3.79

4.90

5.80

1.76

2

0

0

0.71

0.89

0.91

1.74

2.20

2.29

2.87

5.37

2.96

4

0

0

0.56

0.89

0.89

1.42

2.03

2.31

3.67

6.63

1.00

5*

-

-

-

-

-

-

-

-

-

-

-

6

0

0.25

0.64

1.33

1.98

3.81

4.03

4.97

6.63

7.11

1.87

7

0

0

0.35

0.45

0.45

0.60

1.83

2.80

3.36

4.80

1.78

9

0

0

0.82

1.40

1.60

1.80

2.22

2.70

2.78

3.94

1.41

Dial gauge 5 was faulty during testing of Model P4

245

Appendix B

Table B.4d - Deflection measurements during second day of testing


Applied load

(kPa)

0

0.67

1.49

2.11

2.92

3.38

4.4

5.76

7.06

8.39

9.37

11.03

12.92

14.33

15.64

16.04

16.64


1

0

0

0.60

0.90

1.20

1.62

1.94

2.50

3.00

4.65

5.25

6.84

7.20

8.91

10.82

10.90

-

2

0

0

0.11

0.17

0.29

0.39

0.98

1.05

1.54

2.01

2.98

3.31

4.15

4.98

7.10

8.10

-

'4

0

0.25

0.41

0.46

0.69

0.76

1.27

2.00

2.39

2.92

3.66

4.32

5.62

6.73

9.02

9.86

-

5*

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

6

0

0.10

0.36

0.40

0.72

0.81

1.33

1.98

2.49

3.22

4.30

4.38

7.10

8.43

9.49

12.96

-

7

0

0

0.90

1.70

1.70

1.70

2.00

2.00

2.60

3.39

3.51

3.63

4.02

5.00

6.88

7.68

-

9

0

0.10

0.28

0.41

0.60

0.68

1.07

1.65

2.31

2.91

3.48

3.61

4.50

4.62

6.01

6.86

-

246

Table B.4e - Deflection measurements during first day of testing

oi Model P4 (using LVDTs)

Applied load

(kPa)

0

1.37

2.75

4.43

5.71

6.89

8.75

9.72

11.47

12.77

12.70


Centre of panel ABFG

(at location 3)

0

3.00

4.00

4.00

5.10

8.00

9.00

11.50

12.00

16.50

6.00


(at location 8)

0

5.00

5.00

6.00

8.00

8.00

10.00

11.00

13.00

15.50

7.00

247

Appendix B

Table B.4f - Deflection measurements during second day of testing

of Model P4 (using LVDTs)

Applied load

(kPa)

0

0.67

1.49

2.11

2.92

3.38

4.4

5.76

7.06

8.39

9.37

11.03

12.92

14.33

15.64

16.04

16.64


Centre of panel ABFG

(at location 3)

0

2.00

2.50

2.50

3.50

3.50

11.00

13.00

13.50

14.50

16.00

17.50

19.00

21.50

25.00

28.00

132.00


(at location 8)

0

1.00

2.50

2.50

3.00

12.00

12.50

13.50

14.00

16.00

16.50

17.00

18.00

21.50

25.00

27.50

85.00

Appendix C

APPENDIX C

PLOTS OF EXPERIMENTAL MOMENT, SHEAR AND TORSION

RESULTS FOR POST-TENSIONED CONCRETE FLAT PLATE

MODELS

C.l Plots of Experimental Moment, Shear and Torsion Versus Applied Loads at

Corner-connections A and C and at Edge-connection B for Model PI

20

15

9 10

a o

s a

Column A Column B Column C

70

^ ' r 0 5 10 15 20 Applied load (kPa)

• Column A Column B •Column C

0 5 10 15 20 Applied load (kPa)

Fig. CI. 1 - Plot of moment transfer to columns A, B and C versus applied load (Model PI)

Fig. CI.2 - Plot of shear strength around columns A, B and C versus applied load (Model PI)

249

Appendix C

• Column A • Column B - Column C

T i '—r O 5 IO 15 20

Applied load (kPa)

Fig. C1.3 - Plot of torsional moment at columns A, B and C versus

applied load (Model PI)

C.2 Plots of Experimental M o m e n t , Shear and Torsion Versus Applied Loads at

Corner-connections A and C and at Edge-connection B for Model P3 *

25 -

"a 20 -55

m. 1 5 '

a 0

8

5 -

0 -i

— a — t,oiumn A —*— Column B

* — > 1 *-.

jl^P

f*=m | , i"

. .

25 -

f 20" 55

"~ 15 -<=l

B 0 10 j

5 -

0 1?

— ° — Column A —*— Column B — 0 — Column C

3 1 | 1 1

0 IO 20 30 Applied load (kPa)

0 IO 20 30 Applied load (kPa)

(a) Day O n e (b) Day T w o

Fig. C2.1 - Plot of moment transfer to columns A, B and C versus applied load (Model P3)

The stress measurements were recorded over two days of testing of the slab Model P3 due to the sudden failure of some bolted nuts in the whiffle-tree loading system at the near ultimate loading of 21.0 kPa. The model was re-loaded the next day and all measurements were re-taken from the beginning up to failure.

250

Appendix C


0 IO 20 30

Applied load (kPa)

(a) Day One

so -

70 -

60 -

5^, 50 -

g 40 "

8 0 3 0 -

20 -

IO -

O i

A

7 • \ '

— 0 — Column —*— Column —"— Column

1

A B C

O 10 20 30

Applied load (kPa)

(b) Day Two

Fig. C2.2 - Plot of shear strength around columns A, Band C versus applied load (Model P3)

- Column A - Column B - Column C

O IO 20 30

Applied load (kPa)

(a) Day One

• Column A - Column B • Column C

O IO 20 30

Applied load (kPa)

(b) Day Two

Fig. C2.3 - Plot of torsional moment at columns A, B and C versus applied load (Model P3)

251

Appendix C

C.3 Plots of Experimental Moment, Shear and Torsion Versus Applied Loads at Corner-connections A and C and at Edge-connection B for Model P4 **

15 - Column A - Column B Column C

20

' Column A ' Column B • Column C

0 5 10 15 Applied load (kPa)

(a) Day One

n ' i r~ O 5 IO 15 20 Applied load (kPa)

(b) Day Two

Fig. C3.1 - Plot of moment transfer to columns A, B and C versus applied load (Model P4)

DO

50 -

40 -55

^ 3 0 -«H

es u

rtA M 20 -

10 -

0 -t

— a — Column A

— a — Column C

r , , | u. 1

70

0 5 IO 15 Applied load (kPa)

(a) Day One


! 1 ' 1 ' ' 1

0 5 10 15 20 Applied load (kPa)

(b) Day Two

Fig. C3.2 - Plot of shear strength around columns A, B and C versus applied load

(Model P4)

The stress measurements were recorded over two days of testing of the slab Model P4 due to the sudden failure of a threaded steel rod in the whiffle-tree loading system at a loading of 12.7 kPa. The model was re-loaded the next day and all measurements were re-taken from the beginning up to failure.

252

-a— Column A -*— Column B -A— Column C

9

0 5 10 15

Applied load (kPa)

(a) Day One

B 55 ui.1 a o

-u—1

OT »H

o E-H

/

6

5

4

3

7.

1

O

-

-

A 7

— o — Column —*— Column — o — Column

_j_ . _i_iUL - i : 1

A B C

0 5 10 15 20

Applied load (kPa)

(b) Day Two

Fig. C3.3 - Plot of torsional moment at columns A, B and C versus applied load (Model P4)

253

Appendix D

APPENDIX D

TABULATIONS OF EXPERIMENTAL MOMENT, SHEAR AND

TORSION RESULTS FOR POST-TENSIONED CONCRETE

FLAT PLATE MODELS

Table D.l - Shear, moment and torsion at columns A, B and C for Model PI

Applied load

(kPa)

0

4.46

5.93

7.33

8.69

10.18

11.59

13.18

14.89

16.18

Shear (kN) at column

A

0

8.39

10.66

12.93

15.07

17.43

19.66

21.87

25.14

26.93

B

0

24.25

30.20

35.71

40.97

46.86

52.32

58.36

63.35

68.75

C

0

8.23

10.49

12.74

14.96

17.43

19.69

21.88

25.51

27.72


A

0

0.46

1.07

1.72

2.28

2.89

3.52

4.43

6.83

7.39

B

0

1.69

2.92

3.94

5.36

7.38

9.38

11.80

13.62

15.25

C

0

0.52

1.14

1.95

2.91

4.05

5.16

6.32

7.80

8.70


A

0

0.85

1.45

2.04

2.88

-3.75

4.56

5.80

6.84

7.26

B

0

1.24

1.37

1.40

1.45

1.66

1.81

2.12

2.20

2.35

C

0

0.67

1.04

1.60

2.17

2.70

3.17

4.41

5.61

6.14

254

Appendix D

Table D.2 - Shear, moment and torsion at columns A, B and C for Model P2

Applied , load

(kPa)

0

1.37

3.28

4.37

5.69

6.96

9.86

11.27

12.48

13.21

14.69

15.86

16.07

17.23

18.44

19.61


A

0

3.01

4.23

6.11

7.86

9.91

14.41

17.73

19.74

21.18

24.07

26.08

26.64

28.52

30.23

32.02

B

0

10.20

13.79

18.34

23.82

28.89

38.53

43.58

47.96

50.92

56.57

60.73

60.95

65.89

71.01

76.33

C

0

3.36

4.64

6.10

8.32

10.00

14.91

17.65

19.86

20.66

23.27

25.00

25.48

27.39

29.16

32.97


A

0

0.52

0.75

1.20

1.64

2.29

4.49

6.56

7.70

8.68

10.17

11.17

11.57

11.88

11.95

12.01

B

0

2.44

3.41

4.73

6.32

7.90

10.35

12.47

14.62

19.54

23.84

24.75

25.61

26.55

27.51

28.50

C

0

1.52

2.04

2.66

3.50

4.24

5.30

6.04

6.77

7.25

7.94

8.67

8.80

9.69

10.47

11.17


A

0

1.15

1.68

2.34

3.04

3.79

4.85

5.45

5.68

5.87

6.15

6.31

6.59

7.06

7.12

7.35

B

0

0.04

0.05

0.04

0.05

0.07

0.14

0.14

0.10

0.15

0.09

0.14

0.17

0.27

0.32

0.28

C

0

0.05

0.30

0.62

1.14

1.57

3.34

4.59

5.46

5.92

7.49

8.11

8.26

8.71

9.19

9.51

255

Appendix D

Table D.3a - Shear, moment and torsion at columns A, B and C for Model P3

- Day One

Applied load

(kPa)

0

1.40

2.84

4.35

5.69

7.14

8.81

10.04

11.79

13.00

14.03

15.22

17.52

18.16

19.51

20.49


A

0

1.75

3.89

6.12

7.95

10.01

12.32

14.16

16.86

19.40

21.01

22.77

25.17

26.44

28.06

29.63

B

0

4.99

10.83

17.11

22.70

28.81

35.63

40.65

47.39

50.73

54.41

59,20

64.14

66.14

70.19

72.62

C

0

1.86

4.02

6.28

8.27

10.48

13.01

14.91

17.74

20.35

22.08

24.29

28.18

29.42

31.99

33.58


A

0

0.35

0.76

1.11

1.31

1.54

1.99

2.65

3.88

5.36

6.13

7.10

8.19

9.02

10.57

11.27

B

0

0.76

1.59

2.71

3.91

5.61

7.49

8.90

11.01

13.18

15.28

17.65

20.31

28.26

24.69

27.46

C

0

0.51

1.16

2.04

2.94

4.26

5.53

6.48

7.05

7.68

8.37

9.18

9.94

10.94

11.70

12.29


A

0

0.30

0.58

0.88

1.12

1.43

1.75

2.05

2.20

2.63

3.18

3.74

4.14

4.85

5.19

6.11

B

0

0.05

0.06

0.08

0.09

0.10

0.11

0.12

0.13

0.15

0.12

0.15

0.22

0.18

0.25

0.20

C

0

0.35

0.60

0.96

1.21

1.48

1.78

2.01

2.25

2.73

3.27

3.82

4.48

4.97

5.40

6.25

256

Appendix D

Table D.3b - Shear, moment and torsion at columns A, B and C for Model P3 - Day Two

Applied load

(kPa)

0

7.98

14.06

17.36

20.36

21.45

22.42


A

0

10.33

18.25

23.40"

26.60

28.43

30.97

B

0

22.12

44.95

56.26

66.33

70.01

73.35

c 0

10.65

18.61

24.01

27.49

29.70

31.78


A

0

5.14

7.84

9.81

10.53

11.94

12.25

B

0

8.28

15.60

19.60

25.11

30.96

33.01

C

0

5.25

8.46

10.51

12.23

13.11

13.54


A

0

2.38

3.42

5.03

6.54

6.95

7.29

B

0

0.03

0.29

0.44

0.36

0.34

0.28

c 0

3.03

4.18

5.22

6.15

7.88

8.23

Table D.4a - Shear, moment and torsion at columns A, B and C for Model P4

- Day One

Applied load

(kPa)

0

1.37

2.75

4.43

5.71

6.89

8.75

9.72

11.47

12.77

12.70


A

0

2.49

4.94

7.82

9.87

11.66

14.55

16.29

18.70

21.23

21.10

B

0

5.29

10.85

17.51

22.80

27.82

35.74

40.18

47.44

52.21

51.94

C

0

2.12

4.25

6.99

9.16

11.06

14.29

16.11

18.80

21.33

21.24


A

0

0.53

1.14

1.94

2.58

3.36

4.45

5.11

5.83

5.25

5.21

B

0

0.25

0.61

1.24

2.22

3.92

6.44

7.58

10.25

12.66

12.60

C

0

0.37

0.74

1.41

2.10

2.98

4.63

5.18

6.73

7.80

7.77


A

0

0.42^

0.96

1.55

1.97

2.24

2.85

3.17

3.85

5.76

5.71

B

0

0.05

0.03

0.03

0.07

0.15

0.15

0.12

0.12

0.15

0.13

c 0

0.32

0.73

1.35

1.82

2.22

2.89

3.07

3.74

4.15

4.14

257

Appendix D

Table D.4b - Shear, moment and torsion at columns A, B and C for Model P4

- Day Two

Applied load

(kPa)

0

0.67

1.49

2.11

2.92

3.38

4.4

5.76

7.06

8.39

9.37

11.03

12.92

14.33

15.64

16.04

16.64


A

0

0.92

2.28

3.32

4.63

5.38

7.04

9.35

11.49

13.68

15.03

18.00

21.06

23.72

25.86

26.65

27.64

B

0

2.22

5.46

8.01

11.3

13.24

17.38

22.86

28.23

33.71

37.77

44.57

52.16

57.53

62.71

64.05

66.43

C

0

0.84

2.08

3.14

4.47

5.44

7.08

9.34

11.55

13.85

15.84

18.36

21.55

24.03

26.24

27.21

28.22


A

0

0.20

0.55

0.83

1.21

1.46

1.97

2.61

3.28

3.94

4.37

5.10

5.78

6.40

6.90

7.08

7.34

B

0

0.14

0.5

0.89

1.41

1.71

2.64

4.12

5.61

7.15

8.46

10.27

12.47

14.26

15.86

15.83

16.42

C

0

0.11

0.35

0.67

1.11

1.44

2.05

2.84

3.65

4.49

5.23

6.14

7.26

8.07

8.79

9.21

9.55


A

0

0.13

0.44

0.63

0.93

1.09

1.48

2.02

2.54

3.09

3.50

4.23

4.99

5.59

6.07

6.08

6.31

B

0

0.04

0.07

0.09

0.09

0.09

0.09

0.09

0.10

0.10

0.10

0.10

0.09

0.12

0.14

0.12

0.13

C

0

0.13

0.40

0.58

0.81

0.96

1.25

1.66

2.08

2.52

2.88

3.38

3.98

4.45

4.78

4.92

5.10

258

Appendix E

APPENDIX E

FLEXURAL STRENGTH THEORY

E.l Assumptions

The assumptions made in the flexural strength theory are:

• variation of strain on the cross-section is linear, i.e. strains in the concrete and

bonded steel are calculated on the assumptions that plane sections remain plane;

• concrete carries no tensile stress, i.e. tensile strength of the concrete is ignored;

and

• stress in the compressive concrete and steel (prestressed and non-prestressed)

are determined from actual or idealised stress-strain relationships for the

respective materials.

The load at which failure or collapse of a flexural member occurs is called the ultimate

load, and the corresponding moment is termed as the ultimate moment. As the applied

moment increases into the overload range, the neutral axis of the section rises and

eventually the material behaviour becomes non-linear.

259

...

Appendix E

Neutral axis

Centroidal axis

® Decompression moment M 0

@ Cracking moment Mc r

<3> Post-cracking moment Mpc

© Ultimate moment Mu

Cross section Strains at various load stages

'SC

Stage® Stage© Staged) Stage®

Stresses at different load stages

C

Resultant forces

Fig. E.l - Typical strain and stress distribution of a prestressed concrete

cross-section

The depth to the neutral axis of a section is denoted as dn in Fig. E.l . Non-prestressed

steel may yield if its strain est exceeds the yield strain £Sy (= fsylEs). The prestressed

tendon may enter the non-linear part of its stress-strain curve as its strain, Ep , increases.

260

Appendix E

At the ultimate moment state (see Fig. E.l), the stress distribution shows that the

resultant compressive force of magnitude C equals the resultant tensile force T and the

ultimate moment capacity is calculated from the internal couple,

Mu = Cl=Tl ...(E-l)

where / is the lever arm between the internal compressive and tensile resultants.

E.2 Idealised compressive stress block

The idealised stress block at ultimate state, shown in Fig. E.2, is recommended by

ACI318-89 and A S 3600-1988. Both of these Concrete Codes also take the extreme

fibre compressive strain Ecu at ultimate moment to be 8 C M = 0.003, although it has

been mentioned that the actual extreme fibre strain at ultimate may not be close to this

value [Gilbert and Mickleborough 1990]. The use of the idealised compressive stress

block as specified by these two codes is adopted in Section 9.3.2 in the analysis of

flexural strength of slab strips.

The parameter 7" is dependent on the compressive strength of the concrete and is

described in the Australian Standard A S 3600-1988 [SAA, 1988] as shown below:

For f'c < 28 MPa : y" = 0.85 ...(E-2)

For 7% > 28 M P a : y" = 0.85 - 0.007 ( f'c - 28 ) ^ 0 . 6 5 ...(E-3)

261

Appendix E

For the rectangular section shown in Fig. E.2, the hatched area A' ( = y"bdn ) is

therefore assumed to be subjected to a uniform stress of 0.85/c. The resultant

compressive force Cc is the volume of the rectangular stress block.

Cc= 0.85fcy bdn .(E-4)

Neutral axis

~~ Centroidal axis

Cross section Strains at ultimate

a

Jr.

cu

OL

'pu

Jr. Y"d„

3: mt

O,

Q.5Ydn

"*

pu

'sou Cc ~f~f

TP ^ 1

~f

Actual stresses Idealised stress block Resultant forces

Fig. E.2 - Idealised stress block at ultimate m o m e n t conditions

262

Appendix E

Based on Eq.(E-l), the ultimate moment is thus,

Mu = Tplp + Ts h - CSc lsc ...(E-5)

where lp , ls and lsc are the lever arms between the concrete compressive force and the

respective forces at the prestressed and non-prestressed steel levels. Mu may then be

expanded into the following form:

Mu~ vpu\[dp- 1j^]+*suAst[ds- ^1-a.AtY ~dc]

...(E-6)

where opu, osu and ascu are the ultimate stresses in the bonded tendons and non-

prestressed tensile and compressive steel respectively, which are determined from the

consideration of strain compatibility and equilibrium. The various depths shown, ie. dp,

ds and dc are the effective depths of the prestressing steel, the tensile reinforcement and

the compressive reinforcement (if any).

E.3 Prestressed tendon strain components

Unlike non-prestressed steel in reinforced concrete section, the strain in the tendons in

prestressed concrete section is not equal to the concrete at tendon level. The strain in

the bonded prestressing tendon at any stage of loading is equal to the strain caused by

the initial prestress plus the change in strain in the concrete at the tendon level.

263

Appendix E

Centroidal axis of

ia) uncracked section

<fij) o

Fig. E.3 - Strain distribution of prestressed tendon

at three stages of loading

E.3.1 Stage 1

The elastic concrete compressive strain, Ece (denoted by stage la in Fig. E.3) is caused

by effective prestress when there is no externally applied moment. The corresponding

elastic strain in the prestressing tendon, £pe is denoted as stage lb in Fig. E.3 .

E.3.2 Stage 2

The applied moment is sufficient to decompress the concrete at tendon level. The strain

in the tendon is equal to that in stage 1 plus a tensile increment of strain equal in

magnitude to Ece.

264

Appendix E

E.3.3 Stage 3

This stage corresponds to the ultimate load condition. The concrete strain at tendon

level, Ept , may be determined in terms of Ecu ( = 0.003) and dn by applying the

requirements of strain compatibility. This concrete strain at tendon level may also be

measured experimentally via attachment of strain gauges on the prestressed tendons

prior to casting and testing.

265

Appendix F

APPENDIX F

STRAIN MEASUREMENT RESULTS

F.l Model PI

Table F.l - Strain measurements of reinforcing steel for Model PI

Applied load

(kPa)

0

4.46

5.93

7.33

8.69

10.18

11.59

13.18

14.89

16.18

3.62

Strain readings ( x IO"6) at Gauge No.

Rl

1

15

34

56

183

518

828

1126

1742

2088

2050

R2

1

20

48

81

248

606

948

1269

2009

2538

X

R3

0

3

8

13

38

187

387

567

877

937

X

R4

-1

2

8

16

71

180

295

392

770

1009

X

R5

X

X

X

X

X

X

X

X

X

X

X

R6

0

14

30

46

90

310

716

1147

1712

2023

2517

R7

1

9

18

25

41

74

159

405

745

976

933

R8

X

X

X

X

X

X

X

X

X

X

X

R9

1

8

16

26

52

94

289

444

827

1167

1204

R10

0

7

15

23

50

104

194

323

606

888

732

Rll

0

3

6

8

11

22

67

376

769

950

768

R12

0

8

15

22

31

58

184

500

1080

1363

1031

"x" refers to disconnected or faulty strain gauges

266

Appendix F

F.2 Model P2

Table F.2a - Strain measurements of reinforcing steel for Model P2

Applied

load

(kPa)

0

1.37

3.28

4.37

5.69

6.96

9.86

11.27

12.48

13.21

14.69

15.86

16.07

17.23

18.44

19.61

Average strain readings ( x 10-6) at Gauge No.

Rl &R2

0

20

36

62 .

91

147

1221

2107

2542

2845

3370

3995

4201

4967

5834

1952

R3&R4

0

67

84

122

173

264

1107

1477

1693

1879

2210

2541

2693

2981

3360

3579

R5&R6

0

17

10

20

35

48

909

1478

1865

2112

2568

2965

3106

3465

3899

2035

R7&R8

0

27

42

61

97

321

1137

1509

1171

1334

1461

1537

1789

1411

1647

-267

R9 & RIO

0

-73

432

393

470

2051

9196

12033

2554

70

522

X

X

X

X

X

Rll &R12

0

62

95

145

198

241

604

820

910

952

912

885

873

860

927

303


267

Appendix F

Table F.2b - Strain measurements of prestressed tendons for Model P2

Applied load

(kPa)

0

1.37

3.28

4.37

5.69

6.96

9.86

11.27

12.48

13.21

14.69

15.86

16.07

17.23

18.44

19.61


1

0

6

42

71

127

242

1201

2087

2522

2835

3350

3975

4181

4947

5814

1932

2

0

18

50

78

131

224

916

1448

1781

2029

2482

2900

3052

3449

4318

2404

4

0

17

55

106

197

392

1040

1410

1626

1812

2143

2474

2626

2914

3293

3512

5

0

17

53

106

190

350

746

1165

1496

1746

2131 '

2410

2525

2720

2991

2197

8

0

-3

7

20

39

67

1212

1827

2073

2153

2351

2550

2346

2413

1581

1599

9

0

-7

3

18

31

99

982

1461

1848

2095

2551

2948

3089

3448

3882

2049

10

0

14

36

62

109

190

511

851

1020

1180

1411

1586

1384

945

931

871

11

0

15

34

70

294

472

1110

1482

1144

1307

1434

1510

1762

1384

1620

2558

13

0

33

83

136

179

278

542

758

848

890

950

1123

1411

1798

1865

2018

17

0

505

466

543

2124

2254

9269

12106

2627

-3

595

X

X

X

X

X


268

Appendix F

F.3 Model P3

Table F.3a - Strain measurements of prestressed tendons for Model P3 - Day One

Applied load

(kPa)

0 1.40

2.80

4.20

5.60

7.00

8.40

9.80

11.20

12.60

14.00

14.68

15.40

17.61

18.17

19.57

20.27

Strain readings ( x 10-6) at Gauge No.

1 0

94 101 115 134 149 203 265 295 323 389 332 340 258 178 -182

-1008

4 0

51 71 80 64 30 -92 -173

-677

-2598

-3226

-3987

-4550

-5409

-5777

-4904

-3951

6 0

22 27 47 93 191 184 39 -108

-183

-238

-281

-245

-181

-176

6 -37

8 0

85 94 121 130 147 168 254 286 311 357 324 322 194 65 -211

-1178

9 0

27 54 96 116 128 147 199 261 347 159 96 -102

-217

-349

-654

-952

10 0

11 32 74 112 193 612 996 1320

1380

1419

2456

3247

3951

4597

6271

7520

16 0

68 122 175 264 417 623 851 966 1022

1175

1210

2162

3384

4620

6558

8512

18 0

32 81 96 120 238 421 764 848 990 1150

1981

2441

3748

4822

6018

7362

Table F.3b - Strain measurements of prestressed tendons for Model P3 - Day Two

Applied load

(kPa)

0 7.97

14.0

16.77

18.73

20.27

21.52

22.36


1 0

-74 -152

-348

-687

-1157

-2147

-2681

4 0

-120

-254

-514

-649

-1230

-1692

-2157

6 0

-114

-227

-434

-666

-1160

-1584

-2021

8 0

-96 -255

-321

-741

-1548

-2145

-3065

9 0

-117

-254

-406

-612

-1641

-2570

-2963

10 0

121 239 315 653 1488

1912

2422

16 0

98 232 368 814 1678

2597

3119

18 0

132 352 451 962 1547

1963

2751

269

Appendix F

F.4 Model P4

Table F.4a - Strain measurements of prestressed tendons for Model P4 - Day One

Applied load

(kPa)

0 1.37

2.75

4.43

5.71

6.89

8.75

9.72

11.47

12.77


1 0

-67

-29

-37

-155

-41

-59

-61

-149

-139

2 0

179

635

160

162

202

230

334

542

678

4 0

-388

-830

-998

-460

256

-2387

-2630

-2913

-3232

9 0

8

6

12

13

17

51

65

92

157

10 0

-48

8

10

-78

-84

-119

-139

-175

-32

11 0

0

-4

0

2

9

18

32

55

102

14 0

-101

197

297

-2

2

-44

-211

-967

-1160

18 0

-13

695

400

-106

-80

-72

-28

80

32

Table F.4b - Strain measurements of prestressed tendons for Model P4 - Day Two

Applied load

(kPa)

0 0.67

1.49

2.11

2.92

3.38

4.4 5.76

7.06

8.39

9.37

11.03

12.92

14.33

15.64

16.04

Strain readings ( x IO6) at Gauge No.

1 0

13

5

-31

-36

3

-36

-2

19

55

-23

7

-28

22

80

23

2 0

-225

-271

-274

-237

-228

-211

-109

-35

57

256

531

730

499

700

512

4 0

-37

-60

-59

-68

-15

142

237

199

1233

136

210

-380

1185

75

815

9 0

4

7

10

18

19

30

44

57

68

77

91

122

156

194

209

10 0

-44

-73

-74

-70

-80

-84

-98

-106

-255

-145

-118

-73

-393

-588

-519

11 0

4

8

12

18

8

27

41

52

63

72

85

111

153

211

266

14 0

47

37

60

79

114

157

9

28

-404

-146

-167

-42

-1301

-4032

-4772

18 0

287

310

100

27

386

345

266

317

-92

242

139

246

78

-544

-678

270

Appendix F

Table F.4c - Strain measurements of reinforcing steel for Model P4

- Day T w o

Applied load

(kPa)

0

0.67

1.49

2.11

2.92

3.38

4.4

5.76

7.06

8.39

9.37

11.03

12.92

14.33

15.64

16.04

16.64


Rl

0

102

176

266

342

445

554

660

766

896

1031

1114

1251

1366

1438

1437

15747

R2

0

36

46

56

64

83

107

132

160

195

235

265

331

389

434

424

369

R3

0

67

92

122

155

194

253

265

317

375

433

527

651

881

1075

1447

X

R4

0

17

43

74

119

149

242

402

589

780

953

1143

1375

1584

1779

2153

X

R5

0

13

33

59

104

139

200

275

358

455

560

662

781

880

990

1182

X

R6

0

46

59

75

93

118

156

203

253

304

358

403

465

491

519

545

X

R7

0

11

23

39

72

102

159

228

290

325

306

329

376

174

124

124

27

R9

0

35

51

65

81

99

130

194

296

397

505

574

705

946

1117

1328

X

RlOa

0

7

17

23

32

41

92

166

246

325

415

473

578

710

804

918

997

RlOb

0

3

8

14

21

28

43

76

136

196

258

293

372

491

578

699

1129

"x" refers to disconnected strain gauges at failure of model

271

Appendix G

APPENDIX G

SAMPLE CALCULATIONS OF Mj USING MEASURED

TENDON STRAIN

G.l Data for Model P2-B

Model P2-B : at distance d/2 from the front face of edge-column position

p

/;

Es Ep

fsy

fpy b

D

ds dp

dc

=

—

—

=

=

=

—

=

=

=

=

2285 kg/m3

35.7 MPa

200000 MPa

166667 MPa

538 MPa

1500 MPa

400 mm

80 mm

60 mm

44 mm

0

AP

A-st

ASc e

Pi atot

V

PP Zpt.test

=

=

=

=

=

-

=

=

=

39.28 m m 2

314.16 m m 2

0

4mm 46.06 kN (2 tendons)

0

0.2

0.015

0.002197

G.2 Calculations for Mt in Model P2-B

Model P2-B : at edge-column position

(Refer to the flow chart in Fig. 9.2).

272

Appendix G

Effective prestress at the section after allowing for losses (see calculations in Chapter

H):

Pe = 41.89 k N

Eq.(9-1): Mcr = 2.265 k N m

From structural analysis of the model at failure using the simple direct design method,

the applied moment:

M0 = 2.981 k N m

It is a cracked section analysis, since M0 > Mcr.

Eq.(E-3): 7" = 0.80849

Eq.(9-9): 8ce = 0.000049

Eq.(9-10): £pe = 0.006399

From the iterative trial and error procedure, the depth to the neutral axis at ultimate is :

dn = 25 m m

where,

the yield strain for steel is: 8Sy = 0.00269

(Checking Eq. (9-14), indicates that the steel has yielded).

From data: £Pt = 0.002197

Eq. (9-12): £p = 0.008645

The yield strain for tendon is: £py = 0.008999 (i.e. tendons are not at yield).

The compressive forces acting in the section:

273

Appendix G

Eq. (E-4): Cc = 233.17 kN

The corresponding tensile forces:

Tp = 58.92 kN

Ts = 169.02 kN

The error estimate is: EC - ET = 5.23 kN (which is 2.2%).

Therefore,

Eq.(9-17): Mj = 10.790 kNm

274

Appendix H

APPENDIX H

SAMPLE CALCULATIONS FOR PUNCHING SHEAR STRENGTH

USING THE PROPOSED APPROACH (MODIFIED

WOLLONGONG APPROACH)

H.l Sample Calculations for Punching Shear at Corner-column Connection

This section details the typical data obtained (for corner column position Pl-A) from

experimental results in the testing oi Model PI up to failure. It should be noted that this

model was constructed without any spandrel beam nor torsion strip, i.e. it is with a free

edge. This is followed by a sample calculation for punching shear, Vu , at the corner-

column position Pl-A using the Modified Wollongong Approach as introduced in

Chapter 5.

H.l.l Data for Model Pl-A

Model Pl-A : at comer-column position without spandrel beam or torsion strip (i.e.

a free edge).

MC1

MC2 Mj

Mm

bSp

dsp

»1

Ds

=

-

-

-

—

zz

=

=

7.390 kNm

7.260 kNm

7.047 kNm

9.079 kNm

300 mm

60 mm

80 mm

80 mm

^ U ' . V

fly s

f'c Lc

k X

V

= 0

= 512 MPa

= 0

= 35.7 MPa

= 2320 mm

= 410 mm

= 248 mm

= 40 mm

275

Appendix H

de

Ci

c2

PJ

fpy Api

Ap2

nj

n2

" « • / rlnh

=

=

=

=

=

=

=

=

=

60 mm

300 mm

200 mm

23.03 kN

1500 MPa

19.64 mm2

19.64 mm2

1 No.tendon

1 No.tendon

235.62 mm2

Als Jwy

v r u.iest

dp j

dp2

8;

e2 atot

^

PP

=

=

=

zz

=

=

=

=

=

=

157.08 mm2

0

26.93 kN

41 mm

51 mm

0.0253

0.0365

0

0.2

0.015

H.1.2 Calculations for V„ in Model Pl-A

Model Pl-A : at corner-column position

The initial prestress (Pj = 23.03 kN ) was applied at eccentricities of :

ej - I m m (in the direction of M Q J )

ej = 1 1 m m (in the direction of M Q 2 )

Moduli of elasticity of concrete, Ec =

steel, Ep -

30208 M P a

166667 M P a

Prestress losses:

Eq.(5-61): Elastic losses, ^

^2

83.83 N

94.88 N

Eq.(5-62): Residual proportion of prestress after deducting friction losses,

«J>y = 0.99902

0 2 = 0.99940

276

Eq.(5-63): Anchorage Inction losses,

AFL = 460.6 N

Eq.(5-64): Anchoring losses, ALj - 1781.6 N

AL2 = 2263.9 N

Effective pres tresses,

Pel = (Pj.$})-E>}-AF-AL} = 20.68 kN

pe2 = iPj-^2)-^2-AF-AL2 = 20.20 kN

Eq.(5-34):

Effective depth of spandrel, dsp = 56.2 m m

Similarly, effective depth of slab, de= 56.3 m m

PeI.cosQj = 20.67 kN

PeJ.sinQj = 0.523 kN

Pe2.cosB2 = 20.18 kN

Pe2.sinQ2 = ()-737 kN

Spandrel dimensions and strength parameters:

At = xy = 9920 m m 2

Ut = 2 [x + y] = 576 m m

Avt = Aull = 78.54 m m 2

Eq.(5-33):

Eq.(5-20):

Eq.(5-21):

Eq.(5-18):

Eq.(5-19):

Eq.(5-15):

Eq.(5-25):

Mus

CO

co0

a

P 5

Y

3.643 kNr

(.)

446 N/mm

0.40212

5.13792

2.06328

0.28689

Prestress components: Pejx

Pely

Pe2x

Pe2v

277

Appendix H

From Eq.(10.15), the slab restraining factor is calculated as:

y = 2.21981

which is reasonably close to the measured y value shown in Table 10.2 (i.e. Vmeasurej

= 2.22913 ).

The ultimate moment and shear at a distance djl away from the front face of the

column:

Eq.(5-59): For dsp<X2de,

Ml,corner = 6.665 k N m

Eq.(5-42): For dspl de< 1.2,

Vl,corner = 30.54 kN

The -factors used for detenriining Vu are computed as shown below:

Eq.(5-53):

Eq.(5-54):

Eq.(5-48):

Eq.(5-49):

Eq.(5-50):

Eq.(5-51):

Eq.(5-52):

h ~-

h ~-

ki ~-

k2 =

*j =

k4 =

ks =

1.0

30018

-414017

- 394443

12.178

: 0.22086

0

Finally, the punching shear, VH , for Model PI-A is analysed by using the expression

from Eq.(5-47):

278

Appendix H

y - h jjj k4 - k2 k4 + kj k7 kj - k4 k$

which gives:

Vu = 26.35 kN

By comparison with the experimental punching shear, VHJest ( = 26930 N ), it gives a

ratio of the predicted to experimental punching shear:

Vu'Vu.test = 0.978

H.2 Sample Calculations for Punching Shear at Edge-column Connection

Typical data obtained (for edge column position P3-B) from experimental results in the

testing of Model P3 up to failure is illustrated in this section. This model was

constructed with a spandrel beam of 250 m m wide and 120 m m deep. This is followed

by a sample calculation for punching shear, Vu , at this model P3-B using the Modified

Wollongong Approach as shown above in Section 10.3 .

H.2.1 Data for Model P3-B

Model P3-B : at edge-column position with a spandrel beam of 250 mm wide and

120 m m deep.

MC1 = 33.000 k N m

MC2 = 0.280 k N m

Mj = 9.005 k N m

279

Am

fly s

= 28.27 m m 2

= 578 MPa

= 300 m m

Mm

bSp

dsp

Dj

Ds

de

Ci

c2

Pj

fpy

Api

Ap2

nj

n2

"•st sinh

-=

—

-

-

—

-

-

-

-

—

—

=

=

=

—

11.243 kNm

250 mm

94 mm

120 mm

80 mm

60 mm

300 mm

400 mm

23.03 kN

1500 MPa

39.28 mm2

19.64 mm2

2 No.tendons

1 No.tendon

314.16 mm2

/,

Lc

k X

V

*ls

Iwy y ' u.test

dp,

dP2

QJ

e2

°-tot

H

PP

=

=

=

=

=

=

=

—

ur:

=

=

=

=

=

=

47.8 MPa

2320 mm

450 mm

214 mm

84 mm

314.16 mm2

489 MPa

73350 N

41 mm

56 mm

0.0253

0

0

0.2

0.015

H.2.2 Calculations for Vu in Model P3-B

Model P3-B : at edge-column position

The initial prestress (Pj = 23030 N ) was applied at eccentricities of,

e, = 1 m m (in the direction of M^j)

e2 - 16 m m (in the direction of M Q 2 )

Moduli of elasticity of concrete, Ec

steel, Ep

34954 M P a

166667 M P a

Prestress losses:

Eq.(5-61): Elastic losses, t,j

^2

144.90 N

68.89 N

280

Eq.(5-62): Residual proportion ot prestress alter deducting friction losses,

Oy = 0.99902

<t>2 = 0.99208

Eq.(5-63): Anchorage friction losses,

AFL 460.6 N

Eq.(5-64): Anchoring losses, AL,

AL2

1781.6 N

2263.9 N

Effective prestresses:

Pel = iPj.<S>l)-£,!-AF-ALI

Pe2 = (Pj.^2)-t)2-AF-AL2

20.62 kN

20.05 kN

Eq.(5-34):

Effective depth of spandrel,

Similarly, effective depth of slab,

Prestress components: Pejx

Pely

Pe2x

Pely

dsp

4

=

=

=

=

= 87.2

,= 55.3

Pel.cosQj

Pel.&inQj

Pe2.cosQ2

Pe2.s\nQ2

mm

mm

20.61 kN

0.522 kN

20.054 kN

0

Spandrel dimensions and strength parameters:

At =

ut =

Ast ~

Eq.(5-33):

Eq.(5-20):

Eq.(5-21):

Eq.(5-18):

xy

2[x + y]

Alsll

Mus

CO

co0

a

17976 m m 2

596 m m

235.62 m m 2

9.4582 kNm

46.080

446 N/mm

0.90792

281

Appendix. H

Eq.(5-19): {3 = 5.8656

Eq.(5-15): 8 = 8.1492

Eq.(10-18): X = 0.82397

From Eq.(10.13), the slab restraining factor is calculated as:

V = 0.42518

which compares well with the measured \\i value shown in Table 10.2 ( i.e. Vmeasure(j =

0.44078 ).

The ultimate moment and shear at a distance djl away from the front face of the

column:

Eq.(5-44): For dsp> X.2de,

Ml.edge = 9-005 kNm

Eq.(5-46): For dsp/de> 1.2,

Vledge = 51-89 kN

The it-factors used for determining Vlt are computed as shown below:

Eq.(5-53): k6 = 0.95556

Eq.(5-54): k7 = 50846

Eq.(5-56): k{ = 280931

Eq.(5-57): k2 = 12264520

Eq.(5-50): k3 = 21.327

282

Appendix H

Eq.(5-51): . k4 = 0.26655

Eq.(5-52): k5 = 25.0

Finally, the punching shear, Vn , for Model P3-B is analysed by using the expression

from Eq.(5-55):

y _ 2kj kj k4 - k2 k4 + kj k7

kj - k4 k$

which gives:

Vu = 68.83 kN

This gives a ratio of the predicted to experimental punching shear as:

Vu/Vu.test = 0-938

283

Appendix I

APPENDIX I

DETAILS OF SPANDREL BEAMS OR TORSION STRIPS AND

CRITICAL SLAB STRIPS

Table 1.1 - Details of the spandrel beams or torsion strips

Connection

No.

Pl-A P2-A P3-A P4-A

Pl-B P2-B P3-B P4-B

Pl-C P2-C P3-C P4-C

(mm)

300

300

250

300

300

300

250

300

300

300

250

300

(mm)

80

160

120

80

80

160

120

80

80

160

120

80

dspl>

(mm)

60

134

94

60

60

134

94

60

60

134

94

60

Als>

(mm2)

157.08

314.16

314.16

314.16

157.08

314.16

314.16

314.16

157.08

314.16

314.16

314.16

Aws>

(mm2)

0

28.27

28.27

28.27

0

28.27

28.27

28.27

0

28.27

28.27

28.27

(mm)

0

300

300

250

0

300

300

250

0

300

300

250

Ay (MPa)

512

538

578

555

512

538

578

555

512

538

578

555

Jwy )

(MPa)

0

450

489

541

0

450

489

541

0

450

489

541

(mm)

248

248

214

248

248

248

214

248

248

- 248

214

248

y,

(mm)

40

108

84

40

40

108

84

40

40

108

84

40

(mm)

10

10

10

10

10

10

10

10

10

10

10

10

Table 1.2 - Column dimensions of post-tensioned

concrete flat plate models

Column

identification

(for all models)

A B C F G H

Cj,

(mm)

300

300

300

400

400

550

c2,

(mm)

200

400

400

200

400

400

284

Appendix I

-*— ^ —

Table 1.3 - Details of the critical slab strips

Connection

No.

Pl-A P2-A P3-A P4-A

Pl-B P2-B P3-B P4-B

Pl-C P2-C P3-C P4-C

b,

(mm)

200

200

200

200

400

400

450

400

400

400

450

400

(mm)

80

80

80

80

80

80

80

80

80

80

80

80

del'

(mm)

60

60

60

60

60

60

60

60

60

60

60

60

Astl>

(mm2)

235.62

235.62

235.62

235.62

314.16

314.16

314.16

314.16

235.62

235.62

235.62

235.62

Astm <

(mm2)

235.62

235.62

235.62

235.62

235.62

235.62

235.62

235.62

235.62

235.62

235.62

235.62

(mm)

2320

2320

2320

2300

2320

2320

2320

2300

2240

2240

2240

2210

Lj,

(mm)

410

430

450

450

410

430

450

450

410

430

450

450

MCI'

(kNm)

7.39

12.01

12.25

7.34

15.25

28.50

33.00

16.42

8.70

11.17

13.54

9.55

MC2,

(kNm)

7.26

7.35

7.29

6.31

0.24

0.28

0.28

0.13

6.14

9.51

8.226

5.10

Mj,

(kNm)

7.047

7.292

7.131

6.974

10.37

10.79

9.005

10.713

7.529

7.932

7.615

7.773

(kNm)

9.079

9.709

9.325

9.636

11.10

11.731

11.243

11.625

9.749

10.509

9.84

10.029

V u.test '

(kN)

26.93

32.02

30.97

27.64

68.75

76.33

73.35

66.43

27.72

32.97

31.78

28.22

Table 1.4 - Details of prestressing in critical slab strips

Connection

No.

Pl-A P2-A P3-A P4-A

Pl-B P2-B P3-B P4-B

Pl-C P2-C P3-C P4-C

(kN)

23.03

23.03

23.03

23.03

23.03

23.03

23.03

23.03

23.03

23.03

23.03

23.03

fpy '

(MPa)

1500

1500

1500

1500

1500

1500

1500

1500

1500

1500

1500

1500

Apl>

(mm2)

19.64

19.64

19.64

19.64

39.28

39.28

39.28

39.28

19.64

19.64

19.64

19.64

Ap2>

(mm2)

19.64

19.64

19.64

19.64

19.64

19.64

19.64

19.64

19.64

19.64

19.64

19.64

"1 !

(No)

1

1

1

1

2

2

2

2

1

1

1

1

n2,

(No) (mm)

41

41

41

41

41

41

41

41

41

41

41

41

dp2'

(mm)

51

51

51

51

56

56

56

56

51

51

51

51

(rad)

0.0253

0.0253

0.0253

0.0253

0.0253

0.0253

0.0253

0.0253

0.0253

0.0253

0.0253

0.0253

e 2 '

(rad)

0.0365

0.0365

0.0365

0.0365

0

0

0

0

0.0365

0.0365

0.0365

0.0365

atot<

(rad)

0

0

0

0

0

0

0

0

0

0

0

0

M-

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.2

(rad/m)

0.015

0.015

0.015

0.015

0.015

0.015

0.015

0.015

0.015

0.015

0.015

0.015

285

Appendix J

APPENDIX J

LISTING OF SOURCE CODE FOR BASIC COMPUTER

PROGRAMS TO ANALYSE Vu

J.l BASIC Program for the Wollongong Approach [Loo and Falamaki, 1992]

and the Modified Wollongong Approach

10 REM SAVE"TOO.BAS",A 20 Z=2.30258509299# 100 CLS: 103 LOCATE 1,5:PRINT "DATA" 104 GOSUB 9200 105 LOCATE 3,5:PRINT "MODEL ";MODEL$;"AT "COL$;"COLUMN" 106 GOSUB 9200 110 LOCATE 5,5:PRTNT "Mel = ";MCl:LOCATE 5,30:PRINT "Aws = ";AWS 120 LOCATE 6,5:PRINT "Mc2 = ";MC2:LOCATE 6,30:PPJNT "Fly = ";FLY 130 LOCATE 7,5:PRTNT "Ml = ";M1 :LOCATE 7,30:PRTNT "s = ";S 140 LOCATE 8,5:PRINT "Mm = ";MM :LOCATE 8,30:PRTNT "Fc = ";FC1 150 LOCATE 9,5:PRTNT "Bsp = ";BSP:LOCATE 9,30:PRTNT "Lc = ";LC 160 LOCATE 10,5:PRTNT "Dsp = ";DSP:LOCATE 10,30:PRINT "Ll = ";L1 170 LOCATE 11,5:PRINT "DI = ";D1 :LOCATE 11,30:PRINT "x = ";XS 180 LOCATE 12,5:PRINT "Ds = ";DS :LOCATE 12,30:PRTNT "y = ";YS 190 LOCATE 13,5:PRINT "De = ";DE :LOCATE 13,30:PRINT "Als = ";ALS 200 LOCATE 14,5:PRTNT "CI = ";C1 :LOCATE 14,30:PRINT "Fwy = ";FWY 210 LOCATE 15,5:PRINT "C2 = ";C2 :LOCATE 15,26:PRINT "Vu.test = ";VUT 220 PRINT : GOSUB 9200 230 LOCATE 20,5:PRINT "PRESS [ESC] TO PROCEED " 240 KY$=TNPUT$(1): IF KY$=CHR$(27) THEN 245 ELSE GOTO 10 245 CLS:LOCATE 1,5:PRINT "DATA (CONTINUED)..." 246 GOSUB 9200 250 LOCATE 3,5:PRINT "Pj = ";PJ :LOCATE 3,30:PRINT "dpi = ";DP1 260 LOCATE 4,5:PRINT "fpy = ";FPY :LOCATE 4,30:PRINT "dp2 = ";DP2 270 LOCATE 5,5:PRINT "Apl = ";AP1 :LOCATE 5,30:PRINT "atot = ";ALFAT 280 LOCATE 6,5:PRINT "Ap2 = ";AP2 LOCATE 6,30:PRINT "p = ";MIU 290 LOCATE 7,5:PRTNT "nl = ";N1 : LOCATE 7,30:PRINT "6p = ";BETAP 300 LOCATE 8,5:PRINT "n2 = ";N2 :LOCATE 8,30:PRTNT "91 = ";THETA1 310 LOCATE 9,5:PRINT "Ast,slab = ";ASLAB :LOCATE 9,30:PRINT "92 =

";THETA2 350 PRINT : GOSUB 9200 360 LOCATE 15,5:PRINT "PRESS [ESC] TO PROCEED " 370 KY$=TNPUT$(1): IF KY$=CHR$(27) THEN 380 ELSE GOTO 10

286

Appendix J

380 REM CHECK FOR RC OR PC ANALYSIS 390 IF PJ=0 THEN 400 ELSE GOTO 3000 400 REM REINFORCED CONCRETE SLAB ANALYSIS 405 REM COMPUTE At, Ut 410 AT=XS*YS : UT=2*(XS+YS) 420 REM 430 REM INITIAL EQUATIONS *************************************** 440 AST=ALS/2 450MY=(1-(.6*AST*FLY)/(BSP*DSP*FC1))*DSP*AST*FLY 460 REM TRANSVERSE STRENGTH OF SPANDREL

470 IF S=0 THEN W=0 : GOTO 490 480 W=AWS*FWY/S 490 REM ADDTIONAL TRANSVERSE STRENGTH OF SPANDREL

495 W0=446 500 REM 510 IF COL$="CORNER " GOTO 800 520 REM ANALYSIS AT EDGE COLUMN 530 GAMMA=(75/DS)A3 540 ALFA=ALS*FLY/200000! 550 BETA=UT*(W+W0)/50000! 560 DELTA=ALFA*BETA*DSP/DE : X1=GAMMA*DELTA 570 REM 575 REM LET DR=RATIO OF THE EFFECTIVE DEPTHS OF THE SPANDREL

AND SLAB 576 DR=DSP/DE 580 REM EVALUATE VI 590 IF DSP<=DE THEN M1=(600/(2*BSP+C2))A.9*M1 600 IF DR<=1.2 THEN V1=9.350001+3.49*M1*.001/LC ELSE

V1=2.84+5.62*(M1+MM)*.001/LC 610 REM FORMULA FOR SI 620 SIP=2.65-1.71*L0G(X1)/Z 630 REM COMPUTE CONSTANTS 640K6=1-(D1-DS)/(2*L1) 650 IF ALFA<0 OR BETA<0 OR SIP<=0 THEN 6000 660K1=200000!*(ALFA*BETA/SIP)A.5*(SIP-K6*MC2/MY) 670 K2=K6*MC 1 -(DE+BSP)/2* V1 * 1000! -M1 680 K3=AT/UT 690 K4=(DSP/(2*UT))A.5 700K5=(Cl-BSP)/2 710 REM ANALYSIS OF Vu 720K7=2*K3*K4*K1-K4*K2+K3*V1*1000! : K8=K3-K4*K5 730 VU=K7/K8 740 GOTO 1500 750 REM IfiC) R F M *****************************************

800 REM ANALYSIS AT CORNER COLUMN

287

Appendix J

810 ALFA=ALS *FLY/200000! 820 BETA=UT*(W+W0)/50000! 830 DELTA=ALFA*BETA*DSP/DE 840GAMMA=(C2/C1)A3*(75/DS)A.5 : X2=GAMMA*DELTA 850 REM LET DR=RATIO OF THE EFFECTIVE DEPTHS OF THE SPANDREL

AND SLAB 860 DR=DSP/DE 870 REM EVALUATE VI 880 IF DSP<=DE THEN M1=(400/(BSP+C2))A.25*M1 890 IFDR<=1.2 THEN Vl=-3.66+5.04*(Ml+MM)*.001/LC ELSE Vl=-

.5+4.7*(Ml+MM)*.001/LC 900 REM FORMULA FOR SI 910 IF C2>C1 THEN STP=10.7-4.45 *LOG(X2)/Z ELSE SIP=5.04-3.12*LOG(X2)/Z 920 REM 930 REM COMPUTE CONSTANTS 940 K6=1-(D1-DS)/(2*L1) 950 IF ALFA<0 OR BETA<0 OR SIP<=0 THEN 6000 960K1=200000!*(ALFA*BETA/SIP)A.5*(SIP-K6*MC2/MY) 970K2=K6*MC1-(DE+BSP)/2*V1*1000!-M1 980 K3=AT/UT 990 K4=(DSP/(2*UT))A.5 1000 K5=(Cl-BSP)/2 1010 REM ANALYSIS OF Vu 1020 K7=K3*K4*K1-K4*K2+K3*V1* 1000! : K8=K3-K4*K5 1030 VU=K7/K8 1040 REM 1500 REM OPTION FOR RESULTS 1510 CLS:LOCATE 5,10:PRTNT "RESULTS OPTIONS :" 1520 GOSUB 9100 1530 LOCATE 7,10:PRINT "1) SUMMARISED RESULTS " 1540 LOCATE 9,10:PRINT "2) FULL RESULTS " 1550 GOSUB 9100 1560 LOCATE 12,10:INPUT "SELECT 1-2 ";SEL% 1570 IF SEL%=1 THEN 2000 ELSE IF SEL%=2 THEN 1600 ELSE GOTO 1500 1600 IF PJ=0 THEN CHAIN "LOO.BAS"„ALL ELSE CHATN "JOO.BAS"„ALL 2000 REM SUMMARISED RESULTS 2005 CLS:LOCATE 1,10:PRTNT "RESULTS PRINTOUT" 2010 GOSUB 9100 2015 LOCATE 3,10:PRTNT "MODEL ";MODEL$;" - ";COL$:LOCATE

3,38:PRINT "COLUMN" 2020 LOCATE 4,10:PRTNT "LOO & FALAMAKI'S METHOD TO DETERMINE

Vu" 2025 GOSUB 9100 2030 LOCATE 6,10:PRTNT USING "a = ##.######";ALFA:LOCATE

6,40:PRINT USING "6 =##.######" ;BETA 2040 LOCATE 7,10:PRTNT USING "5 = ###.######";DELTA:LOCATE

7,40:PRINT USING > - ##.######";SIP

288

Appendix J

2050 LOCATE 8,10:PRINT USING "dl/d = ##.######";DR:LOCATE 8,40:PRINT USING "7 = ##.######" ;GAMMA

2060 LOCATE 9,10:PRINT USING "VI = ##.####AAAA";V1*1000:LOCATE 9,40:PRINT USING "Ml = ##.####AAAA";M1

2070 LOCATE 10,10:PRTNT USING "Mu.sp = ##.####AAAA";MY:LOCATE 10,40:PRTNT USING "log[78] = ##.######";LOG(Xl)/Z

2080 IF PJoO THEN LOCATE 11,10:PRINT USING "Pel = ######.##";PE1 -.LOCATE 11,40:PRTNT USING "Pe2 = ######.##" ;PE2

2090 IF PJoO THEN LOCATE 12,10:PRINT USING "Lossesl = ###.##";LOSSESl:LOCATE 12,40:PRINT USING "Losses2 = ###.##" ;LOSSES2

2110 GOSUB 9100 2120 LOCATE 14,10:PRINT USING "Kl = ##.####AAAA";Kl:LOCATE

14,40:PRINT USTNG "K4 = ##.######" ;K4 2130 LOCATE 15,10:PRTNT USING "K2 = ##.####AAAA";K2:LOCATE

15,40:PRINT USTNG "K5 = ##.######";K5 2140 LOCATE 16,10:PRINT USING "K3 = ###.######";K3:LOCATE

16,40:PRINT USING "K6 =##.######";K6 2145 IF PJoO THEN LOCATE 17,40:PRINT USING "K7 = ######.##";K7 2150 GOSUB 9100 2160 LOCATE 19,10:PRINT USING "Vu.test = ######.##";VUT 2170 LOCATE 20,10:PRINT USING "Vu,pred = ######.##";VU 2180 LOCATE 21,10:PRINT USING "Ratio Vu,test/Vu,pred = ##.###" ;VUT/VU 2190 GOSUB 9100 2200 REM 2210 REM 2220 LOCATE 23,10:PRINT "[PRESS ESC TO END]" 2230 KY$=TNPUT$(1): IF KY$=CHR$(27) THEN CLOSE #2:GOTO 6500 ELSE

GOTO 2000 2240 REM 2250 REM 3000 REM PRESTRESSED CONCRETE SLAB ANALYSIS 3010 REM INITIAL EOUATIONS ***************************************

3020 AST=ALS/2 3030DSPEFF=(AST*FLY*DSP+AP2*FPY*DP2)/(AST*FLY+AP2*FPY) 3040 DEEFF=(ASLAB*FLY*DE+AP1*FPY*DP1)/(ASLAB*FLY+AP1 *FPY)

3100 REM ECCENTRICITIES 3110 El=DPl-DS/2 : E2=DP2-DS/2 3200 REM ELASTIC LOSSES 3210EPCONC=2400A(1.5)*.043*SQR(FC1) : EPTENDON= 166667! 3220 SIGCPl=(Nl*PJ/(C2*DS))+(Nl*PJ*El*DS/2)/(C2*DSA3/12) :

SIGCP2=(N2*PJ/(Cl*DS))+(N2*PJ*E2*Dl/2)/(Cl*DlA3/12) 3230 ELASTICl=.5*(EPTENDON/EPCONC)*SIGCPl :

ELASTIC2=.5*(EPTENDON/EPCONC)*SIGCP2 3240 ELOSSl=ELASTICl*(APl): ELOSS2=ELASTIC2*(AP2) 3250 REM FRICTION LOSSES 3260 LPAl=(Cl+DEEFF/2)/1000 3270 IF COL$="CORNER " THEN LPA2=C2/1000 ELSE LPA2=2.65

289

Appendix J

3280 FLOSS I =EXP(-MIU*(ALFAT+BETAP*LPA1)): FLOSS2=EXP(-MIU*(ALFAT+BETAP*LPA2))

3300 REM ANCHORAGE FRICTION LOSSES 3310AFLOSS=.02*PJ 3320 REM ANCHORING LOSSES 3330 TW1=AFLOSS/5700: TW2=AFLOSS/3530 3340ZL1=(3*EPTENDON*19.64)/TW1 : ZJ1=SQR(ZLI): ALOSSl=2*ZJl*TWl 3345 ZL2-=(3*EPTENDON*19.64)/TW2 : ZJ2=SQR(ZL2): ALOSS2=2*ZJ2*TW2 3350 REM EFFECTIVE PRESTRESS AND COMPONENTS AFTER ALL LOSSES 3360PE1=(PJ*FLOSS1)-ELOSS1-AFLOSS-ALOSS1 : PE2=(PJ*FLOSS2)-ELOSS2-

AFLOSS-ALOSS2 3365 REM LOSSES 3370LOSSES1=(PJ-PE1)/PJ*100 : LOSSES2=(PJ-PE2)/PJ*100 3380PE1X=PE1*COS(THETA1) : PE1Y=PE1*SIN(THETA1) 3390 PE2X=PE2*COS(THETA2) : PE2Y=PE2*STN(THETA2) 3395 REM GOSUB 9500 3400 REM COMPUTE At, Ut 3410 AT=XS*YS : UT=2*(XS+YS) 3420 REM 3430 RFM SPANDREL STRENGTH *************************************** 3440 REM 3450 MY = ( 1(.6*(AST*FLY + AP2*FPY)) / ( BSP*DSPEFF*FC1) )*DSPEFF*(

AST*FLY + AP2*FPY) 3460 REM TRANSVERSE STRENGTH OF SPANDREL

**************************

3470 IF S=0 THEN W=0 : GOTO 3490 3480 W=AWS*FWY/S 3490 REM ADDTIONAL TRANSVERSE STRENGTH OF SPANDREL

********************

3495 W0=446 3500 REM COMMON EQUATIONS 3502 EF ALS=0 THEN ALFA= 10000/200000! ELSE ALFA=ALS*FLY/200000! 3504 BETA=UT*(W+W0)/50000! : DR=DSPEFF/DEEFF 3506 DELTA=ALFA*BETA*DR 3510 REM CORNER OR EDGE COLUMN? 3520 IF COL$="CORNER " GOTO 3800 3530 REM ANALYSIS AT EDGE COLUMN

**************************************

3540 GAMMA=(75/DS)A3 3550 X1=GAMMA*DELTA 3570 REM 3580 REM EVALUATE VI 3590 IF DSPEFF<=(1.2*DEEFF) THEN M1=(600/(2*BSP+C2))A.9*M1 3600 IF DR<=1.2 THEN V1=9.350001+3.49*M1*.001/LC ELSE

V1=2.84+5.62*(M1+MM)*.001/LC 3610 REM FORMULA FOR SI 3620 SIP=1.98-1.88*LOG(Xl)/Z 3630 REM COMPUTE CONSTANTS

290

Appendix J

3640 K6=1-(D1-DS)/(2*L1): K7=(V1*1000-N1*PE1Y) 3650 REM IF ALFA<0 OR BETA<0 OR SIP<=0 THEN 6000 3660K1=200000!*(ALFA*BETA/ABS(SIP))A.5*(SIP-K6*MC2/MY) 3670K2=(K6*MCl)-Ml-K7*(Cl-(BSP-DEEFF)/2)+(Nl*PElX*El) 3680 K3=AT/(SQR(2)*UT) 3690 K4=(DSPEFF/(2*UT))A.5 3700 K5=(Cl-BSP)/2 3710 REM ANALYSIS OF Vu 3720K8=2*K3*K4*K1-K4*K2+K3*K7 : K9=K3-K4*K5 3730 VU=K8/K9 3740 GOTO 1500 3750 REM 3800 REM ANALYSIS AT CORNER COLUMN

************************************

3840 GAMMA=(C2/C1)A3*(75/DS)A.5 3850 X1=GAMMA*DELTA 3870 REM EVALUATE VI 3880 IF DSPEFF<=(1.2*DEEFF) THEN M1=(400/(BSP+C2))A.25*M1 3890 IF DR<= 1.2 THEN Vl=-3.66+5.04*(Ml+MM)*.001/LC ELSE Vl=-

.5+4.7*(Ml+MM)*.001/LC 3900 REM FORMULA FOR SI 3910 IF C2>C1 THEN SIP=3.5-1.76*L0G(X1)/Z ELSE SIP=1.86-1.58*L0G(X1)/Z

3920 REM 3930 REM COMPUTE CONSTANTS 3940 K6=T-(D1-DS)/(2*L1): K7=(V1*1000-N1*PE1Y) 3950 REM IF ALFA<0 OR BETA<0 OR SIP<=0 THEN 6000 3960K1=200000!*(ALFA*BETA/ABS(SIP))A.5*(SIP-

K6*(MC2+N2*PE2X*L1)/MY) 3970K2=(K6*MCl)-Ml-K7*(Cl-(BSP-DEEFF)/2)+(Nl*PElX*El)

3980 K3=AT/(SQR(2)*UT) 3990 K4=(DSPEFF/(2*UT))A.5 4000 K5=(Cl-BSP)/2 4010 REM ANALYSIS OF Vu 4020 K8=K3*K4*K1-K4*K2+K3*K7 : K9=K3-K4*K5 4030 VU=K8/K9 4040 REM 4100 GOTO 1500

6000 REM ***.** ***************************************************:i::i:*:!:::::i!*****

6010 REM ERROR MESSAGE 6020 CLS 6030 LOCATE 10,10:PRINT "ERROR DURING COMPUTATION !!!' 6040 LOCATE 12,10:IF ALFA<0 THEN PRINT "_ = ";ALFA;" A NEGATIVE

VALUE'" ELSE IF BETA<0 THEN PRINT "6 = ";BETA;" A NEGATIVE VALUE!" ELSE IF SIP<0 THEN PRINT "_ = ";SIP;" A NEGATIVE VALUE!"

ELSE PRINT "_ = ";SIP;" A ZERO!" 6050 LOCATE 14,10:PRINT "[PRESS ESC TO PROCEED]"

291

Appendix J

6100 KY$=INPUT$(1): IF KY$=CHR$(27) THEN CHAIN "ROO.BAS",230,ALL ELSE GOTO 6000

6500 REM SAVE RESULTS? 6510 CLS:LOCATE 10,10:INPUT "DO YOU WISH TO SAVE THE RESULTS (y/n)

";SAVERE$ 6520 IF SAVERE$="y" OR SAVERE$="Y" THEN RECORDL%=RECORD% :

GOTO 7000 ELSE GOTO 8000 7000 REM TO SET RESULTS FILE1 7010 OPEN "R",#2,"PUNS.DAT" 7020 FIELD #2,1 AS FL$,10 AS ML$,10 AS CL$,4 AS ALFA$,4 AS DELTA$,4 AS

DR$,4 AS Vl$,4 AS BETA$,4 AS SIP$,4 AS GAMMA$,4 AS MY$,4 AS Kl$,4 AS K2$,4 AS K3$,4 AS K4$,4 AS K5$,4 AS K6$,4 AS VUTL$,4 AS VU$

7030 REM 7040 LSET ML$=MODEL$ : LSET CL$=COL$ : LSET ALFA$=MKS$(ALFA):

LSET DELTA$=MKS$(DELTA) 7050 LSET DR$=MKS$(DR) : LSET V1$=MKS$(V1) : LSET

BETA$=MKS$(BETA): LSET SIP$=MKS$(SIP) 7060 LSET GAMMA$=MKS$(GAMMA): LSET MY$=MKS$(MY): LSET

K1$=MKS$(K1) : LSET K2$=MKS$(K2) 7070 LSET K3$=MKS$(K3): LSET K4$=MKS$(K4) : LSET K5$=MKS$(K5):

LSET K6$=MKS$(K6) 7080 LSET VUTL$=MKS$(VUT): LSET VU$=MKS$(VU) 7090 LSET FL$=CHR$(255) 7100 PUT#2,RECORDL% 7110 CLOSE #2 7200 REM TO SET RESULTS FELE2 7210 OPEN "R",#2,"TUNS.DAT" 7220 FIELD #2,1 AS FL$,4 AS K7$,4 AS PE1$,4 AS PE2$,4 AS PE1X$,4 AS

PE1Y$,4 AS PE2X$,4 AS PE2Y$,4 AS PLOSSl$,4 AS PLOSS2$,4 AS DSPEFF$,4 AS DEEFF$

7230 REM 7240 LSET K7$=MKS$(K7): LSET PE1$=MKS$(PE1): LSET PE2$=MKS$(PE2) :

LSET PE1X$=MKS$(PE1X) 7250 LSET PE1Y$=MKS$(PE1Y) : LSET PE2X$=MKS$(PE2X) : LSET

PE2Y$=MKS$(PE2Y) 7260 LSET PLOSSl$=MKS$(PLOSSl): LSET PLOSS2$=MKS$(PLOSS2): LSET

DSPEFF$=MKS$(DSPEFF): LSET DEEFF$=MKS$(DEEFF) 7270 LSET FL$=CHR$(255) 7300 PUT#2,RECORDL% 7310 CLOSE #2 7400 REM TO SET RESULTS FILE3 - FOR PARAMETRIC STUDY 7410 OPEN "R",#2,"PARA.DAT" 7420 FIELD #2,1 AS FL$,4 AS AT$,4 AS UT$,4 AS ALFA$,4 AS BETAS,4 AS

MY$ 4 AS PE2X$,4 AS PE1X$,4 AS PE1Y$,4 AS Vl$,4 AS VUS,4 AS DSPEFF$,4 AS DEEFF$,4 AS El$,4 AS DELTA$,4 AS GAMMA$,4 AS Nl$,4 AS N2$,4 AS SIP$

7430 LSET AT$=MKS$(AT): LSET UT$=MKS$(UT): LSET ALFA$=MKS$(ALFA): LSET BETA$=MKS$(BETA)

292

Appendix J

7440 LSET MY$=MKS$(MY): LSET PE2X$=MKS$(PE2X) : LSET PE1X$=MKS$(PE1X)

7450 LSET PE1Y$=MKS$(PE1Y) : LSET V1$=MKS$(V1) : LSET VU$=MKS$(VU) 7460 LSET DSPEFF$=MKS$(DSPEFF) : LSET DEEFF$=MKS$(DEEFF): LSET

E1$=MKS$(E1) 7470 LSET FL$=CHR$(255): LSET DELTAS =MKS$(DELTA): LSET

GAMMA$=MKS$(GAMMA) 7480 LSET N1$=MKS$(N1) : LSET N2$=MKS$(N2): LSET SIP$=MKS$(SIP) 7500 PUT#2,RECORDL% 7510 CLOSE #2 8000 CHAIN "ROO.BAS",130,ALL 9000 END 9100 REM SUBROUTINE FOR LTNE-DRAWING(LONG) 9110 PRINT STRING$(80,196) 9120 RETURN 9200 REM (SHORT) 9210 PRINT STRTNG$(50,196) 9220 RETURN

J.2 BASIC Program for the Australian Standard Procedure, AS 3600-1988 [SAA,

1988]

10 REM SAVE"TAN.BAS",A 20 REM 100 CLS: 103 LOCATE 1,5:PRTNT "DATA" 104 GOSUB 9200 105 LOCATE 3,5:PRTNT "MODEL ";MODEL$;"AT "COL$;"COLUMN" 106 GOSUB 9200 110 LOCATE 5,5:PRTNT "SHEAR HEAD : ";SHEAR$:LOCATE 5,30:PRINT

"CLOSED TIES: ";TIE$ 120 REM 130 LOCATE 6,5:PRINT "Mel = ";MC1 :LOCATE 6,30:PRINT "s = ";S 140 LOCATE 7,5:PRINT "Bsp = ";BSP LOCATE 7,30:PRINT "Fc = ";FC1 150 LOCATE 8,5:PRTNT "Dsp = ";DSP :LOCATE 8,30:PRINT "x = ";XS 160 LOCATE 9,5:PRTNT "DI = ";D1 -.LOCATE 9,30:PRINT "y = ";YS 170 LOCATE 10,5:PRINT "Ds = ";DS :LOCATE 10,30:PRINT "Fwy = ";FWY 180 LOCATE 11,5:PRINT "De = ";DE :LOCATE 11,30:PRINT "DI = ";DL 190 LOCATE 12,5:PRINT "CI = ";C1 LOCATE 12,30:PRINT "Dt = ";DT 200 LOCATE 13,5:PRINT "C2 = ";C2 :LOCATE 13,26:PRINT "Vu.test = ";VUT 210 LOCATE 14,5:PRINT "Aws = ";AWS 215 GOSUB 9200 220 LOCATE 16,5:PRINT "Pj = ";PJ :LOCATE 16,30:PRTNT "nl = ";N1 230 LOCATE 17,5:PRINT "dpi = ";DP1 :LOCATE 17,30:PRINT "n2 = ";N2 240 LOCATE 18,5:PRINT "dp2 = ";DP2 :LOCATE 18,30:PRINT "_tot = ";ALFA

293

Appendix J

250 LOCATE 19,5:PRTNT "Apl = ";AP1 :LOCATE 19,30:PRENT "u = ";MIU 260 LOCATE 20,5:PRINT "Ap2 = ";AP2 :LOCATE 20,30:PRINT "6p = ";BETA 270 GOSUB 9200 280 LOCATE 22,5:PRINT "PRESS [ESC] TO PROCEED " 290 KY$=TNPUT$(1): IF KY$=CHR$(27) THEN 300 ELSE GOTO 10 299 REM 300 REM PRESTRESSING FACTOR **************************** 310 IF PJ>0 THEN 320 ELSE GOTO 400 320 GOSUB 4000 400 REM ANALYSIS *************************************** 410 REM SETTING LIMITS TO ZERO 420 VUMIN=0 : VUMX=0 : DR=DSP/DE 430 REM TO DETERMINE 8h - Cl.9.2.1 ********************* 440 IF C1>=C2 THEN BETAH=C1/C2 ELSE BETAH=C2/C1 450 REM TO DETERMINE a,yl - Cl.9.2.1 ******************* 460A=Cl+DE/2 470 Y1=YS+DL+DT: AS=AWS/S 480 IF Y1=0 THEN RXY=1 : GOTO 500 ELSE ASMTN=.2*Y1/FWY 490 RXY=AS/ASMIN 500 REM CRITICAL PERIMETER ***************************** 510 IF COL$="CORNER " THEN U=C1+C2+DE : BE=C2+DE/2 ELSE

U=2*C1+C2+2*DE : BE=C2+DE 540 REM SHEAR HEAD"? ************************************ 550 IF SHEAR$="AT COLUMN " THEN 600 560 REM NO SHEAR HEAD ********************************** 570FCV=.17*(l+2/BETAH)*SQR(FCl) : FCVMX=.34*SQR(FC1) 580 IF FCV>FCVMX THEN FCV=FCVMX 590 VUO=U*DE*(FCV+.3*SIGCP): GOTO 700 600 REM WITH SHEAR HEAD ******************************** 610 VU0=U*DE*(.5*SQR(FC1)+.3*SIGCP): VUOMX=.2*U*DE*FCl 620 IF VUO>VUOMX THEN VUO=VUOMX 700 REM NO MOMENT TRANSFER ie. MC1=0 - Cl.9.2.3 ******** 710 IF MC 1=0 THEN VU=VUO : GOTO 2000 800 REM MOMENT TRANSFER ie. MC1>0 - Cl.9.2.4 *********** 810 IF TIE$="NONE " THEN 820 ELSE IF TIE$="IN TORSION STRIP"

THEN 850 ELSE GOTO 900 820 REM NO CLOSED TIES - C1.9.2.4(a) ******************* 830VU=VUO/(1+U*MC1/(8*VUT*A*DE)) 840 GOTO 2000 850 REM TORSION STRIP - C1.9.2.4(b) ******************** 860 VUMTN= 1.2* VUO/( 1 +U*MC 1/(2* VUT* AA2)) 870 R=DS/A 880 GOTO 1000 900 REM SPANDREL BEAM - C1.9.2.4(c) ******************** 910VUMIN=1.2*VUO*(D1/DS)/(1+U*MC1/(2*VUT*A*BSP)) 920 R=D1/BSP 1000 REM CHECK FOR ASMIN - C1.9.2.4(d) ***************** 1010 IF AS<=ASMIN THEN VU=VUMIN : GOTO 1100

294

Appendix J

1020 IF AS>ASMIN THEN VU=VUMIN*SQR(RXY) 1100 REM CHECK FOR Vumax ******************************* 1110IFR>1THENR=1/R 1120 VUMX=3*VUMIN*SQR(R) 1130 IF VU>VUMX THEN VU=VUMX 1140 REM 2000 REM RESULTS 2010 CLS:LOCATE 1,10:PRINT "RESULTS PRINTOUT" 2015 GOSUB 9100 2020 LOCATE 3,10:PRINT "MODEL ";MODEL$;" - ";COL$;"COLUMN " 2025 LOCATE 4,10:PRINT "AUSTRALIAN STANDARD AS3600-1988 TO

DETERMINE Vu" 2030 GOSUB 9100 2040 LOCATE 6,10:PRTNT "FEATURES OF MODEL :" 2060 LOCATE 7,10:PRTNT "SHEAR HEAD - ";SHEAR$ 2070 LOCATE 8,10:PRINT "CLOSED TIES - ";TIE$ 2080 GOSUB 9100 2090 LOCATE 10,10:PRTNT "Sh *= ";BETAH:LOCATE 10,40:PRTNT "a = ";A 2110 LOCATE 11,10:PRTNT "u = ";U:LOCATE 11,40:PRINT "fcv = ";FCV 2120 LOCATE 12,10:PRTNT "Vuo = ";VUO:LOCATE 12,40:PRTNT "Vumin =

";VUMTN 2130 LOCATE 13,10:PRTNT "Vumax = ";VUMX:LOCATE 13,40:PPJNT "dl/d =

";DR 2140 IF PJ>0 THEN LOCATE 14,10:PRINT "_cp = " ;SIGCP:LOCATE

14,40:PRINT "Pj = ";PJ 2150 IF PJ>0 THEN LOCATE 15,10:PRINT "Pe = ";PE/1000:LOCATE

15,40:PRINT "losses (%) = ";LOSSES 2160 GOSUB 9100 2170 LOCATE 17,10:PRINT USTNG "Vu.test = ######.##";VUT 2180 LOCATE 18,10:PRTNT USTNG "Vu,pred =######.##";VU 2190 LOCATE 19,10:PRINT USING "Ratio Vu,test/Vu,pred = ##.###" ;VUT/VU 2200 GOSUB 9100 2210 REM 2220 LOCATE 21,10:PRINT "[PRESS ESC TO END]" 2230 KY$=TNPUT$(1): IF KY$=CHR$(27) THEN CLOSE #2: GOTO 6500 ELSE

GOTO 2000 2240 REM 2250 REM 4000 REM SUBROUTINE FOR PRESTRESSING FACTOR 4010 REM PRESTRESS LOSSES 4020 REM ELASTIC LOSSES 4030EPCONC=2400A(1.5)*.043*SQR(FC1): EPTENDON= 166667! : El=DPl-DS/2

: E2=DP2-DS/2 4040SIGCPl=(Nl*PJ*1000/(C2*DS))+(Nl*PJ*1000*El*DS/2)/(C2*DSA3/12):

SIGCP2=(N2*PJ* 1000/(C1*DS))+(N2*PJ* 1000*E2*DS/2)/(Cl *DSA3/12) 4050 ELASTICl=.5*(EPTENDON/EPCONC)*SIGCPl :

ELASTIC2=.5*(EPTENDON/EPCONC)*SIGCP2

295

Appendix J

4060 EL0SS1=ELASTIC1*( API): ELOSS2=ELASTIC2*(AP2): ELOSS=(ELOSS l+ELOSS2)/2

4100 REM FRICTION LOSSES 4110 LPA=(Cl+DE/2)/1000 4120 FLOSS=EXP(-MIU*( ALFA+BETA*LPA)) 4150 REM ANCHORAGE FRICTION LOSSES 4160 AFLOSS=.02*PJ*1000 4200 REM ANCHORING LOSSES 4210 TW1=AFLOSS/5700 : TW2=AFLOSS/3530 : TW=(TWl+TW2)/2 4220 ZL=(3*EPTENDON*19.64)/TW : Z=SQR(ZL): ALOSS=2*Z*TW 4250 REM EFFECTIVE PRESTRESS FORCE PER WIRE AFTER ALL LOSSES 4260 PE=(PJ* 1000*FLOSS)-ELOSS-AFLOSS-ALOSS 4270 LOSSES=(PJ* 1000-PE)/(PJ* 1000)* 100 4300 REM EFFECTIVE NORTH-SOUTH & EAST-WEST SLAB SECTIONAL

AREAS 4310 AREANS=353O*80 : AREAEW=5700*80 4320 REM EFFECTIVE PRESTRESS 4330 PSIGNS=24*PE/AREANS : PSIGEW=32*PE/AREAEW 4340 REM AVERAGE PRESTRESS 4350 SIGCP=(PSIGNS+PSIGEW)/2 4360 RETURN 6500 REM SAVE RESULTS? 6510 CLS:LOCATE 10,10:TNPUT "DO YOU WISH TO SAVE THE RESULTS (y/n)


GOTO 7000 ELSE GOTO 8000 7000 REM TO SET RESULTS FILE 7010 OPEN "R",#1,"TANS.DAT" 7020 FIELD #1,1 AS FL$,10 AS ML$,10 AS CL$,10 AS SH$,16 AS TI$,4 AS

BETAH$,4 AS A$,4 AS U$,4 AS FCV$,4 AS VUO$,4 AS VUMTNS,4 AS VUMX$,4 AS DR$,4 AS VUTL$,4 AS VU$,4 AS SIGCP$,4 AS LPA$

7030 REM 7040 LSET ML$=MODEL$ : LSET CL$=COL$ : LSET SH$=SHEAR$ : LSET

TI$=TIE$ 7050 LSET BETAH$=MKS$(BETAH) : LSET A$=MKS$(A) : LSET U$=MKS$(U):

LSET FCV$=MKS$(FCV) 7060 LSET VUO$=MKS$(VUO) : LSET VUMIN$=MKS$(VUMIN) : LSET

VUMX$=MKS$(VUMX) : LSET DR$=MKS$(DR) 7070 LSET VUTL$=MKS$(VUT): LSET VU$=MKS$(VU): LSET

SIGCP$=MKS$(SIGCP): LSET LPA$=MKS$(LPA) 7080 LSET FL$=CHR$(255) 7100 PUT# 1 ,RECORDL% 7110 CLOSE #1 8000 CHAIN "RAN.BAS",130,ALL 9000 END 9100 REM SUBROUTINE FOR LINE-DRAWING(LONG) 9110 PRINT STRING$(80,196) 9120 RETURN

296

Appendix J

9200 REM (SHORT) 9210 PRINT STRING$(50,196) 9220 RETURN

J.3 BASIC Program for the ACI Method, American Concrete Institute ACI318-

89 [ACI, 1989]

10 REM SAVE"TCI.BAS",A 20 REM 100 CLS: 103 LOCATE 1,5:PRINT "DATA" 104 GOSUB 9200 105 LOCATE 3,5:PRINT "MODEL ";MODEL$;"AT "COL$;"COLUMN" 106 GOSUB 9200 110 REM 120 LOCATE 6,5:PRINT "CLOSED TIES: ";TIE$ 130 LOCATE 7,5:PRINT "Mel = ";MC1 :LOCATE 7,30:PRTNT "De = ";DE 135 LOCATE 8,5:PRINT "Mc2 = ";MC2 :LOCATE 8,30:PRTNT "CI = ";C1 140 LOCATE 9,5:PRTNT "Bsp = ";BSP :LOCATE 9,30:PRINT "C2 = ";C2 150 LOCATE 10,5:PRINT "Dsp = ";DSP :LOCATE 10,30:PRTNT "F'c = ";FC1 160 LOCATE 11,5:PRINT "DI = ";D1 :LOCATE 11,26:PRINT "Vu,test = ";VUT 170 LOCATE 12,5:PRTNT "Ds = ";DS 180 GOSUB 9200 190 LOCATE 14,5:PRINT "Pj = ";PJ :LOCATE 14,30:PRINT "nl = ";N1 200 LOCATE 15,5:PRINT "dpi = ";DP1 :LOCATE 15,30:PRINT "ri2 = ";N2 210 LOCATE 16,5:PRFNT "dp2 = ";DP2 :LOCATE 16,30:PRINT "_tot = ";ALFA 220 LOCATE 17,5:PRINT "Apl = ";AP1 :LOCATE 17,30:PRINT "u = ";MIU 230 LOCATE 18,5:PRINT "Ap2 = ";AP2 :LOCATE 18,30:PRINT "Bp = ";BETA 240 GOSUB 9200 250 LOCATE 20,5:PRINT "PRESS [ESC] TO PROCEED " 260 KY$=TNPUT$(1): IF KY$=CHR$(27) THEN 300 ELSE GOTO 10 270 REM 300 REM ANALYSIS *************************************** 310 REM PRESTRESSING FACTOR 320 IF PJ > 0 THEN 350 ELSE GOTO 400 350 GOSUB 4000 355 IF (Cl/DS) < 4 THEN 400 360 REM SHEAR STRENGTH FOR PRESTRESSED SLAB WHERE MC1=0 370 IF PJ>0 THEN VC=(.29*SQR(FC1)+.3*FPC)*B0*DE 380 GOTO 400 390 REM 400 REM PRELIMINARIES FOR ALL COLUMN POSITIONS 410 REM COLUMN OPTIONS 420 IF COL$="CORNER " THEN 450 ELSE IF COL$="EDGE " THEN 600 ELSE

GOTO 700

297

Appendix J

430 REM 450 REM CORNER COLUMN-PRELIMINARY 460BO=C1+C2+DE 470Bl=C2+DE/2 480 B2=Cl+DE/2 490 ALPHAS= 1.66 500 AC=(C1+C2+DE)*DE 510 CAB=(C1+DE/2)A2*DE/(2*AC) 520 CBD=(C2+DE/2)A2*DE/(2*AC) 530 CCD=Cl+DE/2-CAB 540 CAC=C2+DE/2-CBD 550 JC1 = (DE*(C1+DE/2)A3/12) + ((C1+DE/2)*DEA3/12) +

((C2+DE/2)*DE*CABA2) + (C1+DE/2)*DE*((C1 + DE/2)/2-CAB)A2 560 JC2 = (DE*(C2+DE/2)A3/12) + ((C2+DE/2)*DEA3/12) +

((C1+DE/2)*DE*CBDA2) + (C2+DE/2)*DE*((C2 + DE/2)/2-CBD)A2 570Gl=CCD-Cl/2 580 G2=CAC-C2/2 585 IF JC1>JC2 THEN JC=JC1 ELSE JC=JC2 586 IF G1>G2 THEN G=G1 ELSE G=G2 590 GOTO 800 600 REM EDGE COLUMN-PRELIMINARY 610BO=2*C1+C2+2*DE 620B1=C2+DE 630B2=Cl+DE/2 640 ALPHAS=2.5 650 AC=(2*C1+C2+2*DE)*DE 660 CAB=(C1+DE/2)A2*DE/AC 670 CCD=Cl+DE/2-CAB 680 JC = (2*DE*(C1+DE/2)A3/12) + (2*(C1+DE/2)*DEA3/12) +

((C2+DE)*DE*CABA2) + 2*(C1+DE/2)*DE*((C1 + DE/2)/2-CAB)A2

690G=CCD-Cl/2 695 GOTO 800 700 REM INTERIOR COLUMN-PRELIMINARY 710 B0=2*(C1+C2)+4*DE 720B1=C2+DE 730B2=C1+DE 740 ALPHAS=3.33 750 AC=2*DE*(C1+C2+2*DE) 760 G=0 770JC=DE*(Cl+DE)A3/6+(Cl+DE)*DEA3/6+2*DE*(C2+DE)*(Cl*DE)A2/6 780CAB=Cl/2 790CCD=Cl/2 800 REM OPTIONS FOR REINFORCED AND PRESTRESSED SLABS 802 IF PJ > 0 AND (Cl/DS) > 4 THEN GOTO 900 ELSE GOTO 808 808 REM ALL COLUMNS IN REINFORCED SLABS 810 BETAC=C1/C2 : DR=DSP/DE 820 IF BETAC<1 THEN BETAC=1/BETAC 830 REM SHEAR STRENGTH AT NO MOMENT TRANSFER ie.Mcl=0

298

Appendix J

840 VC 1 =(. 166+.33/BETAC)*SQR(FC 1 )*BO*DE 850VC2=(ALPHAS*DE/BO+.166)*SQR(FC1)*BO*DE 860 VC3=.33*SQR(FCl)*BO*DE 870 REM MINIMUM Vc 880 IF VC1>VC2 THEN VCO=VC2 ELSE VC0=VC1 890 IF VCO>VC3 THEN VC=VC3 ELSE VC=VCO 900 REM CHECK ZERO MOMENT TRANSFER 910 IF MC1=0 THEN VU=VC : GOTO 2000 920 REM MOMENT TRANSFER ie.Mcl>0 930 REM COLUMN POSITION 940 IF COL$="CORNER " THEN 950 ELSE GOTO 1100 950 REM CORNER COLUMN-PUNCHING SHEAR CALCULATIONS 960GAMMAVl=l-l/(l+(2/3)*SQR(Bl/B2)) 970 GAMMA V2= 1 -1/( 1 +(2/3)*SQR(B2/B 1)) 975 IF GAMMAV1>GAMMAV2 THEN GAMMAV=GAMMAV1 ELSE

GAMMAV=GAMMAV2 980VUSTRSA=(VUT/AC)+(GAMMAVl*(MCl-VUT*Gl)*CAB/JCl)-(

GAMMAV2*(MC2-VUT*G2)*CAC/JC2) 990 VUSTRSB=(VUT/AC)+(GAMMAV 1 *(MC 1 -VUT*G 1 )*CAB/JC1 )-(G

AMMAV2*(MC2-VUT*G2)*CBD/JC2) 1000 VUSTRSD=(VUT/AC)+(GAMM AV 1 *(MC 1 -VUT*G 1 )*CCD/JC 1 )-

(GAMMAV2*(MC2-VUT*G2)*CBD/JC2) 1010 REM CRITICAL SHEAR STRESS 1020 IF VUSTRSA>VUSTRSB THEN VUSTRSO=VUSTRSA ELSE

VUSTRSO=VUSTRSB 1030 IF VUSTRSO>VUSTRSD THEN VUSTRS=VUSTRSO ELSE

VUSTRS=VUSTRSD 1040 GOTO 1200 1100 REM EDGE & INTERIOR COLUMN-PUNCHING SHEAR CALCULATIONS 1110 GAMMAV=l-l/(l+(2/3)*SQR(Bl/B2)) 1120 VUSTRSAB=(VUT/AC)+(GAMMAV*(MC1-VUT*G)*CAB/JC) 1130 VUSTRSCD=(VUT/AC)-(GAMMAV*(MC1-VUT*G)*CCD/JC) 1140 IF VUSTRSAB>VUSTRSCD THEN VUSTRS=VUSTRSAB ELSE

VUSTRS=VUSTRSCD 1150 REM 1200 REM CHECK MAXIMUM SHEAR STRESS 1210 VUSTRSMX=.85*VC/(BO*DE) 1220 IF VUSTRS>VUSTRSMX THEN VUSTRS=VUSTRSMX 1230 REM PUNCHING SHEAR ' 1240 VU=VUSTRS*BO*DE 1250 REM 2000 REM RESULTS 2005 CLS:LOCATE 1,10:PRINT "RESULTS PRINTOUT" 2006 GOSUB 9100 2010 LOCATE 3,10:PRINT "MODEL ";MODEL$;" - ";COL$;"COLUMN " 2015 LOCATE 4,10:PRINT "AMERICAN CONCRETE INSTITUTE ACI318-89 TO

DETERMINE Vu" 2020 GOSUB 9100

299

Appendix J

2030 LOCATE 6,10:PRINT "FEATURE OF MODEL :" 2060 LOCATE 7,10:PRTNT "CLOSED TIES - ";TIE$ 2070 GOSUB 9100 2080 LOCATE 9,10:PRINT "Be = ";BETAC:LOCATE 9,40:PRINT "Ac = ";AC 2090 LOCATE 10,10:PRINT "bo = ";BO:LOCATE 10,40:PRINT "g = ";G 2100 LOCATE 11,10:PRINT "_v = ";GAMMAV:LOCATE 11,40:PRFNT "Jc =

";JC 2110 LOCATE 12,10:PRTNT "vu = ";VUSTRS:LOCATE 12,40:PRINT "vumax =

";VUSTRSMX 2120 LOCATE 13,10:PRINT "Vc = ";VC:LOCATE 13,40:PRTNT "dl/d = ";DR 2130 IF PJ>0 THEN LOCATE 15,10.PRTNT "fpe = ";FPC:LOCATE 15,40:PRTNT

"Pj = ";PJ 2140 IF PJ>0 THEN LOCATE 16,10:PRINT "Pe = ";PE/1000:LOCATE

16,40:PRINT "losses (%) = ";LOSSES 2145 IF PJ>0 AND (C1/DS)>4 THEN LOCATE 17,10:PRINT "Cl/DS = ";C1/DS;" >

4 : effect of prestress is considered" ELSE IF PJ>0 AND (C1/DS)<4 THEN LOCATE 17,10:PRINT "Cl/DS = ";C1/DS;" < 4 : effect of prestress is ignored"

2150 GOSUB 9100 2160 IF PJ>0 THEN LOCATE 19,10:PRTNT USING "Vu.test

######.##" ;VUT ELSE LOCATE 15,10:PRINT USING "Vu,test ######.##" ;VUT

2170 IF PJ>0 THEN LOCATE 20,10:PRINT USING "Vu,pred ######.##";VU ELSE LOCATE 16,10:PRINT USTNG "Vu,pred ######.##";VU

2180 IF PJ>0 THEN LOCATE 21',10:PRINT USING "Ratio Vu,test/Vu,pred = ##.###" ;VUT/VU ELSE LOCATE 17,10:PRINT USING "Ratio Vu,test/Vu,pred = ##.###" ;VUT/VU

2190 GOSUB 9100 2200 REM 2210 REM 2220 LOCATE 23,10:PRINT "[PRESS ESC TO END]" 2230 KY$=TNPUT$(1): IF KY$=CHR$(27) THEN 6500 ELSE GOTO 2000

2240 REM 2250 REM 4000 REM SUBROUTINE FOR PRESTRESSING FACTOR 4010 REM PRESTRESS LOSSES 4020 REM ELASTIC LOSSES 4030EPCONC=2400A(1.5)*.043*SQR(FC1): EPTENDON= 166667! : El=DPl-DS/2


SIGCP2=(N2*PJ* 1000/(C 1 *DS))+(N2*PJ* 1000*E2*DS/2)/(C 1 *DSA3/12) 4050 ELASTIC l=.5*(EPTENDON/EPCONC)*SIGCPl :

ELASTIC2=.5*(EPTENDON/EPCONC)*SIGCP2 4060 ELOSSl=ELASTICl*(APl): ELOSS2=ELASTIC2*(AP2) :

ELOSS=(ELOSS l+ELOSS2)/2 4100 REM FRICTION LOSSES 4110LX=(Cl+DE/2)/1000 4120 FLOSS=EXP(-MIU*(ALFA+BETA*LX))

300

Appendix J

4150 REM ANCHORAGE FRICTION LOSSES 4160 AFLOSS=.02*PJ*1000 4200 REM ANCHORING LOSSES 4210 TW1=AFLOSS/5700 : TW2=AFLOSS/3530 : TW=(TWl+TW2)/2 4220ZL=(3*EPTENDON*19.64yrW : Z=SQR(ZL) : ALOSS=2*Z*TW 4250 REM EFFECTIVE PRESTRESS FORCE PER WIRE AFTER ALL LOSSES 4260 PE=(PJ* 1000*FLOSS)-ELOSS-AFLOSS-ALOSS 4270 LOSSES=(PJ* 1000-PE)/(PJ* 1000)* 100 4300 REM EFFECTIVE NORTH-SOUTH & EAST-WEST SLAB SECTIONAL

AREAS 4310 AREANS=3530*80 : AREAEW=5700*80 4320 REM EFFECTIVE PRESTRESS 4330 PSIGNS=24*PE/AREANS : PSIGEW=32*PE/AREAEW 4340 REM AVERAGE PRESTRESS 4350 SIGCP=(PSIGNS+PSIGEW)/2 4360 IF PSIGNS < 1! THEN PSIGNS=1! 4370 IF PSIGNS > 3.5 THEN PSIGNS=3.5 4380 IF PSIGEW < 1! THEN PSIGEW=1! 4390 IF PSIGEW > 3.5 THEN PSIGEW=3.5 4400 REM AVERAGE PRESTRESS 4410 FPC=(PSIGNS+PSIGEW)/2 4420 IF FC1>35 THEN FC1=35 4450 IF COL$="CORNER " THEN B0=C1+C2+DE : GOTO 4500 4440 B0=2*C1+C2+2*DE 4500 RETURN 6500 REM SAVE RESULTS? 6510 CLS:LOCATE 10,10:INPUT "DO YOU WISH TO SAVE THE RESULTS (y/n)

";SAVERE$ 6520 IF SAVERE$="y" OR SAVERE$='Y" THEN RECORDL%=RECORD% :

GOTO 7000 ELSE GOTO 8000 7000 REM TO SET RESULTS FILE 7010 OPEN "R",#3,"TACI.DAT" 7020 FIELD #3,1 AS FL$,10 AS ML$,10 AS CL$,16 AS TI$,4 AS BETAC$,4 AS

AC$,4 AS BO$,4 AS G$,4 AS GAMMAV$,4 AS JC$,4 AS VUSTRS$,4 AS VUSTRSMX$,4 AS VC$,4 AS DR$,4 AS VUTL$,4 AS VU$,4 AS FPC$,4 AS LX$

7030 REM 7040 LSET ML$=MODEL$ : LSET CL$=COL$ : LSET TI$=TIE$ 7050 LSET BETAC$=MKS$(BETAC): LSET AC$=MKS$(AC): LSET

BO$=MKS$(BO) : LSET G$=MKS$(G) : LSET VC$=MKS$(VC) 7060 LSET GAMMAV$=MKS$(GAMMAV): LSET JC$=MKS$(JC) : LSET

VUSTRS$=MKS$(VUSTRS) : LSET DR$=MKS$(DR): LSET VUSTRSMX$=MKS$(VUSTRSMX)

7070 LSET VUTL$=MKS$(VUT): LSET VU$=MKS$(VU) : LSET FPC$=MKS$(FPC) : LSET LX$=MKS$(LX)

7080 LSET FL$=CHR$(255) 7100 PUT#3,RECORDL% 7110 CLOSE #3

301

Appendix J

8000 CHAIN "ACI.BAS",230,ALL 9000 END 9100 REM SUBROUTINE FOR LINE-DRAWING(LONG) 9110 PRINT STRING$(80,196) 9120 RETURN 9200 REM (SHORT) 9210 PRINT STRING$(50,196) 9220 RETURN

J.4 BASIC Program for the British Method, British Standard Institute BS

8110:1985 [BSI, 1985]

10 REM SAVE"TSI.BAS",A 20 REM 100 CLS: 103 LOCATE 1,5:PRTNT "DATA" 104 GOSUB 9200 105 LOCATE 3,5:PRINT "MODEL ";MODEL$;"AT "COL$;"COLUMN" 106 GOSUB 9200 110 REM 120 LOCATE 6,5:PRINT "CLOSED TIES: ";TIE$ 130 LOCATE 7,5:PRINT "Mel = ";MC1 -.LOCATE 7,30:PRINT "CI = ";C1 135 LOCATE 8,5:PRINT "Bsp = ";BSP :LOCATE 8,30:PRTNT "C2 = ";C2 140 LOCATE 9,5:PRINT "Dsp = ";DSP :LOCATE 9,30:PRINT "Fc = ";FC1 150 LOCATE 10,5:PRINT "DI = ";D1 :LOCATE 10,30:PRTNT "Aslab = ";ASLAB 160 LOCATE 11,5:PRINT "Ds = ";DS :LOCATE 11,28:PRINT "Vu.test = ";VUT 170 LOCATE 12,5:PRINT "De = ";DE 180 GOSUB 9200 190 LOCATE 14,5:PRINT "Pj = ";PJ :LOCATE 14,30:PRINT "nl = ";N1 200 LOCATE 15,5:PRINT "dpi = ";DP1 :LOCATE 15,30:PRINT "n2 = ";N2 210 LOCATE 16,5:PRINT "dp2 = ";DP2 :LOCATE 16,30:PRINT "_tot = ";ALFA 220 LOCATE 17,5:PRINT "Apl = ";AP1 :LOCATE 17,30:PRINT "u = ";MIU 230 LOCATE 18,5:PRTNT "Ap2 = ";AP2 :LOCATE 18,30:PRTNT "fip = ";BETA 240 LOCATE 19,5:PRINT "_ = ";DELTA :LOCATE 19,5:PRINT "fpu = ";FPU 250 GOSUB 9200 260 LOCATE 22,5:PRINT "PRESS [ESC] TO PROCEED " 270 KY$=INPUT$(1): IF KY$=CHR$(27) THEN 300 ELSE GOTO 10 280 REM 300 REM ANALYSIS *************************************** 310 REM PRESTRESSING FACTOR 320 IF PJ>0 THEN 350 ELSE GOTO 400 350 GOSUB 4000 360 REM PUNCHING SHEAR FOR PRESTRESSED SLAB 370 IF PJ>0 THEN GOTO 2000 ELSE GOTO 400 400 REM PUNCHING SHEAR FOR NON-PRESTRESSED SLAB

<*

302

Appendix J

405 REM CONVERT CYLINDER fc TO CUBE STRENGTH feu 406 FCU=3.5969+.99686*FC1 410 REM COLUMN OPTIONS-PRELIMINARY 420 IF COL$="CORNER " THEN 450 ELSE GOTO 500 430 REM 450 REM CORNER COLUMN-PRELIMINARY 460UC=C1+C2+3*DE 470BV=C2+1.5*DE 480 GOTO 550 490 REM 500 REM EDGE COLUMN-PRELIMINARY 510 UC=2*C1+C2+6*DE 520 BV=C2+3*DE 540 REM CONCRETE SHEAR STRESS, vc 550 RHO=100*ASLAB/(BV*DE) 560 KHO=400/DE : RATIO=DSP/DE 570 GAMMAM= 1.25 580 IF RHO>3 THEN RHO=3 590 IF KHO<l THEN KHO=l 600 IF FCU>40 THEN FCU=40 610 IF FCU<25 THEN VC=(-79*RHOA(l/3)*KHOA(l/4))/GAMMAM ELSE

VC=(.79*RHOA(l/3)*KHOA(l/4)*(FCU/25)A(l/3))/GAMMAM 620 REM 630 REM CHECK FOR MAX VC 640 VCMX=.8*SQR(FCU) 650 IF VCMX>5 THEN VCMX=5 660 IF VC>VCMX THEN VC=VCMX 670 REM 680 REM CHECK MOMENT TRANSFER 690 VT=VC*UC*DE 700 REM NO MOMENT TRANSFER, Mcl=0 710 IF MC 1=0 THEN VU=VT : GOTO 2000 ELSE GOTO 720 720 REM MOMENT TRANSFER, Mcl>0 770VEFF=1.25*VT 780 REM 790 REM PUNCHING SHEAR Vu 800 VU=VEFF 810 GOTO 2000 820 REM 2000 REM RESULTS 2005 CLS:LOCATE 1,10:PRINT "RESULTS PRINTOUT" 2006 GOSUB 9100 2010 LOCATE 3,10:PRTNT "MODEL ";MODEL$;" - ";COL$;"COLUMN " 2015 LOCATE 4,10:PRTNT "BRITISH STANDARD INSTITUTE BS8110:1985 TO

DETERMINE Vu" 2020 GOSUB 9100 2030 LOCATE 6,10:PRTNT "FEATURE OF MODEL :" 2060 LOCATE 7,10:PRTNT "CLOSED TIES - ";TIE$

303

Appendix J

2070 GOSUB 9100 2080 LOCATE 10,10:PRTNT "bv = ";BV:LOCATE 10,40:PRTNT "Uc = ";UC 2090 LOCATE 11,10:PRINT "_m = ";GAMMAM:LOCATE 11,40:PRINT "dl/d

= ";RATIO 2100 LOCATE 12,10:PRTNT "vc = ";VC:LOCATE 12,40:PRTNT "vcmax =

";VCMX 2110 LOCATE 13,10:PRINT "Vt = ";VT:LOCATE 13,40:PRTNT "Veff = ";VEFF 2120 LOCATE 14,10:PRINT "Steel ratio = ";RHO:LOCATE 14,40:PRINT "feu =

";FCU 2130 IF PJ>0 THEN LOCATE 15,10:PRTNT "fcp = ";FCP:LOCATE 15,40:PRTNT

"Pj = ";PJ 2140 IF PJ>0 THEN LOCATE 16,10:PRINT "Pe = ";PE/1000:LOCATE

16,40:PRINT "losses (%) = ";LOSSES 2150 GOSUB 9100 2160 LOCATE 18,10:PRINT USING "Vu.test = ######.##" ;VUT 2170 LOCATE 19,10:PRINT USING "Vu,pred = ######.##";VU 2180 LOCATE 20,10:PRTNT USING "Ratio Vu,test/Vu,pred = ##.###";VUT/VU 2190 GOSUB 9100 2200 REM 2210 REM 2220 LOCATE 23,10:PRINT "[PRESS ESC TO END]" 2230 KY$=TNPUT$(1): IF KY$=CHR$(27) THEN 6500 ELSE GOTO 2000 2240 REM 2250 REM 4000 REM SUBROUTINE FOR PRESTRESSING FACTOR 4010 REM PRESTRESS LOSSES 4020 REM ELASTIC LOSSES 4030EPCONC=2400A(1.5)*.043*SQR(FC1): EPTENDON= 166667! : El=DPl-DS/2


SIGCP2=(N2*PJ* 1000/(C 1 *DS))+(N2*PJ* 1000*E2*DS/2)/(C 1 *DS A3/12) 4050 ELASTICl=.5*(EPTENDON/EPCONC)*SIGCPl :

ELASTIC2=.5*(EPTENDON/EPCONC)*SIGCP2 4060ELOSS1=ELASTIC1*(AP1) : ELOSS2=ELASTIC2*(AP2) :

ELOSS=(ELOSS l+ELOSS2)/2 4100 REM FRICTION LOSSES 4110LX=(Cl+DE/2)/1000 4120 FLOSS=EXP(-MIU*(ALFA+BETA*LX)) 4150 REM ANCHORAGE FRICTION LOSSES 4160 AFLOSS=.02*PJ*1000 4200 REM ANCHORING LOSSES 4210 TW1=AFLOSS/5700 : TW2=AFLOSS/3530 : TW=(TWl+TW2)/2 4220ZL=(3*EPTENDON*19.64)/TW : Z=SQR(ZL) : ALOSS=2*Z*TW 4250 REM EFFECTIVE PRESTRESS FORCE PER WIRE AFTER ALL LOSSES 4260 PE=(PJ* 1000*FLOSS)-ELOSS-AFLOSS-ALOSS 4270LOSSES=(PJ*1000-PE)/(PJ*1000)*100 4300 REM EFFECTIVE NORTH-SOUTH & EAST-WEST SLAB SECTIONAL

AREAS

304

Appendix J

4310 AREANS=3530*80 : AREAEW=5700*80 4320 REM EFFECTIVE PRESTRESS 4330 PSIGNS=24*PE/AREANS : PSIGEW=32*PE/AREAEW 4340 REM AVERAGE PRESTRESS & OTHER STRESSES 4350 FCP=(PSIGNS+PSIGEW)/2 : FPE=PE/AP1 4360 REM CONVERT CYLINDER fc TO CUBE STRENGTH feu 4370 FCU=3.5969+.99686*FC1 4380 FT=.24*SQR(FCU) 4400 REM SECTION PROPERTIES 4410AG=C2*DS 4420IG=(C2*DSA3)/12 4430 FPT=PE/AG+(PE*El*DS/2)/IG 4450 REM DECOMPRESSION MOMENT 4460 MO=.8*FPT*IG/(DS/2) 4500 REM ANALYSIS OF VU 4510 IF COL$="CORNER " THEN 4550 ELSE GOTO 4600 4550 REM CORNER COLUMN-PRELIMINARY 4560UC=C1+C2+3*DE 4570BV=C2+1.5*DE 4580 GOTO 4700 4600 REM EDGE COLUMN-PRELIMINARY 4610 UC=2*C1+C2+6*DE 4620 BV=C2+3*DE 4700 REM CONCRETE SHEAR STRESS, vc 4710 RHO=100*(ASLAB+AP1)/(BV*DE) 4720 KHO=400/DE : RATIO=DSP/DE 4730GAMMAM=1 4740 IF RHO>3 THEN RHO=3 4750 IF KHO<l THEN KHO=l 4760 IF FCU>40 THEN FCU=40 4770 IF FCU<25 THEN VC=(.79*RHOA(l/3)*KHOA(l/4))/GAMMAM ELSE

VC=(.79*RHOA(l/3)*KHOA(l/4)*(FCU/25)A(l/3))/GAMMAM 4800 REM CHECK FOR MAX VC 4810 VCMX=.8*SQR(FCU) 4820 IF VCMX>5 THEN VCMX=5 4830 IF VC>VCMX THEN VC=VCMX 4900 REM ANALYSIS OF VU 4910VCO=.67*BV*DS*SQR(FTA2+.8*FCP*FT) 4920VCR=(1-.55*FPE/FPU)*VC*BV*DE+MO*(VUT-PE*STN(DELTA))/MC1 4930 IF VCO<VCR THEN VU=VCO ELSE VU=VCR 5000 RETURN 6500 REM SAVE RESULTS? 6510 CLS:LOCATE 10,10:INPUT "DO YOU WISH TO SAVE THE RESULTS (y/n)


GOTO 7000 ELSE GOTO 8000 7000 REM TO SET RESULTS FILE 7010 OPEN "R",#3,"TSI.DAT"

305

Appendix J

7020 FIELD #3,1 AS FL$,10 AS ML$,10 AS CL$,16 AS TI$,4 AS BV$,4 AS UC$,4 AS GAMMAM$,4 AS RATIO$,4 AS VC$,4 AS VCMX$,4 AS VT$,4 AS VEFF$,4 AS RHO$,4 AS VUTL$,4 AS VU$

7030 REM 7040 LSET ML$=MODEL$ : LSET CL$=COL$ : LSET TI$=TIE$ 7050 LSET BV$=MKS$(BV): LSET UC$=MKS$(UC) : LSET

GAMMAM$=MKS$(GAMMAM) 7060 LSET RATIO$=MKS$(RATIO): LSET VC$=MKS$(VC): LSET

VCMX$=MKS$(VCMX): LSET VT$=MKS$(VT) 7070 LSET VUTL$=MKS$(VUT): LSET VU$=MKS$(VU) : LSET

VEFF$=MKS$(VEFF): LSET RHO$=MKS$(RHO) 7080 LSET FL$=CHR$(255) 7100 PUT#3,RECORDL% 7110 CLOSE #3 8000 CHAIN "BSI.BAS",230,ALL 9000 END 9100 REM SUBROUTINE FOR LTNE-DRAWING(LONG) 9110 PRINT STRING$(80,196) 9120 RETURN 9200 REM (SHORT) 9210 PRINT STRING$(50,196) 9220 RETURN

306

PAPERS PUBLISHED BASED ON THIS THESIS

Chiang, C L . and Loo, Y.C. (1993), "Punching Shear Tests of Half-Scale Post-

Tensioned Concrete Flat Plate Models with Spandrel Beams," Proc. 13th

Australasian Conference on Mechanics of Structures and Materials (ACMSM

13), Wollongong, July 1993, V.l, pp.185-192.

Loo, Y.C. and Chiang CL. (1993), "Methods of Punching Shear Strength Analysis of

Reinforced Concrete Flat Plates - A Comparative Study", International Journal

of Structural Engineering and Mechanics, Techno-Press, V.l, No.l, October

1993, pp.75-86.

Loo, Y.C. and Chiang CL. (1994), "Performance of Analytical Methods For Punching

Shear Strength of Reinforced Concrete Flat Plates", Paper No. 124,

Australasian Structural Engineering Conference-1994 (ASEC), to be held in

Sydney Hilton, Sydney, 21-23 September 1994.

307

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