4 Analysis And Design of Steel Deck – Concrete Composite Slabs

ANALYSIS AND DESIGN OFSTEEL DECK – CONCRETE COMPOSITE SLABS

Budi Ryanto Widjaja

Dissertation submitted to the Faculty of the

Virginia Polytechnic Institute and State University

in partial fulfillment of the requirements for the degree of

DOCTOR OF PHILOSOPHY

in

Civil Engineering

W. S. Easterling, Chairman

R. M. Barker

E. G. Henneke

S. M. Holzer

T. M. Murray

October, 1997

Blacksburg, Virginia

Keywords: composite slabs, direct method, iterative method, finite element model, long span, resistance factor

ANALYSIS AND DESIGN OFSTEEL DECK – CONCRETE COMPOSITE SLABS

by

Budi R. Widjaja

Dr. W. S. Easterling, Chairman

Department of Civil Engineering

(ABSTRACT)

As cold-formed steel decks are used in virtually every steel-framed structure for

composite slab systems, efforts to develop more efficient composite floor systems continues.

Efficient composite floor systems can be obtained by optimally utilizing the materials, which

includes the possibility of developing long span composite slab systems. For this purpose, new

deck profiles that can have a longer span and better interaction with the concrete slab are

investigated.

Two new mechanical based methods for predicting composite slab strength and behavior

are introduced. They are referred to as the iterative and direct methods. These methods, which

accurately account for the contribution of parameters affecting the composite action, are used to

predict the strength and behavior of composite slabs. Application of the methods in the

analytical and experimental study of strength and behavior of composite slabs in general reveals

that more accurate predictions are obtained by these methods compared to those of a modified

version of the Steel Deck Institute method (SDI-M). A nonlinear finite element model is also

developed to provide additional reference. These methods, which are supported by elemental

tests of shear bond and end anchorages, offer an alternative solution to performing a large

number of full-scale tests as required for the traditional m-k method. Results from 27 composite

slab tests are compared with the analytical methods.

Four long span composite slab specimens of 20 ft span length, using two different types

of deck profiles, were built and tested experimentally. Without significantly increasing the slab

depth and weight compared to those of composite slabs with typical span, it was found that these

long span slabs showed good performance under the load tests. Some problems with the

vibration behavior were encountered, which are thought to be due to the relatively thin layer of

concrete cover above the deck rib. Further study on the use of deeper concrete cover to improve

the vibrational behavior is suggested.

Finally, resistance factors based on the AISI-LRFD approach were established. The

resistance factors for flexural design of composite slab systems were found to be φ=0.90 for the

SDI-M method and φ=0.85 for the direct method.

In Memory of my Fatherand

In Love of my Mother

ACKNOWLEDGMENTS

I am most grateful to Dr. W. Samuel Easterling for his continuous support, guidance and

friendship throughout my graduate study at Virginia Polytechnic Institute and State University

(Virginia Tech). I would also like to express my sincere appreciation to the members of the

research committee, Dr. R. M. Barker, Dr. E. G. Henneke, Dr. S. M. Holzer and Dr. T. M.

Murray. Special thanks goes to Dr. R. M. Barker for his valuable discussion on the resistance

factors and to Dr. T. M. Murray for his valuable discussion on floor vibrations.

I gratefully acknowledge financial support from the National Science Foundation, under

research grant no. MSS-9222064, the American Institute of Steel Construction, the American

Iron and Steel Institute, Vulcraft and Consolidated System Incorporated. Further, material for

test specimens was supplied by BHP of America, TRW Nelson Stud Welding Division and

United Steel Deck. My sincere thanks is also for the Steel Deck Institute for the Scholarship

Award that I received for my research and very special thanks to Mr. and Mrs. R. B. Heagler for

their very warm hospitality during my visit at the SDI annual meeting in Florida. Mr. Heagler

also keeps me updated with new technical issues and developments in the SDI.

I would also like to thank to Dr. M. Crisinel and Dr. B. J. Daniels for the access to use

the COMPCAL program at the Ecole Polytechnique Federale de Lausanne, Switzerland. They

also allowed me to use the drawings for the elemental tests.

To all my friends in the Civil Engineering Department and especially those at the

Structures and Materials Laboratory of Virginia Tech, I extend my appreciation for their support,

discussion and friendship. I am particularly indebted to Joseph N. Howard for his immeasurable

help in performing the vibration tests on the long span slabs. Special thanks goes to Dennis W.

Huffman and Brett N. Farmer for their constant help and cheerful support during my research

work at the Structures Lab.

Last but certainly not the least, I am thankful to my wife, Surjani, for being a constant

source of inspiration and encouragement. She is a wonderful wife and friend.

vi

TABLE OF CONTENTS

ABSTRACT.......................................................................................................................... iiDEDICATIONS....................................................................................................................ivACKNOWLEDGMENTS .................................................................................................... vTABLE OF CONTENTS...................................................................................................... viLIST OF FIGURES............................................................................................................... ixLIST OF TABLES................................................................................................................ xiLIST OF NOTATIONS ........................................................................................................ xii

Chapter 1. Introduction

1.1. Motivation and Scope of the Research ......................................................................... 1

1.2. Organization of this Report .......................................................................................... 3

Chapter 2. Elemental Tests

2.1. General.......................................................................................................................... 4

2.2. Review of Research on Elemental Tests for Shear Bond and End Anchorages........... 4

2.3. Shear Bond Elemental Tests......................................................................................... 6

2.3.1. Specimen Description and Test Set Up............................................................. 6

2.3.2. Test Procedure................................................................................................... 10

2.3.3. Test Results ....................................................................................................... 10

2.4. End Anchorage Elemental Tests................................................................................... 13

2.4.1. Specimen Description and Test Set Up............................................................. 13

2.4.2. Test Procedure................................................................................................... 15

2.4.3. Test Results ....................................................................................................... 15

2.5. Concluding Remarks..................................................................................................... 17

Chapter 3. Strength and Stiffness Prediction of Composite Slabs by Simple

Mechanical Model

3.1. General.......................................................................................................................... 18

vii

3.2. Review of Methods of Prediction of Composite Slab Strength by Means of

Semi-Empirical Formulations and Simple Mechanical Models................................... 19

3.3. SDI-M Method.............................................................................................................. 27

3.4. Iterative Method............................................................................................................ 27

3.5. Direct Method............................................................................................................... 34

3.6. Comparison of Calculated and Test Results................................................................. 35

3.7 Concluding Remarks..................................................................................................... 40

Chapter 4. Strength and Stiffness Prediction of Composite Slab by Finite

Element Model

4.1. General.......................................................................................................................... 41

4.2. Review of Finite Element Method for Composite Slabs.............................................. 42

4.3. Finite Element Model ................................................................................................... 43

4.3.1. Structure Model................................................................................................. 43

4.3.2. Material Model.................................................................................................. 45

4.4. Method of Analysis....................................................................................................... 48

4.5. Results of Analysis and Discussion.............................................................................. 49

4.6 Concluding Remarks..................................................................................................... 52

Chapter 5. Long Span Composite Slab Systems

5.1. General.......................................................................................................................... 53

5.2. Construction Phase ....................................................................................................... 57

5.3. Service Phase................................................................................................................ 60

5.4. Specimen Description and Instrumentation.................................................................. 60

5.5. Load Test Procedure..................................................................................................... 64

5.6. Test vs. Analysis Results .............................................................................................. 65

5.7. Evaluation of the Floor Vibrations ............................................................................... 67

5.8. Proposed Detailed Connection ..................................................................................... 69


viii

Chapter 6. Reduction Factor, ϕ

6.1. General.......................................................................................................................... 71

6.2. Review of Probabilistic Concepts of Load and Resistance Factor Design .................. 71

6.2.1. Reliability Index................................................................................................ 72

6.2.2. AISC LRFD Approach for the Resistance Factor ............................................. 74

6.2.3. AISI LRFD Approach for the Resistance Factor .............................................. 75

6.3. Statistical Data.............................................................................................................. 76

6.3.1. Material Factor, M ............................................................................................ 77

6.3.2. Fabrication Factor F .......................................................................................... 78

6.3.3. Professional Factor, P........................................................................................ 79

6.3.4. Load Statistic..................................................................................................... 79

6.4. The Resistance Factor................................................................................................... 80


Chapter 7. Conclusions and Recommendations..............................................................83

References............................................................................................................................. 85

VITA..................................................................................................................................... 96

ix

LIST OF FIGURES

2-1. Profile shapes .............................................................................................................. 82-2. Embossment types....................................................................................................... 82-3. Shear bond test ............................................................................................................92-4. Shear bond specimen with frames for lateral force .................................................... 92-5. Shear stress vs. slip of specimen SB2-2-A.................................................................. 122-6. Shear stress vs. slip of specimen SB6-1-B.................................................................. 122-7. Details of the end anchorage specimens ..................................................................... 132-8. End anchorage test ...................................................................................................... 142-9. Load vs. deck to concrete slip of specimen EA1-1-B................................................. 162-10. Load vs. deck to concrete slip of specimen EA2-1-A................................................. 16

3-1. m and k shear bond regression line............................................................................. 203-2. Partial interaction theory (Stark and Brekelmans 1990)............................................. 223-3. Simplified relation between Mp ' and Nb (Stark and Brekelmans 1990) ................. 22

3-4. Boundary curve based on the partial interaction theory ............................................. 243-5. Free body diagram of the forces action in the composite slab section

(Patrick 1990, Patrick and Bridge 1994)..................................................................... 253-6. Plot of M u vs. T (Patrick 1990, Patrick and Bridge 1994)........................................ 263-7. Boundary curve for the ultimate bending moment capacity (Patrick 1990,

Patrick and Bridge 1994) ............................................................................................ 263-8. Reinforcing effects of some devices ........................................................................... 273-9. Forces acting on the cross section............................................................................... 283-10. Shear bond interaction ................................................................................................ 293-11. Concrete bottom fiber elongation, dL, and slip diagrams........................................... 313-12. Additional load carrying capacity from the deck........................................................ 323-13. Forces acting on the cross section for the direct method............................................ 343-14. Test setup .................................................................................................................... 373-15. Test vs. predicted strength .......................................................................................... 383-16. Load vs. mid-span deflection: (a) slab-4, (b) slab-15, (c) slab-21 .............................. 39

4-1. Schematic model of steel deck to concrete slip .......................................................... 444-2. Typical finite element model ...................................................................................... 444-3. Von Mises yield surface in the principal stress space ................................................ 454-4. Concrete failure surface in principal stress space....................................................... 464-5. Concrete uniaxial compressive stress-strain relation.................................................. 474-6. Typical shear bond shear stress vs. slip ...................................................................... 474-7. (a) Shear stud to steel deck interaction, and (b) puddle weld to steel deck

interaction....................................................................................................................484-8. General arc-length method.......................................................................................... 494-9. Slab-4: (a) Load vs. mid-span deflection. (b) Load vs. end-slip................................. 50

x

4-10. Slab-15: (a) Load vs. mid-span deflection. (b) Load vs. end-slip............................... 504-11. Slab-21: (a) Load vs. mid-span deflection. (b) Load vs. end-slip............................... 514-12. Composite slab strength: FEM vs. experimental ........................................................ 51

5-1. Prototype 1 and prototype 2 of Ramsden (1987) deck profiles .................................. 545-2. Innovative light weight and long-span composite floor (Hillman 1990,

Hillman and Murray 1994).......................................................................................... 545-3. Slimflor system (British Steel, Steel Construction Institute 1997)............................. 555-4. 6 in, 4.5 in and 3 in deep profiles................................................................................ 565-5. Yield strength and deflection limit states of the construction

(non-composite) phase ................................................................................................ 585-6. Steel deck weight vs. span length of single span systems........................................... 595-7. Steel deck weight vs. span length of double span systems ......................................... 595-8. System configuration of LSS1 and LSS2.................................................................... 615-9. Strain gage and shear stud schedules of LSS1............................................................ 625-10. Strain gage and shear stud schedules of LSS2............................................................ 635-11. Test set-up ................................................................................................................... 645-12. Map of cracks in LSS1................................................................................................ 655-13. Map of cracks in LSS2................................................................................................ 655-14. Load vs. mid-span deflection of LSS1........................................................................ 665-15. Load vs. mid-span deflection of LSS2........................................................................ 665-16. Normalized relative power vs. frequency of LSS1 ..................................................... 685-17. Normalized relative power vs. frequency of LSS2 ..................................................... 685-18. Proposed beam to girder connection to reduce slab-beam height............................... 70

xi

LIST OF TABLES

2-1. Test parameters ........................................................................................................... 72-2. Summary of shear bond test results ............................................................................ 112-3. Test parameters ........................................................................................................... 142-4. Summary of the end anchorage test results................................................................. 15

3-1. Test parameters ........................................................................................................... 363-2. Prediction vs. test results............................................................................................. 37

4-1. Finite element vs. test results ...................................................................................... 49

5-1. Ratios of actual load capacities and permissible load based onallowable deflection to 50 and 150 psf design live loads ........................................... 57

5-2. Section properties of profiles 1, 2 and 3 ..................................................................... 585-3. Summary of ultimate load capacity and permissible load based on

allowable deflection .................................................................................................... 67

6-1. β vs. pf ....................................................................................................................... 73

6-2. Statistical data of fc ' ,, f f and fy s,max s,min.............................................................. 77

6-3. Statistical data of t....................................................................................................... 786-4. Statistical data of P......................................................................................................796-5. Statistical data of dead and live loads......................................................................... 796-6. Calculated φ factors for SDI-M method (AISI-LRFD Approach) .............................. 816-7. Calculated φ factors for direct method (AISI-LRFD Approach) ................................ 816-8. Calculated φ factors for SDI-M method (AISC-LRFD Approach)............................. 816-9. Calculated φ factors for direct method (AISC-LRFD Approach)............................... 81

xii

LIST OF NOTATIONS

A bf = area of steel deck bottom flange / unit width of slab

A s = steel deck cross sectional area

A webs = area of steel deck webs / unit width of slab

a = depth of concrete stress block

=A fs y

085. ' f bc (Eqn.(3-6))

=F Fs st+

085. ' f bc (Eqn.(3-24))

b = section width

C = resultant of concrete compressive force

c = depth of the neutral axis of composite section

Dn = nominal value of dead load

d = distance of the steel deck centroid to the top surface of the slab (effective depth)

= length of each segment

dL,dLi = elongation of the bottom fiber of concrete slab of segment i

dLc = elongation of the segment at the mid-span

dc = deflection of the partially composite section

ds = deflection of the steel deck

Es = elastic modulus of steel deck

Eo, Esc = initial and secant modulus of concrete

e1, e e2 3, = moment arms of T1, T T2 3, (Eqn.(3-9))

F = minimum anchorage force (Chapter 3) = fA

Aywebs

bf A s − −

2

, (Eqn.(3-8))

= fabrication factor (Chapter 6)

Fm = mean of fabrication factor

xiii

Fs, Fst = tensile force in the steel deck resulted from the effect of shear bond and end

anchorages respectively

Fs it,lim = upper limit of Fs

fanchorage= stress in the steel deck induced by end anchorages

f bond = stress in the steel deck induced by shear bond force, f b

fc ' = concrete compressive strength

fc ',m = mean of concrete compressive strength = µ fc'

fcast = stress in the steel deck induced by concrete casting

fs = shear bond force per unit length

fshore = stress in the steel deck induced by shore removal

f fs,max s,min,

= maximum and minimum of fs

f t = concrete tensile strength

f w = stress in the steel deck induced by puddle welds

f y = steel deck yield stress

f yc = corrected steel deck yield stress due to concrete casting and shoring

fy* = remaining strength of the steel deck

f y,m = mean of steel deck yield stress = µ fy

f1 , f2 = elastic concrete compressive and tensile stress at the extreme fiber

hb = concrete depth above steel deck rib

h1 = depth of the concrete flange (concrete above steel deck rib)

Ieff = effective cross sectional inertia of the slab

Ii = effective cross sectional inertia of a segment

i = sequence number of a segment

L = span length of the slab

L’ = shear span length

Lc = cantilever length

xiv

L n = nominal value of live load

Ls = shear bond length

M = bending moment, general (Chapter 3)

= material factor (Chapter 6)

M et = first yield bending moment

M m = mean of material factor

M m SDI, , Mm,Direct

= means of material factor with regard to the SDI and Direct method, respectively

M nc , M nd = nominal moment capacity: phase-1 and phase-2, respectively

M p = steel deck plastic moment capacity

M u , Mn = nominal bending moment

m = bending moment caused by a unit load

Nb = k fc b h b' (Eqn.(3-3))

Nr = number of shear studs / unit width of slab

n = number of segment from the support to the mid-span

P = professional factor

Pm = mean of professional factor

pf = probability of failure

Qi , Qm = load effect, mean of load effect

Qn = nominal strength of single shear stud

q,qc,qd = load carrying capacity: total, phase-1, phase-2, respectively

R = reduction factor due to insufficient number of shear studs

to provide anchorage = N Q

Fr n

= support reaction

Rn, Rm = nominal resistance, mean of resistance

S = steel deck section modulus

si = total slip at a section

T = resultant of tensile force in steel deck

xv

T1, T T2 3, = forces acting in top flange, web and bottom flange of steel deck

t = steel deck thickness

t m = mean of steel deck thickness = µ t

ud1 = nodal displacement of steel deck beam element in d.o.f.-1 direction (horizontal)

uc1 = nodal displacement of concrete beam element in d.o.f.-1 direction (horizontal)

V, , V VR Q

= coefficients of variation: general, resistance, load effect

VM , , , , ,' V V V V VF P f f tc y

= coefficients of variation of: material, fabrication, professional factors,

concrete compressive strength, steel deck yield stress, steel deck thickness

VM SDI, , VM,Direct

= coefficients of variation of material factor with regard to the SDI and Direct

method, respectively

Vu = ultimate shear capacity

x xi, = distance from the support to the section being investigated

yc = horizontal projection of yd

yd = depth of deck c.g. from concrete c.g.

ys = horizontal slip of steel deck relative to the concrete

y y1 2, = moment arm of Fsand Fst , respectively

β = reliability index

ε ε, cu = concrete strain, concrete strain at the peak compressive stress

εs = steel deck strain

Φ = standard normal probability function

φ = design resistance factor

γ γD , L = dead and live load factors

γ i = design load factor

γD = correction due to diagonal shear cracking

κ = fraction of the support reaction, R in Eqn.(3-11)

xvi

λ λ λ, , R Q = log-normal mean: general, resistance, load effect

µ = coefficient of friction between the deck and concrete

µ fc' = mean of concrete compressive strength = fc ',m

µ fy = mean of steel deck yield stress = fy,m

µ t = mean of steel deck thickness = t m

θ = rotation of cross sectional plane (Chapter 4)

= central safety factor (Chapter 6)

ρ = reinforcement ratio = A bds /

σ σ σ, , P t = standard deviations: general, professional factor, steel deck thickness

τshear bond= shear bond strength

Ψ = γ γDn

nL

D

L+

+

/ 1.05

D

Ln

n1 , (Eqn.(6-18))

ζ ζ ζ, , R Q = log-normal standard of deviation: general, resistance, load effect

Ω i =

M m ds

M m dsi

L

∫

∫ (Eqn.(3-22))

Chapter 1 . Introduction 1

CHAPTER 1

INTRODUCTION

1.1. Motivation and Scope of the Research

Cold-formed steel decks have been widely used in composite slab systems in steel-

framed structures. The system has proven to be very attractive to structural designers because of

many advantages it has over conventional systems of reinforced concrete slabs. These

advantages have been listed by Finzi (1968), Oudheusden (1971), Hogan (1976), Porter and

Ekberg (1976), Fisher and Buettner (1979), Porter (1985), Wright et al. (1987), Evans and

Wright (1988). Among them, elimination or significant reduction of the positive moment

reinforcement and form work for concrete casting are two of the most important ones. This is in

contrast to the early use (before 1950) of the steel deck-concrete floor, where the concrete is used

only as a filling material (Dallaire 1971).

The knowledge of this composite interaction as well as elemental behavior involved in

the system has progressed rapidly during the past two decades. Much effort has been put forth to

better understand and model the behavior of the system. Research on the subject has been

conducted worldwide (U.S., Canada, Europe and Australia). Motivations for the research can be

summarized as follows:

1. To develop an efficient composite floor system that optimally utilizes the material and thus

yields an economical design.

2. To avoid the dependency on many full-scale experiments which are expensive and time

consuming.


3. To provide structural designers with analytical means by which they can verify design

calculations.

Efficient composite floor systems can be obtained by optimally utilizing materials, which

includes the possibility of developing long span composite slab systems. These long span

systems require investigation of new deck profiles that can be used to provide an adequate

interaction with the concrete slab. However, with the dependency of steel manufacturers on full-

scale slab tests, a substantial number of tests have to be performed to develop a new deck profile.

Therefore, from the manufacturer point of view, an alternative that can reduce the required

number of full-scale tests is desirable. This can be achieved by using analytical means supported

by elemental tests that are less expensive than the full-scale tests. Many kinds of analytical

means are now being made available due to development in the past decade, particularly in the

area of nonlinear analysis. By the same means, structural designers will have analytical tools to

cross-examine the design calculations. Current design formulations, such as the m and k method

(Schuster 1970, Porter et al 1976), do not sufficiently describe the physical behavior of

composite slabs. The only way structural designers can verify the design calculation based on

load tables generated by the m and k method is to look back into the experimental test results.

Depending upon the application, these analytical tools may range from a simple hand calculation

to a special purpose nonlinear finite element code.

As a continuation of on-going research in the area of composite slabs, with the same

motivations as mentioned above, this study has been conducted. New deck profiles, which

enable the deck to span longer than the typical spans currently used, are investigated. By

introducing a longer span floor system some filler beams can be eliminated along with their

connections to the girders. This results in more economical floor systems.

To establish a profile suitable for long spans, analytical models are developed to predict

the behavior of the new slab prior to any experimental tests. Two mechanical based models and

a finite element model are introduced. These models require knowledge of interaction properties

of some components of composite slabs. Hence, elemental tests for the shear bond and end

anchorages are performed. These analytical models, along with the elemental tests, offer an

alternate solution to the full scale tests that are required for the current design procedures.

Additionally, resistant factors, φ, for flexure design of composite slabs are also sought. The

current resistant factors, φ, for composite slab design (Standard for 1992) were taken from the


steel or concrete design specifications. Therefore, it is desired to obtain these factors based on

test results and refined analytical studies of composite slabs.

1.2. Organization of this report

This report is organized as follows. Following Chapter 1, elemental tests for shear bond

and end anchorages that were performed are described in Chapter 2. Results of these elemental

tests were used in the analytical methods of prediction for the composite slab strength and

stiffness using simple mechanical and finite element models that are presented in Chapter 3 and

Chapter 4, respectively. In these chapters, by using the afore-mentioned methods, predicted

strength and stiffness of experimentally tested composite slabs were compared to test results.

Chapter 5 discusses the investigation toward the long span composite slab systems. New deck

profiles are introduced for these long span systems and the methods described in Chapter 3 were

applied to predict the strength and behavior of the slab. In Chapter 6, φ factors for flexural

design of composite slabs are derived and discussed. Finally, conclusions and recommendations

for future research are presented in Chapter 7. Note that pertinent literature is reviewed in each

chapter.

Chapter 2 . Elemental Tests 4

CHAPTER 2

ELEMENTAL TESTS

2.1. General

Composite slab behavior is a function of interactions among the components of the slab.

Two of the most important interactions that significantly affect the slab behavior are: (1) the

shear bond interaction at the interface of steel deck and concrete and (2) the interaction among

the concrete, steel deck and end anchorages at the supports. Therefore, two types of elemental

tests were conducted in this study: shear bond and end anchorage. The purpose of these tests is

to study more closely the strength and behavior of shear bond interaction and end anchorages.

These tests will also provide interaction data required for the numerical analysis that will be

described in detail in Chapter 3 and Chapter 4. Elemental tests used in this study are similar to

the push-out and pull-out tests by Daniels (1988).

2.2. Review of Research on Elemental Tests for Shear Bond and End Anchorages

The shear bond, or m-k, method requires a substantial number of performance tests for

the shear bond regression line, plus additional flexure tests if flexural failure occurs within the

range of parameters tested. The problem becomes more pronounced with the recent findings of

other parameters that have significant impact on the strength of composite slabs, such as load

pattern, end anchorages and additional reinforcing bars. This finding drastically increases the

number of performance tests the manufacturers have to perform (Daniels and Crisinel 1987,

1993; Patrick 1990; Patrick and Bridge 1990; Patrick and Poh 1990; Bode and Sauerborn 1992).


This fact motivates research toward an alternative solution which can reduce the number of

performance tests required or replace them with smaller elemental tests that are less expensive.

Such elemental tests were set forth, such as the pull-out test (Daniels 1988; Daniels and Crisinel

1993; Sonoda et al. 1994), slip-block test (Patrick 1990; Patrick and Poh 1990), concrete-block

bending test (An and Cederwall 1992; An 1993), push-test (Veljkovic 1993), push-out test

(Tagawa et al. 1994). These elemental test results are to be used in the analysis for the slab

strength and stiffness. One may argue that these elemental test results may not directly represent

the actual behavior of the composite slabs because all the affecting parameters are inseparable

with each other, such as clamping force and curvature of the slab. Analytical models using shear

bond elemental test results, however, have shown good agreement with the full-scale test results,

which indicates that those elemental tests are applicable.

Another parameter that significantly affects the strength of composite slab systems is the

end anchorage. The presence of end anchorages over the support has a favorable effect on the

strength of the composite slabs because these end anchorages tend to block the relative slippage

of the concrete to the steel sheeting (Stark 1978; Crisinel et al. 1986a, 1986b; O’Leary et al.

1987; Jolly and Lawson 1992). End anchorages can be in one of the following forms: headed

shear studs welded through the deck to the supporting beams, hot rolled angles welded to the

beams, or cold formed members, such as pour stops. Porter and Greimann (1984) reported an

increase of 8% to 33% in composite slab strength when stud end anchorages are used.

The strength expression for the headed shear studs has been established by Ollgaard et

al. (1971) and has been used in the AISC Specification. This expression, however, was derived

in conjunction with composite beam design in which both the concrete and the steel deck slip

toward the same direction relative to the supporting beam. This is not the case with composite

slab action in which the concrete moves relatively to the steel sheeting. Therefore, elemental

tests for this type of end anchorages are of interest.

When a longitudinal slip occurs in the composite slabs, the steel deck is being pulled-out

from between the supporting beam and the concrete. The strength of the anchorage for the steel

sheeting is therefore a function of the sheeting strength and thickness, and also the clamping

force provided by the concrete and the steel beam due to the support reaction. Hence, elemental

tests for the end anchorage are needed to determine the force provided by the anchorage to the

steel deck. Very detailed and extensive elemental tests for the end anchorages were carried out


by Daniels (1988). The results from both the pull-out (shear bond) tests and the push-out (end

anchorage) tests were input to finite element analyses. The analytical results were reported to

compare favorably to the results of full-scale slab tests.

2.3. Shear Bond Elemental Tests

The shear bond interaction at the interface of steel deck and concrete can be separated

into three components, namely, the chemical bonding, mechanical interlocking, and friction

between the two materials. The first component is the type of bond that is developed through a

chemical process as the concrete cured. This component of interaction is brittle in nature, and

once it is broken it can not be restored. The mechanical interlocking gains its strength from the

interlocking action between the concrete and the steel deck due to the embossments. This action

is directly affected by the embossment shape and steel deck thickness. Finally, the presence of

the friction between the concrete and the steel deck is due to the presence of internal pressure

between the two materials. Unlike the first, the last two components are always present although

they may change in magnitude. The shear bond elemental tests were designed to obtain as much

information as possible about these three components.

2.3.1. Specimen Description and Test Set Up

The specimens were cast in a horizontal position so as to simulate the actual casting

position for a composite slab. The size of the specimen was made 1 ft wide by 2 ft long such that

it has at least one complete typical shape of the deck profile. To prevent the deck from being

bent during handling, which may result in the loss of the chemical bonding, each piece of deck

was fastened to a steel plate. Concrete cover above the deck was at least 2 in. to provide enough

bearing area for testing. After the concrete had cured, the specimens were coupled back to back.

Finally, banding strips were used to keep the concrete from falling off from the deck during

handling and storing. These strips were removed before the test.

Test parameters considered were concrete compressive strength, steel deck strength,

thickness, rib height, profile shape and embossment type as given in Table 2-1. The profile

shapes and embossment types that were used are illustrated in Fig. 2-1 and Fig. 2-2, respectively.

A single test frame was designed to handle both the shear bond and end anchorage tests.

The test set up for the shear bond test as shown in Fig 2-3, is intended to apply axial force, i.e., to


pull the steel deck out from the concrete. This axial force was applied through a ram that was

operated manually. The magnitude of the load applied was measured through a loading rod that

was instrumented with strain gages and calibrated as a tensile load cell.

Table 2-1. Test Parameters

ID# Concrete Steel Deck Internalfc' fy Thicknss Rib ht. Profile Emboss. Pressure

(psi) (ksi) (in) (in) Shape type (psf) **SB1-1 3850 50.3 0.031 2.00 1 3 500SB1-2 3850 50.3 0.031 2.00 1 3 300SB1-3 3850 50.3 0.031 2.00 1 3 100SB2-1 3850 45.4 0.034 2.00 1 1 500SB2-2 3850 45.4 0.034 2.00 1 1 300SB2-3 3850 45.4 0.034 2.00 1 1 100SB2-4 3850 45.4 0.034 2.00 1 1 300*SB2-5 3850 45.4 0.034 2.00 1 1 100*SB3-1 3850 46.5 0.047 2.00 1 2 500SB3-2 3850 46.5 0.047 2.00 1 2 300SB3-3 3850 46.5 0.047 2.00 1 2 100SB3-4 3850 46.5 0.047 2.00 1 2 300*SB4-1 4710 55.5 0.034 3.00 2 3 500SB4-2 4710 55.5 0.034 3.00 2 3 300SB4-3 4710 55.5 0.034 3.00 2 3 100SB5-1 4710 52.1 0.056 3.00 2 3 500SB5-2 4710 52.1 0.056 3.00 2 3 300SB5-3 4710 52.1 0.056 3.00 2 3 100SB6-1 4710 50.8 0.034 2.00 3 _ 500SB6-2 4710 50.8 0.034 2.00 3 _ 300SB6-3 4710 50.8 0.034 2.00 3 _ 100SB7-1 3840 48.2 0.056 6.00 4 _ 300SB7-2 3840 48.2 0.056 6.00 4 _ 100SB8-1 3840 49.6 0.057 4.50 5 _ 300SB8-2 3840 49.6 0.057 4.50 5 _ 200SB8-3 3840 49.6 0.057 4.50 5 _ 128

* initial pressure, no further adjustment** internal pressure at the interface of steel deck-concrete

For profile shapes and embossment types , refer to F ig. 2-1 and F ig. 2-2,respectively

For shear bond tests, a pair of additional frames is added to induced lateral force (Fig. 2-

4). The lateral force is applied by tightening the nuts in the rods. This lateral force is to simulate

internal pressure that is developed on the interface between the deck and the concrete. Load

cells were installed in the lateral frames, as indicated in Fig. 2-4, to measure the magnitude of the

lateral load applied.


Figure 2-1. Profile shapes (all dimensions are in inches)

Figure 2-2. Embossment types

type 1 type 2 type 3

57

12

2.5

2

4.757.25

12

2.5

3

1.5

0.5

6

12

2.5

2

9.25

3.75 7.125 1.5 0.5

1

3

6

12.875

9

1.5 9 1.125 0.375

1

2.5

4.5

12

(1)

(2)

(3)

(4)

(5)


Figure 2-3. Shear bond test

Figure 2-4. Shear bond specimen with frames for lateral force

load cell

nuts to adjustinternal pressure

specimen

hydraulic ram

tension rodinstrumented withstrain gages

clevis

specimenLoad cells


2.3.2. Test Procedure

The test was performed by applying lateral and axial forces simultaneously. The lateral

force is to produce internal pressure between the steel deck and concrete. The values of these

internal pressures are listed in Table 2-1. These internal pressures were obtained by adjusting

the nuts on the threaded rod of the lateral frame. The pressures were monitored from the load

reading obtained from the load cells placed in the lateral frame as shown in Fig. 2-5.

The axial force was applied by using a hydraulic ram. This axial force pulls the steel

deck out of the specimen, and therefore produces shearing stress on the interface between the

concrete and steel deck.

At the bottom side of the specimen, slip transducers were placed to measure the slip

between the concrete and the steel deck. The loads applied along with the corresponding slip

were recorded. The tests were stopped after 1 in. of slip was reached where a relatively constant

plateau was achieved.

2.3.3. Test Results

A summary of the test results is listed in Table 2-2. Plots of shear stress vs. slip for

specimen SB2-2-A and SB6-1-B are shown in Figs. 2-5 and 2-6. In specimen SB2-2-A, as shown

in Fig. 2-5, chemical bond can be observed as the vertical line at zero slip. At a shear stress level

of 6.86 psi, this chemical bond failed which caused a sharp drop in the shear stress value.

Beyond this point, the strength was due to the mechanical interlocking and friction. Some

specimens, however, did not show a clear chemical bond response. This has been caused by a

loss of chemical bond during handling.

The typical characteristic of this shear bond interaction is that after the loss of the

chemical bond, the strength increases until the ultimate (peak) shear stress value and it is

followed by a descending curve until it reaches a relatively long horizontal plateau at the end of

the descending curve. The ascending and the descending curve represent the action of the

mechanical interlocking, when the concrete tries to over-ride the embossment. In this action, the

steel deck stiffness that is characterized by the thickness and rib height plays an important role.

After the concrete completely over-rides the embossment of the deck, the resistant to the slip is

relatively constant, in which case a horizontal plateau is resulted.

In the case with un-embossed deck, the response is very brittle as shown in Fig. 2-6. A


horizontal plateau was obtained directly after the failure of chemical bond. This is because of

the absence of mechanical interlocking due to the lack of embossments. The plateau is due to

friction between the steel deck and concrete as observed from the test results that an increase in

the lateral pressure results in a higher shear bond strength, in particular, in the plateau portion of

the response. This fact is due to the friction between the two materials.

Table 2-2. Summary of shear bond test results

ID# Max. Shear Constant Slip at Max. Shearbefore slip (psi) after Shear(chemical bond) slip (plateau) (in)

A B (psi) (psi) A BSB1-1 6.87 8.10 8.25 5.95 0.160 0.102SB1-2 6.03 5.81 6.67 4.32 0.124 0.154SB1-3 7.07 7.07 6.23 2.25 0.208 0.041SB2-1 4.44 5.75 7.19 3.92 0.094 0.093

SB2-2 6.86 5.09 6.49 3.60 0.119 0.167SB2-3 2.85 3.22 5.02 2.03 0.159 0.067SB2-4 4.40 4.15 5.60 3.19 0.151 0.129SB2-5 4.87 2.87 4.80 2.73 0.085 0.142SB3-1 9.78 9.36 12.17 8.78 0.064 0.104SB3-2 9.88 6.81 11.92 6.10 0.062 0.166SB3-3 7.97 9.83 8.67 3.27 0.126 0.226SB3-4 7.07 6.02 10.45 6.21 0.094 0.103SB4-1 7.05 9.18 10.59 6.20 0.031 0.072SB4-2 6.91 7.03 8.79 3.70 0.057 0.057SB4-3 3.38 5.53 6.93 1.50 0.061 0.088

SB5-1 4.46 5.43 15.17 6.00 0.064 0.067SB5-2 3.88 8.06 15.44 4.50 0.078 0.095SB5-3 3.73 2.89 12.76 2.00 0.112 0.062SB6-1 11.59 10.54 10.54 6.50 0.102 0.004SB6-2 9.50 8.91 10.00 5.00 0.699 1.001SB6-3 7.48 7.38 7.38 5.80 0.005 0.216SB7-1 10.78 17.66 21.12 7.66 0.431 0.235SB7-2 17.99 12.49 17.99 8.70 0.195 0.476SB8-1 11.34 13.18 11.34 5.27 0.108 0.082SB8-2 11.65 11.40 11.40 4.06 0.065 0.106

SB8-3 10.30 14.53 15.55 3.99 0.004 0.204

A and B indicate the two specimen halves


0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

0.00 0.20 0.40 0.60 0.80 1.00

SLIP (in)

SH

EA

R S

TR

ES

S (

psi)

Figure 2-5. Shear stress vs. slip of specimen SB2-2-A

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

0.00 0.20 0.40 0.60 0.80 1.00

SLIP (in)

SH

EA

R S

TR

ES

S (

psi)

Figure 2-6. Shear stress vs. slip of specimen SB6-1-B


2.4. End Anchorage Elemental Tests

Three types of end anchorages were tested: headed shear studs, pour stops, and a

combination of the two.

2.4.1. Specimen Description and Test Set Up

Similar to the shear bond specimens, the end anchorage specimens were cast in a

horizontal position. The width of the specimens was 3 ft and the concrete cover above the deck

was at least 2 in. to provide enough bearing area for testing. Details of end anchorage tested are

illustrated in Fig. 2-7. In specimens EA2 and EA3, the deck was puddle welded to the beam and

fillet welds were used for the pour stop. After the concrete had cured, the specimens were

coupled back to back. Parameters of the tests are listed in Table 2-3.

For end anchorage tests, the shear bond test frame was used with a slight modification.

A pair of rods was used to pull the deck out from the specimens. Figure 2-8 shows the test set

up. A hydraulic ram, operated by an electric powered hydraulic pump, was put on top of the load

cells and an additional frame, as shown in Fig. 2-8, was added to hold the ram. A load beam,

made from a box section, was placed on top of the ram. In the space between the two specimens,

several displacement transducers were placed to measure the relative slip of the concrete to the

deck, and the deck to the beam.

Figure 2-7. Details of the end anchorage specimens

EA1 EA2 EA3

puddlewelds


Figure 2-8. End anchorage test

Table 2-3. Test Parameters

ID# Concrete Deck End No. of No. of Fillet Weldfc' fy Thicknss Emboss. Rib ht. Profile Anchor. Studs Puddle Welds on pour stop

(psi) (ksi) (in) Type (in) Type Type /side on deck*EA1-1 4050 45.4 0.034 2 2.00 1 S 2 _ _EA1-2 4050 45.4 0.034 2 2.00 1 S 2 _ _EA2-1 4050 45.4 0.034 2 2.00 1 PS 2 4 1" - 12"EA2-2 4050 45.4 0.034 2 2.00 1 PS 2 4 1" - 12"EA3-1 4050 45.4 0.034 2 2.00 1 P _ 4 1" - 12"EA3-2 4050 45.4 0.034 2 2.00 1 P _ 4 1" - 12"

End anchorage types: S=shear studs, P=pour stop, PS=pour stop and shear studsEmbossment and profile type, refer to Fig. 2-2 and 2-1, respectively* Puddle weld: 3/4" visible diameter

load beam

hydraulic ram

load cell

tension rod

specimen


2.4.2. Test Procedure

In this test, there was no lateral force applied to the specimens. The axial force from the

ram was incremented with an interval of 5 minutes to allow the system to settle. The load and

the corresponding slips were recorded and the test was stopped when failure occurred as

indicated by a consistently decreasing resistance to load.

As shown in Fig. 2-8, the ram pushes the load beam upward during the load test and the

two rods held by this beam will pull the steel deck out of the specimens. The concrete part of the

specimen is sustained by the frame.

2.4.3. Test Results

A summary of the test results is given in Table 2-4. Figure 2-9 and 2-10 show load vs.

deck to concrete slip for specimen EA1-1-B (shear stud end anchorages) and specimen EA2-1-A

(shear stud and pour stop). The failure mode in the later specimen is deck tearing around the

weld, which is typical for other specimens with deck welded to the beam. The shear studs in this

case do not give significant contribution to the strength because they were not welded through

the deck.

Table 2-4. Summary of the end anchorage test results

ID# Max. Load Computed Strength

per Stud or Stud Weld

Weld (k) (k) (k)EA1-1 10.45 26.59 _

EA1-2 9.90 26.59 _EA2-1 6.87 26.59 3.03

EA2-2 7.16 26.59 3.03EA3-1 5.86 _ 3.03

EA3-2 5.70 _ 3.03

In EA1 group of specimens, in which the studs were welded through the deck, the typical

response of load vs. slip shows relatively ductile plateau. The failure was due to steel deck

tearing and pilling in front and behind the studs, respectively. In EA2 group of specimens, the

fact that strength of the specimens was considerably lower than in the EA1 was because the studs

were not welded through the deck. Another cause was the relatively short distance of the steel

deck puddle weld to the end of the deck (1.5 in). Therefore, the behavior of EA2 specimens are

similar to those of EA3, where ductile plateau can not be maintained as soon as the deck tearing


propagates to the edge.

0.0

2.0

4.0

6.0

8.0

10.0

12.0

0.00 0.05 0.10 0.15 0.20 0.25 0.30

SLIP (in)

LOA

D (

kips

)

Figure 2-9. Load vs. deck to concrete slip of specimen EA1-1-B

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

0.00 0.05 0.10 0.15 0.20 0.25 0.30

SLIP (in)

LOA

D (

kips

)

Figure 2-10. Load vs. deck to concrete slip of specimen EA2-1-A


2.5. Concluding Remarks

Based on the results of the shear bond test, it can be concluded that the shear bond

strength is influenced by the internal pressure developed between the deck and the concrete. A

more accurate determination of the internal pressure will lead to a more accurate shear bond

strength prediction. This raises new issues on the relation of the internal pressure to the shear

bond strength as well as the determination of the internal pressure.

From the comparison shown in Table 2-4, it can be noted that the strength of the puddle

welds that were resulted from the tests are approximately double to the computed single weld

strength values (LRFD Cold-Formed, 1991). The strength of the anchorage by the shear stud,

however, is less than half of the single stud strength computed by using the AISC (1993)

specifications. In the first case, the higher strength was suspected due to the clamping effect on

the deck between the concrete and the steel beam. In the later case, the lower strength was

caused by the deck tearing rather than the stud shearing.

Chapter 3 . Strength and Stiffness Prediction - Simple Mechanical Model 18

CHAPTER 3

STRENGTH AND STIFFNESS PREDICTIONS OF

COMPOSITE SLABS BY SIMPLE MECHANICAL MODEL

3.1. General

One of the purposes of developing simple mechanical based methods for composite slab

strength is to provide tools suitable for design purposes. Methods based on this model have been

developed worldwide in the past two decades (Stark 1978, Patrick 1990, Stark and Brekelmans

1990, Heagler et al. 1991, Bode & Sauerborn 1992, Easterling and Young 1992, Patrick and

Bridge 1994). Despite the complex nature of interactions inside composite slab systems, the

methods have demonstrated good performance in predicting the slab strength. In contrast to the

so-called m-k method, these methods do not rely heavily on full-scale test results, which becomes

the main advantage of the methods.

In this study, two new methods based on simple mechanical model are developed. The

methods are based on partial connection theory. Unified formulation for the studded and non-

studded slabs and inclusion of shear bond strength at the steel deck-concrete interface offer

advancements to the SDI method (Heagler et al. 1991). In comparison to the method developed

by Patrick (1990), the remaining strength of the steel deck beyond the shear bond transfer

strength is considered. On the other hand, clamping forces at the supports are neglected due to

the fact that at the supports, the slab rests on the tip of the supporting beams.

The first of the two new methods is an iterative procedure, in which the slab strength is

calculated based on the location of the critical cross section, i.e., the location of the concrete

crack that initiates shear bond failure. With this method, the ultimate strength and response


history of the slab can be obtained. A computer program is required to perform the iterations.

The method is referred to as the iterative method.

The second method is one in which simple expressions are used in the formulation.

Thus, it is suitable for hand computation. The method is referred to as the direct method.

Along with these two new methods, a modified version of the SDI method is presented.

This method is referred to as the SDI-M method. The modifications include a corrected yield

stress due to concrete casting and omission of the shoring effect to the steel deck yield stress.

Modifications were introduced because the SDI method often yields unconservative results if the

casting stresses are not introduced and it may give very unconservative results if the shoring

stresses are included using the simple approach.

3.2. Review of Methods for Prediction of Composite Slab Strength by Means of

Semi-Empirical Formulations and Simple Mechanical Models

Although the use of cold-formed steel decks in the U.S. began as early as the 1920’s, the

standard design procedures for composite steel deck-concrete slabs were not formulated until

much later. A landmark research program that led to a design specification for composite slabs

was initiated in 1966 at Iowa State University (ISU) under the sponsorship of the American Iron

and Steel Institute (Ekberg and Schuster 1968; Porter and Ekberg 1971, 1972). The results of the

research led to design recommendations for composite slabs, which later became the basis for an

American Society of Civil Engineers design standard for composite slabs (Standard for 1992).

These design recommendations were based on two limit states, namely, the flexural and the shear

bond limit states. Determination of the slab strength based on shear bond requires a series of full

scale-tests.

The flexure limit state is characterized by the achievement of the flexural capacity, M u

(ASCE nomenclature), of the cross section at the maximum positive bending moment location,

although slip between the steel deck and concrete may occur anywhere in the slab including at

the end of the slab. The shear bond limit state is characterized by the occurrence of slip such that

it limits the capability of a section to reach its flexural capacity. Yielding of the steel deck

section, however, may occur prior to the failure.

The shear bond limit state was found to be the governing limit state in most composite

slab tests conducted at ISU, as well as in other research programs. The formulation of the design


method, which is commonly referred to as the m-k method, was chosen to follow the shear

equation from the ACI Building Code (Building Code 1995). The expression was developed by

Schuster (1970) and refined by Porter and Ekberg (1971). The equation for the limit state is

given by:

Vu = bd m d

L'+ k fc

ρ'

(3-1)

where Vu = ultimate shear capacity obtained from experimental test, b = unit width of the slab, d

= slab effective depth, measured from the compression fiber to the centroid of the steel deck,

ρ = A bds , L' = shear span length, fc ' = concrete compressive strength, A s = steel deck

cross sectional area per unit width, m and k are parameters shown in Fig. 3-1, obtained by

regression on the values obtained from full scale tests.

Figure 3-1. m and k shear bond regression line

k

REGRESSIONLINE

REDUCEDREGRESSIONLINE

k’

m

m’V

bd fu

c '

ρd

L f c '


Because shear bond was found to be the predominant failure mode of composite slabs,

the focus of recent research in this area has been to study more closely the behavior of this shear

bond action and to improve the performance of this action with or without adding other devices

such as end anchorages. Three components were identified in the shear bond action: chemical or

adhesion bonding, mechanical interlocking, and surface friction. The afore-mentioned m-k

method does not explicitly reflect the action of these components. To substantiate the effects of

these actions, tests have been performed and semi-empirical formulations have been developed

separately by Schuster and Ling (1980), Luttrell and Prasanan (1984), and Luttrell (1987a,

1987b).

The natures of those design procedures previously described are semi-empirical which

rely heavily on full-scale tests. This fact raises some problems as to how to incorporate more

parameters without significantly increase the number of full-scale test required and how to cross-

examine the design calculations analytically. In 1978, Stark introduced a partial interaction

theory similar to that used for composite beam design. The method was developed further by

Stark and Brekelmans (1990), in which they view the ultimate bending moment capacity of the

slab as built up from two components: (1) the contribution of the normal force of the steel sheet

and (2) the contribution of the reduced plastic moment Mp' of the deck. The formulation is

given by:

M d Mu p = Nb. '+ (3-2)

N h bb b = k. fc ' . . (3-3)

Mfpy

'.

= 1.25M 1-N

A Mp

b

sp

≤ (3-4)

where M u = ultimate bending moment capacity, Mp = steel deck plastic moment capacity, b =

slab unit width, fy = steel deck yield stress, d = repeat definition, k, and hb are explained in

Fig. 3-2. Equation (3-4) is a bi-linear simplification of a nonlinear relation between Mp' and


N b illustrated in Fig. 3-3.

Figure 3-2. Partial interaction theory (Stark and Brekelmans 1990)

Figure 3-3. Simplified relation between Mp' and N b

(Stark and Brekelmans 1990)

In 1991, the Steel Deck Institute (SDI) launched an alternative formulation to predict the

strength of composite slabs for design purposes (Heagler et al. 1991, 1992, 1997, Easterling and

Young 1992). These design procedures were based on research conducted at Virginia

Polytechnic Institute and State University and West Virginia University sponsored by the SDI.

k f c. '

M

Mp

p

'

fy

Mfpy

'= 1.25 M 1-N

A Mp

b

sp

≤

fy

hb

N

A fb

s y

N b

0.0

z

0.2

Na

0.4

M p '

0.6

0.8

1.0

0.2 0.4 0.6 0.8 1.0


The advantage of the SDI procedure is that the effect of end anchorages can be taken into

account in a simple manner. In the procedure, there is no distinction between ductile and brittle

behavior of the slab, however, it recognizes the studded and non-studded slab condition in which

generally, the studded shows ductile behavior and the non-studded sometimes has brittle

behavior. The nominal moment capacity is calculated based on the expression for a singly

reinforced concrete section, given by:

M = R.A . f d -a

2 n s y

(3-5)

where

a = A f

0.85f b

s y

c'

(3-6)

R Q

Fn=

Nr (3-7)

F A

Awebsbf= f Ay s − −

2

(3-8)

with M n = nominal moment, A s, fc ' , fy , b, and d are previously defined, N r = number of

studs per unit width of the slab, Qn = nominal shear stud strength, A webs, A bf = area of the

webs and bottom flange of the steel deck, respectively, per unit width of the slab. In the non-

studded slabs, the bending capacity of the slabs is predicted by using the moment at first yield,

which is given by:

( )M = T e T e T eet 1 1 2 2 3 3+ + (3-9)

where T1,T2 ,T3 are the total forces of the top flange, web and bottom flange of the deck,


respectively, and e e e1 2 3, , are the corresponding moment arms ofTi ’s to the centroid of the

compression side of concrete.

Linear interpolation between the full nominal moment capacity and the first yield

moment for slabs that do not have sufficient number of shear studs to provide full anchorage was

introduced to the method based on the research by Terry and Easterling (1994). With this

interpolation, the studded and non-studded cases can be unified.

Following the development by Stark and Brekelmans (1990), Bode and Sauerborn (1992)

developed a method based on the same partial interaction theory that can include the shear bond

effect explicitly. To determine the strength of a composite slab, a boundary curve of the slab

nominal bending moment resistance vs. the shear bond length for the particular slab for various

degree of partial interaction need to be generated (see Fig. 3-4). The expression for the shear

bond length is given by:

Ls = N

b.b

shear bondτ(3-10)

where Ls = shear bond length, N b = normal force developed in the concrete slab (see Fig. 3-4),

b = slab unit width, τshear bond = shear bond strength at the interface between the steel deck and

concrete. In this case, the shear bond strength is determined from full-scale composite slab tests.

Figure 3-4. Boundary curve based on the partial interaction theory(Bode & Sauerborn 1992)

M

LA LB Ls

Nb = 0

Nb Nb max

LA

LB


Patrick and Bridge (1990, 1994) developed a partial shear connection method, which is

also based on partial interaction theory. In this method, the effect of the end anchorages and

clamping forces over the support as well as the shear bond strength can be taken into account.

Similar to the ASCE procedure, the principle of a singly reinforced rectangular concrete section

is used to obtain the nominal bending moment, M n . The normal force, T, in the steel deck,

which can be viewed as the reinforcing force in a concrete section, can be determined from the

free body diagram shown in Fig. 3-5:

( )T = f x + L D R f As c y s− + ≤γ µκ (3-11)

where f s= shear bond force per unit length, x = distance from the support to the section being

investigated, Lc = cantilever length, γ D = correction due to diagonal shear cracking, µ =

coefficient of friction between the deck and concrete, R = support reaction, and κ = fraction of R

that has some contribution in T through a frictional action. With the T value calculated from

Eqn. (3-11), the corresponding M n value can be determined. However, because the shear bond

force varies along the slab, then a plot of M n vs. T (reinforcing force provided by the shear

bond, end anchorages, etc.) needs to be generated, as shown in Fig. 3-6, in order to form the

boundary curve for the slab load carrying capacity (Fig. 3-7). This concept is very similar to the

one introduced by Stark and Brekelmans (1990) (compare Fig. 3-6 to Fig 3-3) and Bode and

Sauerborn (1992) (compare Fig. 3-7 to Fig. 3-4). The critical section is then found by matching

up the boundary curve to the bending moment diagram due to the applied load, and the first point

to intersect with the bending moment capacity diagram is the critical location.

Figure 3-5. Free body diagram of the forces acting in the composite slab section(Patrick 1990, Patrick and Bridge 1994)

f s M

T T

MC

x

µκ R


Figure 3-6. Plot of M n vs. T (Patrick 1990, Patrick and Bridge 1994)

Figure 3-7. Boundary curve for the ultimate bending moment capacity(Patrick 1990, Patrick and Bridge 1994)

The procedure offers a good means that can take into account the shear bond and end

anchorage effect in the determination of the bending moment capacity based on the critical cross

T

δ = 0 0.

δ = 10.

M

Distance from the support, x

A

B

A

B

M n


section. In the procedure, however, the remaining strength in the steel deck, i.e. the reduced

plastic moment of the deck in the method by Stark and Brekelmans (1990), is omitted and on the

other hand, as can be noted from Eqn. (3-11), the clamping force at the support due to support

reaction is accounted for. In their research, the shear bond strength used for the procedure was

obtained from the slip block test instead of the full-scale tests.

3.3. SDI-M Method

The SDI-M method is a modified version of the SDI design procedure. All the equations

given by Eqns. (3-5) to (3-9) apply. The modifications are introduced by: (1) replacing fy

(original steel deck yield stress) in Eqns. (3-5) to (3-9) with fyc (corrected steel deck yield stress

due to concrete casting), and (2) omission of the construction shoring effect in the fyc , thus in

this case, the slab is treated as if it were unshored. Tests on shored composite slabs revealed that

unconservative predictions using the SDI method could be resulted when the shoring effect was

included in this simple model.

3.4. Iterative Method

The method utilizes a singly reinforced concrete beam section as the basis for the

approach. All effects that help the concrete resists cracking in the positive moment regime are

considered as reinforcement as indicated in Fig. 3-8. Such effects come from shear bond action

( f s), end anchorages (Fst ), reinforcing bars, etc.

Figure 3-8. Reinforcing effects of some devices

Two phases are considered in the analysis: phase-1, analysis of a composite cross section

in which the steel deck acts as a tensile member reinforcing the slab, and phase-2, analysis of the

steel deck as a flexural member. Phase-1 can be regarded as the composite action while phase-2

qc

Fstf s


as the non-composite action of the system.

In phase-1, analysis is performed exactly in the same manner as one treats a singly

reinforced concrete section. Two equilibrium equations are considered: equilibrium of forces

and equilibrium of moments on the cross section. Assumptions used in the procedure therefore

follow directly from the concrete beam section procedure, with one exception. Because in this

procedure one wants to obtain the response of the slab through the entire loading history, the

Whitney stress block (equivalent rectangular stress block) for the concrete is replaced by an

elasto-plastic model of the stress distribution. This is illustrated in Fig. 3-9 in which, Fs and Fst

are forces resulting from the effect of shear bond and end anchorages, respectively. Additional

effects of welds or pour stop can be added in a way similar to Fs and Fst .

Figure 3-9. Forces acting on the cross section

Two independent variables have to be solved to determine the stress distribution on the

cross section. In Fig. 3-9, c and f1 are chosen as the independent variables. They can be solved

from the two equilibrium equations on the cross section: equilibrium of forces and equilibrium of

moments. The magnitude of Fs and Fst , however, depends upon the value of slip between the

concrete and steel deck which in turn depends on concrete strain at locations where these two

forces are acting. The result is a nonlinear relation between Fs or Fst and the concrete strain,

such that c and f1are coupled together in a nonlinear system of equations. Therefore, an iterative

procedure is needed to solve for c and f1. The iterations are performed for each cross section for

a given load level. The greater the number of cross sections considered the more accurate the

prediction of the location of the critical section.

f1fc'

C c f( , )1

T c f( , )1

M

F c fs( , )1F c fst( , )1

ch1

ftf c f2 1( , )


The afore-mentioned shear bond force, Fs, is computed as follows. Consider the

schematic illustration of the shear bond interaction in Fig. 3-10. Figure 3-10a shows a typical

relation between shear bond force per-unit length, f s , versus slip at the interface of steel deck-

concrete. This relationship is obtained from elemental tests. In general, at a certain load level,

the distribution of fs along the slab is not uniform due to the difference in the amount of slip at

different cross sections. This is illustrated by different values of fs A, and fs B, in Fig. 3-10b.

The shear bond force, Fs, acting on a cross section is the sum of f s from the end of the slab to

the particular cross section (represented by the shaded area in Fig. 3-10b). Figure 3-10c shows

the distribution of Fs along the slab. In the case of high strength shear bond, Fs can not be

greater than the strength of the steel deck, f Ayc s.

Figure 3-10. Shear bond interaction

Partial interaction between the deck and the concrete is accounted for by limiting the

deck contribution to the capacity of the shear bond, such that after a certain phase, the steel deck

and concrete no longer have the same amount of strain at the interface. Hence, at any loading

point, strength contribution of the deck can not be greater than Fs as shown in Fig. 3-10c, so

that, as reinforcement for the concrete, the steel deck strength can be expressed as:

slip

fs diagram

(a)

(b)

(c)

Fs limitf Ayc s.

Fs

fs f s A,

fs A,

fs B,

fs B,


F = Fs s s,limitε . .E As s ≤ (3-12)

where Fs = shear bond force, εs, Es and A s are, respectively, the strain, elastic modulus and

cross sectional area of the steel deck, Fs it,lim = limitation on the shear bond force based on the

shear bond force per unit length vs. slip data obtained from the elemental tests. Note that Fs it,lim

for a cross section does not have a constant value through the loading history, rather, forms a

function of slip at that location. Once the maximum normal stress in the steel deck reaches a

value of F As it s,lim / , slip starts to occur. Again, Fs it,lim can not exceed the strength of the steel

deck, and hence we can state:

Fs it,lim f .Ayc s≤ (3-13)

with fyc as the corrected steel deck yield stress.

The effect of the end anchorages, Fst , can be obtained upon the determination of the slip

of the slab relative to the beam at the location of the anchorages, i.e., at the support. Slip values

can be obtained by summing the elongation of the bottom fiber of the concrete for each element

or segment from the mid-span to the support, neglecting axial deformation of the steel deck.

To this end, both shear bond and end anchorage forces require determination of slip

along the slab. This creates a problem because the slip is not known in advance. Two

approaches can be pursued to overcome the problem. One is to apply a forward iteration

scheme, in which, the analysis proceeds by utilizing the values obtained from the last convergent

state. These values might not be correct for the current state, however, the forward iteration

scheme does not require additional iteration. The second approach is to use a backward iteration

scheme. In this scheme an additional iteration loop is introduced inside the current iteration loop

for c and f1 . Computationally, the approach is expensive.

In this study, a forward iteration scheme is applied with an assumed distribution of

bottom fiber elongation of the concrete slab along the length to reduce error introduced by this

integration scheme. The actual distribution of this elongation will have a parabolic shape as

shown in Fig. 3-11b. A simplified distribution by using a linear distribution as shown in Fig. 3-


11d is used. In this case, the elongation of the bottom fiber of a segment located at xi from the

support can be written as:

dL dLi c = x

L / 2i (3-14)

in which, L = the span of the slab, dLi = elongation of bottom fiber of segment-i and dLc =

elongation of bottom fiber at the mid-span. Using Eqn. (3-14), the total slip at location xi can be

expressed as:

Figure 3-11. Concrete bottom fiber elongation, dL, and slip diagrams

( ) ( )s x xdL

L

dL

Li i nc c = dL = x = i + (i + 1)+...+n d i i

i=1

n+ + +∑ +1 2 2

.../ /

(3-15)

where si = slip at location xi , n = total number of segments from the support to the mid-span, i

= sequence number of segment counted from the support, and d = length of each segment.

Substituting Eqn. (3-14) into Eqn. (3-15) for dLc , and replacing ( )i i n+ + + +( ) ...1 in Eqn. (3-

L

L+dL

(a)

(b)

(c)

(d)

dL diagram

slip diagram

simplifieddL diagram

d

L/2 dLidLc

xi


15) by ( )[ ]1 2 1 2 1+ + + − + + + −... ) ( ... ( )n i , the slip at a cross section can be expressed in terms of

elongation of that particular segment as follow:

si i

idLi i =

n(n + 1)

2− −

( )1

2

1(3-16)

In phase-2 of the analysis, the remaining strength of the deck beyond its strength that has

been used for shear bond transfer is considered. This strength of the deck contributes additional

load carrying capacity and it is assumed that this action occurs through a non-composite type of

action. For this purpose, a deflection compatibility condition is assumed between the deck and

the concrete as illustrated in Fig. 3-12:

Figure 3-12. Additional load carrying capacity from the deck

d = ds c (3-17)

in which, ds = steel deck deflection, and dc= composite slab deflection. Additional strength

stemming from phase-2 of the analysis is contributed from the flexural strength of the deck and it

can be significant. The stress developed in the steel deck in conjunction with this additional

strength, however, can not be greater than the remaining strength available in the steel deck given

by:

f = f f f f - fy*

y cast shore bond anchorage− − − − f w (3-18)

qc

qd

dc

ds

(a) composite action

(b) non-composite action


where fcast, fshore, f bond, fanchorage, f w = stress in the steel deck induced by concrete casting,

shore removal, Fs (shear bond force), Fst (end anchorage force), and weld force, respectively. If

qd denotes the additional load carrying capacity, then the total load carrying capacity is simply:

q = q + qc d (3-19)

in which qc = load carrying capacity from phase-1 of the analysis (partially composite action).

Beyond this value, the deck is yielded and it deforms plastically without adding any contribution

on the load capacity.

Deflection of the slab can be computed simultaneously with the strength calculation. In

this part of analysis, however, there are additional assumptions required. The modulus of

elasticity of concrete is assumed unchanged and equal to its initial value, even though the

concrete is in an inelastic state in certain cross sections. Similar to the strength procedure, the

portion of the concrete stressed beyond the tensile stress limit is considered to be ineffective.

Therefore, the cross sectional inertia of the concrete varies along the slab. The contribution of

steel deck stiffness to the slab stiffness is proportional to the degree of interaction between the

deck and the concrete. This degree of interaction is represented by the ratio of steel deck stress

to the corrected steel deck yield stress at the beginning of the analysis (after concrete casting and

shore removal). With this, the slab will have a non-prismatic effective cross section. The

deflection can then be computed by utilizing the unit load method for which the integration can

be performed numerically. The effective cross sectional inertia can be computed from:

δ = ds = ds + ds + ... + dsL 1 2 nMm

EI

M m

EI

M m

EI

M m

EIeff

n n

n∫ ∫ ∫ ∫1 1

1

2 2

2(3-20)

where δ is the mid-span deflection of the slab, M and M i ’s are moment functions along the slab

and at segment-i, respectively, due to the applied load, m and mi are moment functions along the

slab and at segment-i, respectively, due to a unit load at the mid-span, I i is the effective inertia

of segment-i and I eff is the average of the effective inertia of the slab. By assuming that the

cross sectional inertia does not vary within each segment, then Eqn. (3-20) can be reduced to:


1 1

1

2

2I I I Ieff

n

n= + + +

Ω Ω Ω... (3-21)

where

Ω ii

L

M m ds

M m ds=

∫

∫(3-22)

with i∫ = integration over the segment,

L∫ = integration over the entire length of the slab, M =

bending moment function along the slab, and m = weighting function (bending moment caused

by the unit load).

3.5. Direct Method

The direct method shares the same basic concept as the iterative method. In fact, the

direct method is just one point, namely the ultimate load point of the iterative analysis, therefore,

all assumptions of the iterative method are applicable. In this case, a fully plastic condition of

the cross section is assumed and the Whitney stress block for the concrete is utilized. The stress

distribution is illustrated in Fig. 3-13.

Figure 3-13. Forces acting on the cross section forthe direct method

The main advantage of the direct method is that the procedure of computation is non-iterative,

thus it is convenient for hand computation. The effects of shear bond and end anchorages can

085. ' fc

y1y2

C

Fst

Fs

M


still be taken into account. Partial interaction between the deck and concrete is also considered

as in the iterative procedure. The nominal moment capacity provided by the composite action of

the steel deck and the concrete is expressed as:

M = F y + F ync s 1 st 2 (3-23)

where y1, y2 = the moment arm length of Fs and Fst , respectively, to the center of the

compressive stress block. The depth of the stress block is obtained from:

a = F F

0.85f bs st

c'

+(3-24)

Equation (3-23) constitutes phase-1 of the analysis. Phase-2 of the analysis, the effect of the

flexural deck strength, is given by:

M = f Snd y* (3-25)

where fy* = the remaining deck strength, defined by Eqn. (3-18), and S = section modulus of the

steel deck. In contrast to the iterative method, the response history of the system can not be

obtained. The result only gives the nominal moment capacity. From Eqns. (3-23) to (3-25), it

can be noted that there is no distinction in the formulations whether the slab is studded or not.

The fact that the steel deck strength is limited to the shear bond action in the composite action

(phase-1) and the inclusion of the remaining strength of the deck represent a more realistic

physical interaction in composite slab. This gives a more accurate account for the changes in

steel deck strength such as shoring effect during the construction, etc.

3.6. Comparison of Calculated vs. Test Results

Predicted values of the slab strength were made by using the iterative, direct and SDI-M

methods. They were compared to experimental results. The tests were performed using several

different deck profiles, embossment patterns and steel thicknesses. Different span lengths, slab


depths, end anchorages and concrete strengths were also incorporated in the tests. The width of

the specimens was 6 ft. Loading was applied through an air bag to the top surface of the concrete

slab to produce a uniformly distributed load. The test setup is shown in Fig. 3-14. Table 3-1 lists

main parameters of the specimens and computed values using previously described methods are

listed in Table 3-2. Test data are taken from Terry and Easterling (1994), and Widjaja and

Easterling (1995, 1996).

Table 3-1. Test parameters

SLAB DECK RIB STEEL EMBSM. OVER- SPAN END TOTAL DECK SHORING CONCR# PROF. HT. THCK. TYPE HANG LENGTH ANCHR. DEPTH CONT. fc' (in) (in) (ft) (ft) TYPE (in)1 1 2 0.0345 1 1 9 S-5 4.5 C N 31802 1 2 0.0345 1 1 9 S-4 4.5 C N 31803 1 2 0.0345 1 1 9 S-3 4.5 C N 51704 1 2 0.0345 1 1 9 S-2 4.5 C N 51705 1 2 0.0345 1 1 9 W-7 4.5 C N 33406 1 2 0.0345 1 _ 9 W-7,P 4.5 C N 33407 1 2 0.0345 1 1 9 W-7 4.5 D N 37708 1 2 0.0345 1 _ 9 W-7,P 4.5 D N 37709 1 2 0.0470 2 1 9 S-3 4.5 C N 5300

10 1 2 0.0470 2 1 9 S-5 4.5 C N 530011 2 3 0.0355 3 1 10 S-3 5.5 C N 375012 2 3 0.0355 3 1 10 S-5 5.5 C N 375013 2 3 0.0355 3 1 10 W-7 5.5 D N 337014 1 2 0.0470 2 1 9 W-7 4.5 D N 337015 3 2 0.0335 _ 1 9 S-3 5.0 C Y 380016 3 2 0.0335 _ 1 9 S-6 5.0 C Y 380017 3 2 0.0335 _ 1 13 S-4 6.0 C Y 278018 3 2 0.0335 _ 1 13 W-6 6.0 D Y 278019 2 3 0.0339 3 1 9 W-7 5.5 D Y 390020 2 3 0.0339 3 1 9 W-7 5.5 D N 390021 2 3 0.0558 3 1 12 W-7 5.5 D Y 512022 2 3 0.0558 3 1 12 W-7 5.5 D Y 455023 2 3 0.0558 3 1 12 W-7 5.5 D N 455024 4 6 0.0560 _ 1 20 S-6 8.5 D N 307025 4 6 0.0560 _ 1 20 S-6 8.5 D N 307026 5 4.5 0.0570 _ 1 20 S-6 7.0 C N 233027 5 4.5 0.0570 _ 1 20 S-6 7.0 C N 2330

Note* End anchorages: S=stud, P=pour stop, W=puddle weld* The number following S and W is the number of studs or welds installed* Deck continuity: C=continuous over the support, D=discontinuous* Deck profiles and embossment types: refer to Fig. 2-1 and 2-2, respectively

From Table 3-2, it can be observed that the iterative and direct methods predicted the

capacity of the slab reasonably well. The SDI-M method tends to give conservative predictions.

A graphical comparison of the test vs. predicted strengths using the iterative and direct methods

are shown in Fig. 3-15.

A comparison of the experimental and iterative method response histories for slab-4

(studded slab with trapezoidal deck profile), slab-15 (studded slab with re-entrant deck profile)


and slab-21 (non-studded slab) are shown in Fig. 3-16.

Figure 3-14. Test Setup

Table 3-2. Prediction vs. test results

SLAB SDI-M ITER. DIRECT TEST RATIO OF SLAB SDI-M ITER. DIRECT TEST RATIO OF# TEST/PREDICTED # TEST/PREDICTED psf psf psf psf SDI-M ITER. DIRECT psf psf psf psf SDI-M ITER. DIRECT1 503 673 755 730 1.45 1.08 0.97 15 1153 798 908 1017 0.88 1.27 1.122 503 637 657 700 1.39 1.10 1.07 16 1158 1218 1256 1185 1.02 0.97 0.943 521 633 657 600 1.15 0.95 0.91 17 627 411 572 565 0.90 1.37 0.994 515 507 519 600 1.16 1.18 1.16 18 353 251 352 368 1.04 1.47 1.055 351 366 337 490 1.40 1.34 1.45 19 507 515 533 523 1.03 1.01 0.986 348 510 534 590 1.69 1.16 1.11 20 507 495 478 523 1.03 1.06 1.097 297 346 321 375 1.26 1.08 1.17 21 425 510 486 467 1.10 0.91 0.968 293 487 519 490 1.67 1.01 0.94 22 422 421 485 494 1.17 1.17 1.029 751 766 802 900 1.20 1.18 1.12 23 422 421 431 507 1.20 1.20 1.18

10 753 1008 1060 900 1.20 0.89 0.85 24 476 638 654 621 1.30 0.97 0.9511 595 672 658 750 1.26 1.12 1.14 25 476 638 654 559 1.17 0.88 0.8512 595 876 881 870 1.46 0.99 0.99 26 294 445 464 498 1.69 1.12 1.0713 357 443 388 480 1.34 1.08 1.24 27 294 445 464 455 1.54 1.02 0.9814 461 457 528 500 1.08 1.09 0.95

AIR BAG


0

200

400

600

800

1000

1200

1400

0 200 400 600 800 1000 1200 1400

ITERATIVE (psf)

TE

ST

(ps

f)

-15%

15%

non-studded

studded

0

200

400

600

800

1000

1200

1400

0 200 400 600 800 1000 1200 1400

DIRECT (psf)T

ES

T (

psf)

-15%

15%

non-studdedstudded

0

200

400

600

800

1000

1200

1400

0 200 400 600 800 1000 1200 1400

SDI-M (psf)

TE

ST

(ps

f)

-15%

15%

non-studdedstudded

Figure 3-15. Test vs. predicted strength


0

100

200

300

400

500

600

0.0 1.0 2.0 3.0 4.0 5.0

MID-SPAN DEFLECTION (in)

LOA

D (

psf)

test

iterative analysis

0

200

400

600

800

1000

1200

0.0 1.0 2.0 3.0 4.0 5.0


LOA

D (

psf)

test

iterative analysis

0

100

200

300

400

500

0.0 1.0 2.0 3.0 4.0 5.0


LOA

D (

psf)

test

iterative analysis

Figure 3-16. Load vs. mid-span deflection: (a) slab-4, (b) slab-15, (c) slab-21

(a) (b)

(c)



From the comparison and discussion presented, it can be concluded that the iterative and

direct methods generally predict the slab strength reasonably well. The methods are simple to

perform and because they are based on a mechanical model rather than an empirical one, they are

able to take into account parameters such as shear bond interaction and end-anchorages.

Therefore, the methods can offer an alternate solution to performing full size slab tests. The

SDI-M method, while not as accurate, provides a conservative design approach that is very

simple to apply.

Chapter 4 . Strength and Stiffness Prediction - Finite Element Model 41

CHAPTER 4

STRENGTH AND STIFFNESS PREDICTIONS OF

COMPOSITE SLABS BY FINITE ELEMENT MODEL

4.1. General

Successful use of the finite element method in many studies involving complex

structures or interactions among structural members has been one of the motivations for applying

the method in this study. To compare with simple mechanical models discussed in the previous

chapter, finite element models may offer more accurate analyses because of the ability to model

the material and interaction of each part of the system in more detail. Further, the response

history of virtually any part of the model can be obtained. In this method, element and material

model types play an important role for the entire analysis. Selection of element and material

model types for the analysis is based on the structural system and any specific need or emphasis

of the study.

In this study, because the main concern is behavior of one-way composite slabs with a

large ratio of length to the cross sectional dimensions in a typical width of the slab, then the

choice of beam and spring elements for a finite element model is the most effective one. The

model is similar to the one proposed by An (1993) with modifications such as the inclusion of

end anchorages and a concrete fracture model for concrete in tension. With this concrete fracture

model, the mesh sensitivity problem in finite element analysis involving concrete (brittle)

material can be removed (Fracture 1992; Karihaloo 1995). Descending curves of end anchorages

and shear bond interaction are also included. ABAQUS is used to conduct the analyses.


4.2. Review of Finite Element Method for Composite Slabs

Due to the complex nature of interactions within composite slab systems, finite element

modeling has become a powerful tool in predicting the slab strength and stiffness. The power

rests in the ability to locally model each different part or interaction of the system and

systematically integrate contributions of those parts or interactions to represent the whole system.

For composite slabs, various models have been proposed. The selection of model types depends

on the physical system of the slabs and specific need of the study.

Daniels et al. (1989, 1990), Ren and Crisinel (1992) and Ren et al. (1993) used plane-

beam elements to model one-way composite slabs. Special ten-degree of freedom beam elements

that can take into account nonlinear slip behavior between the steel deck and concrete slab was

used. For this purpose, a special finite element code was developed.

By using ABAQUS, a commercial general-purpose finite element code, two-node plane

Timoshenko beam elements were used by An (1993) for one-way slab systems. Two series of

beam elements were generated, one for the concrete slab and the other for the steel deck. Shear

bond interaction was modeled by using series of spring elements and additional set of equations

to the stiffness equations to prescribe imposed relations among the degree of freedoms of the

spring, concrete beam and steel deck beam nodes.

Three dimensional brick elements were used for a two-way composite slab system.

Some difficulties concerning numerical convergence was reported in the 3D model (An 1993).

The problem was thought to be due to mesh sensitivity in relation to the tension-stiffening model

of the concrete material. Because of this problem, the concrete material model was replaced by

two different J2 (metal) plasticity models, each of which is representing the tension and

compression parts of the concrete separately.

Other 3D models using brick elements were proposed by Veljkovic (1993, 1994, 1996)

and Oloffsson et al. (1994). In their study, DIANA, a general-purpose finite element code was

used. It was reported that some trials for concrete tension stiffening functions were needed in

some cases before a stable numerical result can be obtained.


4.3. Finite Element Model

4.3.1. Structure Model

A simply supported beam configuration is chosen as a typical model of the system. In

the case with continuous deck over an interior support, a rotational spring element is added to the

continuous end. The stiffness of this spring represents an elastic rotational stiffness of the

adjacent span at the common support. This type of configuration (simply supported beam) is

based on observations during experimental tests. Because of the absence of negative

reinforcement over interior supports, negative cracks along these supports were developed at a

relatively low load level. Therefore, the assumption that the concrete slab is discontinuous over

the interior support will not have any significant effect to the analysis.

Two series of Euler-Bernoulli beam elements with 12 in. typical length were generated.

One series is for the concrete slab and the other is for the steel deck. Only a single typical

longitudinal slice of the slab is considered in the model. Vertical nodal displacements of these

two series of beam elements are forced to be the same. This is based on previous study (An and

Cederwall 1994) which concluded that the effect of vertical separation between the two parts is

minor.

End anchorages and shear bond interactions at the interface of the concrete and steel

deck are modeled by using horizontal spring elements. In the case of the shear bond interaction,

the spring elements are placed along the slab. One end of each spring element is attached to the

steel deck beam element and the other to the concrete beam element. Both are at the steel deck

centroid elevation. This means that the attachment of the spring elements to the concrete beam

element is not at the centroid of concrete beam elements. In ABAQUS, this can be assigned by

using the EQUATION option in which the magnitude of a certain degree of freedom can be made

equal to scalar multiplications of any other degree of freedoms. This compatibility condition is

shown schematically in Fig. 4-1 and can be expressed as:

y = y sin y c d dθ θ≅ (4-1)

y = u u + y s 1d

1c

d− θ (4-2)


Figure 4-1. Schematic model of steel deck to concrete relative slip

where yc = horizontal projection of yd , yd = depth of deck c.g. from concrete c.g., θ = rotation

of cross sectional plane, ys= horizontal slip of steel deck relative to the concrete, u1d = nodal

displacement of steel deck beam element in d.o.f.-1 direction (horizontal), u1c = nodal

displacement of concrete beam in d.o.f.-1 direction (horizontal).

For end anchorages, spring elements are placed at the supports to produce resistance to

horizontal movements of the concrete slab and steel deck relative to the support. The spring is

attached to the bottom surface of the deck. A schematic diagram of the model is shown in Fig. 4-

2.

Figure 4-2. Typical finite element model

CONCRETE

STEEL DECK

SHEAR BOND

STUD-CONCRETEINTERACTION

STUD-DECKINTERACTION

WELD-DECKINTERACTION

IMPOSED EQUATIONFOR HORIZONTAL SLIP

c.g.c

IMPOSED EQUATIONFOR VERTICAL DSPL.

c.g.sys

ud1

uc1

yc

ydθ

Planeof

reference


4.3.2. Material Model

Incremental plastic flow theory is applied for the steel and concrete materials whereas

nonlinear elasticity theory is applied for end anchorages and shear bond interaction. J2 -

plasticity (von Mises) with associative flow rule is used for the steel material of the steel deck.

In this case, the yield surface is independent of the hydrostatic component of the stress vector as

shown in Fig. 4-3. Although top flange buckling at the maximum positive moment region was

observed in some specimens, no buckling is assumed in the model.

Figure 4-3. Von Mises yield surface in the principal stress space

The concrete material on the other hand, is pressure dependent. The general shape of

failure surface for concrete material is illustrated in Fig. 4-4. ABAQUS uses the Drucker-Prager

failure surface, a two-parameter model, for concrete material (Drucker and Prager 1952). This

model is valid only for problems with low confining pressures (Hibbitt 1987). For a high

confining pressure, many finer models of concrete failure surfaces are available, such as the

Ottosen four parameter model (Ottosen 1977), Hsieh-Ting-Chen four parameter model (Hsieh et

al. 1982), Willam-Warnke five parameter model (Chen 1982), etc. The Drucker-Prager model,

however, is sufficient for one-way composite slabs. Moreover, because of the conical shape of

the failure surface, singularity is only at the apex. Multi-vector return stress based on Koiter’s

(1953) approach is a common method to handle such singularity. Other methods such as a

multiple single vector return (Widjaja 1997b) may improve the accuracy of the former method.

Recent developments in the application of fracture mechanics to concrete, in particular,

failure surface

π plane

hydrostatic axis

σ2

σ3


concrete in tension, enabled a fracture mechanics model to be used for the tensile portion of the

concrete. This model can avoid the mesh sensitivity effect of a tension-stiffening model.

Further, in this study, an associative flow and isotropic work hardening rule is assumed.

Figure 4-4. Concrete failure surface in principal stress space

The uniaxial stress-strain relation for concrete in compression is modeled using the

Saenz equation up to the peak value (Saenz 1964). This model has been successfully used by

Razaqpur and Nofal (1990) to model a composite bridge. The expression of Saenz equation is

given by:

σε

εε

εε

= E

E

E

o

o

sc cu cu1 2

2

+ −

+

(4-3)

where σ and ε are the stress and the corresponding strain of the concrete respectively, Eo and

Esc are the initial and the secant modulus of elasticity, respectively, εcu = concrete strain at the

peak compressive stress. The descending branch of concrete-stress-strain relation is omitted in

this beam model configuration to preserve stability of the system when compressive strength of

concrete is approached. Figure 4-5 shows the concrete stress strain relation.

failure surface

π plane

deviatoric plane

hydrostatic axis

σ2

σ3

σ1

ε

ρθ


Figure 4-5. Concrete uniaxial compressive stress-strain relation

The backward Euler integration scheme is used in the plastic analysis. The scheme

assumes that the return of the stress state to the yield surface is normal to the final yield surface

(note that the yield surface keeps changing to follow the work hardening rule when plastic flow

occurs).

Finally, a nonlinear elastic model is used to model end anchorages (welds or shear studs)

and shear bond interaction. The force-displacement relation of these end anchorages and shear

bond interactions were obtained from elemental tests as presented in Chapter 2. Typical shear

bond interaction is shown in Fig. 4-6 and typical shear stud to steel deck and puddle weld to steel

deck interactions, respectively, are shown in Figs. 4-7(a) and (b).

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

0.0 0.2 0.4 0.6 0.8 1.0

SLIP (in)

SH

EA

R S

TR

ES

S (

psi)

ACTUAL

SIMPLIFIED

Figure 4-6. Typical shear bond shear stress vs. slip

σ

εεcu

Esc

Eo


0.0

2.0

4.0

6.0

8.0

10.0

12.0

0.0 0.1 0.2 0.3 0.4

SLIP (in)

FO

RC

E (

kips

)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.0 0.2 0.4 0.6 0.8

SLIP (in)

FO

RC

E (

kips

)

Figure 4-7. (a) Shear stud to steel deck interaction,and (b) puddle weld to steel deck interaction

4.4. Method of Analysis

Among the three sources of nonlinearity: material, geometrical and boundary, only the

first two are applicable to composite slab problems in this study. Both the material and

geometrical nonlinearity were applied in the analyses. The integration scheme used to trace the

equilibrium path was the arclength method with a cylindrical constraint surface as suggested by

Crisfield (1981). The cylindrical constraint surface converges much faster than the general

spherical one. Despite the problem with inconsistency in the physical units used in its constraint

equations (Yang and McGuire 1985; Chen and Blandford 1993), no serious problem related to

this inconsistency was reported. However, Widjaja (1997a) shows that the method is sensitive to

the selection of physical units used. A choice of units that make the order of magnitude of each

d.o.f. type (rotation, translational, etc.) comparable may improve the performance of the method.

Other problems were indicated by Carrera (1994), such as no convergence due to a relatively

large load step, very slow or no convergence due to oscillation near the equilibrium path, or no

real root that satisfies the constraint surface. These later problems can be overcome by avoiding

the use of large step sizes. Figure 4-8 illustrates the method with a general constraint surface.

(a) (b)


Figure 4-8. General arclength method

4.5. Results of Analysis and Discussion

Finite element analyses have been performed to simulate composite slab tests and results

are listed in Table 4-1. Parameters of each slab specimens are listed in Table 3-1. Among the

analysis results, load vs. mid-span deflection and load vs. end-slip response histories of slab-4

(studded slab with trapezoidal deck profile), slab-15 (studded slab with re-entrant deck profile)

and slab-21 (welded slab, shored during the construction) are shown in Figs. 4-9, 4-10 and 4-11,

respectively.

Table 4-1. Ultimate slab capacity: finite element vs. test results

SLAB FEM TEST RATIO SLAB FEM TEST RATIO

# TEST/ # TEST/ psf psf FEM psf psf FEM1 627 730 1.16 15 985 1017 1.032 617 700 1.13 16 1037 1185 1.143 577 600 1.04 17 506 565 1.124 543 600 1.10 18 264 368 1.40

5 480 490 1.02 19 537 523 0.976 565 590 1.04 20 496 523 1.057 293 375 1.28 21 456 467 1.038 480 490 1.02 22 441 494 1.129 775 900 1.16 23 408 507 1.24

10 790 900 1.14 24 534 621 1.1611 733 750 1.02 25 534 559 1.0512 799 870 1.09 26 353 498 1.4113 409 480 1.17 27 353 455 1.2914 364 500 1.37

equilibrium path

λ = load proportionality factor

displacement, u

load, F

FnλF1λ

nu

1u

∆L


0

100

200

300

400

500

600

0.0 1.0 2.0 3.0 4.0 5.0


LOA

D (

psf)

testfinite element

0

100

200

300

400

500

600

0.00 0.10 0.20 0.30 0.40

END SLIP (in)

LOA

D (

psf)

test

finite element

Figure 4-9. Slab-4: (a) Load vs. mid-span deflection. (b) Load vs. end-slip

0

200

400

600

800

1000

1200

0.0 1.0 2.0 3.0 4.0 5.0


LOA

D (

psf)

test

finite element

0

200

400

600

800

1000

1200

0.00 0.10 0.20 0.30 0.40 0.50

END SLIP (in)

LOA

D (

psf)

testfinite element


(a) (b)

(a) (b)


0

50

100

150

200

250

300

350

400

450

500

0.00 0.02 0.04 0.06 0.08

END SLIP (in)

LOA

D (

psf)

testfinite element

0

100

200

300

400

500

0.0 1.0 2.0 3.0 4.0 5.0


LOA

D (

psf)

test

finite element


Figure 4-12 shows graphical comparison of predicted vs. test values of slab strength. It

can be seen from the figure, the predicted values for studded slabs fall within ±15% margin.

For non-studded slabs, predicted values tend to be more conservative. This fact may be caused

by the exclusion of clamping force to the steel deck and friction at steel deck-concrete interface

at the supports.

0

200

400

600

800

1000

1200

1400

0 200 400 600 800 1000 1200 1400

FEM (psf)

TE

ST

(ps

f)

-15%

15%

non-studdedstudded

Figure 4-12. Composite slab strength: FEM vs. experimental

(a) (b)



Comparison of the finite element results to those of the tests for a relatively wide range

of parameters demonstrates the ability of the method in predicting composite slab strength and

behavior. This ability may reduce the number of expensive full-scale experimental tests for

composite slabs. Further, the stress-strain response history of virtually any point in the system

can be obtained.

In comparison to the iterative method, the nonlinear finite element method offers some

advantages, such as the possibility to obtain stresses and strains at virtually any location of the

slab. The method, however, requires more advanced user’s knowledge than the iterative method.

Therefore, the iterative method is more suitable for design purposes.

Chapter 5 . Long Span Composite Slab Systems 53

CHAPTER 5

LONG SPAN COMPOSITE SLAB SYSTEMS

5.1. General

The maximum span length of unshored single span composite slabs used in the U.S.

based on available steel deck floor in the market is around 13 ft. The choice of unshored systems

is very common because these systems can save construction cost and time. If the span length

can be increased by a factor of, for example, 1.5 or 2, significant cost savings can be expected

from elimination of some intermediate beams and their connections to the girders. These

potential advantages have motivated research in the area of long span slab systems. In this case,

long span slab systems that do not cause any significant increase in the depth and weight of the

slabs compared to regular span slabs are particularly attractive. This has been one of the main

objectives of this part of the study.

Research in this area has been carried out by other researchers. Notable among these

are, the investigations by Ramsden (1987), the innovative lightweight floor system by Hillman

and Murray (Hillman 1990, Hillman and Murray 1990, 1994) and the slimflor system (British

Steel, Lawson et al. 1997). Ramsden (1987) conducted a study on two new prototypes of deck

profiles that can span a distance up to about 24 ft (7.5 m). The prototypes have holes in the web

to ensure the composite action between the deck and the concrete. The second prototype is an

improved version of the first one. These two prototypes are shown in Fig. 5-1. Because of the

shape of the profile, the concrete slab is virtually a solid slab with a thickness of 5 in. to 6 in.,

which is disadvantageous because of it selfweight. There is no mention in the paper whether

shoring of the slab during the construction was provided.


Figure 5-1. Prototype 1 and prototype 2 of Ramsden (1987) deck profiles

An innovative composite slab floor system design was developed and reported by

Hillman (1990) and Hillman and Murray (1990, 1994). The floor system developed was not only

lightweight but also able to span up to 30 ft without any intermediate beams. Figure 5-2 shows

schematically the design of the composite slab.

Figure 5-2. Innovative light weight and long-span composite floor(Hillman 1990, Hillman and Murray 1990, 1994)

Prototype 1

Prototype 2

Concrete slab

Perpendicularsteel decks


The slimflor system, marketed by the British Steel, utilizes deep deck sections of

ComFlor 210 (210 mm, approximately 8.25 in. deep) and SlimDek 225 (225 mm, approximately

8.86 in. deep) sections. With lightweight concrete, the ComFlor 210 deck section can span up to

6 m (approximately 19.7 ft). Figure 5-3 shows schematic view of this slimflor system.

Figure 5-3. Slimflor system (British Steel, Steel Construction Institute 1997)

In the current study, two 16 ga, deep steel deck profiles are investigated. The first

profile, referred to as profile 1, has a 6 in. rib height. The profile is currently not available in the

market so it was designed and manufactured by a press-brake process for this project. Because

of this, the length of the deck was limited to 25 ft. For long span slab specimens, the length is

only enough for a single span configuration. The second profile, i.e. profile 2, is a currently

available roof deck profile whose stiffness, as discussed later in this chapter, satisfies the

requirements for a long span slab in a double span configuration. This section was manufactured

through a cold-rolling process. Profile shapes of these sections are shown in Fig. 5-4. Note that

neither of these shapes incorporated embossments. This is because neither are currently

available composite deck profiles. For comparison, a 3 in. deep trapezoidal section is also

included in the figure.

Two design phases have to be considered in the development of these new deck profiles

for long span composite slab systems, namely the construction (non-composite) phase and

service (composite) phase. The construction phase considers the strength and stiffness of the

steel deck as a working platform that is subject to concrete self-weight and construction loads.

This phase is important in the determination of the required deck stiffness. It is shown later that

Concrete

Support beam

Steel deck


when a long span system is involved, the deflection (stiffness) limit state becomes very crucial.

Figure 5-4. 6 in., 4.5 in. and 3 in. deep profiles

The service phase deals with a composite section of steel deck-concrete slab that is

subject to occupancy loads. Studies on composite slabs with typical span lengths (Terry and

Easterling 1994, Widjaja and Easterling 1995, 1996, 1997) revealed that the actual load capacity

of the slabs are very high compared to the standard design live loads (50 to 150 psf). Table 5-1

shows that the ratios of actual load capacities (from the tests) to a 150 psf design live load range

from 2.45 to 7.90. At these (ultimate) load capacities, however, the slabs have undergone

excessive deflections. If the allowable deflection is limited to L/360 (SDI 1992), then, the

permissible loads based on this allowable deflection will be much lower than the ultimate load

capacities. The ratios of these permissible loads to a 150 psf design live load, as shown in Table

5-1, range from 1.37 to 3.11.

These ratios suggest that the service phase rarely governs the design of composite slabs.

However, this is not always the case for long span composite slabs as latter shown by the

analysis and test results. For long span slabs, both the construction and service phase have an

equal change to govern the design.

9.25

3.75 7.125 1.5 0.5

1

12.875

6

9

1.5 9 1.125 0.375

1

4.5

12

4.757.2512

3

profile 1

profile 2

profile 3


Table 5-1. Ratios of actual load capacities and permissible load based onallowable deflection to 50 and 150 psf design live loads.

slab ultimate load load at allow. test load / 50 test load / 150# capacity deflection *) ultimate load load at allow. ultimate load load at allow.

(psf) (psf) capacity deflection capacity deflection1 730 345 14.60 6.91 4.87 2.302 700 326 14.00 6.52 4.67 2.173 600 238 12.00 4.76 4.00 1.594 600 223 12.00 4.47 4.00 1.495 490 310 9.80 6.20 3.27 2.076 590 316 11.80 6.32 3.93 2.117 375 301 7.50 6.01 2.50 2.008 490 320 9.80 6.41 3.27 2.149 900 374 18.00 7.49 6.00 2.50

10 900 388 18.00 7.76 6.00 2.5911 750 352 15.00 7.04 5.00 2.3512 870 418 17.40 8.36 5.80 2.7913 480 399 9.60 7.98 3.20 2.6614 500 389 10.00 7.78 3.33 2.5915 1017 407 20.34 8.14 6.78 2.7116 1185 466 23.70 9.32 7.90 3.1117 565 301 11.30 6.02 3.77 2.0118 368 303 7.36 6.07 2.45 2.0219 523 396 10.46 7.93 3.49 2.6420 523 445 10.46 8.91 3.49 2.9721 467 229 9.34 4.57 3.11 1.5222 494 206 9.88 4.11 3.29 1.3723 507 246 10.14 4.92 3.38 1.64

*) based on L/360

5.2. Construction Phase

As previously mentioned, this design phase considers the strength and stiffness of steel

deck due to the fresh concrete weight. For typical span lengths, the flexural strength limit state is

generally the governing condition in the design. For a longer span length, the governing

condition is shifted toward the stiffness or deflection limit state. This condition is schematically

shown in Fig. 5-5. The deflection is limited to l/180, as required in the SDI Composite Deck

Design Handbook (Heagler et al 1997). The 0.75 in. maximum deflection limitation was not

used because it is considered to be to restrictive for long span slabs.

For the purpose of this study, the construction phase is utilized to determine the profile

shape of the steel deck that can be used for a desired span length. This was performed by

generating charts of steel deck weight vs. span length as shown in Fig. 5-4, for various types of


profile shapes. For the profiles shown in Fig. 5-4 with a 2.5 in. concrete cover in a single span

system, Fig. 5-6 gives the plots of the steel deck weight vs. span length. Figure 5-7 shows similar

plots for double span (continuous) condition. The steel deck weight was calculated based on the

deck thickness that corresponds to the required moment of inertia for a certain span length with a

given concrete self-weight plus the construction load.

0

5

10

15

20

25

30

6 9 12 15 18 21 24

SPAN LENGTH (ft)

ST

EE

L D

EC

K W

EIG

HT

(lb

/ft2)

DEFLECTIONLIMIT STATE

YIELD STRENGTHLIMIT STATE

Figure 5-5. Yield strength and deflection limit statesof the construction (non-composite) phase

It can be observed from Figs. 5-6 and 5-7, that for a same weight of steel deck, profiles 1

and 2 allow one to have a longer span than that of profile 3. This indicates that profiles 1 and 2

are more efficient than profile 3. Therefore, for a long span slab system of 20 ft, only the 4.5 in.

(profile 2) and 6 in. (profile 1) sections are considered in this study.

Table 5-2. Section properties of profiles 1, 2 and 3

Profile Thickness Area Inertia yp weight slab# weight

(in) (in2/ft) (in4/ft) (in) (lb/ft2) (lb/ft2)1 0.056 1.694 10.54 3.197 5.8 61.82 0.056 1.380 4.70 2.610 4.8 48.63 0.056 0.895 1.49 1.500 3.1 51.4


0

2

4

6

8

10

6 9 12 15 18 21 24

SPAN LENGTH (ft)

ST

EE

L D

EC

K W

EIG

HT

(lb

/ft2)

16 ga

18 ga

20 ga

16 ga

18 ga

20 ga16ga

18 ga20 ga

profile 1

profile 2

profile 3

Figure 5-6. Steel deck weight vs. span length ofsingle span systems

0

2

4

6

8

10

6 9 12 15 18 21 24

SPAN LENGTH (ft)

ST

EE

L D

EC

K W

EIG

HT

(lb

/ft2)

16 ga

18 ga

20 ga

16 ga

18 ga

20 ga16ga

18 ga20 ga

profile 1

profile 2

profile 3

Figure 5-7. Steel Deck weight vs. span length ofdouble span systems


From Table 5-2, by comparing values of profiles 1 and 3, it can be seen that the steel

deck moment of inertia of profile 1 is approximately 7 times higher than that of profile 3, which

corresponds to an ability to span 1.6 (= 74 ) times further. The steel deck self-weight is almost

double the one of profile 3. For a 20 ft long piece of deck with only one typical rib of profile 1,

the piece weighs about 116 lb and it can be handled by two people in the construction site.

For profile 2, the increase of the moment of inertia is about 3 times of that of profile 3,

and it corresponds to an ability to span 1.3 times further. The total weight of the slab, for the

same 2.5 in concrete cover above the rib, is slightly lighter than the slab with profile 3 as the

steel deck.

5.3. Service Phase

In the service phase, predicted maximum test loads of composite slabs can be calculated

using various ways. The iterative, direct, SDI-M and finite element methods were used to predict

the capacities of the specimens built using profiles 1 and 2 in this study. However, only the

iterative and finite element methods can provide response histories of load vs. deflection of the

slabs. In the SDI-M and direct methods, Iavg is used with an elastic analysis to obtain

permissible loads based on deflection limit state. The analysis was performed in the same ways

as those with typical span length.

5.4. Specimen Descriptions and Instrumentation

Long span slab 1 (LSS1) has a configuration of two single deck spans of 20 ft each and 1

ft cantilever as shown in Fig. 5-8 (a). The total slab depth was 8.5 in. (2.5 in. concrete cover

above the 6 in. deck rib height). Six 3/4 in. diameter, 8-3/16 in. tall shear studs were used at each

end of the slab, spaced at 1 ft on center as shown in Fig. 5-9.

For long span slab 2 (LSS2), a two-span system was used with 20 ft span lengths. The

configuration is shown in Fig. 5-8 (b). The total depth of the slab was 7 in. (2.5 in. concrete

cover above 4.5 in. rib height). Six 3/4 in. diameter, 6-3/16 in. tall shear studs were used at the

support, spaced at 1 ft on center as shown in Fig. 5-10.


Figure 5-8. System configurations of LSS1 and LSS2

Strain gages were placed at the bottom surface of the deck to measure the steel deck

strain during concrete casting and the load test. Three cross sections were monitored in each

span of the slab: the exterior support, interior support and mid-span. A set of six strain gages

was used at each of those cross sections. The schedules of these strain gage and shear stud

locations are shown in Figs. 5-9 and 5-10 for LSS1 and LSS2, respectively. In addition to these

strain gages, potentiometers were also placed at each end of the slab to measure the slip between

the concrete and the deck. Several displacement transducers were also used to measure vertical

displacements.

No shoring was provided during the construction of the slabs. The measured mid-span

deflections of the steel deck during concrete casting were 0.695 in. and 0.685 in. for LSS1 and

LSS2, respectively. Concrete compressive strength at 28 days were 3060 psi and 2330 psi for

LSS1 and LSS2, respectively.

concrete slabsteel deck steel deck

concrete slabsteel deck

1′ 20′ 20′ 1′

SLAB 1

SLAB 2


Strain gage locations

Shear stud locations

Figure 5-9. Strain gage and shear stud schedules of LSS1

strain gage locations

Section A-A

A A A A A A

A A A A A A

12 9 111 111 9 9 111 111 9 12

8.5

Section B-B

B B B B

B B B B

12 240 240 12


Strain gage locations

Shear stud locations

Figure 5-10. Strain gage and shear stud schedules of LSS2

strain gage locations

Section A-A

A A A A A A

A A A A A A

12 9 111 111 9 9 111 111 9 12

7

Section B-B

B B B

B B B

12 240 240 12


5.5. Load Test Procedure

A uniform load configuration was used for the load tests. An air bag, placed on the top

surface of the slab, was used for this purpose, and the load was applied by gradually increasing

the pressure in the air bag. The air bag has a capacity of 20 psi in a fully constrained condition.

The view of the test set-up is shown in Fig. 5-11. Each span was tested separately and in an

attempt to prevent development of negative cracks into the adjacent span, crack inducers were

placed along the interior supports of LSS1 and LSS2. The crack inducers were groves,

approximated 0.5 in. deep, and made along the interior supports on the top surface of the

concrete when it was still wet after the casting.

Figure 5-11. Test set-up

At the beginning of each load test, the tested span was preloaded with approximately

0.35 psi (50 psf) to settle the system and check the instrumentations. The slab was unloaded

afterward and the loading was restarted and continued until a permanent set in the system was

obtained. This permanent set can be observed from the presence of the nonlinear relation of the

load versus mid-span displacement. Load increments of approximately 0.25 psi (36 psf) was

applied with a pause, of approximately two minutes before any data recording, to allow the

system to settle. When a permanent set had been noted, the system was once again unloaded

completely. The loading was then restarted until failure or excessive deflection was obtained.

In the inelastic region where the stiffness of the slab had decreased considerably,

displacement control loading was used with a displacement increment of approximately 0.5 in.

air bag


The test was terminated after 7 in. (LSS1) or 8 in. (LSS2) deflection was obtained.

5.6. Test vs. Analysis Results

Before the load tests, fine cracks through the depth of the concrete were observed on the

sides of the slabs over the interior supports. During the load test, as the load was increased,

flexural cracks developed within the tested span. In LSS2, because of the continuity of the steel

deck over the interior support, cracks appeared in the adjacent span on the top surface of the slab.

Maps of the cracks of LSS1 and LSS2 after the test are shown in Figs. 5-12 and 5-13. In Fig. 5-

13, cracks indicated by x are cracks that were developed during the test of the adjacent span.

Figure 5-12. Map of cracks in LSS1

Figure 5-13. Map of cracks in LSS2

1st test 2nd test

2nd test 1st test

x x x x

x x x x

x x x x


Flexural cracks in the positive moment regime appeared on the side of the slabs tend to

turn horizontally approximately at the level of the top flange of the steel deck. This may indicate

some separation of the slab portion (concrete cover) from the beam portion (concrete rib) of the

concrete.

Load vs. mid-span deflection response from the tests and analyses of LSS1 and LSS2 are

compared in Figs. 5-14 and 5-15. It can be observed from these figures, that the response of the

second test of each LSS was relatively weaker and softer compared to the first. This may be

caused by damage that occurred in the adjacent span (first test), so that less (horizontal) restraint

was resulted. In LSS2, the occurrence of the negative cracks before the test on the second span

may have increased this effect.

0

100

200

300

400

500

600

700

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0


LOA

D (

psf)

SDI-M

Direct

IterativeTest

0

100

200

300

400

500

600

700

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0


LOA

D (

psf)

SDI-M

Direct

Iterative

Test

(a) 1st test (b) 2nd test

Figure 5-14. Load vs. mid-span deflection of LSS1

0

50

100

150

200

250

300

350

400

450

500

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0


LOA

D (

psf)

SDI-M

DirectIterativeTest

0

50

100

150

200

250

300

350

400

450

500

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0


LOA

D (

psf)

SDI-M

Direct

Iterative

Test

(a) 1st test (b) 2nd test

Figure 5-15. Load vs. mid-span deflection of LSS2


Predicted responses using the iterative method, as shown in Figs. 5-14 and 5-15, show

reasonable agreement to those of the tests, particularly the first test of each slab. In terms of the

slab strength, the direct method also shows relatively good agreement to the test results. The

SDI-M method, however, predicted rather low strength (very conservative). This is due to the

very low values of the reduction factor, R, based on the required anchorage forces. These were

0.545 and 0.447 for LSS1 and LSS2, respectively. Finally, a summary of the maximum test load

capacity and permissible load based on the allowable deflection is given in Table 5-3.

Table 5-3. Summary of maximum test load and permissible loadbased on allowable deflection

slab ultimate load load at allow. test load / 50 test load / 150# capacity deflection *) ultimate load load at allow. ultimate load load at allow.

(psf) (psf) capacity deflection capacity deflectionLSS1a 621 245 12.42 4.90 4.14 1.63LSS1b 559 210 11.18 4.20 3.73 1.40LSS2a 498 163 9.96 3.26 3.32 1.09LSS2b 455 121 9.10 2.43 3.03 0.81

*) based on L/360

From the above table, it can be noted that for LSS2, the permissible loads based on the

allowable deflection are relatively low compared to those of typical span slabs and LSS1.

Therefore, in the case of long span composite slab, it is important to check the deflection limit

state.

5.7. Evaluation of the Floor Vibrations

Vibration tests on LSS1 and LSS2 were conducted prior to the load tests to determine the

frequency of the fundamental mode of these slabs. For LSS1, the frequency of the fundamental

mode was 10.63 Hz., and it was 8.13 Hz. for LSS2. Plots of the frequency spectra in terms of the

normalized relative power vs. the frequency resulting from the tests are shown in Figs. 5-16 and

5-17.


0.0

0.2

0.4

0.6

0.8

1.0

1.2

4 6 8 10 12 14 16

FREQUENCY (Hz)

NO

RM

ALI

ZE

D R

ELA

TIV

E P

OW

ER

Figure 5-16. Normalized relative power vs. frequency of LSS1

0.0

0.2

0.4

0.6

0.8

1.0

1.2

4 6 8 10 12 14 16

FREQUENCY (Hz)

NO

RM

ALI

ZE

D R

ELA

TIV

E P

OW

ER

Figure 5-17. Normalized relative power vs. frequency of LSS2


Analytical calculations were made to determine if the frequencies satisfy the acceptance

criteria for human comfort (Murray et al. 1997). Two criteria were considered in this case.

LSS1 was classified as a footbridge with 6 ft effective width. The estimated peak acceleration

was 3.13% g. This estimated peak acceleration is higher than the specified value of 1.5% g and

thus the slab can not be considered satisfactory. The effective width and the occupational load of

the slab influence the vibration performance of the slab. For an effective width of 15 ft and for

office and residential use of the same slab, the estimated peak acceleration becomes 0.31% g,

which is lower than the maximum peak acceleration limit of 0.50% g. The slab stiffness

requirement was also satisfactory (5.89 k/in, experimental, compared to the minimum

requirement of 5.70 k/in). Therefore, in the later case, the slab can be considered satisfactory.

These estimations, however, are rather approximate, and further investigation is necessary.

The vibration response of LSS2 was not as good as those of LSS1. The estimated peak

acceleration for a footbridge condition is 10.2% g compared to the maximum peak acceleration

limit of 1.5% g and the experimental slab stiffness was 2.48 k/in which is below the minimum

required stiffness of 5.7 k/in. For the condition with an effective width of 15 ft for office and

residential purpose, the estimated peak acceleration is 0.95%, and again is greater than the

specified value of 0.50% g. Further evaluations are necessary based on these preliminary

evaluations of the composite slabs.

5.8. Proposed Detail Connection

The total depth of composite floor system using steel deck profiles as described in this

study is relatively shallow. In comparison with the 3 in. trapezoidal deck profile using a same

thickness of concrete cover, profiles 1 and 2 will result in 3 in. and 1.5 in., respectively, of

additional slab depth. Therefore, typical beam to girder connection for composite slabs with

regular span length can be used without adding any significant height to most structures.

However, should this additional structure height be objectionable, it can be reduced or eliminated

by using a beam to girder connection as shown in Fig. 5-18.


Figure 5-18. Proposed beam to girder connection to reduce slab-beam height.


A study on long span composite slab systems has been presented and two steel deck

profiles have been investigated. The study, verified by experimental tests, shows very promising

results on the use of relatively slim slabs (8.5 in. and 7 in. total slab depth), with almost the same

concrete volume or weight as of the typical span slabs. With the proposed beam to girder

connection, the slab-beam depth may be reduced to a total floor depth comparable to currently

used floors. This feature of slab depth and weight promise potential advantages over the slimflor

systems that are now used in European countries.

The design method for the development of the deck profile by generating charts of the

steel deck weight vs. the span length, and the analytical methods for the prediction of the

composite strength and stiffness of the slab were shown to be good tools. These methods of

analyses are very promising for the development of new deck profiles before any experimental

tests. They can also reduce the number of full-scale tests needed.

Permissible loads based on the deflection limit state of the service phase may become the

governing limit state in the case of long span composite slabs. This limit state rarely governs the

design in typical span slab systems. Therefore, in the case of long span slab systems, both the

construction (non-composite) and service (composite) phases have to be evaluated carefully.

Results of the evaluation of floor vibrations suggest further study be required to improve

the performance of the slabs with respect to the floor vibration criteria. A deeper slab thickness

with a little sacrifice in span length could be considered to give higher slab stiffness, which may

improve the vibration characteristics.

girderbeam

Chapter 6 . Resistance Factor 71

CHAPTER 6

RESISTANCE FACTOR FOR

THE DESIGN OF COMPOSITE SLABS

6.1. General

Probability-based design criteria in the form of load and resistance factor design (LRFD)

are now applied for most construction materials. The design requirements have to insure

satisfactory performance of structures. The main advantage of the approach is the ability to

achieve a uniform level of reliability for structural members, or to impose a certain level of

reliability (higher or lower) of some certain parts of the structures. This gives a strong rationale

to the load and resistance factors as compared to the design safety factors of the allowable stress

design. Additionally, a unified design strategy as to setting up common load combinations and

load factors can be obtained.

In this part of the study, resistance factors, φ, for the flexural design of composite slabs

were evaluated based on test data of 39 full scale composite slab specimens. The tests were

performed at the Structures and Materials Laboratory of Virginia Polytechnic Institute and State

University, Blacksburg, Virginia. The φ factors evaluated correspond to the SDI-M method and

direct method described in Section 3.

6.2. Review of Probabilistic Concepts of Load and Resistance Factor Design

Discussions on the probabilistic concepts of the LRFD approach are given in detail by

many sources (Cornell 1969, Lind 1971, Ang and Cornell 1974, Galambos and Ravindra 1977,

Ravindra and Galambos 1978, Ellingwood et al. 1980, Load and 1986, Hsiao et al. 1990,


Geschwindner et al. 1994, Commentary on 1996, Barker and Puckett 1997).

In principle the following inequality applies:

φ γ R Qii

in ≥ ∑ (6-1)

in which, Rn = nominal resistance, Qi = load effect, φ = resistance factor and γ i = load factor.

The probability of failure can be expressed by:

( )pf = 1− Φ β (6-2)

where Φ is the standard normal probability function, and β is the reliability index.

6.2.1. Reliability Index

The reliability index, β, used in Eqn. (6-2), in a log-normal format, can be expressed by:

βλ λ

ζ ζ

ln R

V

m

R2

=−

+≈

+

R Q

R Q

m

Q

Q

V2 2 2(6-3)

where λ, ζ and V, respectively, denote the log-normal mean, log-normal standard deviation and

the coefficient of variation. Subscript R and Q denote the resistance and the load effect,

respectively. Rm and Qm are the means of resistance and load, respectively. Introduce a

linearization given by:

( )V V VR Q Q2 2+ + = VRα (6-4)

then Eqn. (6-3) can further be approximated as:


( )βα

=

ln R

Qm

m

+V VR Q(6-5)

where according to Lind (1971), for 3 V/V 3/1 RQ ≤≤ , α = 0.75 gives a good approximation

with ±6% maximum error. Equation (6-5) forms the basis equation for the AISC and AISI load

and resistance factor design specification for structural steel and cold-formed steel. From Eqn.

(6-5), the central safety factor can be expressed as:

( )θ αβ =

R

Q = em

m

V VR Q+(6-6)

By minimizing the error of this central safety factor, Galambos and Ravindra (1977) suggested a

value of α = 0.55 which was later adopted in AISC LRFD. The reliability index, β , can be

determined from Eqn. (6-5). As an illustration, the following table shows some β values and

the corresponding probability of failure, pf .

Table 6-1. β vs. pf

β pf

5.0 -710 x 9.2

4.0 32. x 10-5

3.0 14. x 10-3

2.0 2 3. x 10-2

AISC-LRFD uses the following β values:

β = 3.0, for members, under DL + LL or Snow

β = 4.5, for connections, under DL + LL or Snow

β = 2.5, for members, under DL + LL + Wind

β = 1.75, for members, under DL + Earthquake


whereas the AISI-LRFD uses the following β values:

β = 2.5, for members

β = 3.5, for connections

Galambos et al. (1982) give β values for various structural members under conditions of ratio of

basic specific live load to normal value of dead load equal to 1, 2 and 5. Ranges of β values

were also given by Ellingwood et al. (1980). These values of β range between 1.9 - 3.5 for

reinforced concrete members and 3.0 - 4.5 for steel members.

6.2.2. AISC LRFD Approach for the Resistance Factor

Using the central safety factor given in Eqn. (6-6), the following inequality can be written:

Rm ≥ θ Qm (6-7)

which leads to:

RmV

mR e Q e

VQ− ≥αβ αβ(6-8)

or,

φ γ R Qn n≥ (6-9)

in which, Rn and Qn are nominal values of resistance and load,

φ αβ = R

Rm

ne VR− (6-10)

γ αβ =

Q

Qm

ne

VQ (6-11)

Further, the mean resistance, Rm , can be expressed in terms of the nominal resistance and

statistical parameters that represent the variability of material strength and stiffness, M,


fabrication, F, and the uncertainties involved in the assumptions of the engineering design

equation, P (Ravindra and Galambos 1978):

( )R R F Pm n m m= M m (6-12)

where M m, F and Pm m are the means of M, F, and P, respectively. Accordingly, the coefficient

of variation of the resistance can be approximated by using:

V V V VR M F P≈ + +( ) ( ) ( )2 2 2 (6-13)

in which, VM , V and VF P are, respectively, the coefficients of variation of M, F and P. Here,

Eqn. (6-13) assumes independent relations among M, F and P variables. By using Eqn. (6-12),

the resistance factor given by Eqn. (6-10) can be modified to:

( )φ αβ = M emF Pm mVR− (6-14)

6.2.3. AISI LRFD Approach for the Resistance Factor

The AISI specification for cold-formed steel structures follows a different approach in

determining the resistance factor, φ. The approach is based on the research by Hsiao et al.

(1990). Instead of using Eqn. (6-10), it starts by expressing the effective resistance in terms of

the nominal loads and load factors multiplied by a deterministic coefficient, c, that relates the

load intensities to the load effect and is given by:

( )φ γ γ γ γ R = c D + L = D

L+ c LD n n D

n

nnn L L

(6-15)

where γ γD and L are the dead and live load factors, and Dn and Ln are the nominal values of

the dead and live load. Similarly, the mean of the load effect can be expressed as:


Qm = 1.05 D

L + c Ln

nn1

(6-16)

Notice that in the last equation, D Lm n= =105. D and Ln m were used (based on load statistic

by Ellingwood et al. 1980). From Eqns. (6-15) and (6-16), one obtains:

R

Qm

m =

R

Rm

n

ψφ

(6-17)

with,

ψ γ γ = D

L + / 1.05

D

L + 1D

n

nL

n

n

(6-18)

By combining Eqns. (6-3), (6-12) and (6-17), an expression of the resistance factor can be

obtained:

( )φ ψβ

= M em- VR

2F Pm m

VQ+ 2(6-19)

Using this equation, determination of the α coefficient can be avoided. However, the coefficient

of variation of the load has to be known.

6.3. Statistical Data

Evaluation of the resistance factor, φ, as given by Eqn. (6-14) or (6-19) requires

statistical values of the parameters involved. These data are available from the lab tests

conducted on the composite slab specimens previously mentioned. However, larger sets of

database are preferred to give more representative values of means, standard deviations, and

coefficients of variation of the afore-mentioned parameters. Therefore, statistical values from

other sources that were based on larger sets of database were used. These values are the

statistical values of the concrete compressive strength, fc ' , which was based on the study by


MacGregor (1997), and steel deck yield stress, fy , which was based on the study on cold-formed

steel members by Hsiao et al. (1990). For these two parameters, the data obtained from the lab

tests from the composite slab specimens were used as a comparison only.

Data obtained from the lab tests, which are not available elsewhere from larger sets of

database, were used for the determination of the resistance factor. These data are the statistical

data of deck thickness, t, maximum and minimum shear bond strength at the interface of steel

deck - concrete, f and fs,mins,max , respectively.

6.3.1. Material Factor, M

The material factor, M, represents the variability of the strength and stiffness of the

material. In this case, M is affected by the variability of fc ' , fy , f and fs,mins,max . Statistical

data of these parameters are listed in Table 6-2.

Table 6-2. Statistical data of fc ' , fy , f and fs,mins,max

µ σ Vfc' (MacGregor, 1997) 3940 psi 615 psi 0.156fc' (test) 3867 psi 878 psi 0.227fy (Hsiao et al. 1990) 1.100 fy 0.121 fy 0.110fy (test) 1.002 fy 0.058 fy 0.058fs,max 0.999 fs,max 0.035 fs,max 0.035fs,min 1.001 fs,min 0.073 fs,min 0.073Note: µ = mean, σ = standard of deviation, V = coefficient of variation

Assuming that those parameters are statistically independent, coefficients of variation of

the material factor can be approximated by:

V VM SDI fy, ' = V = 0.191fc2 2+ (6-20)

for the SDI-M method while for the Direct method:

V V V VM Direct f f fy s, s,, ' max min = V = 0.208fc

2 2 2 2+ + + (6-21)


for the SDI-M or direct design procedure, respectively. The mean values of M for the SDI-M

and direct design procedures can be evaluated from:

M m SDIm fc

,, ''

' ' =

f

f

f

f =

f

f = 1.445c

c

y,m

y c

f

y

y

µ µ(6-22)

( ) ( )M m Direct

m,

,'

' =

f

f

f

f

f

f

f

fc

c

y,m

y

s,max m

s,max

s,min m

s,min

M m Directf fc s

,'

' ',max =

f

f

f

f = 1.397

c

f

y s,max

f

s,min

y s,minµ µ µ µ

(6-23)

6.3.2. Fabrication Factor, F

The fabrication factor, F, represents the variability of the manufacturing process. In this

case, the variability of the steel deck thickness, t, is considered. The statistical data for this steel

deck thickness are listed in Table 6-3. These data were based on the measurement conducted on

the steel decks that were used for the composite slab specimen tests.

Table 6-3. Statistical data of t

µ t σt Vt0.966 t 0.030 t 0.313

Based on the above statistical values, VF and Fm can be computed as follow:

0.313 = V = V tF (6-24)

Fm = t

t =

t = 0.966,m tµ

(6-25)

6.3.3. Professional Factor, P

The professional factor, P, takes into account the uncertainties of the design equation.

This professional factor is defined as (Ravindra and Galambos 1978, Geschwindner et al. 1994):


P = test

prediction(6-25)

The prediction is the resistance of the slab as predicted by the design equation based on the

measured (actual) values of its parameters. Based on the lab tests performed on the afore-

mentioned full-scale composite slab specimens, the following statistical data is obtained:

Table 6-4. Statistical data of P

µ p = Pm σp VpSDI 1.193 0.244 0.205

Direct 1.071 0.183 0.172

6.3.4. Load Statistic

Information regarding statistical data of the load in terms of the coefficient of variation,

VQ , is needed for the AISI approach as shown in Eqn. (6-19). For this reason, statistical data of

dead and live loads were taken from a special publication of the National Bureau of Standards

(Ellingwood et al. 1980). These data are summarized in Table 6-5. Dn and Ln denote the

nominal dead and live loads.

Table 6-5. Statistical data of dead and live loads

µ σ VD 1.05 Dn 0.105 Dn 0.10L 1.00 Ln 0.250 Ln 0.25

For the combination of the dead and live loads given by:

Q = D + LD Lγ γ (6-26)

the mean and standard of deviation of this combination can be expressed by:

µ γ µ γ µQ = + D D L L (6-27)


σ γ σ γ σQ = Var(Q) = + D2

D2

L2

L2 (6-28)

assuming that the distribution of D and L are statistically independent. In Eqn. (6-28), Var(Q)

denotes the variance of Q. By substituting values from Table 6-5 into Eqn. (6-27) and Eqn. (6-

28), the coefficient of variation of Q can be obtained as:

VQ L = 0.011 D

L + 0.063 / 1.05

D

L + D

2 n

n

2

Dn

nLγ γ γ γ

2 (6-29)

6.4. The Resistance Factor

In this study, the AISI-LRFD approach is adopted. The AISC-LRFD approach is used to

give a comparison. Considering the fact that composite slabs are generally used in steel framed

structures, a β (reliability index) value greater than 3.0 is not considered necessary (β=3.0 for

steel members). Therefore, β=3.0 is chosen as the target reliability index (AISI uses β=2.5 as the

basic case). The final result of φ factors, however, is rounded to the closest 0.05 and hence, the

actual β values used will not be exactly 3.0. A minimum limit of β=2.5 is used.

A load combination with γ D = 1.2 and γ L = 1.6 as given in the SDI Composite Deck

Design Handbook (Heagler et al. 1997) is used as the basic load case. The combination using

γ D = 1.4 and γ L = 1.0 is not considered because the ratio of dead to live load is typically < 3.0

for composite slabs. A range of dead to live load ratios between 0.5 (short to normal span slabs

with relatively heavy live load, approximately 100 psf) and 1.5 (long span slabs up to 20 ft with

relatively light live load, approximately 50 psf) is considered.

Based on the statistical data presented in section 6.3 and equations given in section 6.2, φ

factors for several values of D/L (0.5, 1.0 and 1.5) were computed and the results are listed in

Table 6-6 and Table 6-7 for the SDI-M and direct design procedures, respectively. Again, these

results are based on the AISI-LRFD approach presented in section 6.2.3.


Table 6-6. Calculated φ factors for SDI-M method(AISI-LRFD Approach)

βD/L 3.00 2.75 2.500.5 0.8790 0.9558 1.03931.0 0.8773 0.9497 1.02821.5 0.8704 0.9403 1.0158

Table 6-7. Calculated φ factors for direct method(AISI-LRFD Approach)

βD/L 3.00 2.75 2.500.5 0.8112 0.8801 0.95481.0 0.8109 0.8757 0.94581.5 0.8052 0.8677 0.9350

Based on the results in Tables 6-6 and 6-7, φ = 0.90 is chosen for the SDI-M method and φ = 0.85

is selected for the direct method. For comparison, φ factors computed by using the AISC-LRFD

approach are listed in Table 6-8 for the SDI-M method and Table 6-9 for the direct method for

several combinations of α and β values. This later approach is not influenced by the ratio of the

dead to live load (D/L). As shown in these tables, the choice of α between 0.65 to 0.75 show

relatively close results to the AISI approach.

Table 6-8. Calculated φ factors for SDI-M method(AISC-LRFD Approach)

αβ 0.55 0.65 0.75

3.00 1.046 0.961 0.8832.75 1.087 1.006 0.9312.50 1.130 1.053 0.982

Table 6-9. Calculated φ factors for direct method(AISC-LRFD Approach)

αβ 0.55 0.65 0.75

3.00 0.957 0.882 0.8132.75 0.993 0.922 0.8562.50 1.031 0.963 0.900



Resistance factors for the flexural design of composite slabs based on the SDI-M and

direct methods have been presented. The AISI LRFD approach for evaluating the resistance

factor was adopted. By this approach, the determination of the α coefficient is not necessary. A

target reliability index β=3.0 and minimum limit of β=2.5 were used. This choice was based on

the target reliability β=3.0 for steel members (AISC-LRFD), and the lower bound β=2.5 used in

AISI-LRFD for the basic load case. The resulting resistance factors are φ=0.90 for the SDI-M

method and φ=0.85 for the direct method. These φ values were based on a range of dead to live

load ratios between 0.5 and 1.5, which is representative of typical composite slab designs.

Chapter 7 . Conclusions and Recommendations 83

CHAPTER 7

CONCLUSIONS AND RECOMMENDATIONS

A study of the strength and behavior of composite slabs in general, with a particular

investigation of the use of long span composite slab systems, has been carried out analytically

and experimentally. Two new methods of predicting composite slab strength and stiffness based

on simple mechanical models have been developed. The methods, which are supported by

experimental data obtained from elemental tests of shear bond and end anchorages, offer an

alternative solution to the m-k method, which requires a number of full-scale tests. Experimental

test results conducted on full-scale composite slab specimens reveal that the methods predict the

slab strength more accurately than the SDI-M method. This is due to the ability of these methods

to include the effects of shear bond strength, weld strength, end anchorage strength and any

remaining strength of the deck.

The nonlinear finite element method was used to model the complex nature of composite

slabs. From this analysis, a response history of virtually any point of the system can be obtained.

The development and use of a special purpose finite element code, which is particularly designed

for composite slabs and incorporates a concrete plasticity model with three or higher number of

parameters for the concrete failure surface and an energy based path following technique is

recommended. This is based on the fact that the concrete material is one of the most sensitive

aspects of the composite slab analysis, particularly when the concrete is in tension. The

suggested energy based path following technique is due to the inconsistency of the physical units

in the arc length method, which may lead to numerical problems.

Chapter 7 . Conclusions and Recommendations 84

Application of the methods of analysis described earlier shows a promising ability in

providing analytical tools and an alternate solution to performing a large number of full-scale

tests. These later tests can be replaced by elemental tests of shear bond and end anchorages,

which are less expensive.

The study on the long span composite slab systems indicates that the system can be used

without significantly increasing the depth or weight of a floor system. This promises a potential

advantage over the slimflor systems that are now used in European countries. With the long span

systems, more efficient use of the material can be expected as some of the filler beams and their

connection to the girders can be eliminated, and therefore, less construction work is required.

More detailed study regarding the economy of the system is recommended for future research.

Further study on the vibration behavior of the long span system is also recommended for future

research.

Finally, the study on the resistance factor, φ, for flexure design of composite slabs

concluded that φ=0.90 and φ=0.85 can be used for the SDI-M and direct method, respectively.

As more databases on the shear bond strength become available, further study of these φ factors

is suggested. It is also recommended to extend the study to φ factors for other limit states of the

design used for composite slab.

85

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V I T A

Budi R. Widjaja was born on May 9, 1961 in Semarang, Indonesia. He obtained his B. S. degree

in civil engineering from Parahyangan Catholic University in Bandung, Indonesia, in April 1985.

After a year working in a contracting company, he joined with Parahyangan Catholic University

as a part time teaching assistant. Simultaneously he worked as a structural engineer in a

consulting firm in Bandung, Indonesia, until Fall of 1990, at which time he entered the graduate

program in civil engineering at Virginia Polytechnic Institute and State University. He

completed his Master of Science degree in May 1993 and continued pursuing a Ph.D. degree in

civil engineering. He worked as a research assistant at the Structures and Materials Laboratory

during his doctoral study.

4 Analysis And Design of Steel Deck – Concrete Composite Slabs

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