ANALYSIS AND DESIGN OF STEEL 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
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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
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
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
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
Chapter 2 . Elemental Tests 13
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
Chapter 2 . Elemental Tests 14
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
Chapter 2 . Elemental Tests 15
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
Chapter 2 . Elemental Tests 16
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
Chapter 2 . Elemental Tests 17
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
Chapter 3 . Strength and Stiffness Prediction - Simple Mechanical Model 19
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
Chapter 3 . Strength and Stiffness Prediction - Simple Mechanical Model 20
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 '
Chapter 3 . Strength and Stiffness Prediction - Simple Mechanical Model 21
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
Chapter 3 . Strength and Stiffness Prediction - Simple Mechanical Model 22
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
Chapter 3 . Strength and Stiffness Prediction - Simple Mechanical Model 23
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,
Chapter 3 . Strength and Stiffness Prediction - Simple Mechanical Model 24
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
Chapter 3 . Strength and Stiffness Prediction - Simple Mechanical Model 25
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
Chapter 3 . Strength and Stiffness Prediction - Simple Mechanical Model 26
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
Chapter 3 . Strength and Stiffness Prediction - Simple Mechanical Model 27
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
Chapter 3 . Strength and Stiffness Prediction - Simple Mechanical Model 28
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( , )
Chapter 3 . Strength and Stiffness Prediction - Simple Mechanical Model 29
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,
Chapter 3 . Strength and Stiffness Prediction - Simple Mechanical Model 30
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-
Chapter 3 . Strength and Stiffness Prediction - Simple Mechanical Model 31
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
Chapter 3 . Strength and Stiffness Prediction - Simple Mechanical Model 32
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
Chapter 3 . Strength and Stiffness Prediction - Simple Mechanical Model 33
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:
Chapter 3 . Strength and Stiffness Prediction - Simple Mechanical Model 34
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
Chapter 3 . Strength and Stiffness Prediction - Simple Mechanical Model 35
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
Chapter 3 . Strength and Stiffness Prediction - Simple Mechanical Model 36
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)
Chapter 3 . Strength and Stiffness Prediction - Simple Mechanical Model 37
and slab-21 (non-studded slab) are shown in Fig. 3-16.