two-way steel floor system using open-web joists · using open-web joists by john a. schaad, b.s. a...

Two-Way Steel Floor System

Using Open-Web Joists

by

John A. Schaad, B.S.

A Thesis submitted to the Faculty of the Graduate School,

Marquette University, in Partial Fulfillment of

the Requirements for the Degree of Master of Science

Milwaukee, Wisconsin August, 2005

i

Preface

Minimizing floor-to-floor heights in mid- to high-rise buildings is a concern held by both

engineers and architects. Many attempts have been made in steel construction to adopt

design philosophies that utilize structural floor members with high span/depth ratios.

These designs, however, have been limited to floor systems that predominantly span in

one direction. The primary objective of this thesis is to investigate the structural

feasibility of interlocking open-web steel joists to form a panelized two-way steel floor

system.

This thesis includes a detailed discussion of the fabrication of the proposed floor

system. A complete description of a proposed construction sequence is also presented.

Finally, the structural behavior of the floor system is demonstrated under static loading.

Conclusions were made discussing the benefits of this new system as well as

recommendations for the direction of further investigation.

ii

Acknowledgements

First and foremost, I would like to thank my advisor, Dr. Chris Foley, for his time,

guidance, and willingness to listen as my proposed floor system design evolved. I want

to extend my appreciation to my other thesis committee members: Dr. Stephen Heinrich

and Dr. Sriramulu Vinnakota. I am also extremely grateful for the guidance and

suggestions that I received from David Samuelson at Nucor.

In addition, I want to extend my gratitude to my parents, John and Annette

Schaad, for their continued support and encouragement during my pursuit of a graduate

degree in civil engineering. Finally, I offer heartfelt thanks to my fiancé, Erin Morin, for

her support and patience while I have been away in Milwaukee, Wisconsin.

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Table of Contents

List of Figures…………...……………………………………………………………….vi

List of Tables…………...……………………………………………………………...…x

Chapter 1: Introduction and Literature Review……………………...…….1

Section 1.1 Introduction…………………………………………………….….1

Section 1.2 Stub-Girder System………………………………………………..2

Section 1.3 Girder-Slab……….………………………………………………..6

Section 1.4 AISC Multi-Story Residential Construction Competition………...8

Section 1.4.1 Structural Steel/Autoclaved Aerated Concrete (AAC)

Composite Floor System…………………………..……....9

Section 1.4.2 Stiffened Plate Floor Deck……………………………….12

Section 1.5 Staggered-Truss………………………………………………….14

Section 1.6 Open-Web Steel Joist Construction………………………...........18

Section 1.6.1 Dry Floor Construction…………………………..............18

Section 1.6.2 Composite Steel Joists…………………………...............20

Section 1.6.3 Composite Girders….........................................................23

Section 1.7 Space Trusses….…………………………………………………24

Section 1.8 Synthesis of Past Literature and Direction for Present

Research…………………………………………………………28

Chapter 2: Proposed System Fabrication and Erection…....……..............30

Section 2.1 Introduction……………………………………………………....30

Section 2.2 Fabrication ……………………………..………………………..31

iv

Section 2.3 Erection Sequence……………….……………………………….35

Chapter 3: Structural Behavior……………………...……………..………41

Section 3.1 Introduction….………………...………………………………....41

Section 3.2 Plate Bending…………………………………………..………...41

Section 3.3 Two-Way Concrete Floor Systems...…………………………….45

Section 3.4 K-Series Joist Selection (One-Way System)…….………………47

Section 3.5 Structural Analysis……………………………………………….50

Section 3.5.1 FEA Element Types and Modeling Assumptions………..50

Section 3.5.2 Traditional Joist Analysis and Results…..……………….52

Section 3.5.3 Proposed System Panel Design…………………………..59

Section 3.5.4 Modeling the Proposed Design…………………………..64

Section 3.5.5 Connection Design ……..………………………………..73

Section 3.6 Girder Design ……..……………………………………………..80

Chapter 4: Composite Design………………………...……………..………85

Section 4.1 Introduction….………………...…………………………………85

Section 4.2 Proposed System’s Composite Capabilities………………….......86

Section 4.3 Composite Girders..……...………………………………………93

Chapter 5: Conclusions and Recommendations…...…..…………..………96

Section 5.1 Summary….………………...…………………………………....96

Section 5.2 Conclusions….………………...…………………………………97

Section 5.3 Recommendations for Future Research………………………...100

References…………………………………………………..…...…..…………..……..102

Appendices………………………………………………….…...…..…………..……..107

v

Appendix A: Results of Proposed System’s Member Sizes……………………107

Appendix B: Calculations for Non-Composite Girders…………………...……112

Appendix C: Calculations for Composite Girders…………………...……..…..118

vi

List of Figures

Figure 1.1 - Stub-Girder System (Chien 1993)................................................................... 2

Figure 1.2 - Conventional and Modified End Details (Ahmad 1992) ................................ 5

Figure 1.3 - Girder-Slab System (Girder-Slab 2005a)........................................................ 6

Figure 1.4 - “Gooseneck” Connection Detail (Girder-Slab 2005b).................................... 8

Figure 1.5 - Typical AAC Floor System Details (Itzler 2004) ........................................... 9

Figure 1.6 - Possible Composite AAC Assembly Details (Adapted From Original

Proposal Figures (Itzler 2004)) ......................................................................................... 11

Figure 1.7 - Stiffened Plate Floor Deck (Hassett 2004).................................................... 13

Figure 1.8 - End Connection (Hassett 2004) .................................................................... 13

Figure 1.9 - Connection of Modules (Hassett 2004)......................................................... 14

Figure 1.10 - Staggered-truss System (Scalzi 1971)......................................................... 15

Figure 1.11- Transfer of Lateral Loads to Trusses (Scalzi 1971) ..................................... 16

Figure 1.12 - Example of Dry Floor System (Adapted from (Newman 1966)) ............... 19

Figure 1.13 - Gypsum-Plank Details (Fang 1968)............................................................ 19

Figure 1.14 - Composite Steel Joist System (Samuelson 2002) ....................................... 20

Figure 1.15 - Composite Joist Flexural Model (Adapted From (Samuelson 2002)) ........ 21

Figure 1.16 - Composite Girder System with Open-Web Joist Framing (Rongoe 1984). 23

Figure 1.17 - Example of End Fittings and Node Complexity (El-Sheikh 1993)............. 25

Figure 1.18 - Top Joint Shown (Composite Option) and Member Shear Stud (El-Sheikh

2000) ................................................................................................................................. 26

Figure 1.19 - Layout of Catrus Truss Model (El-Sheikh 2000)........................................ 27

Figure 2.1 - Non-Dominant Joist ...................................................................................... 31

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Figure 2.2 - Dominant Joist .............................................................................................. 33

Figure 2.3 - Connection Elements .................................................................................... 34

Figure 2.4 - Connection Detail.......................................................................................... 34

Figure 2.5 - Phase 1 of General Erection Sequence.......................................................... 35





Figure 3.1 - Plate Geometry.............................................................................................. 42

Figure 3.2 - Moment Distribution in a Simply-Supported Square Plate........................... 45

Figure 3.3 - Variation of Positive Moment Across the Width of Critical Sections

Assumed in Two-Way Concrete Design (Adapted From (Nilson 1991)) ........................ 46

Figure 3.4 - Sample of Moment Coefficient Table (Adapted from (Nilson 1991)) ......... 47

Figure 3.5 - K-Series Joist Selection................................................................................. 48

Figure 3.6 - Joist Self-weight............................................................................................ 49

Figure 3.7 - Frame Element Internal Forces and Moments (Adapted from (CSI 2004)) . 51

Figure 3.8 - Axes about Which Buckling Can Occur ....................................................... 52

Figure 3.9 - Reduction of K Factor Due to Slab Presence................................................ 53

Figure 3.10 - Traditional Joist Model ............................................................................... 55

Figure 3.11 - Traditional 16K9 Member Numbering ....................................................... 56

Figure 3.12 - Axial Force Distribution in a the 16K9 Model ........................................... 56

Figure 3.13 - Moment Distribution in a the 16K9 Model................................................. 56

Figure 3.14 - Preliminary System Designs ....................................................................... 61

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Figure 3.15 - Preliminary Joist Intersections .................................................................... 62

Figure 3.16 - Plan View of Proposed Design ................................................................... 63

Figure 3.17 - Proposed System’s Joist Models................................................................. 64

Figure 3.18 - 3D View of SAP2000 Proposed System Model ......................................... 65

Figure 3.19 - String Action Axial Forces.......................................................................... 66

Figure 3.20 - Deformed Panel........................................................................................... 68

Figure 3.21 - Axial Force Distribution in Proposed System............................................. 68

Figure 3.22 - Spring Analogy for Non-Dominant Joists................................................... 70

Figure 3.23 - Moment Distribution in Proposed Model ................................................... 71

Figure 3.24 - Moment Distribution in a Dominant Joist with Additional Web Members

Removed ........................................................................................................................... 72

Figure 3.25 - Cruciform Connection................................................................................. 74

Figure 3.26 - Alternate (Welded) Connection .................................................................. 75

Figure 3.27 - Chosen Connection Detail........................................................................... 77

Figure 3.28 - Connection Loads........................................................................................ 77

Figure 3.29 - Bolt Location............................................................................................... 78

Figure 3.30 - Load Cases .................................................................................................. 81

Figure 3.31 - Variation in Maximum Moment ................................................................. 81

Figure 3.32 - Reactions from Proposed System................................................................ 82

Figure 4.1 - Partially Composite System .......................................................................... 87

Figure 4.2 - Fully Composite Two-Way Joist System...................................................... 88

Figure 4.3 - Fully Composite System Using Two-Way Steel Decking............................ 90

Figure 4.4 - Supporting Girder Moments ......................................................................... 91

ix

Figure 4.5 - Proposed FE Model....................................................................................... 93

Figure 4.6 - Composite Girder Details.............................................................................. 94

Figure A.1 - Joist Labeling ............................................................................................. 107

Figure B.1 - Traditional Girder Loading......................................................................... 113

Figure B.2 - Girder Supporting the Dominant Joists ...................................................... 114

Figure B.3 - Girder Supporting Non-Dominant Joists.................................................... 116

Figure B.4 - Moment Diagram of Girder Supporting Non-Dominant Joists .................. 117

Figure C.1 - Composite Section...................................................................................... 119

Figure C.2 - Girder Loads............................................................................................... 119

Figure C.3 - Loading on the Girder Supporting the Dominant Joists............................. 123

Figure C.4 - Loading on the Girder Supporting the Non-Dominant Joists..................... 128

x

List of Tables

Table 3.1 - Top Chord Results for a Traditional 16K9..................................................... 57

Table 3.2 - Bottom Chord Results for a Traditional 16K9 ............................................... 58

Table 3.3 - Web Tension Member Results for a Traditional 16K9 .................................. 58

Table 3.4 - Web Compression Member Results for a Traditional 16K9.......................... 58

Table 3.5 - Joist Self-weight ............................................................................................. 73

Table 3.6 - Girder Selection.............................................................................................. 84

Table 4.1 - Resulting Girder Sizes.................................................................................... 95

Table 5.1 - Summary of Final Floor Panel Configurations .............................................. 98

Table 5.2 - Comparison of System Self-weights .............................................................. 99

Table A.1 - Chord Member Design for Outer Non-Dominant Joists ............................. 108

Table A.2 - Chord Member Design for Outer Non-Dominant Joists ............................. 108

Table A.3 - Chord Member Design for Dominant Joists................................................ 109

Table A.4 -Top Chord Results for Dominant Joists (Double Angle) ............................. 109

Table A.5 - Web Tension Members (Outer Non-Dominant Joists)................................ 110

Table A.6 - Web Tension Members (Inner Non-Dominant Joists) ................................ 110

Table A.7 - Web Tension Members (Dominant Joists) .................................................. 110

Table A.8 - Web Compression Members (Outer Non-Dominant Joists)........................ 111

Table A.9 - Web Compression Members (Inner Non-Dominant Joists) ........................ 111

Table A.10 - Web Compression Members (Dominant Joists) ........................................ 111

Table C.1- Components of a Fully Composite Girder (Traditional System).................. 120

Table C.2 - Deflections Along Various Load Stages (Traditional System) ................... 121

Table C.3 - Plastic Section Components (Traditional System) ...................................... 122

xi

Table C.4 - Components of Fully Composite Girder (Supporting Dominant Joists) ..... 124

Table C.5 - Deflections Along Various Load Stages (Girder Supporting Dominant Joists)

......................................................................................................................................... 125

Table C.6 - Plastic Section Components (Girder Supporting Dominant Joists)............. 127

Table C.7 - Components of Fully Composite Girder (Supporting Non-Dominant Joists)

......................................................................................................................................... 129

Table C.8 - Deflections Along Various Load Stages (Girder Supporting Non-Dominant

Joists) .............................................................................................................................. 129

Table C.9 - Plastic Section Components (Girder Supporting Non-Dominant Joists)..... 131

1

Chapter 1 Introduction and Literature Review

Section 1.1 Introduction

When floors are being designed in commercial or residential settings, there has to be

consideration on the part of the designer to identify the impact that the design will have

on the entire building. The depth of floor design chosen may profoundly affect the

overall height of mid- to high-rise buildings. Therefore, a desire to minimize floor depths

in buildings, based on innovative floor assemblages, is present for both architects and

engineers.

The definition of the phrase “floor-to-floor” height is the distance from the top of

a floor to the top of the next floor. This height is made up of three main components: the

depth of the floor construction, the “sandwich,” and the distance from the top of the floor

to the top of the architectural ceiling. The “sandwich” area contains parts of the

electrical, communication, fire protection, HVAC (HVAC ductwork takes up the most

space), and plumbing systems. It should also be noted that the depth of construction

includes floor slab thickness and the depth of the supporting members. Designers should

attempt to individually minimize one of the three main components or allow them to

share functions in the same space (Kirmani 2000).

The structural methodologies that attempt to minimize floor-to-floor heights,

which are reviewed in this paper, will be limited to steel systems or systems compatible

with structural steel framing. The construction topics discussed include stub-girders,

Girder-Slab, two recent AISC competition proposals, staggered-truss, composite steel

joist, and space truss floor systems. While each system offers a unique set of advantages,

2

it is the author’s opinion that an additional design alternative should be investigated.

This new alternative is a two-way steel floor system using open-web joists.

Section 1.2 Stub-Girder System

One floor system capable of minimizing floor-to-floor heights is the stub-girder system

(Figure 1.1). Colaco (1972) developed the stub-girder floor system to address the

problems that conventional floor framing systems had in accommodating mechanical

ducts. A traditional system places the duct-work under the supporting beams, and in

some circumstances, penetrations are made in the beams or girders. In a stub-girder

system, duct-work can be incorporated between the girders and the deck-slab.

Figure 1.1 - Stub-Girder System (Chien 1993)

Stub members are inserted into the interstitial space between the main girders and

a floor slab. The stubs are welded to the main girders and connected via shear

connections to the concrete slab. Floor beams run transversely on top of the beams,

carrying the weight of the concrete slab. These floor members are designed as cantilever

beams and drop-in beams (or “Gerber” members) are inserted to fill the discontinuity.

3

The stubs provide a relatively small addition of material that increases the

distance between the compressive concrete slab and the supporting girder. This increase

leads to a greater moment of inertia of the stub girder to resist bending (Wang 1995).

Interestingly, Chien (1993) notes that because tension and shear generally govern the

girder sizing, its required depth is not particularly span dependent (unlike conventional

systems).

Colaco (1972) concluded in his original paper that the advantages of the stub-

girder system when compared to a conventionally framed panel are:

1) A reduction in steel required in the girder due to the greater depth.

2) A reduced amount of steel in the floor beams due to continuity. There is also a

simplification of the end connection details of the floor beams due to lower shear

values.

3) An estimated 25 % reduction in the structural steel in the floor and approximately

15 % of the structural cost of the floor system.

4) A drop in total depth of approximately 8 in. between the top of the slab and the

ceiling. This results in a lower floor-to-floor height and additional material

savings in the exterior window wall system for the building.

Chien (1993) reflects upon several changes that have been made to the original

proposed stub-girder system, namely:

- Reduction in girder depth.

- Use of partial-height end plate stiffeners rather than full-height fitted stiffeners or

elimination of stub stiffening by using them only when required.

4

- Reduction in stub welding.

- Increased emphasis on slab reinforcement.

- The truncation of girder bottom chord to accommodate services near supports.

To gain perspective on typical system dimensions, Chien (1993) explains that in

Canadian construction, stub-girders span about 12 m (39.4 ft.), often from core to exterior

wall in conventional office buildings. The secondary floor beams would then span 8-12

m (26.2 – 39.4 ft.), with 9 m (29.5 ft.) being the most common span length. Floor beams

typically range in depth from about 0.3 to 0.5 m (12.2 to 18.1 in.), placed on 2.5 to 3.5 m

(8.2 to 11.5 ft.). This, of course, depends upon the structural module and deck span. A

typical deck-slab system consists of a 75 mm (3.0 in.) deep wide-rib profile deck with

approximately 75 mm (3.0 in.) of normal density (ND) concrete or 85 mm (~ 3.5 in.) of

semi-low density (SLD) concrete on top.

Two accepted methods of modeling a stub-girder include using either a finite-

element analysis or modeling the system as a Vierendeel Truss. In the Vierendeel truss

model, the deck-slab serves as a flexural-compression top-chord, while the full-length

steel girder acts like a flexural-tensile bottom-chord. The steel stubs, in turn, serve as the

shear stubs in the Vierendeel girder (Chien 1993). Refer to Wang (1995) for a

description of a nonlinear ultimate strength analyses, using both the vierendeel and finite

element methods.

As mentioned earlier, one of the changes to the original stub-girder system is the

truncation of the girder near the supports (Figure 1.2). This modification provides

several advantages over the traditional detail including (Ahmad 1992):

5

1) Crack control of the concrete slab is provided near the column connection.

2) More room available for stud connection, eliminating an overcrowding of shear

studs.

3) A wider duct space is introduced below the end stub at the girder end. This

consequentially provides a bigger space for utility services such as fire sprinkler

mains.

4) An end detail offering an option of using a single-angle connection between the

column and deeper end stub, therefore eliminating any coping required in a

conventional end detail.

Figure 1.2 - Conventional and Modified End Details (Ahmad 1992)

Modifying the end detail creates some structural concerns. If the stub-girder

configuration is being designed for negative moment over the columns, the conventional

stub-girder detail would be more appropriate due to the drop in moment of inertia of the

modified stub-girder detail at the ends (Ahmad 1992). Another concern is the welded

connection between the bottom chord and the extended end stub. Stress concentration in

this area can lead to connection failure. Test results demonstrated that performance of

6

this connection was excellent during the ultimate load stage; however, the authors

cautioned that further finite-element analysis should be performed (Ahmad 1992).

Section 1.3 Girder-Slab

Girder-Slab is another relatively recent floor system in which engineers set out to

minimize floor-to-floor heights. The idea behind its development is to use a prestressed,

precast concrete plank and steel system (Figure 1.3) that could replace plank and bearing

wall construction (Naccarato 2000). More specifically, the technology creates a

monolithic structural assembly using precast hollow-core slabs with an integral steel

girder (Cross 2003).

Figure 1.3 - Girder-Slab System (Girder-Slab 2005a)

In 1993, there was a revision to the Building Officials and Code Administrators

(BOCA) Seismic Section that deterred the use of block bearing walls in low- to mid-rise

structures. In light of these changes, Constanza Contracting Company, Fisher Steel, and

O’Donnell & Naccarato Inc. joined together and created Girder-Slab Technologies, L.L.C

(Naccarato 1999). Initial testing of the Girder-Slab system used a dissymmetric cross-

7

section consisting of a lower tee cut from a W8x40 and an upper tee from an S3x7.5.

After testing, it was concluded that composite action was developed between this section,

the grout, and the precast slab. Formal independent testing followed at Drexel

University, where it was decided that by castellating a W-section (W10x49 in this case)

and welding a 1 in. x 3 in. continuous flat bar (serving as the upper flange), a

dissymmetric beam could be derived to serve as a more efficient steel bearing member

for precast slabs (Naccarato 2001).

Referring to Figure 1.3, the dissymmetric beam (known as the D-BeamTM Girder)

acts compositely with the concrete planks, enabling the floor to better support residential

live loads. The composite action is accomplished by first breaking small 8 in. sections

called “knockouts,” scribed at the end of each plank core. Next, debris from this action is

shoved back into the core to form a dam. Then, reinforcing bar is run through the open-

web of the D-Beam, placed into the hollow-core openings, and grouted into place (Cross

2003).

The resulting slab thickness of this structural arrangement is 8 in. An 8 in. slab is

supported by a DB-8TM girder while a DB-9TM girder is necessary if an additional 2 in.

concrete topping is added. Slab span lengths for this system are reported to span as much

as 28 ft. (Cross 2003). The steel girder spans are shorter, with lengths of up to 15 ft. If a

girder span increase is necessary due to the building layout, a “gooseneck” connection

detail can be introduced at the columns (Figure 1.4). By welding the connection to the

columns, a span increase of up to 20 ft. or 22 ft. is possible (Veitas 2002).

8

Figure 1.4 - “Gooseneck” Connection Detail (Girder-Slab 2005b)

In addition to being able to provide structural bays in the proximity of 20 ft. by

28 ft., developers of the Girder-Slab system state that there are other advantages of this

floor system. Naccarato (2001) contends that the typical structural arrangement provides

an 8’-8” minimum floor-to-floor height (8’-0” floor-to-ceiling height can be maintained

because the slab underside can become a finished ceiling) and the slab is non-

combustible. Cross (2003) reports that, in residential construction, Girder-Slab’s

efficiency can lead to a 25% reduction in the construction schedule and has the capability

of maintaining equivalent floor-to-floor heights with cast-in-place concrete construction.

Section 1.4 AISC Multi-Story Residential Construction Competition

In March of 2004, the American Institute of Steel Construction (AISC) held a

competition in which participants entered innovative floor system solutions that

addressed minimizing floor-to-floor heights (AISC 2004). The systems needed to consist

of structural steel or be compatible with structural steel framing. The following two

sections of this report are summaries of the two award winning proposals.

9

Section 1.4.1 Structural Steel/Autoclaved Aerated Concrete (AAC)

Composite Floor System

The first place prize recipient in the AISC competition was Itzler (2004). The idea of this

proposal is to develop a floor system where autoclaved aerated concrete (AAC) floor

panels and structural steel act compositely to provide a light weight solution to

minimizing floor-to-floor heights. Dr. Axel Eriksson, in Sweden, invented the building

material in 1924, while its introduction to the United States took place around the early

1990’s. AAC has been used in precast floor and roof panels for many years. However,

the idea of this material acting compositely with structural steel to achieve floor spans of

up to 40 ft. is a new concept.

A typical section illustrating an AAC floor system is shown in Figure 1.5. The

main advantage of choosing AAC over conventional stone concrete lies in the self-weight

of the material.

Figure 1.5 - Typical AAC Floor System Details (Itzler 2004)

AAC has a 73 to 77 % lighter specific weight than that of stone concrete (assuming 150

lb/ft3). This difference in construction self-weight becomes apparent in typical floor

10

construction. An 8 in. precast hollow core slab weighs 70 psf, while an 8 in. AAC slab

weighs only 30 psf.

Schematic studies have demonstrated economical benefits of choosing AAC

construction over other residential building slab systems. For instance, the reduced dead

loads can create savings when sizing foundation members (e.g. piles or drilled piers),

transfer girders, and columns. Significant reduction in dead loads also inherently reduces

the seismic loading on the lateral load resisting system of a building. Additionally, AAC

construction can limit requirements for building insulation (in certain climatic conditions)

because the material’s “R” values are much higher than those describing normal weight

concrete.

Figures demonstrating the various ways in which a composite assemblage can be

detailed are shown in Figure 1.6. Composite action would eliminate the need to field

weld AAC floor panels to the structural steel. The proposal states that both composite

action between concrete topping and AAC panels, as well as composite action between

structural steel and topped AAC panels, will need to be tested to determine performance

ability. The proposal goes on to suggest that by assuming composite action, supporting

steel beams placed between 14 ft. and 20 ft. on center would likely allow the floor system

to achieve 40 ft. clear spans (one of the AISC competition requirements). The

corresponding depth of the floor construction could be as low as 12 in. plus concrete

topping (assuming assemblage Detail IV or V in Figure 1.6).

11

Figure 1.6 - Possible Composite AAC Assembly Details (Adapted From Original

Proposal Figures (Itzler 2004))

Important characteristics of AAC in current construction that would also be

applicable to a composite system include, but are not limited to:

12

1) AAC panels are easy to set in place with small cranes or forklifts, and in some

cases, by hand.

2) Field modifications can easily be made by cutting, drilling, or routing with

wood tools. As an example, a 6 in. diameter hole can be cut or drilled in a two-

foot wide panel without the addition of special reinforcement.

3) AAC panels typically operate at relatively low stresses, reducing vibration

concerns. Adding concrete topping would further reduce the vibrations in the

proposed composite floor system.

4) Full-scale testing of an AAC Structure (at the University of Texas at Austin)

revealed that untopped AAC floors can provide adequate diaphragm strength

and stiffness in typical residential applications.

5) AAC has excellent sound insulation properties. Sound Test Reference (STC)

ratings as high as 51 for 8 in. thick AAC panels are comparable to systems

consisting of multiple layers of drywall and double stud construction.

6) AAC panels require no fire protection and can serve as finished ceilings once

visible joints are skim coat plastered. With a topping added in the proposed

system, a high quality, level floor can also be provided.

7) Mechanical, electrical, and plumbing (MEP) conduit can be placed in the grout

keys, or panels can be routed to accept the conduit.

Section 1.4.2 Stiffened Plate Floor Deck

The second place prize recipient in the AISC competition was Hassett (2004). This

proposal describes a floor system composed of closed ribs welded on the underside of a

thin plate (Figure 1.7). The 1/4 in. bent plate flutes help in stiffen the 3/8 in. steel floor

13

deck. Shear flow stresses are low at the rib to plate junction, so the fillet or partial joint

penetration (PJP) welds could be a continuous or intermittent AISC minimum weld size.

The space between the ribs could accommodate electrical and plumbing conduit. The

total steel depth would be approximately 10 in. and the modules could be fabricated in 7’-

6” x 40’ units.

Figure 1.7 - Stiffened Plate Floor Deck (Hassett 2004)

The floor system has the advantage of allowing the top of steel (TOS) of the floor to be

the same as the TOS of the supporting steel girder. Gravity load transfer and diaphragm

shear transfer are accomplished at the ends via welding seat angles (Figure 1.8).

Figure 1.8 - End Connection (Hassett 2004)

14

If field modifications had to be made, workers could accomplish the changes by

field burning or field welding. Along the length of the floor modules, shear is transferred

by welding flat bars as shown in Figure 1.9.

Figure 1.9 - Connection of Modules (Hassett 2004)

In addition to the ribs and the plate, gypcrete is placed on top of the plate to

provide a finished surface as well as sound and vibration dampening and fire rating. On

the bottom of the floor arrangement, “hat” or “z” channels could be spot welded or

screwed to the flutes to accept drywall and to furnish additional sound dampening.

Finally, 5/8 in. gypsum board would provide more sound and vibration dampening as

well as fireproofing for the underside of the floor. For a 40 ft. span, the total floor

thickness would become approximately 13 in. With a camber of less than one inch in 40

ft., the D + LL deflection is 1.6 in., which is less than L/240 = 2 in.

Section 1.5 Staggered -Truss

The staggered-truss is another building system that can minimize floor-to-floor heights.

It is a double-planar system of steel framing developed by a team of architects and

engineers from the Departments of Architecture and Civil Engineering at M.I.T. The

system consists of exterior columns supporting story-deep trusses spanning the full

15

transverse width of the building on adjacent column lines (Figure 1.10). The real benefit

of a truss system is that the entire building weight is mobilized in resisting the

overturning moment (Cohen 1986).

Figure 1.10 - Staggered-Truss System (Scalzi 1971)

The paths that the lateral loads follow in the transverse direction, to reach the

foundation, make the system very unique. The staggered-truss system will collectively

behave like a cantilever beam under this loading. Lateral load collected from the

building cladding will make its way to the floors. The floors mimic deep beams, or

diaphragms, taking half of the lateral load on each adjacent side of a truss and

transferring this loading to the top chord of that truss (Figure 1.11). The truss is now

acting like a shear wall and, in the absence of another truss directly below, will send the

loading back through the floor system attached to its bottom chord. This back and forth

pattern continues until the loading has made its way to the lateral resisting system at the

building’s base. Building drift becomes a function of slab and truss stiffness and column

cross-sectional areas (Cohen 1986). The three main components of the system, namely

the columns, trusses, and floors, will be discussed in detail.

16

Figure 1.11- Transfer of Lateral Loads to Trusses (Scalzi 1971)

Columns

The task of the column members is to resist wind loading in both the longitudinal and

transverse directions as well as to support gravity loads. A nice feature of the building

layout is that, with two column lines, it may only be necessary to pour two strip footings

(Scalzi 1971). In the longitudinal direction, columns should be oriented about their

strong axis and rigidly attached to the spandrel beams. Drift is therefore controlled by

either portal frames or braced bays (Cohen 1986).

Trusses

Trusses in the staggered-truss system must span the total dimension of the building,

resisting both gravity and lateral loads. Two examples of truss styles include the

Vierendeel and the Pratt. Design details should provide the truss with an opening, near

center span, to serve as a corridor. Secondary bending moments, due to panel shear in

these openings, must be evaluated to aid the design of the chords and web members. Care

must be taken to restrict the width of the chord members to allow minimum wall

17

dimensions while still providing adequate bearing for the floor construction. It should be

noted that, in a simplistic sense, all truss members are subjected to axial forces only,

allowing designers to take advantage of using high strength steel members (Scalzi 1971).

In the field, the trusses are connected at both chords, thus local bending (at the

bottom chord connection) occurs in the columns. However, research has shown that this

bending is usually negligible; however, a computer analysis should verify that decision

(Scalzi 1971). Posts and hangers placed on the panel points of trusses can provide

additional loading when locations in the building prohibit the placement of a truss in the

typical staggered arrangement (Scalzi 1971).

Floors

The floor systems can consist of steel deck with infill concrete, steel joists with concrete

topping, concrete slabs, or concrete planks, provided that the system can adequately

handle the gravity and lateral loads from wind. In terms of gravity loading, the floor

systems can be modeled series of continuous spans or simple spans over two column

spacings. Code requirements may allow a reduction in the design live load because large,

clear spans are typically present.

Floor panels are also subjected to lateral loads and must exhibit enough in-plane

diaphragm strength and stiffness to transfer these lateral loads to the trusses. Direct

welding (if a steel deck is used) or welded shear plates (if concrete slabs or planks are

used) are used as shear connections to transfer the in-plane shear. In some instances, the

height of the building may be limited by the shear capacity of the floor. Mechanical

requirements may also dictate the type of floor system to be used. However, floor depths

18

can be minimized because the floor spans may be short bay lengths, providing two

column bay spacings for room arrangements (Scalzi 1971).

Section 1.6 Open-Web Steel Joist Construction

The first open-web steel joist, fabricated in 1923, was a Warren truss configuration. The

top and bottom chords were round bars with the web of the joist formed from a

continuous, bent bar. In 1928, the first specifications were adopted, and in the following

year, the first load table was introduced (SJI 2005).

Section 1.6.1 Dry Floor Construction

An early attempt to develop an innovative steel floor system, using open-web steel joist

to compete with flat plate concrete slab construction, was the Dry Floor system (Figure

1.12). A Dry Floor system was proposed to address problems in high-rise apartment

construction including: structural borne sound, noise transmission, impact noise, interior

partition cracking, seasonal limitations, labor-material balance, and economics. For a

quantitative description on the performance of the Dry Floor system in these categories,

refer to Newman (1966).

The steel bar joists were capable of acting compositely with the steel-edged,

gypsum planks to obtain sufficient diaphragm strength to resist lateral loads. Full scale

testing of a two-bay portion of floor area (between two column lines), in a typical

apartment building, was conducted at the U.S. Steel Applied Research Laboratory

19

Figure 1.12 - Example of Dry Floor System (Adapted from Newman (1966))

(Fang 1968). The gypsum planks used were precast units 2 in. thick, 15 in. wide, and 10

ft. long. The edges of the planks were reinforced with 22 ga. galvanized-steel tongue-

and-groove edges to form mating joints (Figure 1.13). The results of the testing

concluded that a very small magnitude of deflections resulted from the testing of the

gypsum planks in place. Furthermore, it was apparent that the gypsum deck provided

nearly all of the resistance to horizontal movement with very little shear contributed by

the stiffness of the frame (Fang 1968).

Figure 1.13 - Gypsum-Plank Details (Fang 1968)

20

Section 1.6.2 Composite Steel Joists

With an increase in the availability of steel decking, concrete slabs, supported by ribbed

steel decks bearing on the joists, became the mainstream open-web steel joist floor

system. In an attempt to further minimize floor-to-floor heights, the concept of a

composite steel joist system has been introduced by the joist industry (Figure 1.14).

Figure 1.14 - Composite Steel Joist System (Samuelson 2002)

The term “composite” implies that the joist top chord and overlying concrete slab will act

as an integral unit once the concrete has cured. The main components of the system are

the steel joists, metal deck, and concrete slab (encasing welded wire fabric). The joists

are made of hot-rolled or cold-formed steel. Welded shear studs or specially designed

truss top chords must be provided to ensure adequate transfer of shear; this allows the

concrete slab to act as a compression flange. In industry, shear connections include

(Samuelson 2002):

- Specially rolled cold-formed steel “s” shaped top chords (Hambro 2005).

- Specially embossed back-to-back double angle top chords (Vescom 2005).

- Shear studs welded through the metal deck (Canam 2005), (SMI 2005), and

(Vulcraft 2005).

21

The composite action between the slab and joists, not only increases the depth of

the cross-section, but also increases the size of the resisting compression flange. The

effective width, “be” shown in Figure 1.15, is the sum of the effective widths of each side

of the joist center-line, each of which shall not exceed the smallest of: 1/8 of the joist

span (center-to-center of supports), one-half the distance to the center-line of the adjacent

joist, or the distance to the edge of the slab (AISC 2001b). It should be noted that in a

composite cross-section, the top chord of the truss contributes little to the moment

capacity of the section (much like compression reinforcement in doubly reinforced

concrete beams). The force balance model, present at ultimate loads, becomes very

similar to the ductile failure model used in reinforced concrete design (Figure 1.15).

Figure 1.15 - Composite Joist Flexural Model (Adapted From (Samuelson 2002))

Research has shown that joint eccentricity, as well as web shear deformation,

reduces the theoretical moment of inertia of a joist. When the span-to-depth ratio of a

joist is approximately 18, the full non-composite moment of inertia should be multiplied

by 0.85 (Samuelson 2002). Full-scale joist load tests (Murray 1997) demonstrated how

this adjustment factor can vary from 0.5 to 0.9 when the span-to-depth ratio equals 6 and

24 respectively.

22

Since the mid 1960’s, many research efforts have been made in testing composite

joists (Samuelson 2002). Among this list of testing includes the work of: Lembeck

(1965), Galambos (1970), Atkinson (1972), Atkinson (1972), Azmi (1972), Robinson

(1978), Leon (1987), Curry (1988), and Brattland (1992).

Quantifying the benefits of composite joists in a standardized manner is a

relatively recent task. In 1996, the ASCE Task Committee on Design Criteria for

Composite Structures in Steel and Concrete published a “Proposed Specification and

Commentary for Composite Joists and Composite Trusses (ASCE 1996).” Design topics

in this proposal include: the design of the top chord, bottom chord, web elements, and

shear connections as well as flexural capacity calculations and serviceability criteria

(Samuelson 2002). Finally, an inaugural Steel Joist Institute specification publication

date is anticipated to occur in the near future.

Depending on the type of project, benefits from choosing composite over non-

composite joists may include (Samuelson 2003):

- A more efficient and stiffer composite design makes it possible to support a given

load with a shallower joist.

- Weight savings from the joist design reduces building costs.

- Simplified erection, faster connections, and minimal crane lifts occur. With fewer

and simpler connections, ironworkers don’t have to align a large number of bolt

holes.

- Large column-free areas give tenants maximum flexibility on floor layouts.

Composite joists have been used successfully in floors with spans exceeding 100’.

23

- Customized composite joist designs can be created for any given loading and

serviceability requirements.

The two main advantages of composite construction are the high speed of

construction and the economy of composite joists at long spans. Samuelson (2003)

explained that using composite joists for spans around 35 ft. to 45 ft. or longer definitely

demonstrate economical construction.

Section 1.6.3 Composite Girders

In addition to a composite steel joist design, the supporting girder of a joist supported

slab system can also be composite (Figure 1.16). When open-web joists bear on a

supporting girder, the girder flange is not in direct contact with the concrete slab, rather

they are separated by a distance equivalent to the joist seat depth (usually 2 1/2 in). In a

system illustrated by Rongoe (1984), this void is filled intermittently with a tee connector

welded to the girder flange. These pieces, which are analogous to stubs in a stub-girder

system, provide a surface to which shear studs can be welded. Therefore, the system

employs the economics of composite action between the concrete slab and the steel

girder.

Figure 1.16 - Composite Girder System with Open-Web Joist Framing (Rongoe

1984)

24

Based upon a full-scale load test load test performed at the Berlin Construction

Company, Rongoe (1984) concluded that:

- Lower floor-to-floor heights are achieved by dropping the girder depths on the

order of 4 to 6 in. for girders spans of 20-35 ft.

- Girder weights are lowered by replacing non-composite girders with composite

girders (aW18 x 35 was reduced to a W14 x 22 in this test).

- Composite action created a stiffer floor system, diminishing floor vibrations.

- A cost analysis comparing three different 30 ft. girders configurations confirmed

that a cheaper installation cost is obtained.

- Special fittings, techniques, and training workmen are not required.

- Several combinations of connectors and studs are possible to meet material,

equipment, labor, and local regulations requirements.

Section 1.7 Space Trusses

Space trusses can serve as floor systems, consisting of highly indeterminate three-

dimensional lattice networks. This type of floor system relies upon disciplined member

repetition and geometric modularity in order to span long distances in two directions.

Therefore, this system, though not as common as the other floor systems mentioned in

this report, offers a designer the opportunity to create a steel floor system that spans in

two directions.

Generally speaking, problems have become apparent with these types of

structures. In non-composite trusses, there is the possibility of brittle failure caused by

the successive buckling of a series of critical compression-chord members. A

25

progressive collapse mode of failure can be attributed to residual forces that are a result

of member lack-of-fit, experimental scatter in peak loads of compression, and the

stiffness of the member-node joints (El-Sheikh 1993). A study conducted by El-Sheikh

(1993) concluded that forming a composite concrete top chord was more effective than

providing overstrengthened top chord members in order to improve failure behavior.

Another problem with space structures is the high cost, that results from often using

contributed by the often complicated node connectors and member end fittings in

assemblage (El-Sheikh 2000). An example of such a connection, used in the testing done

by El-Sheikh (1993), is shown in Figure 1.17.

Figure 1.17 - Example of End Fittings and Node Complexity (El-Sheikh 1993)

An innovative space truss system proposed to provide low cost and sound

structural behavior is the Catrus Space Truss (El-Sheikh 2000). Three main features of

the system’s jointing arrangement include:

- Continuous top and bottom chord members are located across the joints.

- Single bolts are used to directly bolt together the members (eliminating the use of

node connectors).

26

- Diagonal and chord members are stacked above each other, thus producing joint

eccentricity, but allowing chord member continuity.

Member sections include rectangular hollow sections (top chords), flat bars (bottom

chords), and circular hollow sections (diagonal members). As shown in Figure 1.18, the

composite option has a top nut and bolt that combine to serve as a shear stud encased in a

concrete deck. Not shown is the non-composite top joint detail and an alternative deck

detail using timber boards. Benefits of this system over traditional space trusses include:

- Simple jointing using direct bolting instead of complicated node connectors

leading to an easy fabrication and erection process.

- High strength/weight and stiffness/weight ratios.

- Ductile failure.

- Adequate ability to cover areas with different sizes with flexibility of support

locations.

- Easy attachment of cladding and false ceilings.

Figure 1.18 - Top Joint Shown (Composite Option) and Member Shear Stud (El-

Sheikh 2000)

27

An experimental program conducted between 1994 and 1996, assessed five

different models of the Catrus system (El-Sheikh 2000). Truss models with overall

dimensions of 4 x 4 x 0.6 m were used, as shown in Figure 1.19. The results

demonstrated that the system distributed the forces away from affected areas

exceptionally well while retaining good joint stability. Noticeable sagging and top chord

deformations supplied clear, ample warning of failure in all tests. Furthermore, benefits

of composite action included higher strength and stiffness with better overall ductility.

The composite action also provided economical savings in truss top chord members,

while preventing buckling of these top members.

Figure 1.19 - Layout of Catrus Truss Model (El-Sheikh 2000)

28

Section 1.8 Synthesis of Past Literature and Direction for Present

Research

Each of the systems previously discussed offers an inventive way to minimize floor

depths in structural steel framing. However, each system comes with its own short-

comings. The author believes that the biggest problem encountered with systems that

strive to achieve large span-to-depth ratios is the accumulation of undesirable excessive

member self-weight. This is especially true for the systems that rely on concrete plank

construction. These systems also have trouble achieving spans beyond 30 feet. Systems

that become much lighter, namely composite open-web steel joist systems, still have

problems because they predominantly span in one direction. The two supporting girders

likely set the floor-to-floor height of this construction arrangement unless a prodigious

price is paid to select a shallower member size.

In this thesis, the author will attempt to devise a steel floor system that contains

the following characteristics:

- Large span-to-depth ratios.

- Low member self-weight.

- Unshored construction.

- Load distribution that spreads predominantly in two directions.

- Ample interstitial space available for MEP systems.

- Easy on-site assembling.

A system with these characteristics can be achieved by slightly modifying the fabrication

of open-web steel joists to obtain a system in which two directions of joists are oriented

orthogonally with respect to one another in a given floor bay. This unorthodox approach

29

of using steel joists results in structural behavior that allows a reduction in supporting

girder depths, which in turn reduces the floor-to-floor heights of a given story. This

thesis will show a detailed construction sequence as well the structural feasibility of the

proposed system.

30

Chapter 2 Proposed System Fabrication and Erection


Fabricating open-web steel joists is a labor-intensive assembly line process, but is

necessary to create a system in which each individual member is efficiently used. This is

especially true for the web members (round bar or crimped angles). The reduction in a

joist’s web material, compared to the amount of web material present in a rolled W-

section, for example, is significant, but comes with the price of individually welding each

web diagonal to the chord “flange” members. Through years of experience,

manufacturers have created assembly processes that have evolved, using techniques that

efficiently assemble joists. As a result, less effort is needed in the field to install joists.

The author’s goal is to continue this fabrication philosophy and take joist efficiency into a

new phase of two-way design.

The design floor bay investigated in this thesis is 30 ft. x 30 ft. with 55 psf DL

and 65 psf LL (typical office loadings). A 4 in. thick slab using 1.5 VL steel decking

(Vulcraft 2005) was selected to pass standard fire rating standards outlined in the current

steel manual (AISC 2001a). Minimizing the depth of construction within the panel was

the driving force used to select a joist configuration with an overall depth of 16 in. The

proposed system utilizes non-composite joist construction, but has promising composite

action capabilities (see chapter 4). A total of eight joists (four in each direction) are used

in the study. All joist chord members are double angles taken from cold-formed steel

shapes. The rationale behind these decisions is discussed in Chapter 3.

31

Section 2.2 Fabrication

There are two different types of joists used in the proposed system. One direction of

joists has dominant joists (16 in. depth), while the other direction consists of non-

dominant joists (14 in. depth). Both joist designs mimic standard K-series joist

dimensions and member sizes as closely as possible. Top and bottom chord members are

2L2x2 angles, and the web members are 3/4 in. diameter round bars (with the exception

of 7/8 in. diameter round bar used at the joist ends). Joist seat details (2 1/2 in. depth) in

the proposed system are the same as those describing a traditional configuration.

Figure 2.1 shows a non-dominant joist. The depth of 14 in. is 2 in. less than the

shallowest joist depth available in the standard selection tables (SJI 2005), given a span

of 30 ft.

Figure 2.1 - Non-Dominant Joist

A

A

32

This reduction in depth is necessary to facilitate the coexistence of the joist top chord

with the top chord present in the transverse (dominant joist) direction. A total of four

“special” panel points (10’-3” and 11’-10” inward from the joist ends) are needed along

the top chord to form this connection.

Inverting the bottom chord is vital to address clearance issues brought about by

intersecting the joists. This inversion reduces the chord’s section modulus, but is

necessary to ensure that the flanges of the non-dominant joist bottom chord do not come

into contact with the web members in the dominant direction. A distance between panel

points of 19 in. was selected because it reflects a typical panel dimension used when the

joist depth is 14 in. Increasing this panel dimension to 24 in. would cause an appreciable

lack of moment of inertia of the cross section. The joist manufacturer has the option of

cambering the non-dominant joists because both the top and bottom chord members are

continuous. The joist system in this study, however, did not take advantage of this

opportunity.

Dominant joist design (Figure 2.2) in the proposed system deviates much further

from traditional design than non-dominant joist design. The core of the member is

adapted from a 16K9 joist (a 24 in. panel length was maintained and the chord sizes were

very similar). The most pronounced adaptation is the discontinuous top chord member.

A cut in the top chords is made every 6 ft. to accommodate the non-dominant joists.

Additional web members are added within the vicinity of each cut; this includes

four small angles (2L1.5x1.5x0.113) and two vertical round bars (3/4” dia.). There are

two primary functions of the web angles welded to the outside of each chord. The web

33

angles are stiff enough to ensure that the non-dominant joist does not deflect an amount

great enough to cause contact between the bottom chords.

Figure 2.2 - Dominant Joist

The web angles also “calm” the moment distribution in the top chords by forming

triangles (discussed in section 3.5.4). It should be noted that the web angles are coped at

their upper ends to allow a fastening tool to enter unobstructed.

Figure 2.3 shows the different connecting elements needed to form the

orthogonal intersections of the two rows of joists. The piece shown in Figure 2.3a

is welded to the flanges of the non-dominant joist’s top chord (also see Figure 2.4). The

“C” channel formed by cutting a HSS cross section in half may have to be substituted

with another structural piece (possibly 3 plates welded together) if clearance becomes an

issue as the chord member size increases with load demand. The two chamfered plates in

A

A

Coping Of Angle Flange

34

Figure 2.3b serve as stiffening elements, restraining the vertical portions of the HSS

shape from acting as small cantilever beams. Finally, the plate in Figure 2.3c is

welded to the top chord of the dominant joist. Bolting is accomplished in the field, and

welding is done in the shop.

Figure 2.3 - Connection Elements

Figure 2.4 - Connection Detail

(a) (b)

(c)

6”

2”

CUT HSS SECT. 6”x6”x3/16”x5 3/4”

PLATE 5/8” A 325-N ASTM BOLT

3/16” PLATE W/ 3/8” CHAMFER 3/4” DIA. ROUND BAR WEB

3”

1”

E701/8

5 3/4”x3”x3/16”

Non-Dom. Joist

Dom. Joist

35

Section 2.3 Erection Sequence

The general panelized erection sequence begins with the delivery of the joists to the job

site. The dominant joists have discontinuous top chord members; therefore, temporary

restraint is provided at 6 ft. intervals. Otherwise, excessive lateral and torsional

deformations may take place during construction. The temporary restraints will likely be

sacrificial dowels placed through the bolt holes of the connection plates. When the joists

are picked up and moved, workers may elect to handle them “up-side-down” so that the

continuous bottom chord (now on top) is the member in compression.

A flat spot needs to be established on the job site (on the ground or perhaps on a

floor bay already formed in the building). The dominant joists are then arranged in a

parallel manner, held in place with some sort of jig (e.g. 2 x 4 framing) that inhibits roll-

over (Figure 2.5). Traditional lateral bridging could be attached at 6’ intervals to the

joists’ top chords during this phase of erection.

Figure 2.5 - Phase 1 of General Erection Sequence

36

Phase 2 of the general erection sequence entails removing the temporary top

chord restraints and setting the non-dominant joists into place (Figure 2.6). Four 5/8 in.

diameter bolts are fastened at each top chord intersection. The author feels that bolting is

faster and more economical than welding. It should be noted that a small vertical void

(on the order of 1/8 in.) exists between the two bottom chords at the joist intersections.

This demands that the load transfer from one joist direction to the other takes place only

through the top chords.


A

A

37

In phase 3 of the general erection sequence, the interlocked joists (together

weighing approximately 2800 lbs) are hoisted into the air with a crane and set onto the

awaiting steel girders (Figure 2.7). The absence of structural members in the corners of

the panels allows crane operators and iron workers to easily maneuver the system. The

joist seats are either bolted or welded to the girders in a manner no different from the

manner in which traditional open-web joists are connected.


A

A

Upper Chords

Lower Chords

38

Steel decking is welded into place in phase 4 of the general erection sequence. A

contractor may decide to have some of the decking attached to the joists prior to setting

the system onto the girders. As shown in Figure 2.8, the steel decking runs perpendicular

to the non-dominant joists and is welded to the top chords of these joists at increments

consistent with traditional joist construction.


A

A

39

The decking will “bubble-up” a small amount in the vertical direction at the joist

intersections due to the presence of the connection elements. This misalignment,

equivalent to the thickness of the HSS piece (3/16 in.), is assumed to be negligible in the

design.

Direct contact is assumed to be non-existent between the dominant joists and the

decking. In other words, the dominant joists only receive loading via the top chord

connections to the non-dominant joists. Note that the steel decking in Figure 2.8 is

shown to be terminated at the girders. The figure is illustrated in this manner for clarity.

In actuality, the deck is continuous over the girders because the panel is located in an

interior bay.

The proposed system uses the girders (which run parallel to the non-dominant

joists) to directly carry some of the decking. In other words, each of these girders will

have point loads from dominant joist reactions as well as uniform line loading from a 6 ft.

tributary width of deck (assuming that a symmetrical adjacent bay is present). To

facilitate the bearing of the deck, a small steel shape with a depth of 2 1/2 in. (to match

the depth of the joist seat) needs to be welded to the top of the girder. A cold-formed

steel channel is shown in Figure 2.8, but a variety of options are available depending on

the contractor’s preference. The type of detail used depends on whether or not the girders

are designed for composite construction. If composite construction is desired, using a

structural tee (Rongoe 1984) may be preferred to provide a more direct load path

(through the stem of the tee) from the shear stud to the girder flange.

Similar to a traditional system, the girders occupying the orthogonal column line

do not directly carry the steel decking. Due to the load distribution of the system, these

40

girders (running parallel to the dominant joists) will be smaller than the girders in the

other direction. If a member is needed to fill the void between the girder flange and the

deck (such would be the case if a bearing wall was placed directly over the girder), a

concrete pour stop detail could be used (detailed no differently than a traditional system).

The final phase of the general construction sequence is shown in Figure 2.9. A

mat of welded wire fabric is set into place and a 4 in. concrete slab is poured over the

decking. Normal-weight concrete was assumed in the design of the proposed system.


A

A

41

Chapter 3 Structural Behavior Section 3.1 Introduction

This chapter begins with an overview of concepts drawn from basic plate theory, and

later sections compare and contrast the behavior of the proposed two-way floor system

with the structural behavior of a two-way plate and a traditional one-way joist system.

The reader is then guided through the detailed process of arriving at the final proposed

floor system. This entails sifting through preliminary panel configurations, creating a FE

model, sizing joist members using the current specification (SJI 2005), and producing a

sufficient connection design. The influence of the joist arrangement on the supporting

steel girders is also illustrated.

Section 3.2 Plate Bending

In order to gain an understanding of the load distribution in a two-way steel floor system,

a brief derivation, accompanied by a numerical example, using classical thin-plate theory

is provided. In effect, a plate is a two-dimensional beam having bending about two in-

plane axes with twisting moment. The plate analysis provided uses expressions

consistent with Kirchoff’s plate theory of bending for isotropic, homogeneous, thin

plates. Fenster (2003) outlines the assumptions as follows:

1) The deflection of the midsurface is small in comparison with the thickness of the

plate. The slope of the deflected surface is much less than unity.

2) Straight lines initially normal to the midsurface remain straight and normal to that

surface subsequent to bending. Therefore, deflection is associated only with

42

normal bending strains.

3) No midsurface straining, in-plane straining, stretching, or contracting occurs as a

result of bending. Also, no membrane forces are present.

4) The component of stress normal to the midsurface is negligible. To describe the

moment distribution of a simply supported square plate with a uniform loading

(see Figure 3.1), Fenster (2003) starts with general stress, curvature, and moment

relations arriving at

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

−= 2

2

2

2

x ywv

xwDM

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

−= 2

2

2

2

y xwv

ywDM (3.1)

yxwv)D(1M

2

xy ∂∂∂

−−=

where

)v12(1EtD 2

3

−= (3.2)

Equation 3.2 defines the flexural rigidity of the plate. Fenster (2003) points out that a

unit width of plate exhibits greater stiffness than a narrow beam by a factor of 1/(1-v2) or

by about 10 %.

Figure 3.1 - Plate Geometry

a

bz, w

y

x

p(x,y)

t

43

After further tedious mathematical derivations, Fenster (2003) goes on to provide an

equation for the deflection of the plate’s surface described as

[ ] bynsin

axmsin

)/((m/a)

pπ

1w1m 1n

222

mn4

ππ∑∑∞

=

∞

= +=

bnD (3.3)

where

dydxb

ynπsina

xmπy)sinp(x,ab4p

b

0

a

0mn ∫ ∫= (3.4)

However, for a uniform load po, equation 3.4 reduces to

otherwise.,0p

odd;nm,,mnπ

16pp

mn

2o

mn

=

= (3.5)

Substituting this expression into equation Eq. 3.3, and realizing that pmn = 0 for even

values of m and n, gives the deflection

[ ]∑∑∞ ∞

=+

=m n

2226o 1,3,5,...nm,

(n/b)(m/a)mn

y/b)x/a)sin(nπsin(mπDπ

16pw (3.6)

Finally, expressions for the moments Mx and My can be formulated by substituting Eq.

3.6 into 3.1 resulting in

[ ] byn

axm

bnammnbnvamp

Mm n

ox

πππ

sinsin)/()/()/()/(16

222

22

4 ∑∑∞ ∞

+

+=

[ ] byn

axm

bnammnbnamvp

Mm n

oy

πππ

sinsin)/()/()/()/(16

222

22

4 ∑∑∞ ∞

+

+= (3.7)

1,3,5,...nm,for =

It should be noted that Eq. 3.6 will converge more rapidly than Eq. 3.7. Fenster

(2003) notes that after only four terms, the maximum deflection (located at midspan of

the plate) is wmax = 0.0443 po(a4/Et3). Another author (Boresi 1993) tabulates the

44

coefficient of this equation as C = 0.047. Boresi (1993) comments that this coefficient is

reduced to C = 0.016 when the edge restraints are clamped (i.e. the deflection of the

center of a clamped plate is about one third of the value found for a plate with simply

supported edges). In terms of moment at midspan, Boresi (1993) shows a Mclamped value

≈ 40% less than that of a simply supported plate.

In order to gain a feel for how the moment distribution varies over a slice of

simply supported square plate, consider the following input for a steel plate.

a = 360 in b = 360 in

v = .3 E = 29,000,000 psi t = 6 in

po = 2.5347 psi (sum of 120 psf super-imposed loading and 245 psf self-weight)

Using Eq. 3.7, allowing x to vary from 0 to 360 in. while holding y constant at 180 in.,

the distribution of Mx is show in Figure 3.2. The solution was obtained after truncating

the series at m and n equal to 11 (recall that only the odd terms are used) resulting in a

percent difference of 0.6% when compared to mid-span moment with m and n equal to 9.

The maximum moment is determined to be 0.0479 poa2 which, when taken over a unit

width, is notably less than the Mmax = 1/8 (w/2)l2 = 0.0625wl2 if only bending were

present. This illustrates that the twisting moments relieve the orthogonal axis bending

moments by approximately 25% (Nilson 1991). Upon studying the distribution in Figure

3.2, it becomes apparent that the majority of the moment resistance supplied by the plate

comes from the middle half of the plate. The moment only drops by 19% within this

middle portion. The remaining half of the plate (the two outer portions) carries less of

45

the plate panel loading, exhibited by the moment dropping rapidly to zero around the

perimeter.

Figure 3.2 - Moment Distribution in a Simply-Supported Square Plate

Section 3.3 Two-Way Concrete Floor Systems

Unlike steel construction, two-way floor systems (slabs) are prevalent in concrete

structures. As noted by Nilson (1991) however, “the precise determination of moments

in two-way slabs with various edge conditions is mathematically formidable and not

suited to design practice.” Therefore, the ACI (2002) code permits the use of a

coefficient method when designing floor slabs, provided certain assumptions are met.

One of the main assumptions is that the supporting girders are infinitely stiff. With the

help of tabulated moment “coefficients,” a designer can estimate the amount of moment

reinforcement needed (steel rebar) based upon the panel’s aspect ratio and assumed edge

46

restraints. The coefficients are based on elastic analysis, but also account for inelastic

redistribution (Nilson 1991). Therefore, the design moment is smaller than the elastic

maximum moment (by an appropriate amount) in a given direction. For example, in the

case of a simply supported square slab, the method allows a design moment of 0.036wl2.

That’s 25% less than the actual theoretical elastic maximum moment discussed earlier.

In the coefficient method, the panel is fictitiously separated into a middle strip

(inner half of panel) and two edge strips (combining to form the remaining half of the

panel). From Figure 3.3, the design procedure states that the entire middle strip is to be

designed for the full tabulated design moment. The edge strips moments, on the other

hand, carry less moment and can be conservatively assumed to carry an average moment

equal to two thirds of the corresponding middle strip moment.

Figure 3.3 - Variation of Positive Moment Across the Width of Critical Sections

Assumed in Two-Way Concrete Design (Adapted From (Nilson 1991))

47

Returning to the numerical plate problem, let’s assume that the plate is made of

reinforced concrete instead of solid steel. Using a coefficient table (a portion of the table

shown in Figure 3.4), the positive moment in the middle strip (assuming the load was

factored appropriately) can be calculated as

ft)/ft(lb11826sf)(30ft)0.036(365pM 2pos −== (3.8)

while the positive moment in an edge strip is conservatively taken as

ft)/ft(lb788411826)(32M pos −== (3.9)

With a brief overview of basic plate theory and a reflection upon a common design

procedure used in concrete floor slabs, one can start to develop a design philosophy to

create a two-way steel floor system using steel joists.

Figure 3.4 - Sample of Moment Coefficient Table (Adapted from (Nilson 1991))

Section 3.4 K-Series Joist Selection (One-Way System)

The term “open-web steel joists k-series” refers to open-web, parallel-chord, load-

carrying members used for the direct support of floors utilizing hot-rolled or cold formed

steel (SJI 2005). Similar to other mainstream steel floor systems, open-web joists

Ratio

b

a

llm =

Case 1 Case 2

1.00 Ca,dl Cb,dl

0.036

0.036

0.95 Ca,dl Cb,dl

48

predominantly span in one direction. The selection of appropriate member sizes is a

process that has been highly refined by joist manufactures. For example, manufactures

can pick from a group of cold-formed angles varying in increments on the order of 1/64

in. To compare how well the proposed two-way system utilizes material, a design is

provided for a floor arrangement composed of K-series joists (given the same bay

dimensions and loading that is used in the proposed system). The 30 ft. x 30 ft. interior

floor bay used in this study has the following loading:

SI DL = 40 psf (concrete slab & deck)

15 psf (plumbing & electrical)

SI LL = 50 psf (office)

15 psf (partition loading)

Figure 3.5, shows data taken from the K-series ASD selection table (SJI 2005).

Joist Spacing For Non-Composite K-Series JoistsGiven a 30' Span With 65 psf SI LL & 55 psf SI DL

16K316K4

16K616K7

16K9

18K318K4

18K518K6

18K7

18K9

18K10

20K320K4

20K520K6

20K7

20K9

20K10

22K422K5

22K622K7

22K9

22K10

24K424K5

24K624K7

24K8

26K526K6

26K7 28K6

n = 14

n = 11n = 10n = 9n = 8n = 7

n = 6

n = 5

n = 4

n = 3

n = 30

30K724K1024K9 26K8

26K928K7

28K8

16K2

16K5

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

9.00

K-Series Joist

Jois

t Spa

cing

(ft)

n = # of joists required

controlled by total loadcontrolled by LL deflection

Figure 3.5 - K-Series Joist Selection

49

It should be noted that 22K11, 24K12, 26K10, 26K12, 28K9, 28K19, 28K12, 30K8,

30K9, 30K10, 30K11, and 30K12 joists were excluded from the data pool, because there

are lighter joists preceding these selections with the same load capacity. Each K-series

joist has two data points aligned vertically. One point gives the number of joists in the

panel and the corresponding spacing based upon total permissible loading, while the

other point gives the information based on permissible live load deflection.

With this design aid, one can quickly determine whether total load or live load

deflection governs the selection of a joist. Given the depth of a joist group, the highest

numbered section (16K9, 18K10, etc.) will produce the fewest number of joists in the

panel. This won’t necessarily produce the lightest system for a given depth. Figure 3.6

shows the best joist configuration for a given depth.

Joist Selection Based On Self-Weightof Non-Composite K-Series Joists

Given a 30' Span With 65 psf SI LL & 55 psf SI DL

16K2

16K

316

K416

K516

K616

K716

K918

K3

18K4

18K5 18

K618

K7

18K9

18K1

020

K320

K420

K5

20K

620

K7 20K9

20K1

022

K422

K522

K622

K7 22K9

22K1

024

K424

K5 24K6

24K7

24K8

24K9

24K1

026

K526

K626

K726

K826

K9 28K

628

K7 28K

830

K7

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

K-Series Joist

Jois

t Wei

ght (

psf) Select a 16K9

w/ 10 joists @ 2.727' O.C.

Select a 18K10w/ 9 joists @ 3' O.C.

Select a 20K10w/ 6 joists @ 4.286' O.C.

Select a 22K7 w/ 8 joists @ 3.333' O.C.





Figure 3.6 - Joist Self-weight

50

While maintaining the goal of minimizing floor-to-floor heights, this study desires a

depth based upon achieving the maximum span to depth ratio. Therefore, the shallowest

one-way system possible under the given loading at a span of 30 ft. is ten 16K9 joists at

2.727 ft. on center.

Section 3.5 Structural Analysis

The commercial software package SAP2000 (CSI 2004) was used to analyze the

proposed two-way floor system. A finite element analyses was performed assuming

static linear elastic structural behavior. The analysis began with a model of a single

traditional 16K9 steel joist. Upon determining the distribution of member forces, the

current joist specification (SJI 2005) was used to size both chord members and web

members using allowable stress design. Next, a FE model of the proposed system was

constructed. The behavior of this model was then compared and contrasted to both

traditional (one-way) joists and two-way plate behavior. Again, the member sizes were

determined using allowable stress design. Finally, the current steel specification (AISC

2001b) was used to determine an adequate connection design.

Section 3.5.1 FEA Element Types and Modeling Assumptions

A steel joist has two main components: the chord members and the web members. Each

chord member is commonly made of two back-to-back double angles separated by a

distance equivalent to the web member thickness. A chord member primarily carries

axial load, but it will also carry some bending moment (beam-column effect). Therefore,

a typical frame element from SAP2000’s element library was used in the model (Figure

51

3.7). Given the orientation of the steel deck, the direction of the applied loading, and the

addition of bridging placed between the steel joists, some of the forces become

negligible, namely torsion and weak-axis bending moment. The web members (steel

round bar) were also entered into the model as frame elements. Major and minor axis

moments were released at both ends and torsion was considered negligible; therefore, the

web elements behave as two-force members.

Figure 3.7 - Frame Element Internal Forces and Moments (Adapted from (CSI

2004))

Positive Moment and Shear in the 1-3 Plane

Positive Axial Force and Torque

Axis 2

Axis 3

Axis 1 P T

T P

Axis 2

V2 Axis 1

Compression Face M3

Axis 3

V2

M3

Ten. Face

Positive Moment and Shear in the 1-2 Plane

Axis 2

M2 Axis 1

Compression Face

V3

Axis 3

M2

V3

Ten. Face

52

As with any finite element model, consideration of the discretization of a system

needs to be addressed. For convenience, all elements between panel points in the models

were further divided by the same amount. The author believes that dividing the chord

members between panel points into 16 elements (equating to 1.5 in. and 2.125 in.

elements for 24 in. and 34 in. panel lengths respectively) is sufficient to capture the

appropriate distribution of bending moment. Further effort in studying the accuracy of

the convergence of a more refined mesh is unwarranted because of other assumptions

built into the model. Web members, carrying only axial load, were comprised of only

one element.

Section 3.5.2 Traditional Joist Analysis and Results

Open-web joist design makes use of very slender angles; therefore, acknowledging

different buckling modes of the top chord becomes important. There are three possible

axes about which a single angle can buckle (Figure 3.8).

Figure 3.8 - Axes about Which Buckling Can Occur

The attachment of the deck to the joist’s top chord plays a critical role in determining

which buckling mode will control. The current joist specification (SJI 2005) section 5.8e

states that the spacing for deck attachments along the top chord shall not exceed 36 in. In

Xbar

Ybar

Z

Z

X X

Y

Y

53

order to provide additional restraint, top chord fillers are commonly inserted between the

two top chords at mid panel (used in the panels located near joist midspan where axial

load is highest). The following KL/r ratios must be checked (largest controlling) for axial

loading (KxLx/rx is used for bending)

x

xx

rLK

y

yy

rLK

z

zz

rLK w/o filler or

z

zz

rLK w/ filler (3.10)

Two cases need to be checked (using Eq. 3.10) to determine which scenario controls the

design. This includes a check using properties and loading reflecting a single angle as

well as a check using two angles (back-to-back, separated by a distance equivalent to the

web member thickness). These expressions are consistent with the joist specification (SJI

2005), however, they are conservative.

One can argue that Kx or Kz factors are closer to the values shown in Figure 3.9.

A lower K factor can be obtained as a result of the adjacent panels forcing the top chord

to buckle into the slab (either about the x-axis or z-axis). The resulting K factors likely

reduce to K = 0.5 for an interior panel and K = 0.7 for an exterior panel. As a

conservative measure, these reductions in K values will not be taken advantage of in the

analysis of the joist chords (neither the traditional nor the proposed design).

Figure 3.9 - Reduction of K Factor Due to Slab Presence

K→1.0

L

K→1.0

K→ 0.5

I.P I.P

Lend

K→0.7

54

Following the specification (SJI 2005), the following ASD equations were used to

determine the adequacy of the web members. It should be noted that these equations

were designed for hot rolled steel shapes. However, most joist manufactures make use of

cold-formed shapes because the member thickness increments are smaller. To avoid

deviating into all the design idiosyncrasies that come from cold-formed steel design, it is

assumed that the chord member design (though using member sizes reflecting cold-

formed steel shapes) can be accomplished by using “hot-rolled” equations.

The ASD equations governing axial loading are as follows:

fa ≤ Fa = 0.6 Fy (Tension Members) (3.11)

yF

QF

aa F0.6580.6QFf e

y

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡=≤

⎟⎟⎠

⎞⎜⎜⎝

⎛

(Comp. Members) (3.12)

where

fa = applied axial stress

Fa = allowable axial stress

Fy = yield stress (50 ksi)

Q = form factor (= 1 for round bar)

Fe = elastic buckling stress 2

2

rKL

Eπ

⎟⎟⎠

⎞⎜⎜⎝

⎛

The ASD equations for the chord members (beam columns) become

at the panel point: fa + fb ≤ 0.6 Fy (3.13)

at the mid panel for (fa/Fa ≥ 0.2) :

55

1.0QF

F1.67f

1

fC98

Ff

be

a

bm

a

a ≤

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−

+ (3.14)

where

fb = applied bending stress

Fb = allowable bending stress (equivalent to 0.6 Fy assuming full lateral bracing)

Cm = Moment Reduction Factor (Conservatively = 1.0 for uniform load)

Q = form factor, EF

tbQQ y

s ⎟⎠⎞

⎜⎝⎛−== 76.0340.1 (AISC 2001b)

The SAP2000 model used for the 16K9 joist is shown in Figure 3.10. While the

depth of the joist is 16 in., the depth of the model takes into consideration the location of

the chords’ elastic neutral axes. This reduces the depth of the model to 14.88 in. The

joist is simply supported with a uniformly distributed loading. The magnitude of the

loading is the maximum permissible superimposed loading (taken from the standard load

table (SJI 2005)) equal to 345 lb/ft (355 lb/ft – 10 lb/ft).

Figure 3.10 - Traditional Joist Model

Design Length = 356 in.

22 @ 12 in. 46 in.

56

The labeling shown Figure 3.11 takes advantage of symmetry to identify each top

chord, bottom chord, and web member. The resulting axial force diagram is shown in

Figure 3.12.

Figure 3.11 - Traditional 16K9 Member Numbering

Figure 3.12 - Axial Force Distribution in a the 16K9 Model

The corresponding moment diagram becomes

Figure 3.13 - Moment Distribution in a the 16K9 Model

When qualitatively examining Figure 3.12 and Figure 3.13, a number of factors

stand out. The axial forces located in both the top and bottom chords increase until

1 2 3 4 5 6

8 9 10 11 12 13

14 15

16

17 19

18

21 23 25

20 22 24 26

4.683 K-in 2.610 K-in

1.640 K-in

31.07 K (C)

31.35 K (T) 15.37 K (T)

5.07 K (C)

57

reaching a maximum value at the joist midspan (where the internal couple required to

resist the bending moment is at a maximum value). The peak axial force in the web

members occurs at the ends (member 14 in tension), and becomes nearly zero at the

center of the joist. This is expected because the shear developed in the uniformly loaded

joist is resisted by the web members and theoretically goes to zero at the midspan.

With respect to bending moment, all joist panels experience positive curvature

and thus positive moment, with the exception of the development of negative moment

located a few panel points in from the ends. Because the top chord accepts uniform

transverse loading between panel points, the moment will vary quadratically. The bottom

chord, with no loading between panel points, has a linear variation of moment.

The results of each member are tabulated in Table 3.1 through Table 3.4. The

values of axial force and bending moment, tabulated in Table 3.1 and Table 3.2, were

half of the total forces necessary to reflect appropriate values for a single angle capacity

check. The analysis also assumes that the six innermost panels have top chord fillers.

Mem Area S Axial (1) fa Mom.(1) fb Axial(2) Bend.(3) Fea(4) Feb

(5) Cm Q Axial Bend. Tot. # Description in2 in3 (kips) (ksi) (k-in) (ksi) KL/r KL/r (ksi) (ksi) Cap. Cap. Cap.1 2L2x2x.176 0.673 0.457 7.313 10.87 2.3415 5.13 86.21 54.94 38.5 94.8 1 0.981 0.63 0.19 0.822 2L2x2x.176 0.673 0.457 7.0205 10.43 1.046 2.29 60.852 38.78 77.3 190.3 1 0.981 0.46 0.08 0.543 2L2x2x.176 0.673 0.457 9.857 14.65 0.815 1.78 60.852 38.78 77.3 190.3 1 0.981 0.65 0.06 0.714 2L2x2x.176 0.673 0.457 12.126 18.02 1.1005 2.41 60.852 38.78 77.3 190.3 1 0.981 0.80 0.09 0.885 2L2x2x.176 0.673 0.457 13.841 20.57 1.1355 2.49 38.78 38.78 190.3 190.3 1 0.981 0.78 0.09 0.876 2L2x2x.176 0.673 0.457 14.965 22.24 1.2585 2.76 38.78 38.78 190.3 190.3 1 0.981 0.84 0.10 0.957 2L2x2x.176 0.673 0.457 15.534 23.08 1.305 2.86 38.78 38.78 190.3 190.3 1 0.981 0.87 0.11 0.98

Comments1) Appropriate values for a single angle2) This value is for the axis w/ the largest slenderness ratio regardless of bending3) This is always w/ respect to the x-axis (axis of bending)4) The subscript "a" is for axial5) The subscript "b" is for bending and this is the value used in the amplification of the bending moment

Table 3.1 - Top Chord Results for a Traditional 16K9

58

Mem Description Area S Axial(1)T/A Mom.(1) fb Axial Bend. Tot.

# in2 in3 (kips) (ksi) (k-in) (ksi) Cap. Cap. Cap.8 2L2x2x.148 0.57 0.398 8.6105 15.11 0.6055 1.52 0.50 0.05 0.5549 2L2x2x.148 0.57 0.398 11.129 19.52 0.6255 1.57 0.65 0.05 0.703

10 2L2x2x.148 0.57 0.398 13.129 23.03 0.6305 1.58 0.77 0.05 0.82111 2L2x2x.148 0.57 0.398 14.543 25.51 0.812 2.04 0.85 0.07 0.91812 2L2x2x.148 0.57 0.398 15.389 27 0.822 2.07 0.90 0.07 0.9713 2L2x2x.148 0.57 0.398 15.675 27.5 0.82 2.06 0.92 0.07 0.985

Comments1) Appropriate values for a single angle

Table 3.2 - Bottom Chord Results for a Traditional 16K9

Ten. Description Area Axial fa Fa AxialMem # in2 (kips) (ksi) (ksi) Cap.

14 7/8" Round 0.601 15.4 25.6 30 0.8517 23/32" Round 0.406 3.97 9.8 30 0.3319 23/32" Round 0.406 3.21 7.9 30 0.2621 9/16" Round 0.249 2.27 9.1 30 0.3023 9/16" Round 0.249 1.34 5.4 30 0.1825 9/16" Round 0.249 0.46 1.9 30 0.06

Table 3.3 - Web Tension Member Results for a Traditional 16K9

Comp. Area KL Axial Axial fa Fe Fa AxialMem # Description in2 in. KL/r (kips) (ksi) (ksi) (ksi) Cap.

15 23/32" Round 0.406 19.12 106.4 0.93 2.30 25.29 13.114 0.1816 23/32" Round 0.406 19.12 106.4 5.07 12.48 25.29 13.114 0.9518 23/32" Round 0.406 19.12 106.4 4.02 9.91 25.29 13.114 0.7620 23/32" Round 0.406 19.12 106.4 3.20 7.87 25.29 13.114 0.6022 9/16" Round 0.249 19.12 135.9 2.24 8.98 15.49 13.584 0.6624 9/16" Round 0.249 19.12 135.9 1.35 5.43 15.49 13.584 0.4025 9/16" Round 0.249 19.12 135.9 0.45 1.80 15.49 13.584 0.13

Table 3.4 - Web Compression Member Results for a Traditional 16K9

As stated earlier, joist chord members are primarily axial load carrying members.

This is evident from the axial percent capacity tabulated in the analysis results. Section

4.4 of the specification (SJI 2005) states that the top chord can be designed for only axial

compressive stress when the panel length does not exceed 24 in. Referring to Table 3.1,

a smaller chord size may have been chosen if the chord member was treated solely as an

axial loaded member. Similarly, section 4.4 of the specification (SJI 2005) states that the

bottom chord shall be designed as an axially loaded tension member. Therefore, the

59

bottom chord member size in Table 3.2 could also have been lowered, neglecting the

presence of bending moment.

Section 3.5.3 Proposed System Panel Design

Joist location is critical in the panel design because it mandates the load path of the

system. The load starts over a tributary width, located above a joist included in the group

of joists carrying the slab. These joists in turn “hang” on the joists running in the

perpendicular direction which do not directly carry the slab. Therefore, the loading will

inherently redistribute itself, attempting to mimic basic two-way plate behavior through

deformation compatibility. Unlike a plate however, the proposed system lacks the

material and, in turn, stiffness to carry a twisting moment (Mxy in the previous plate

example) across the floor bay.

Many preliminary designs were investigated, each considering a different manner

in which the joists could be arranged and spaced in a 30 ft. x 30 ft. interior bay. The

author designed a two-way joist arrangement that would use less than 10 joists (the

number of K-series joists needed in a traditional design), while utilizing chord sizes that

deviate little from the sizes used in the 16K9 joist design. This will establish a fair

comparison of how much steel is needed in each system to carry the prescribed loading.

It should be noted that the girders running perpendicular to the direction of the deck are

assumed to directly carry a portion of the deck, i.e. half the tributary width to the first

deck-supporting joist. Another critical assumption is that symmetrical adjacent floor

bays are present on all four sides of the “interior” design bay. Preliminary designs were

discarded on the basis of providing poor joist efficiency, fabrication problems, or

60

inefficient distribution of the loading to the supporting girders. Recall from Chapter 2

that the deeper joists are labeled dominant joists, while the shallower joists are considered

non-dominant joists.

The amount in which a joist can shift horizontally is restricted by the panel length

of the joists positioned in the orthogonal direction. This enables dominant joists to move

in 19 in. increments while non-dominant joists move in 24 in. increments. Another

restriction is the system’s inability to contain an odd number of non-dominant joists

(while keeping a symmetrical arrangement of these joists), which is prohibited by the

alignment of the panel points in the dominant joists.

Figure 3.14a shows the first configuration considered for the proposed design, in

which the joist spacing was divided evenly across each supporting girder. The biggest

problem with this design is its inefficiency. The load distribution will mimic that of a

two-way plate (i.e. the load is channeled towards the center of the bay in both directions).

Therefore, each joist would have to be designed individually, with the joists near the

edges being the smallest and increasing to a maximum size at the center of the panel.

Figure 3.14b places the joists near the center of the panel in an attempt to increase

joist efficiency without varying the joist’s chord sizes from one joist to the next. This

design is sufficient for the dominant joists because they don’t “directly” carry the deck.

The non-dominant joists, however, start developing unfavorable, and essentially

unreliable, loading because the slab over the middle joists experiences uplift.

The idea of using the configuration shown in Figure 3.14c was eliminated because

of limitations with the decking material. On a positive note, the design eliminated uplift,

and the joists are placed in the center of the slab where most of the load is channeled.

61

However, the design violates the maximum distance in which unshored decking can span

(≈11 ft.).

Figure 3.14 - Preliminary System Designs

The design philosophy of Figure 3.14d was to space the non-dominant joists close

to the maximum permissible unshored deck span while maintaining an even joist spacing,

in this case, 10 ft. The idea was that this arrangement would not only make very efficient

use of the deck, but also of the joist. Joist efficiency would increase due to the use of

Key Non-Dominant Dominant

Direction of Deck Span

a) b)

c) d)

e) f

62

fewer overall joists while requiring only two different joist sizes (one size for the

dominant and one size for the non-dominant directions). The problem with this system is

that it does not evenly distribute the loading to the supporting girders. In other words, the

girder size necessary to support the dominant joists only dropped a few sizes from that of

a W-section needed to support a traditional one-way floor system.

The floor assembly illustrated in Figure 3.14e inverted the placement of both the

dominant and non-dominant joists. Now the dominant joists “directly” carry the deck.

The author feels that, even though temporary dowels are in place, a temporary lack of

lateral and torsional stiffness would be present during field assembly (Figure 3.15). This

design may also create an overly congested connection detail at certain locations along

the joist bottom chord because a total of five web members are confined to one location

(two additional web angles are present behind the two that are shown, and are also

welded to the bottom chord).

Figure 3.15 - Preliminary Joist Intersections

The floor assembly shown in Figure 3.14f is similar to the final design. This

arrangement was altered, however, because a more favorable distribution of point loads

(joist reactions) could be obtained for the girders supporting the dominant joists. The

Congested Connection

Lack of Lateral/Torsional Stiffness Temporary Dowel

63

final design moved each dominant joist an additional 19 in. away from the center of the

slab. This reduced the moment in the supporting girders while allowing the joists to

remain within the middle strip (the inner third of the panel).

The final design is a compromise of all the aforementioned competing factors

which influence the behavior of the deck, joists, and supporting girders. The design

provides adequate deck and joist efficiency, while maintaining a sufficient balance of

two-way behavior which allows an appreciable reduction in girder sizes (Figure 3.16). A

total of 8 joists are used: (2) Outer Non-Dominant Joists (ONDJ), (2) Inner Non-

Dominant Joists (INDJ), and (4) Dominant Joists (DJ). The dominant joists could have

been further divided into two categories of inner and outer joists. This was avoided for

simplicity by using an inner joist (which carries more loading) as a gauge for which to

conservatively assign member sizes.

Figure 3.16 - Plan View of Proposed Design

ONDJ

INDJ

ONDJ Trib. Width

DJ

Girder Trib. Width

INDJ Trib. Width

6’

6’

6’

6’

6’

30’

30’

10’-3”

19”

64

Section 3.5.4 Modeling the Proposed Design

Figure 3.17 shows the models constructed in SAP2000 to represent both the dominant

and non-dominant joists.

Figure 3.17 - Proposed System’s Joist Models

Also for simplicity, sixteen elements (the same number used in the traditional K-series

joist model) were chosen to discretize the chord members between two panel points.

This equates to having 1.625 in., 1.1875 in., 2.125 in., and 1.5 in. elements present for 26

in., 19 in., 34 in., and 24 in. panel lengths. The crudest element (2.125 in.) was adequate

for capturing an appropriate distribution of bending moment.

Figure 3.17a illustrates the model used to represent both the inner and outer non-

dominant joists. The two types differ only in the inner non-dominant joist’s larger chord

member sizes. Both top and bottom chords are continuous while all of the web members

have full moment releases at both ends of the members (torsional restraint was present to

prevent the creation of an unstable structure). The bottom chords are inverted to allow

passage of the non-dominant joists through the web members of the dominant joists

without interruption. The element type chosen was SAP2000’s frame element (discussed

(a) Non-Dominant Joist


30 @ 9.5 in.35.5 in.


22 @ 12 in.46 in.

6 in.

q

(b) Dominant Joist

65

earlier). The interior panel dimension is 19 in., a typical panel dimension for 14 in. deep

K-series joists. A superimposed uniform total service line loading of (6 ft.)(120 psf) =

720 lb/ft was placed onto each non-dominant joist.

Figure 3.17b shows the model used for the dominant joists (DJ). The joist

dimensions are very similar to those describing a 16K9, except for the added web

members used to provide extra support at the joist intersections. The dominant joists

have discontinuous top chords, but were modeled as having continuous members. The

connection detail, which is assumed to transfer negligible amounts of weak or major axis

bending moment, provides the missing piece to make this assumption possible.

Combining the four dominant and four non-dominant joists together forms the

SAP2000 model shown in Figure 3.18. Rotation is permitted on all four ends of the

model. Two sides of the system have all three translations restrained (ball and socket

joint) while the other two sides only have two translations prohibited. If the system were

modeled inaccurately by using all pins and no rollers, “string action” axial forces would

occur (Figure 3.19). In this case, membrane forces predominate and the system

experiences stress stiffening. This behavior becomes most apparent at the ultimate load

stage.

Figure 3.18 - 3D View of SAP2000 Proposed System Model

AA

B C

x y z

A x, y, and z Node Type Translation Restrained

B y and z C x and z

66

Figure 3.19 - String Action Axial Forces

Because the system is statically indeterminate, calculating adequate member sizes

is an iterative process. For simplicity, all web member sizes (except web angles and web

round bar used at the joist ends) were 3/4 in. round bar. Further refinement by varying

the web member sizes in additional iterations would increase joist efficiency. However,

reducing member sizes based on strength would reduce the stiffness of the system.

Therefore, member reduction is limited by serviceability constraints, i.e. L/360 service

live load deflection limit.

When analyzing a three-dimensional structure, one must determine the role that

torsion has in the structural behavior of the members. In this two-way joist system,

curvature developed in a row of joists causes the joists in the perpendicular direction to

rotate. The amount of torsion developed, as a result of top chord rotation, can be

considered negligible as long as a few key assumptions are met. First, the difference in

slopes between two adjacent joists must be small. The closer the joists are located to the

q dx

T T

M

V V

M

L q

dx

67

center of the panel, the more negligible torsion becomes. Another key assumption is that

the joist seats are assumed to rotate freely about a joist’s longitudinal axis. Finally, the

bottom chords need to translate laterally uninhibited. This is why the pieces of bridging,

placed every 6 ft. on the dominant joists, are attached along the top chords only. The

author feels that all of the assumptions were met; therefore, torsion was neglected in the

structural models.

The remaining criteria for selecting member sizes was axial loading as well as

combined axial loading and bending when applicable (used to select chord member

sizes). Once again, the ASD method was chosen to calculate appropriate member sizes

because this method is widely used among joist manufactures. The full tabulated results

of the analysis are located in Appendix A. To reflect appropriate values for a single

angle capacity check, the tabulated values of axial force and bending moment present in

the chord members were determined to equal one half of the total forces from the FE

model.

Figure 3.20 shows the deformed shape of the system after SAP2000 compiled a

static linear elastic analysis. The rollers placed at the aforementioned locations moved

inward in response to the loading, an indication that the model was working properly.

The dish shape that resulted from the deflection is consistent with that of a two-way plate.

The maximum deflection found in the panel, under superimposed service live loading,

was 0.973 in. (L/360 = 1 in.).

Figure 3.21 shows the distribution of axial forces, under full service level loading,

developed in the three main joist categories (DJ, ONDJ, and INDJ respectively).

68

Figure 3.20 - Deformed Panel

Figure 3.21 - Axial Force Distribution in Proposed System

16.84 K (C)

7.11 K (C)

17.65 K (T)

38.43 K (C)

38.39 K (T)

11.85 K (C)

15.45 K (C) 14.45 K (C)

14.91 K (T) 15.71 K (T)

12.51 K (T)

3.38 K (C)

15.95 K (C) 29.66 K (C)

30.12 K

16.84 K (T)

5.37 K (C)

a) Dominant Joist

b) Outer Non-Dominant Joist

18.43 K (T)

12.94 K (T)

21.50 K (T)

25.90 K (T)

26.68 K (T)

A

C

E

F B

G

δA = 0.973 in.

δB = 0.962 in.

δD = 0.904 in.

δC= 0.922 in. δG = 0.629 in.

δE = 0.793 in.

δF = 0.745 in.

D

c) Inner Non-Dominant Joist

69

The dominant joists (Figure 3.21a) develop higher axial loads in both the top and bottom

chords when compared to both types of non-dominant joists. This was expected because

the dominant joists have larger depths, and load is attracted to stiffness. In a simplistic

sense, the manner in which a dominant joist receives loading along its length at each

intersection is analogous to a beam with four equally spaced point loads (though not

equal in magnitude). As a result, the internal couple (and hence the chord forces)

continuously increases to a maximum value at midspan.

The primary function of the web members (round bar) is to accommodate the

shear forces throughout the joist. When a joist behaves as a simply supported beam with

a uniformly distributed load (i.e. as a K-series joist), the maximum shear is at the ends. A

dominant joist, reflecting this behavior, has its largest web axial forces (both tension and

compression) residing at the ends of the joist (Figure 3.21a).

The distribution of axial forces in both the outer and inner non-dominant joists is

more complicated than the distribution found in the dominant joists. The joist

intersections play a much different role for the non-dominant joists. Instead of applied

point loads, a non-dominant joist acquires “springs” roughly at the third points of its span

(Figure 3.22a). As these springs become infinitely stiff, the joist behaves like a 3-span

continuous uniformly loaded beam (Figure 3.22b). This means negative moment

develops over the “supports,” which would translate to a load reversal in a joist at these

locations (compression in the bottom chord and tension in the top chord). Because the

dominant joists are far from being infinitely stiff, the “springs” won’t cause a load

reversal, but rather they will slightly lower the axial force in the chords.

70

Figure 3.22 - Spring Analogy for Non-Dominant Joists

This “spring” effect is most pronounced in the outer non-dominant joists. Axial

forces in both the top and bottom chords are reduced at the joist intersections (shown as

dashed lines in Figure 3.21b). In fact, the axial forces run out of room to recover

(moving left to right) and the maximum chord forces no longer occur at joist midspan.

Because the inner non-dominant joists are closer to the center of the panel, the “spring”

stiffness provided by the dominant joists will be lower. This is why the axial forces,

though still reduced by a small amount at the joist intersections, recover afterwards and

are hightest at the joist midspan (Figure 3.21).

Figure 3.23 shows the major-axis moment distributions, under full service level

loading, developed in the three main joist categories. It should be noted that because all

the loading received from the panel is in the vertical direction only, minor-axis bending

(though included in the model) was considered negligible. The dominant joist does not

have uniform loading between panel points. Therefore, the highest order moment present

in either the top or bottom chord is linear (Figure 3.23a).

b)

K1 K2 K1 & K2 inf.

a)

K1 K2

Compression

Tension

71

Figure 3.23 - Moment Distribution in Proposed Model

The maximum moment occurs at the end panels because these panels only have

one adjacent panel to help counterbalance the moments. The location of maximum

moment at the top chord end panels is consistent in all of the joist categories. It should

be noted that “spikes” in bending moment, shown at the top chord intersections, are not

consistent with the actual connection geometries. In other words, the model does not take

4.72 K-in 2.25 K-in

1.65 K-in 3.09 K-in

5.13 K-in 2.07 K-in

1.11 K-in 1.25 K-in

6.18 K-in

1.76 K-in

3.09 K-in

2.09 K-in

*

*Moment distribution is not reflective of actual connection geometry.

a) Dominant Joist

b) Outer Non-Dominant Joist

c) Inner Non-Dominant Joist

Positive Moment Negative Moment

2.07 K-in

2.34 K-in

72

into consideration the presence of a “stiffening plate” which allows the connection to

undergo shear deformations rather than behave like a short cantilevered beam.

The decision to add additional web members at the joist intersections allowed the

web to retain a favorable “triangle” pattern of intersecting members. Even though

appreciable amounts of bending moment develop in the bottom chords at these locations,

the added web members “calm” the distribution of bending moment in the chords. Figure

3.24 shows a more chaotic distribution of moment that results from the absence of

additional web reinforcement. Also, notice how the magnitude of the moments becomes

quite large.

Figure 3.24 - Moment Distribution in a Dominant Joist with Additional Web

Members Removed

Figure 3.23b shows the distribution of moment found in an outer non-dominant

joist. Because this joist receives loading directly from the deck, the distribution of

moment in the top chord varies quadratically, while the bottom chord (indirectly

receiving loading) has a linearly varying moment. A noticeable reduction in moment (in

both the bottom chord and top chord) occurs near the joist intersections (shown as dashed

lines). This is caused by the “spring” effect discussed earlier.

7.19 K-in

13.46 K-in 7.80 K-in

Positive Moment

Negative Moment

73

The moment distribution in the inner non-dominant shown in Figure 3.23c

illustrates similarities between the structural behavior of the proposed system and that of

a two-way plate. The closer a joist is positioned to the center of a slab the more curvature

it develops. Moment is proportional to curvature; therefore, it’s no surprise that the

magnitudes of the moments found between chord panel points are larger in the inner non-

dominant joists. The “spring effect” also influences the moment distribution in the chord

members, though to a lesser degree than in the outer non-dominant joists.

Table 3.5 shows how the self-weight of the joists used in the proposed system

differs little from the self-weight of a 16-K9 joist. SAP2000 doesn’t take into account

additional material from round bar fillets and joist seats, nor the full lengths of top and

bottom chords. Therefore, tabulated values from SAP2000 slightly underestimate the

true member self-weight. Connection hardware will slightly increase the weight of the

non-dominant joists.

Table 3.5 - Joist Self-weight

Section 3.5.5 Connection Design

Different preliminary connection details were generated to connect the joists along their

top chords. The author believes that a connection joining intersecting top chords would

Top Chord Bottom Chord Self Wt. (lbf)

2L2x2x0.176 2L2x2x0.148 9.4 16K9

Joist Description

INDJ

ONDJ

DJ

2L2x2x0.187 2L2x2x0.176 10.9

2L2x2x0.148 2L2x2x0.137 9.2

2L2x2x0.205 2L2x2x0.187 12.6

74

be more accessible in the field than a bottom chord (tension) connection. Two of the

preliminary designs are described in detail and illustrated in this section.

The “cruciform” connection piece shown in Figure 3.25a was designed as a single

piece that could be bolted into place, permitting joists in both directions to be 16 in. deep

(Figure 3.25b). Concrete then encases the connection, allowing the additional depth of 2

in. without penalizing the depth of construction as shown in Figure 3.25c.

Figure 3.25 - Cruciform Connection

The “cruciform” connection caused numerous problems. Continually cutting the

deck and forming appropriate pour stops looked much less attractive than having a

system with flush top chords. Also, special details at the joist ends would have to be

75

fabricated unless the tops of the girders were aligned at different elevations. Another

problem with the connection is that the common node, shared by both directions of joists,

is markedly eccentric from adjacent panel points. The connection would have to be

rather stiff, requiring excessive material in order to avoid considerable bending

deformations. The connection would have trouble passing all failure modes and limit

states with 30 + kips of loading using one or two bolts in double shear, while meeting all

of the applicable fastening clearance requirements. Also, reaming the bolt holes in

preparation for the fasteners is more difficult in the “cruciform” configuration than in the

connection in the final design (Figure 3.27). This is especially true if camber is desired in

the dominant joists.

An alternate (welded) connection detail, illustrated in Figure 3.26, provided a

system in which the centroids of the top chords from both directions were concentric.

Figure 3.26 - Alternate (Welded) Connection

Location Of Field Welds

Pieces Of Angle Welded In The Shop

76

The connection was designed to be welded instead of bolted in the field. The idea of

shifting from a bolted to a welded connection was abandoned because welding needs to

be performed by highly skilled workers. Another lingering problem is the excessive

eccentricity of the connection with respect to adjacent panel points. Measures taken to

alleviate this problem in the final design could also be applied to this connection. Part of

this solution would entail shifting the joists so that the dominant joists intersect a non-

dominant panel point. A new problem would arise from this maneuver, however,

facilitating over 30 kips of compressive load through the gap in the non-dominant joist

top chords. If the non-dominant joist was shifted “out-of-phase,” the loading could pass

through a top chord filler, welded between the chords in the shop. This would require the

load to pass through the fillet of the bent round bar web; this scenario would likely be

insufficient to handle such loading. This could be verified with either FE modeling or

experimental testing.

Figure 3.27 represents the connection used in the proposed system. The main

element responsible for transmitting the compressive axial loading across the

discontinuous top chord is the HSS piece. The thickness of the piece was chosen to

match, as closely as possible, with the thickness of the top chord of joist “A.” Referring

to Table J2.4 (AISC 2001b) shows that the minimum weld size, given a material

thickness of 1/4 in. or less, is a 1/8 in. fillet weld. Therefore, the fillet welds connecting

the top of HSS section with the top chords of the non-dominant joist will have a SMAW

1/8” weld leg. This will prevent local buckling of the HSS section as the compression

force Pu is transferred horizontally through the connection. Also aiding in the distribution

of the compressive chord force are the two stiffening plates.

77

Figure 3.27 - Chosen Connection Detail

Aligning these plates concentrically with the dominant joist should keep the bending

deformation of the HSS shape to a negligible magnitude. The stiffening plates will be

held in place by 1/8” fillet welds on three sides.

Figure 3.28 qualitatively illustrates how the compressive force arriving and

ending through the top chord of the dominant joist will likely find its way through the

intermediate HSS shape (Figure 3.28a).

Figure 3.28 - Connection Loads

BA

Pu

Vu B A

e

a)

b) Note: Web Angles Removed For Clarity “A” Dominant and “B” Non-Dominant Joist

6”

2”

CUT HSS SECT. 6”x6”x3/16”x5 3/4”

2L2”x2”

PLATE

5/8” A 325-N ASTM BOLT

3/16” PLATE 3/8” CHAMFER

3/4” ROUND BAR WEB

3” 1”

E70 1/8

5 3/4”x3”x3/16”

78

The eccentricity developed along this load path should be rather small, because the

centroid of the top chord already hovers near the top of the member (about 1/2 in. below

the top). Figure 3.28b demonstrates how the transfer of shear “pries” up on the HSS

connection element while it pushes down on the web angle (removed for clarity) of the

dominant joist.

The current LRFD specification (AISC 2001b) was used to guide the connection

design. This design assumes that the primary function of the connection is to transfer

transverse shear. Given the load combination of 1.2D + 1.6L, the maximum force present

at the interface between the dominant and non-dominant joist is Vu = 2.2 kips (the value

from the SAP2000 finite element analysis). Within the panel, maximum shear transfer

occurs at the intersection of the 2L2x2x0.205 dominant joist and the 2L2x2x0.148 non-

dominant joist.

Figure 3.29 - Bolt Location

Weld Design

Referring to Figure 3.29 and using equation 6.19.8 (Vinnakota 2005)

1 7/32”

1 19/32"

1”

1” 11/16” DIA. HOLE

A 3/16” Plate

2 7/8”

Outer Edge Of Web Angle (Web Member Removed For Clarity)

1 3/16"

3/16”

3”

79

Rdw = 0.45FexxteLw

Including the transverse weld, but conservatively treating it as a longitudinal weld (i.e.

50% additional strength is not considered) gives

Rdw = 2 [(0.45)(70)(.707)(0.125”)((2”-.205”) + 2”)] = 21.1 kips

Rdw >> Vu Okay

Bolt Design

From Table 6.7.1 (Vinnakota 2005), the shear strength of a 5/8 in. dia. A325-N bolt in

single shear is

Bdv = 11 kips

Check to see if shear tear-out governs design

Lce = Le – 0.5dh = (1 19/32”) – (0.5)(11/16”)

Lce = 1.25” ≤ 2d = 1.25” (Shear tear-out governs)

From equation 6.8.5 (Vinnakota 2005)

Bdte = 0.9Fup(Le – 0.5dh)t

Bdte = 0.9(65 ksi)[(1 19/32”) – 0.5(11/16”)](3/16”) = 13.71 kips

From equation 6.8.5 (Vinnakota 2005)

Cd = min[N Bdv, N Bdte] = min[(2)(11 kips), (2)( 13.71 kips)] = 22 kips

22 kips >> Vu Okay (future designs may take advantage of a smaller dia. bolt)

Check Edge Distance

From Section J3.4 (AISC 2001b)

Ls = 1 3/16” > 1 1/8” Okay

From Table 7-3a (AISC 2001a)

Fmin = 1” Okay

80

Section 3.6 Girder Design

Perhaps the most critical dimension of a floor panel is the depth of the supporting girders.

The depth of the girders is included in the depth of the floor construction; therefore

minimizing this dimension is crucial for reducing the floor-to-floor heights of a building.

In open-web steel joist construction, the top of steel (TOS) of the girders is 2-1/2 in.

lower than the TOS of the joists. Decreasing the distance a girder extends below the

bottom of the joists can markedly increase the necessary girder self-weight.

In non-composite construction, manipulating the placement of the loading on the

girders is the most effective way to decrease a girder size in a one-way system. In other

words, the manner in which the total girder loading “Q” (approximately equal to one-half

of the total loading acting over the entire panel) is distributed over the length of the girder

determines the magnitude of the maximum moment in the member. Figure 3.30 lists 16

different cases of load distribution, and the corresponding variations in maximum

bending moment developed in the supporting girder are shown in Figure 3.31. Each

arrow in Figure 3.30 represents a point load “P” (unless otherwise labeled). The worst

case scenario arises from concentrating all the loading at the center of the girder (see

Case 1, in which Mmax = (0.25)QL). As equally spaced point loads become closer to each

other, the maximum moment approaches Case 16 (where Mmax = (0.125)QL).

Recall the traditional (one-way) system studied using ten 16K9 joists (see Appendix B

for calculations). This translates to the load distribution shown as Case 13 in Figure 3.30.

The next illustration, Figure 3.31, demonstrates that spacing the joists in this manner

creates a favorable magnitude of maximum moment, with only a 9% difference from the

best case scenario (a uniform loading on the girder). Therefore, the biggest problem with

81

this joist configuration doesn’t reside with the way the joist reactions are dispersed along

the girder, but rather in the way in which the load arrives there in the first place, i.e. one-

way action.

Figure 3.30 - Load Cases

0

0.05

0.1

0.15

0.2

0.25

0.3

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Case #

QL

coef

ficie

nt

Figure 3.31 - Variation in Maximum Moment

Case 1

Case 2

Case 3

Case 4

Case 5

Case 6

Case 7

Case 8

Case 9

Case 10

Case 11

Case 12

Case 13

Case 14

Case 15

Case 16

¾ P ¾ P P P

P

q

L

82

The proposed system offers both two-way action and an opportunity to maneuver

joists to decrease girder sizes. An efficient system is created when all four girders

participating in supporting the gravity loads creates an efficient system, especially if two

additional girders were already present along the transverse column lines (likely part of

the lateral load resisting frame). The joist reactions in the proposed system are outlined

in Figure 3.32.

Figure 3.32 - Reactions from Proposed System

Outer Non-Dominant Joist

Inner Non-Dominant Joists

Outer Non-Dominant Joist

Dominant Joists

Girders R3 R4 R4 R3

R1

R2

R2

R1

Girders

Full Service (1.2 D + 1.6 L) Adjacent Bay Present LRFD (LRFD X 2)

4792 lbs 6768 lbs 13536 lbs 6256 lbs 8830 lbs 17660 lbs

5405 lbs 7611 lbs 15222 lbs

5587 lbs 7876 lbs 15734 lbs

R1

Joist Reaction

R2 R3 R4

83

Figure 3.32 reveals that the location and relative magnitude of the loading

transferred to the girder (supporting the non-dominant joists) from the joist reactions are

closely described by case 7 in Figure 3.30. Choosing a system with four non-dominant

joists, rather than two (Figure 3.14d), not only drags more loading to those supporting

girders, but also distributes it more advantageously once it arrives (case 4 vs. case 7 in

Figure 3.31). With the magnitudes of the reactions from the dominant joists nearly equal,

case 3 in Figure 3.30 comparatively represents the manner in which the reactions are

dispersed.

The design shown in Figure 3.14f was discarded because moving the dominant

joist an additional 19 in. away from the panel center decreased the moment, caused by the

joist reactions, by approximately 12.5% in the supporting girder (case 2 vs. case 3 in

Figure 3.31). This was only a portion of the total loading the girder received because it

also accepted loading directly from the slab (case 3 coupled with case 16). Designers

must find a balance between joist design and girder design because if the dominant joists

deviate beyond the confines of a fictious “middle strip,” the stiffness of the system

diminishes drastically. The calculations used to select adequate W-sections to serve as

girders for the proposed system are located in Appendix B.

Shifting the panel design from a one-way to a two-way system resulted in an

appreciable drop in girder sizes. Table 3.6 contains a modified excerpt taken from Table

5-3 (AISC 2001a). Given the same loading on a 30 ft. x 30 ft. floor panel, the girder

sections necessary to support the panel dropped from a W21x68 (one-way system) to a

W18x60 (girder supporting the dominant joists) and to a W16x50 (supporting the non-

dominant joists). As long as the base of the top flange exceeded 7 in., it was assumed

84

that two joist seats (typically 4 inches long) could be accommodated with only slight

modification to the end detail. Selected girders passed all strength and live load

deflection requirements.

Table 3.6 - Girder Selection

Shape Zx (in3) Ix (in

4) bf (in)W24x68 177 1830 8.97W16x89 176 1310 10.4

W14x99†† 173 1110 14.6W21x73 172 1600 8.3W12x106 164 933 12.2W18x76 163 1330 11

W21x68 160 1480 8.27W14x90†† 157 999 14.5

W24x62 154 1560 7.04W16x77 151 1120 10.3W12x96 147 833 12.2W10x112 147 716 10.4W18x71 146 1170 7.64

W21x62 144 1330 8.24W14x82 139 882 10.1

W24x55 135 1360 7.01W18x65 133 1070 7.59W12x87 132 740 12.1W16x67 131 963 10.2W10x100 130 623 10.3W21x57 129 1170 6.56

W21x55 126 1140 8.22W14x74 126 796 10.1W18x60 123 984 7.56W12x79 119 662 12.1W14x68 115 722 10W10x88 113 534 10.3

W18x55 112 890 7.53

W21x50 111 989 6.53W12x72 108 597 12

W21x48†† 107 959 8.14W16x57 105 758 7.12W14x61 102 640 9.99W18x50 101 800 7.5W10x77 97.6 455 10.2

W12x65†† 96.8 533 12

W21x44 95.8 847 6.5W16x50 92 659 7.07W18x46 90.7 712 6.06W14x53 87.1 541 8.06W12x58 86.4 475 10W10x68 85.3 394 10.1W16x45 82.3 586 7.04

W18x40 78.4 612 6.02W14x48 78.4 485 8.03W12x53 77.9 425 9.99W10x60 74.6 341 10.1

One-Way System

Two-Way System (Supporting DJ)

Two-Way System (Supporting NDJ)

85

Chapter 4 Composite Design


Taking advantage of composite action, when designing one-way open-web steel joist

floor systems, has become increasingly popular in recent decades. Composite

construction is attractive because it allows shallower floor depths, consequently reducing

the costs of fireproofing, foundations, etc. A designer has the option of making either the

open-web joists or the supporting girders composite (or both composite). Joist members

are designed to be “fully composite” but it is usually more economical, in the case of

using wide-flange steel sections, to design the girders as “partially composite” members.

The proposed two-way steel floor system was designed as a non-composite

system but can also offer composite capabilities to the designer. This chapter discusses

three different ways in which the joists and steel decking can be combined to achieve

composite action. Also included is a discussion on how the load distribution of the

system could be impacted by developing steel decking to force the slab to predominantly

span in two directions. This is followed by a suggestion explaining how one can

construct an FE model to predict the composite behavior of the proposed system. Finally,

composite girder options are investigated and the results of a composite design are

summarized to illustrate how the member sizes differ from those of non-composite

construction.

86

Section 4.2 Proposed System’s Composite Capabilities

When limiting the floor-to-floor height is the controlling design factor, it is not

uncommon for a traditional one-way steel floor system to have a span-to-depth ratio as

high as 30 (Samuleson 2005). This equates to 12 in. deep composite joists that are likely

spaced 6 ft. on center, assuming 1.5 in. steel decking is used to comply with fire rating

requirements found in Table 2.6 of the AISC manual (AISC 2001a). The author’s goal is

to shed light on how a two-way steel floor system could be constructed to compete with

traditional one-way floor systems’ span-to-depth ratios while simultaneously dropping

girder sizes.

The proposed steel joist system, as described thus far, exhibits non-composite

behavior. The panel loading is initially carried by the deck, which spans in one direction

until it reaches the interlocked joists. Joists distribute the loading in two directions within

a panel supported by four girders. Continuing to use traditional one-way decking will

limit the ways in which a two-way composite system can be created.

Two composite options exist, a partially composite or a fully composite system.

The term “partially composite” commonly means that not enough shear connectors are

present to completely develop uniform compressive stress corresponding to 0.85fc’ in the

concrete (in the compression flange). However, in the context of this thesis, the term

“partially composite” will be used when only one direction of joists becomes composite

with the overlying slab (Figure 4.1).

87

Figure 4.1 - Partially Composite System

In a two-way partially composite floor system, shear studs are placed between the

ribs of the steel decking (using 1.5 VLR (Vulcraft 2005)), alternating between the top

chord angles (similar to how they are attached in a one-way joist system). Once again,

the dominant joists will not directly carry the load from the slab. Rather, they will be

forced into service through deformation compatibility. The composite joists become

much stiffer, and cause a larger percentage of the load to be transferred to their

supporting girders, compared with the load distribution of the non-composite two-way

system. Taking advantage of the additional stiffness that the slab contributes and

utilizing the same spacing of joists used in the two-way non-composite system results in a

span-to-depth ratio of the partially composite system that likely matches or exceeds the

ratio of the traditional composite system. The decision to shift to composite construction

is often based on the cost of furnishing and installing shear connectors; therefore, creating

a system in which only half of the joists are composite may be a viable economical

solution.

88

In order to take further advantage of the overlying slab, a fully composite two-

way steel joist system can be employed. Both the non-dominant and dominants joists

will receive loading directly from the slab. The choice of decking as well as the direction

in which the decking spans is important to ensure that the shear studs can adequately

attach to the top chords of the dominant joists. As shown in Figure 4.2, the dominant

joists must be aligned in increments of 12 in. so that the shear studs can be attached

through the decking. This limits the panel lengths of the non-dominant joists to two

choices: 12 in. or 24 in. With a span-to-depth ratio likely ≥ 30, choosing a panel length

of 12 in. is the most logical choice. If the joists where to employ a 24 in. panel length,

there would be a considerable reduction in the joist moment of inertia due to excessive

web shear deformations. The dominant joist panel length would also be smaller than the

dimension chosen in the non-composite two-way system (24 in.). If the bottom chord of

one joist comes into contact with an orthogonal joist web member as a result of adjusting

the joist panel lengths, modifications can be made during fabrication (chords shifted

vertically, web fillets increased, etc.).

Figure 4.2 - Fully Composite Two-Way Joist System

4 ½”

2 VLI Deck 12”

2”

Approx. 1/8 in. Void

89

The manner in which loading is distributed throughout a composite system

completely changes when joists spanning in both directions carry the slab. A major

complication arises when trying to quantify the magnitude of loading that each joist

receives from the concrete slab. When carrying wet concrete, the slab will predominantly

span in one direction (parallel to the deck flutes). When the concrete cures the slab will

try to span in two directions depending on the internal reinforcement configuration. The

degree to which it spans in each direction is difficult to quantify in the absence of

experimental testing and significant analytical effort. If the load distribution cannot be

computed within a reasonable degree of certainty, the spacing and sizing of the joists

cannot be determined.

One possible solution to this dilemma is to employ the use of two-way steel

decking. The design philosophy is simple: create steel decking that has ribs aligned

orthogonally so that the deck has ample stiffness and strength in two orthogonal

directions. Overcoming the difficulty of the fabrication of this type of product has yet to

occur. If and when the Steel Deck Institute (SDI) develops two-way decking, a

composite-two-way steel joist system can easily make use of the new technology.

Figure 4.3 shows a plan view of a possible layout of two-way steel decking

overlying a fully composite two-way steel joist system. This configuration assumes that

the two-way steel decking can span 10 ft. in both directions by using unshored

construction. The load distribution is markedly different from the rectangular tributary

areas assumed in a four joist, one-way joist system. The joists in the two-way system

directly carry 66% of the total panel loading, while the joists of a traditional carry 80% of

the panel loading.

90

Figure 4.3 - Fully Composite System Using Two-Way Steel Decking

Therefore, the two-way composite joists can essentially be designed for a smaller panel

loading. Less loading, however, results in less mass on the composite two-way joists to

aid in damping the system. The author believes that the system’s ability to dissipate

vibrational energy in two directions will make up for this loss of damping ability. If the

two-way floor system exhibits poor vibrational performance, vibration mitigation

measures are available and would need to be implemented.

Composite systems push the span-to-depth ratio of joists farther than non-

composite systems; therefore, the supporting girders play an even bigger role in

establishing floor-to-floor heights. Aside from making the girders composite with the

overlying slab, changing the magnitude and distribution of the loading will have the

biggest impact on specifying an appropriate girder depth. For simplicity, assume that the

proposed system has the same floor depth (joist depth) in both directions. This

Direction 1 Joists

Direction 2 Joists

Girder

Tributary Areas

3 @ 10’- 0”

3 @ 10’- 0”

Direction 1 Joists

Direction 2 Joists

Girders

91

assumption allows the proposed system with a two-way deck to distribute the load evenly

to all four girders. With the total load (in pounds) on the floor panel assigned as “Q”,

Figure 4.4 illustrates the impact that two-way decking has on the maximum moment in a

girder. Using the proposed system rather than a traditional system drops the maximum

moment in a supporting girder from (0.06285)QL to (0.02777)QL + (0.01080)QL =

(0.03857)QL (Figure 4.4a - Figure 4.4c). This translates to a 39% difference in

maximum girder moment when a design takes advantage of four girders supporting the

panel loading.

Figure 4.4 - Supporting Girder Moments

Designing a fully composite two-way joist system that uses two-way steel

decking is somewhat challenging. In a traditional one-way composite joist system, a

designer has the option of bypassing the determination of the actual non-linear

distribution of bending stresses in the concrete compression flange. This is commonly

P P

P P P P

h

L/3 L/3 L/3

L/5 L/5 L/5 L/5 L/5

L/3 L/3 L/3

P=(1/12)Q Mmax = .02777QL

P = (.1)Q Mmax = 0.06284 QL

h = Q/(6L) Mmax = 0.01080 QL

b) Load from 2 joists

a) Load From 4 joists

c) Load Directly From Deck

92

done by isolating the joist from the structural system and replacing the actual slab with a

narrower or effective slab subject to a state of uniform stress modeled using the Whitney

stress block. The concrete slab in the two-way system will behave like a two-way plate;

therefore, using the effective flange width approximation may be inappropriate. This is a

critical issue needing resolution. FE modeling of the two-way system could be done.

Figure 4.5 illustrates a possible FE model that could be used under service level

loading. To model the composite behavior of the system, supplementary elements can be

added between the plate elements and the joist top chord (beam elements), as

schematically shown in Figure 4.5. These elements will allow the modeling of shear

transfer from the plate elements to the joist elements.

An infinitely stiff vertical “stub” element can be entered into the model to avoid

vertical displacement of the slab with respect to the joist top chord. The primary function

of the supplementary diagonal elements is to transfer shear from the plate elements to the

joist top chord elements. The axial stiffness entered into the model for these elements is

critical for modeling the internal couple that results from positive curvature of the

composite joist.

93

Figure 4.5 - Proposed FE Model

Section 4.3 Composite Girders

Forming composite action between the overlying deck and the supporting floor girders is

an additional step taken to reduce the depth of the floor construction. There are a variety

of ways to detail the composite mechanism and a few of these options are illustrated in

Figure 4.6. The details shown in Figure 4.6a through Figure 4.6c describe the bearing

condition of both a traditional system and a two-way system (in which a girder supports

the non-dominant joists) using one-way steel decking. One possibility, represented in

Figure 4.6a, uses a structural tee welded to the girder flange to fill the 2 1/2 in. void

created by the joist seats (Rongoe 1984). The detail that offers the shallowest depth of

floor construction is the shear-gusset connection shown in Figure 4.6b. A shear

connection attached to the web of the girder allows the TOS of the joists to match the

Active Slab Thickness

Supplementary Elements

Plate Element Centroid

V

V

Major And Minor Axis Moment Release Supplementary Element To

Model Stud Load-DeformationBehavior

Joist Elements

94

TOS of the girders. Another possibility uses specially prepared pour stops to form

concrete haunches over the girder (Figure 4.6c). The remaining option, shown in Figure

4.6d, shows a modified Rongoe system used to support the dominant joists in the

proposed two-way floor system (notice how the decking runs in the perpendicular

direction). Recall that these girders have to directly carry a tributary width of decking. If

the girders are non-composite, a contractor may elect to use pieces of channel to fill the

void because channel can easily be welded to the girder (see chapter 2). However, if

composite action is desired, a channel section may be replaced with a structural tee; this

configuration allows a more direct load path to exist for the shear transferred between the

stud and girder top flange.

Figure 4.6 - Composite Girder Details

a) Rongoe System

c) Concrete Haunches With Pour Stop

d) Modified Rongoe System Note: Deck Span Parallel To Joists

b) Flush Joists With Shear Gusset Plate

95

To illustrate the impact that composite action has on dropping girder depths, an

analysis was performed using the girder loads from both the traditional and two-way non-

composite joist designs. The analyses for the girders in a traditional system and the

girders supporting the non-dominant joists in the proposed two-way system assumed that

shear studs were welded directly into the girders and surrounded by formed concrete

haunches (Figure 4.6c). During service and ultimate loading stages, the presence of the

additional 2 1/2 in. of concrete below the deck was neglected to simplify the calculations.

The girders supporting the dominant joists in the proposed two-way system were

assumed to use the detail shown in Figure 4.6d. All girder depths were set at 14 in.,

creating a scenario in which the bottoms of the girders are (approximately) flush with the

bottom of 16 in. deep joists. Fifty percent composite action was assumed to take place

between the interface of the steel and concrete. The full analysis can be found in

Appendix C. The drop in girder sizes is summarized in Table 4.1.

Non-Composite Size W21x68 W18x60 W16x50

Composite Size W14x61 W14x48 W14x34

Two-Way Joist System (Supporting Dominant

Joists)Girder Type Traditional

Joist System

Two-Way Joist System

(Supporting Non-Dominant Joists)

Table 4.1 - Resulting Girder Sizes

96

Chapter 5 Conclusions and Recommendations

Section 5.1 Summary

The literature review allowed the reader to reflect upon the more common ways in which

engineers and architects have set out to minimize floor-to-floor heights in commercial

and residential structural steel buildings. In particular, the AISC competition drew

attention to the reality that steel fabricators, erectors, and designers are looking for floor

systems that are shallower and, at the same time, can be categorized as light and easily

able to accommodate MEP conduits. The information in the literature review helped to

mold some of the design concepts used to design the proposed two-way steel joist

system.

A description of the proposed system began with outlining some of the fabrication

issues as well as a general erection sequence that a non-composite two-way floor system

will undertake. Two categories of joists used by this system were defined: dominant

(deeper) and non-dominant (shallower). Three-dimensional solid models of the system

were presented, enabling the reader to quickly survey the critical constituent elements of

the two-way floor system. The author contends that the more simplistic a floor

assembling process becomes (especially in the case of connections), the more willingly

fabricators and erectors will adapt to significant changes in steel floor construction.

The quintessence of the study was examining the structural behavior of the

proposed system, starting with basic plate theory and arriving at the structural behavior of

a non-composite two-way floor system. This chapter outlined the modeling and design

efforts used to select the joist chord and web members, interlocking connections, and

97

supporting girders. The chapter also explained the rationale behind discarding

preliminary system configurations.

The chapter on composite design explained the proposed system must be capable

of becoming composite with the overlying concrete slab in order to emerge as a more

economical alternative to current steel floor construction systems. Three different

composite designs were discussed: partially composite, fully-composite, and fully-

composite with two-way steel decking. Composite girder design was addressed and

accompanied with results illustrating the change in girder depth that occurs when

composite action is used.

Section 5.2 Conclusions

Current joist construction, both composite and non-composite, makes efficient use of

each individual member. Composite construction, in particular, gives a designer the

opportunity to achieve large span-to-depth ratios without developing excessive member

self-weight. The supporting girders in a traditional system, however, can benefit by

changing the manner in which they receive loading, namely, changing the panel from

predominantly spanning in one direction to spanning in two directions. This change is

important because the depth of a girder is included within the depth of the floor

construction.

This study verified that it is structurally feasible to interlock open-web steel joists,

arranged orthogonally within a floor panel, to form a floor system that predominantly

spans in two directions. The system exhibited some structural behavior analogous to

traditional steel joists, while other behavior reflected a uniformly loaded two-way simply

98

supported plate. The results of the two systems are summarized in Table 5.1. When the

two floor systems are compared (Table 5.2), the proposed non-composite system drops

supporting girder sizes while using less steel within the panel. Less material was needed

in the proposed non-composite joist system despite the fact that further design iterations

would have further reduced the proposed system’s member sizes.

Proposed System Studied Traditional System Studied

Depth of the Floor Construction (Non-Composite Joists with Composite Girders)

The depth of the floor construction was set at 14 in. + 2.5 in. + 4 in. = 20.5 in. However, the girders in the two-way system were lighter.

The depth of the floor construction was also 20.5 in.

The depth of the floor construction is deeper in the traditional system, equating to 21 in. + 2 1/2 in. + 4 in. = 27.5 in. Recall that the girders in both systems were selected by calculating the required plastic section modulus and then choosing the lightest W-section with a value that meets or exceeds the required modulus.

Using non-composite construction, the depth of the floor construction is controlled by the girders supporting the dominant joist. This depth comes to 18 in. + 2 1/2 in. + 4 in. = 24.5 in.

Depth of the Floor Construction (Non-Composite Joists and Non-Composite Girders)

A W18x60 girder was used to support the dominant joists while a W16x50 was used to support the non-dominant joists.

A W21x68 girder was used to support the traditional joist system.

The decking in the system was changed to a 1.5 VLR. The girder depths were set at a constant 14 in. depth and the details are as shown in chapter 4. For simplicity, only a 3.25" concrete slab thickness was assumed to be present under service level loads (regardless of the detail). The girder size needed to support the dominant joists was a W14x48 while the girder size dropped to a W14x34 supporting the non-dominant joists.

Composite girder design

The same assumptions for loading and design, used in the proposed system, were also used for the traditional design. Again, the composite detail used was shown in chapter 4. The girder size needed to support the (10) 16K9 joists was a W14x61.

Non-composite girder size

The slab was 4 in. thick and used 1.5 VL steel decking.Floor Slab

The slab was 4 in. thick and used 1.5 VL steel decking.

Description of the Joist System within the 30' x 30' interior floor bay

This system used a total of (8) non-composite joists, interlocked to form a two-way system. The deeper joists (16 in.) were labeled "dominant" and the shallower joists (14 in.) "non-dominant." A total of (4) girders supported the system.

This system was composed of (10) non-composite 16K9 joists. A total of (2) girders supported the system

Table 5.1 - Summary of Final Floor Panel Configurations

99

Table 5.2 - Comparison of System Self-weights

Compared to a traditional one-way (non-composite joist) system general

advantages and disadvantages are:

Advantages

- The depth of the floor construction, established by the height of a non-composite

girder, is decreased by 3 in.

- Lighter girders can be used (composite girder or non-composite girder

construction) as a result of carrying the panel loading in two-directions.

- Fewer joists are needed to carry the prescribed panel loading. Overall, less joist

material is used within the panel.

- The stiff, interlocked joist system is safer for iron-workers to walk on before it

becomes permanently attached to the girders.

- The proposed system is a big step towards joist panelized construction.

- Due to the deflected “dish shape” that the proposed system undertakes, ponding

(developing as a result of pouring wet concrete) is less of a concern.

Disadvantages/Limitations

- More fabrication time and material is needed to build each individual joist.

- Altering the layout of the joists is limited by the panel dimensions of the joists.

- One-way steel decking restricts the two-way behavior of the system.

Inner Non-Dominant Joist (INDJ ) 10.9 327 2 654Outer Non-Dominant Joist (ONDJ ) 9.2 276 2 552

Dominant Joist (DJ) 12.6 378 4 1512Connection Hardware N/A 3 16 48

2766

Traditional Non-Composite

Proposed Non-Composite

Total (lbs)

16K9 9.4 282 10 2820

Description of Joist MemberSystem Estimated Self Wt. (lbf)

Estimated Self Wt.

(lbs.)

Quantity (Within the

Panel)

100

Section 5.3 Recommendations for Future Research

This study is a first step in a new direction for steel joist floor construction. The focus of

this study was to investigate the structural feasibility of slightly modifying the fabrication

of open-web joists to produce a steel floor system capable of spanning in two directions.

Many additional studies can be undertaken to further verify the practicality of this new

design philosophy.

This study examined a 30’x 30’ ( 1LL

αshort

long == ) floor bay under typical office

loadings. Further studies could provide insight into how varying α from 1≤ α ≤ 2

influences the behavior of the system. Different floor loadings, panel locations (exterior

bay, corner bay, etc.), and floor openings can be investigated to determine the impact of

varying these design parameters.

To act upon the discussion set forth in chapter 4, additional finite element analysis

and experimentation is needed to evaluate composite design. Composite design is likely

to have a significant influence on the behavior of the floor system and the floor system

depth. FE analysis could be used to study load distribution, vibration characteristics,

connection design, and deflections. Experimental work is needed to validate all

analytical efforts. This could involve testing the bolted connection components or going

as far as testing the entire system at full-scale.

A floor system must exhibit satisfactory vibrational performance in order to gain

acceptance among architects and contractors. This study examined only the static

structural characteristics of the proposed system. Further analysis is required to

determine whether or not the system would meet vibrational performance standards under

dynamic loading.

101

Finally, and perhaps most importantly, the results of an economical evaluation are

necessary for the development of this system. This evaluation should not only include

comparisons with open-web steel joist, but also with other floor framing systems

(including concrete floor systems). Once the design of a two-way steel floor system

using open-web steel joists becomes established, computer algorithms could be

developed to determine the most effective joist configurations.

102

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Appendix A

The following tables and figures illustrate the joist member sizes chosen for the proposed

two-way joist system that uses allowable stress design. Taking advantage of symmetry,

the labeling scheme is shown in Figure A.1. The numbering shown in Figure A.1a takes

advantage of symmetry to identify each top chord, bottom chord, and web member in the

proposed system. Even though the design distinguishes between two types of non-

dominant joists (outer non-dominant joists and inner non-dominant joists), Figure A.1b is

valid for both types. The equations used were outlined in Chapter 3.

Figure A.1 - Joist Labeling

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19

20 21 22

23

24

25 26

27

28

29 30 31 32

33

34

35 36

37

38

39 40

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17

19 18 21 23 25 27 29 31 33

20 22 24 26 28 30 32 34

a) Dominant Joist

b) Non-Dominant Joist

108

The following tables show the results of the chord member analysis. The

tabulated results reflect the loading and equations used for a single angle member. It

should be noted that the dominant joist top chords assume there are filler plates welded in

the middle of each 24 in. panel (and at 3rd points at the end panels). This insures that

yielding, not lateral torsional buckling, controls the design.

Top Chord Design For ONDJ (Outer Non-Dominant Joist)Mem Area S Axial fa Mom fb ra rb (KL)a (KL)b Axial Bend. Fea Feb Axial Bend. Tot.

# Description in2 in3 (kips) (ksi) (k-in) (ksi) in in in in KL/r KL/r (ksi) (ksi) Q Cap. Cap. Cap.1 2L2x2x.148 0.57 0.40 5.92 10.39 2.56 6.44 0.40 0.62 26.00 26.00 65.66 41.74 66.40 164.28 0.91 0.51 0.23 0.742 2L2x2x.148 0.57 0.40 5.42 9.51 1.04 2.60 0.40 0.62 19.00 19.00 47.98 30.50 124.33 307.63 0.91 0.40 0.09 0.493 2L2x2x.148 0.57 0.40 7.01 12.31 0.89 2.23 0.40 0.62 19.00 19.00 47.98 30.50 124.33 307.63 0.91 0.52 0.08 0.604 2L2x2x.148 0.57 0.40 7.72 13.55 1.08 2.71 0.40 0.62 19.00 19.00 47.98 30.50 124.33 307.63 0.91 0.58 0.09 0.675 2L2x2x.148 0.57 0.40 7.53 13.21 1.06 2.66 0.40 0.62 19.00 19.00 47.98 30.50 124.33 307.63 0.91 0.56 0.09 0.666 2L2x2x.148 0.57 0.40 6.45 11.31 0.86 2.17 0.40 0.62 19.00 19.00 47.98 30.50 124.33 307.63 0.91 0.48 0.07 0.567 2L2x2x.148 0.57 0.40 5.69 9.98 0.66 1.65 0.40 0.62 19.00 19.00 47.98 30.50 124.33 307.63 0.91 0.43 0.06 0.488 2L2x2x.148 0.57 0.40 6.33 11.11 0.87 2.19 0.40 0.62 19.00 19.00 47.98 30.50 124.33 307.63 0.91 0.47 0.08 0.559 2L2x2x.148 0.57 0.40 7.22 12.67 1.04 2.60 0.40 0.62 19.00 19.00 47.98 30.50 124.33 307.63 0.91 0.54 0.09 0.63

NoteThe subscript "a" is for axialThe subscript "b" is for bending and this is the value used in the amplification of the bending moment

Bottom Chord Design For ONDJ (Outer Non-Dominant Joist)Mem Area S* Axial fa Mom fb Axial Bend. Tot.

# Description in2 in3 (kips) (ksi) (k-in) (ksi) Cap. Cap. Cap.10 2L2x2x.137 0.53 0.14 6.47 12.23 0.62 4.39 0.41 0.15 0.5511 2L2x2x.137 0.53 0.14 7.59 14.34 0.62 4.39 0.48 0.15 0.6212 2L2x2x.137 0.53 0.14 7.86 14.85 0.58 4.09 0.50 0.14 0.6313 2L2x2x.137 0.53 0.14 7.21 13.62 0.58 4.09 0.45 0.14 0.5914 2L2x2x.137 0.53 0.14 5.69 10.76 0.37 2.63 0.36 0.09 0.4515 2L2x2x.137 0.53 0.14 5.67 10.72 0.38 2.67 0.36 0.09 0.4516 2L2x2x.137 0.53 0.14 7.00 13.23 0.55 3.86 0.44 0.13 0.5717 2L2x2x.137 0.53 0.14 7.45 14.09 0.55 3.90 0.47 0.13 0.60

Note* The section modulus is reduced due to member orientation

Table A.1 - Chord Member Design for Outer Non-Dominant Joists Top Chord Design For INDJ (Inner Non-Dominant Joist)

Mem Area S Axial fa Mom fb ra rb (KL)a (KL)b Axial Bend. Fea Feb Axial Bend. Tot. # Description in2 in3 (kips) (ksi) (k-in) (ksi) in in in in KL/r KL/r (ksi) (ksi) Q Cap. Cap. Cap.1 2L2x2x.187 0.71 0.48 7.97 11.18 3.09 6.47 0.39 0.62 26.00 26.00 66.02 42.11 65.66 161.39 1.00 0.51 0.22 0.732 2L2x2x.187 0.71 0.48 7.55 10.59 1.59 3.32 0.39 0.62 19.00 19.00 48.25 30.77 122.95 302.22 1.00 0.42 0.10 0.523 2L2x2x.187 0.71 0.48 10.30 14.45 1.01 2.11 0.39 0.62 19.00 19.00 48.25 30.77 122.95 302.22 1.00 0.57 0.07 0.644 2L2x2x.187 0.71 0.48 12.15 17.04 1.40 2.94 0.39 0.62 19.00 19.00 48.25 30.77 122.95 302.22 1.00 0.67 0.10 0.775 2L2x2x.187 0.71 0.48 13.09 18.36 1.42 2.96 0.39 0.62 19.00 19.00 48.25 30.77 122.95 302.22 1.00 0.73 0.10 0.826 2L2x2x.187 0.71 0.48 13.15 18.44 1.34 2.80 0.39 0.62 19.00 19.00 48.25 30.77 122.95 302.22 1.00 0.73 0.09 0.827 2L2x2x.187 0.71 0.48 13.12 18.40 1.17 2.45 0.39 0.62 19.00 19.00 48.25 30.77 122.95 302.22 1.00 0.73 0.08 0.818 2L2x2x.187 0.71 0.48 13.94 19.55 1.36 2.85 0.39 0.62 19.00 19.00 48.25 30.77 122.95 302.22 1.00 0.77 0.09 0.879 2L2x2x.187 0.71 0.48 14.83 20.80 1.54 3.23 0.39 0.62 19.00 19.00 48.25 30.77 122.95 302.22 1.00 0.82 0.11 0.93


Bottom Chord Design For INDJ (Inner Non-Dominant Joist)Mem Area S* Axial fa Mom fb Axial Bend. Tot.

# Description in2 in3 (kips) (ksi) (k-in) (ksi) Cap. Cap. Cap.10 2L2x2x.176 0.67 0.18 9.22 13.69 0.88 4.91 0.46 0.16 0.6211 2L2x2x.176 0.67 0.18 11.44 16.99 0.88 4.91 0.57 0.16 0.7312 2L2x2x.176 0.67 0.18 12.86 19.10 0.94 5.21 0.64 0.17 0.8113 2L2x2x.176 0.67 0.18 13.34 19.82 0.94 5.21 0.66 0.17 0.8314 2L2x2x.176 0.67 0.18 12.96 19.25 0.83 4.59 0.64 0.15 0.7915 2L2x2x.176 0.67 0.18 13.28 19.73 0.85 4.75 0.66 0.16 0.8216 2L2x2x.176 0.67 0.18 14.60 21.70 1.03 5.75 0.72 0.19 0.9217 2L2x2x.176 0.67 0.18 15.06 22.38 1.04 5.80 0.75 0.19 0.94

Note* The section modulus is reduced due to member orientation

Table A.2 - Chord Member Design for Outer Non-Dominant Joists

109

Top Chord Design For DJ (Dominant Joist)Mem Area S Axial fa Mom fb ra rb (KL)a (KL)b Axial Bend. Fea Feb Axial Bend. Tot.



Bottom Chord Design For DJ (Dominant Joist)Mem Area S Axial fa Mom fb Axial Bend. Tot.

# Description in2 in3 (kips) (ksi) (k-in) (ksi) Cap. Cap. Cap.10 2L2x2x.187 0.71 0.48 10.75 15.08 1.25 2.61 0.50 0.09 0.5911 2L2x2x.187 0.71 0.48 11.20 15.71 1.22 2.56 0.52 0.09 0.6112 2L2x2x.187 0.71 0.48 14.48 20.30 1.33 2.78 0.68 0.09 0.7713 2L2x2x.187 0.71 0.48 14.03 19.68 1.33 2.78 0.66 0.09 0.7514 2L2x2x.187 0.71 0.48 16.10 22.58 0.70 1.47 0.75 0.05 0.8015 2L2x2x.187 0.71 0.48 18.10 25.39 1.50 3.13 0.85 0.10 0.9516 2L2x2x.187 0.71 0.48 18.56 26.02 1.50 3.13 0.87 0.10 0.9717 2L2x2x.187 0.71 0.48 19.59 27.48 1.54 3.23 0.92 0.11 1.0218 2L2x2x.187 0.71 0.48 19.16 26.87 1.54 3.23 0.90 0.11 1.0019 2L2x2x.187 0.71 0.48 19.20 26.92 0.82 1.72 0.90 0.06 0.95

Table A.3 - Chord Member Design for Dominant Joists

The dominant joist top chords have top chord fillers; therefore, another set of

checks may control the design (if two angles are considered to act together). The two

slenderness ratios compared are (KyLy/ry) and (KxLx/rx). The ratio (KxLx/rx) = 39 controls

the design and the results in Table A.4 reflect the loading and equations applicable for

two angles acting together.

Top Chord Design For DJ (Dominant Joist)Mem Area S Axial fa Mom fb ra rb (KL)a (KL)b Axial Bend. Fea Feb Axial Bend. Tot.



Table A.4 -Top Chord Results for Dominant Joists (Double Angle)

The following tables show the results of the web member design. Using the same

round bar size (3/4 in. diameter) throughout the design simplified the calculations. This

is why the values of percent capacity are quite small (especially in the tension members).

Further iterations could produce a more economical design.

110

Tension Member Design For ONDJ (Outer Non-Dominant Joist)Ten. Area Axial fa Fa Axial

Mem # Description in2 (kips) (ksi) (ksi) Cap.18 7/8" Round 0.60 12.51 20.81 30.00 0.6921 3/4" Round 0.44 1.75 3.96 30.00 0.1323 3/4" Round 0.44 4.53 10.25 30.00 0.3426 3/4" Round 0.44 1.04 2.35 30.00 0.0828 3/4" Round 0.44 2.44 5.52 30.00 0.1829 3/4" Round 0.44 0.00 0.00 31.00 0.0030 3/4" Round 0.44 0.06 0.15 30.00 0.0031 3/4" Round 0.44 2.14 4.83 30.00 0.1633 3/4" Round 0.44 0.73 1.64 30.00 0.05

Table A.5 - Web Tension Members (Outer Non-Dominant Joists)

Tension Member Design For INDJ (InnerNon-Dominant Joist)Ten. Area Axial fa Fa Axial

Mem # Description in2 (kips) (ksi) (ksi) Cap.18 7/8" Round 0.60 16.84 28.01 30.00 0.9321 3/4" Round 0.44 3.51 7.95 30.00 0.2623 3/4" Round 0.44 2.31 5.22 30.00 0.1725 3/4" Round 0.44 0.77 1.74 30.00 0.0628 3/4" Round 0.44 0.63 1.41 30.00 0.0529 3/4" Round 0.44 0.55 1.25 30.00 0.0431 3/4" Round 0.44 2.14 4.84 30.00 0.1633 3/4" Round 0.44 0.73 1.65 30.00 0.05

Table A.6 - Web Tension Members (Inner Non-Dominant Joists)

Tension Member Design For DJ (Dominant Joist)Ten. Area Axial fa Fa Axial

Mem # Description in2 (kips) (ksi) (ksi) Cap.20 7/8" Round 0.60 17.65 29.37 30.00 0.9821 3/4" Round 0.44 0.40 0.90 30.00 0.0323 3/4" Round 0.44 1.12 2.53 30.00 0.0825 3/4" Round 0.44 5.46 12.36 30.00 0.4128 3/4" Round 0.44 1.16 2.62 30.00 0.0929 3/4" Round 0.44 3.39 7.66 30.00 0.2631 3/4" Round 0.44 3.27 7.40 30.00 0.2533 3/4" Round 0.44 1.21 2.73 30.00 0.0935 3/4" Round 0.44 1.88 4.25 30.00 0.1438 3/4" Round 0.44 1.13 2.55 30.00 0.0939 3/4" Round 0.44 0.14 0.33 30.00 0.0140 3/4" Round 0.44 0.03 0.06 30.00 0.00

Table A.7 - Web Tension Members (Dominant Joists)

111

Compression Member Design For ONDJ (Outer Non-Dominant Joist)Comp. Area KL Axial Axial fa Fe Fa AxialMem # Description in2 in. KL/r (kips) (ksi) (ksi) (ksi) Cap.

19 3/4" Round 0.44 15.30 81.59 1.63 3.68 43.00 18.44 0.2020 3/4" Round 0.44 15.30 81.59 3.39 7.66 43.00 18.44 0.4222 3/4" Round 0.44 15.30 81.59 1.84 4.16 43.00 18.44 0.2324 3/4" Round 0.44 15.30 81.59 0.42 0.96 43.00 37.71 0.0325 3/4" Round 0.44 15.30 81.59 1.06 2.40 43.00 37.71 0.0627 3/4" Round 0.44 15.30 81.59 2.43 5.50 43.00 37.71 0.1532 3/4" Round 0.44 15.30 81.59 2.13 4.82 43.00 37.71 0.1334 3/4" Round 0.44 15.30 81.59 0.74 1.67 43.00 37.71 0.04

Table A.8 - Web Compression Members (Outer Non-Dominant Joists)

Compression Member Design For INDJ (Inner Non-Dominant Joist)Comp. Area KL Axial Axial fa Fe Fa AxialMem # Description in2 in. KL/r (kips) (ksi) (ksi) (ksi) Cap.

19 3/4" Round 0.44 15.30 81.59 1.38 3.11 43.00 18.44 0.1720 3/4" Round 0.44 15.30 81.59 5.37 12.15 43.00 18.44 0.6622 3/4" Round 0.44 15.30 81.59 3.63 8.21 43.00 18.44 0.4524 3/4" Round 0.44 15.30 81.59 2.27 5.13 43.00 37.71 0.1426 3/4" Round 0.44 15.30 81.59 0.79 1.79 43.00 37.71 0.0527 3/4" Round 0.44 15.30 81.59 0.62 1.40 43.00 37.71 0.0430 3/4" Round 0.44 15.30 81.59 0.49 1.12 43.00 37.71 0.0332 3/4" Round 0.44 15.30 81.59 2.13 4.82 43.00 37.71 0.1334 3/4" Round 0.44 15.30 81.59 0.74 1.68 43.00 37.71 0.04

Table A.9 - Web Compression Members (Inner Non-Dominant Joists) Compression Member Design For DJ (Dominant Joist)

Comp. Area Q KL Axial Axial fa Fe Fa AxialMem # Description in2 in. KL/r (kips) (ksi) (ksi) (ksi) Cap.

22 3/4" Round 0.44 1.00 15.30 81.59 7.11 16.07 43.00 18.44 0.8724 2L1.5x1.5x.113 0.33 0.92 17.40 58.60 1.75 5.36 83.35 21.91 0.2426 3/4" Round 0.44 1.00 15.30 81.59 4.97 11.25 43.00 37.71 0.3027 2L1.5x1.5x.113 0.33 0.92 17.40 58.60 1.72 5.27 83.35 21.91 0.2430 3/4" Round 0.44 1.00 15.30 81.59 3.21 7.26 43.00 37.71 0.1932 3/4" Round 0.44 1.00 15.30 81.59 3.10 7.02 43.00 37.71 0.1934 2L1.5x1.5x.113 0.33 0.92 17.40 58.60 1.76 5.39 83.35 21.91 0.2536 3/4" Round 0.44 1.00 15.30 81.59 1.42 3.20 43.00 37.71 0.0837 2L1.5x1.5x.113 0.33 0.92 17.40 58.60 1.66 5.10 83.35 21.91 0.23

Table A.10 - Web Compression Members (Dominant Joists)

112

Appendix B The following calculations are for a joist and non-composite girder design of a 30 ft. x 30

ft. traditional interior floor bay panel.

Description:

The girder supports reactions from two adjacent floors bays composed of (10) 16 K9 non-

composite joists. The panel has a 4 in. concrete (4 ksi) floor slab. All girders are grade

50 steel.

Loadings:

SI DL = 40 psf (4 in. slab using 1.5 VLI composite decking (Vulcraft 2005))

15 psf (plumbing, electrical, and ceiling)



Assumptions:

- At the ultimate load stage (girder design), the load combination 1.2 D + 1.6 LL

governs the design.

- The LL deflection limit is L/360 (1 in.).

Joist Design:

Assuming minimum depth governs the floor joist design, (10) 16K9’s were considered @

2.727 ft. O.C.

(120 psf)(2.727 ft.) = 327.24 lb/ft

From Standard Joist Load Table (SJI 2005)

TLallow = 355 lb/ft

327.24 lb/ft + 10 lb/ft = 337.2 lb/ft < 355 lb/ft okay

113

Check Deflections

Iapprox = (26.767)(178 lb/ft)(29.666 ft)3(10-6) = 124.4 in4

4

DL 6 4

5(160lb/ft)(29.666ft) 1728δ = = 0.7724 in.384(29x10 psi)(124.4 in )

≤ L/240 = 1.5 in. okay

4

LL 6 4

5(177.3 lb/ft)(29.666ft) 1728δ = =0.855 in.384(29x10 psi)(124.4 in )

< L/360 = 1.0 in. okay

Joist size and spacing is adequate.

Girder Design:

The reactions from the joists will produce the following (factored) loading on the

supporting girder (Figure B.1).

Figure B.1 - Traditional Girder Loading

Pu = 2 [1.2((10 lb/ft)(15 ft.) + (55 psf)( 2.727 ft.)(15 ft.)) + 1.6(65 psf)( 2.727’)(15 ft)]

Pu = 14268 lb

Mumax = 7004 k-in (This does not include girder self-weight)

3reqd

7004k-inZx = =155.6in(0.9)(50ksi)

From Table 5-3 (AISC 2001a), try a W21x68 as a trial section where Zx = 160 in3

Including the self-weight of the member, the moment becomes

2

additional(1.2)(68 lb/ft)(30 )M = =9180 lb-ft or 110.16 k-in

8

(10) 16 K9 Reactions

30’

Pu Pu Pu Pu Pu Pu Pu Pu Pu Pu

114

3 3reqd

7114.16k-inZx = =158.1in < 160 in(0.9)(50ksi)

Okay

Lb = 2.727 ft. < Lp = 6.36 ft. (Full plastic moment capacity is reached) Mdx = ΦbMpx = 600 k-ft > 592.85 k-ft okay

Check Shear

ΦvVn = 245 k >> Vu

Check Service Load Level Deflections

P = 2[(65 psf)( 2.727’)(15 ft)] = 5318 lb

Ix = 1480 in4

ΔLLmax = .8225 in > 1.0 in (L/360) okay

Select a W21 x 68 Girder

The following calculations illustrate the girder size needed for a girder supporting

dominant joists, as a result of a two-way distribution of loading. The design assumptions

(panel loading from two adjacent symmetrical interior floor bays, deflection limits, etc.)

are the same as those chosen for the one-way system. With values taken from the SAP

2000 finite element analysis, Figure B.2 illustrates the loading that will be present on the

girders (point loads are from joist reactions).

Figure B.2 - Girder Supporting the Dominant Joists

19”

30’

10’-3”

wu

P1 P2 P3 P4

P1, P4 = 15222 lb P2, P3 = 15734 lb

6’-4” 10’-3”19”

115

The girder shown in Figure B.2 will also support a uniform loading directly from the slab

equivalent to

wu = 1.2(55 psf)(6 ft.) + 1.6(65 psf)(6 ft.) = 1020 lb/ft

Girder Member Sizing

From the loading in Figure B.2

Mumax = 5483 k-in (girder self-weight not included)

3reqd

5483 k-inZx = =121.8 in(0.9)(50ksi)

From Table 5-3 in the steel manual (AISC 2001a), try a W18x60 as a trial section where

Zx = 123 in3. The girder is braced by the floor slab and joists; therefore, it will reach its

full plastic moment capacity. Including the self-wt of the member, additional uniform

loading becomes

1.2(65 k/ft) = 72 lb/ft

This results in a maximum moment of

Mu = 465 k-ft

because

Mdx = ΦbMpx = 461 k-ft ≈ 465 k-ft

the moment strength is deemed adequate.

Check Shear

ΦvVn = 204 k >> Vu


Live load loading becomes

wserv = (65 psf)(6 ft.) = 390 lb/ft

P1 = P4 = 5628 lb and P2 = P3 = 5812 lb

116

Ix = 984 in4

δLLmax = 0.952 in < 1 in (L/360) Okay

Select a W18x60 Girder

The following calculations illustrate the girder size needed for a girder supporting

the non-dominant joists as a result of a two way distribution of loading. The same design

criteria, used for the two previous girder designs, are exercised here. With values taken

from the SAP2000 finite element analysis, Figure B.3 illustrates the loading, assuming

the load combination of 1.2D + 1.6L, that will be present on the girders (point loads are

from joist reactions).

Figure B.3 - Girder Supporting Non-Dominant Joists

Mumax = 3518 k-in (girder self-weight not included)

3reqd

3518 k-inZx = =78.2 in(0.9)(50ksi)

From Table 5-3 in the steel manual (AISC 2001a), try a W16x50 as a trial section where

Zx = 92 in3

P1, P4 = 13536 lb

P2, P3 = 17660 lb

30’

P1 P2 P3 P4

6’ 6’ 6’ 6’ 6’

117

2

additional(1.2)(50 lb/ft)(30 )M = =6750 lb-ft or 81 k-in

8

The resulting moment diagram is shown in Figure B.4.

Figure B.4 - Moment Diagram of Girder Supporting Non-Dominant Joists

3 3reqd

3599 k-inZx = =80 in < 92 in(0.9)(50ksi)

okay

Check adequacy at midspan

Lb= 6 ft. Cb = 1.0

Mdx = Min [ΦbMpx; Cb[ΦbMpx-BF(Lb-Lp)]]

1.0[(345 k-ft)-10.1 k(6 ft. - 5.62 ft.)] = 341.2 k-ft

Mdx = 341.2 k-ft > 300 k-ft okay

Check Shear

ΦvVn = 167 k >> Vu


LL becomes

P1 = P4 = 5090 lb

P2 = P3 = 6612 lb

Ix = 659 in4

δLLmax = .929 in < 1 in (L/360) Okay

Select a W16x50 Girder

Lb = 6’

30’

300 k-ft

118

Appendix C

The following is a composite girder design for a 30 ft. W-section in a traditional interior

bay.

Description:

The girder supports reactions from two adjacent floors bays composed of (10) 16 K9 non-

composite joists. The panel has a 4 in. (4 ksi) concrete floor slab. All girders are grade

50 steel.

Loadings:

Construction = 10 psf

SI DL = 44 psf (4 in. concrete slab using 1.5 VLR composite decking (Vulcraft 2005))

15 psf (plumbing, electrical, and ceiling)



Assumptions:

- Composite action is accomplished by forming concrete haunches over the girder

top chord. The added concrete located below the slab thickness (filling the 2 ½”

void) will be neglected, however, with respect to both added weight and added

strength.

- The fraction of “compositeness” is assumed to be 50%.

- The slab dimensions are as shown in Figure C.1.

- The deflection criteria is as follows:

δconstruction ≤ L/240 (Construction Load Stage)

δLL ≤ L/360 (Service Load Stage)

119

- Under factored loading, the load combination 1.2 D +1.6 LL governs.

Figure C.1 - Composite Section

Solution:

Supporting (10) 16K9’s @ 2.727 ft. O.C., Figure C.1 shows that the reactions (point

loads) will be present on the girder (a symmetrical adjacent bay is assumed to be present).

Figure C.2 - Girder Loads

Select a W14x61 as a trial size.

Construction Load Stage:

RD = (30 ft.)([(44 psf)(2.727 ft.)] + 10 lb/ft) = 3900 lb

Beff = 90 in.

14”

3.25”

3.25”

∑Qn/Cf = 0.50

(10) 16 K9 Reactions

30’

R R R R R R R R R R

120

RLL = (30 ft.)[(10 psf)(2.727 ft.)] = 818 lb

wdead = 61 lb/ft

Service Load Stage:

RDL = (30 ft.)([(44 psf + 15 psf)(2.727 ft.)] + 10 lb/ft) = 5127 lb

RLL = (30 ft.)[(50 psf + 15 psf)(2.727 ft.)] = 5318 lb

wdead = 61 lb/ft

From equation C-I3-6 (AISC 2001b)

Ieff = )I(I)/C(QI StrfnS −+ ∑

where

Is = moment of inertia for steel shape acting alone

Itr = fully composite moment of inertia

Qn = compressive force for the partially composite case

Cf = compressive force in the concrete for the fully composite condition (the smaller of

AsFy and 0.85fc’Ac)

∑Qn/Cf = 0.5 (from problem statement)

Table C.1 outlines the components of the fully composite section.

Component A (in2) y (in) Ay (in3) Ibar (in4) d (in) Ibar+Ad2

Concrete 36.56 1.6 59.4 32.2 3.9 584.4W14x61 17.90 13.5 240.8 640.0 7.9 1768.1

SUM 54.46 300.2 2352.5

ybar (in) 5.51

Table C.1- Components of a Fully Composite Girder (Traditional System)

121

This results in the following data table describing the deflections at the different load

stages (Table C.2). Deflection of 1.45 in. under construction stage loading controls the

design (just under L/240 = 1.5 in.). The floor construction should be arranged to

accommodate the 1.97 in. (L/183) displacement that occurs from long-term deflections.

This deflection was obtained by multiplying the sustained dead loading by λ = 2.0 (to

account for creep) and then adding the superimposed transient live loading.

Description Label MagnitudeMOI of steel W-section Isteel (in

4) 640MOI of fullycomposite cross-section Itr (in

4) 2353MOI of 50% composite cross-section Ieff (in

4) 1851Construction Load Stage Deflections

DL (decking, wet concrete, W-sect. Self Wt.) δdead (in) 1.45LL (10 psf construction live load) δconsLL (in) 0.29Total δTL (in) 1.75Service Load Stage

Deflection from steel self-wt., wet concrete, and SI 15 psf loadings δdead (in) 0.65Deflection from superimposed live loading δconsLL (in) 0.66Total δTL (in) 1.31Long term deflection from sustained dead loading δDLLongterm (in) 1.31Long term deflection from sustained dead loading plus SI LL δDLLT + LL (in) 1.97

Table C.2 - Deflections Along Various Load Stages (Traditional System)

Check Strength

For non-composite action during construction loading

Pu = (1.2)(3900 lb) + (1.6)(818 lb) = 5989 lb

wu = (1.2)(61 lb/ft) = 73.2 lb/ft

Mu = [0.13636(10Pu)(30 ft.)] + (1/8)(wu)(30 ft.)2 = 245 k-ft + 8.2 k-ft = 253.2 k-ft

From Table 5-3 in the steel manual (AISC 2001a)

ΦbMpx = 383 k-ft > 253.2 k-ft okay

For a partially composite section

122

Pu = (1.2)(5127 lb) + 1.6(5318 lb) = 14661 lb

wu = (1.2)(61 lb/ft) = 73.2 lb/ft

Mu = [0.13636(10Pu)(30 ft.)] + (1/8)(wu)(30 ft.)2 = 599.8 k-ft + 8.2 k-ft = 608 k-ft

For fully composite action, the compressive force, C, in the concrete is the smaller of

AsFy = (17.9 in2)(50 ksi) = 895 kips (controls)

0.85fc’Ac = 0.85(4 ksi)((90 in.)(3.25 in.)) = 994.5 kips

The problem stated that the fraction of “compositeness” equals 0.5.

Vhorz = 895/2 = 447.5 kips

The plastic neutral axis (PNA) is located in the steel. Check to see if it is located in the

top flange:

Pyf = bfTfFy = (10 in.)(0.645 in.)(50 ksi) = 322.2 kips

(AsFy – Pyf) – Pyf = 895 kips – 2(322.2 kips) = 251 kips

251 kips < 447.5 kips (the PNA is in the flange of the W-section)

Setting tprime equal to the depth of compression in the flange

895 kips – 2(10 in.)(tprime)(50) = 447.5 kips

tprime = 0.4475 in.

Component A (in2) y (in) AyW14x61 17.9 6.95 124.405

Flange Segment -4.48 0.22 -1.00Sum 13.43 123.40

ybar (in) 9.19

Table C.3 - Plastic Section Components (Traditional System)

Moment arm for the concrete compressive force

in1.46)0.85(4)(90

kips447.5a ==

123

Using ybar obtained from Table C.3

ybar + t – a/2 = (9.19 in) + (6.5 in.) – (1.46 in / 2) = 14.96 in.

Moment arm for the compressive force in the steel is

ybar – (tprime/2) = 9.19 – (0.4475 in. / 2) = 8.97 in.

Taking moments about the tensile force:

Mn = (447.5 kips)(14.96 in.) + (0.4475 in.)(10 in)(50)(8.97 in.)

Mn = 8701.6 k-in or 725 k-ft

ΦMn = 0.85(725 k-ft) = 616 k-ft > 608 k-ft (okay)

Select a W14x61

The following is a composite girder design for a 30 ft. W-section in the proposed

system (Figure C.3). This girder will support the dominant joists, as well as a direct

portion of the floor slab. The same panel loading, assumptions, and deflection criteria

used in the traditional composite girder design will be used here. Again, the loading on

the girders assumes that a symmetrical adjacent bay is present. The reactions from the

joists were determined using SAP2000.

Figure C.3 - Loading on the Girder Supporting the Dominant Joists

Select a W14x48 as a trial section

Construction Load Stage

19”

30’

10’-3”

w

P1 P2 P2 P1

6’-4” 10’-3” 19”

124

At this load stage, loading on the girder comes from wet concrete, joist self-weight,

girder self-weight, and a construction load. Therefore:

P1 = 4230 lb P2 = 4379 lb w = (44 psf)(6 ft.) = 264 lb/ft (wet concrete and joists)

P1 = 865.7 lb P2 = 894 lb w = (10 psf)(6 ft.) = 60 lb/ft (10 psf construction load)

wSW = 48 lb/ft (Self-weight)

Service Load Stage

P1 = 5627 lb P2 = 5812 lb w = (65 psf)(6 ft.) = 390 lb/ft (LL: 50psf + 15psf)

P1 = 5528 lb P2 = 5720 lb w = (44 psf + 15 psf)(6 ft.) = 354 lb/ft (sustained dead

loads)

wSW = 48 lb/ft (Self-weight)

Maximum deflection will occur at girder mid-span and is governed by the following

equation

δmax = 1 20.058016P 0.062868P+I I

The components of the fully composite section are outlined in Table C.4.


Concrete 36.56 1.6 59.4 32.2 3.3 424.8W14x48 14.10 13.4 188.9 484.0 8.5 1502.2

SUM 50.66 248.4 1927.1

ybar (in) 4.90

Table C.4 - Components of Fully Composite Girder (Supporting Dominant Joists)

This table leads to the following data describing the deflections at the different load

stages (Table C.5). The design is governed by the deflection of 1.43 in. during the

construction load stage (just under L/240 = 1.5 in.). Once again, the floor construction

should be arranged to accommodate the 1.82 in. (L/198) displacement that occurs from

125

long-term deflections. This deflection was obtained by multiplying the sustained dead

loading by λ = 2.0 and then adding the superimposed transient live loading.







Table C.5 - Deflections Along Various Load Stages (Girder Supporting Dominant

Joists)

Check Strength


P1uD = (1.2)(4230 lb) = 5076 lb P2uD = (1.2)(4379 lb) = 5254.8 lb

wDu = (1.2)(264 lb/ft + 48 lb/ft) = 374.4 lb/ft

P1uLL = (1.6)(865.7 lb) = 1385 lb P2uLL = (1.6)(894 lb) = 1430.4 lb

wDLL = (1.6)(60 lb/ft) = 96 lb/ft

P1u = 6461 lb P2u = 6685.2 lb wu = 470 lb/ft

From a matrix analysis

Mu = 199 k-ft

From Table 5-3 in the steel manual (AISC 2001a)

126

ΦbMpx = 294 k-ft > 199 k-ft okay


P1uD = (1.2)(5627 lb) = 6752.4 lb P2uD = (1.2)(5812 lb) = 6974.4 lb

wDu = (1.2)(390 lb/ft + 48 lb/ft) = 525.6 lb/ft

P1uLL = (1.6)(5528 lb) = 8844.8 lb P2uLL = (1.6)(5720 lb) = 9152 lb

wDLL = (1.6)(354 lb/ft) = 566.4 lb/ft

P1u = 15597.2 lb P2u = 16126.4 lb wu = 1092 lb/ft


Mu = 474 k-ft




The problem statement said that the fraction of “compositeness” equals 0.5.

Vhorz = 705/2 = 352.5 kips


top flange:

Pyf = bfTfFy = (8.03 in.)(0.595 in.)(50 ksi) = 239 kips

(AsFy – Pyf) – Pyf = 705 kips – 2(239 kips) = 227 kips

227 kips < 352.5 kips (the PNA is in the flange of the W-section)


705 kips – 2(8.03 in.)(tprime)(50 ksi) = 352.5 kips

tprime = 0.439 in.

127

Component A (in2) y (in) AyW14x48 14.1 6.9 97.29


ybar (in) 9.13

Table C.6 - Plastic Section Components (Girder Supporting Dominant Joists)

Moment arm for the concrete compressive force is

352.5 kipsa 1.15 in.0.85(4)(90)

= =

Using ybar derived in Table C.6

ybar + t – a/2 = (9.13 in.) + (6.5 in.) – (1.15 in. / 2) = 15.06 in.


ybar – (tprime/2) = 9.13 – (0.439 in. / 2) = 8.91 in.


Mn = (352.5 kips)(15.06 in.) + (0.439 in.)(8.03 in.)(50 ksi)(8.91 in.)

Mn = 6879 k-in or 573 k-ft

ΦMn = 0.85(573 k-ft) = 487 k-ft > 474 k-ft (okay)

Select a W14x48

The following is a composite girder design for a 30 ft. W-section in the proposed

system (Figure C.4). This girder will support the non-dominant joists. The same panel

loading, assumptions, and deflection criteria used in the traditional composite girder

design will be used here. Again, the loading on the girders assumes that a symmetrical

adjacent bay is present. The reactions from the joists were determined using SAP2000.

128

Figure C.4 – Loading on the Girder Supporting the Non-Dominant Joists

Select a W14x34 as a trial section (assuming that bf = 6.75 in. does not create a joist

bearing problem)

Construction Load Stage

At this load stage, loading on the girder comes from wet concrete, joist self-weight,

girder self-weight, and a construction load. Therefore:

P1 = 3633 lb P2 = 4781 lb (wet concrete and joists)

P1 = 783 lb P2 = 1017 lb (10 psf construction load)

wSW = 34 lb/ft (self-weight)

Service Load Stage

P1 = 5090 lb P2 = 6612 lb (LL: 50psf + 15psf)

P1 = 4807 lb P2 = 6307 lb (sustained dead loads)

wSW = 34 lb/ft (self-weight)

Maximum deflection will occur at girder mid-span and is governed by the following

equation

δmax = 1 20.03808P 0.06328P+I I

The components of the fully composite section are shown in Table C.7.

30’

P1 P2 P2 P1

6’ 6’ 6’ 6’ 6’

129


Concrete 36.56 1.6 59.4 32.2 2.6 270.0W14x34 10.00 13.5 135.0 340.0 9.3 1209.5

SUM 46.56 194.4 1479.5

ybar (in) 4.18

Table C.7 - Components of Fully Composite Girder (Supporting Non-Dominant

Joists)







Table C.8 - Deflections Along Various Load Stages (Girder Supporting Non-

Dominant Joists)

As shown in Table C.8, the design is governed by a deflection of 1.36 in. during the

construction load stage. Once again, the floor construction should be arranged to

accommodate the 1.59 in. (L/226) displacement that occurs from long-term deflections.

This deflection was obtained by multiplying the sustained dead loading by λ = 2.0 and

then adding the superimposed transient live loading.

130

Check Strength


P1uD = (1.2)(3633 lb) = 4360 lb P2uD = (1.2)(4781 lb) = 5737 lb

wDu = (1.2)(34 lb/ft) = 40.8 lb/ft

P1625eun825uLL = (1.6)(783 lb) = 1253 lb P2uLL = (1.6)(1017 lb) = 1627 lb

P1u = 5613 lb P2u = 7364 lb wu = 40.8 lb/ft


Mu = 127 k-ft

From (AISC 2001a) table 5-3

ΦbMpx = 203 k-ft > 127 k-ft okay


P1uD = (1.2)(4807 lb) = 5769 lb P2uD = (1.2)(6307 lb) = 7568 lb

wDu = (1.2)(34 lb/ft) = 40.8 lb/ft

P1uLL = (1.6)(5090 lb) = 8144 lb P2uLL = (1.6)(6612 lb) = 10580 lb

P1u = 13913 lb P2u = 18148 lb wu = 40.8 lb/ft


Mu = 306 k-ft




The problem stated that the fraction of “compositeness” equals 0.5.

Vhorz = 500/2 = 250 kips

131


top flange:

Pyf = bfTfFy = (6.75 in.)(0.455 in.)(50 ksi) = 153.6 kips

(AsFy – Pyf) – Pyf = 500 kips – 2(153.6 kips) = 193 kips

193 kips < 250 kips (the PNA is in the flange of the W-section)


500 kips – 2(6.75 in.)(tprime)(50 ksi) = 250 kips

tprime = 0.370 in.

Component A (in2) y (in) AyW14x34 10 7 70


ybar (in) 9.27

Table C.9 - Plastic Section Components (Girder Supporting Non-Dominant Joists)

Moment arm for the concrete compressive force is

250 kipsa 0.817 in.0.85(4)(90)

= =

From the ybar derived in Table C.9

ybar + t – a/2 = (9.27 in.) + (6.5 in.) – (0.817 in. / 2) = 15.36 in.


ybar – (tprime/2) = 9.27 in. – (0.370 in. / 2) = 9.09 in.


Mn = (250 kips)(15.36 in.) + (0.370 in.)(6.75 in)(50 ksi)(9.09 in)

Mn = 4976 k-in or 414.7 k-ft

ΦMn = 0.85(414.7 k-ft) = 352 k-ft > 306 k-ft (okay)

Select a W14x34

two-way steel floor system using open-web joists · using open-web joists by john a. schaad, b.s. a...

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