RIBBE SL BS UN E C E TR TED l 5 - IDEALS

t

J.

! l

IU Ll'iA 3q~

UILU-ENG .. 73-2005

~~ 3CIVIL ENGINEERING STUDIES STRUCTURAL RESEARCH SERIES NO. 398

UN E RIBBE SL BS

C E TR TED l

Civil BI06 C. t;" Univc:I'd:L Urbana 9

By

T. ATTAJARUSIT

W. L. GAMBLE

Conducted by

5

THE STRUCTURAL RESEARCH LABORATORY

DEPARTMENT OF CIVil ENGINEERING

ENGINEERING EXPERIMENT STATION

UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN

UNIVERSITY OF ILLINOIS

URBANA, IlUN.OIS

MAY 1973

BIBLIOGRAPHIC DATA 11. Report No. SHEET UILU-ENG-73-2005

Ribben Slabs Under Concentrated Loads

7. A.uchor(s)

T. Attajarusit and W. L. Gamble 9. Pcrfllrll1ing Organization Name and Address

Department of Civil Engineering University of Illinois 2209 Civil Engineering Building Urbana, Illinois 61801

3. Recipient's Accession No.

5. Report Date May 1973

6.

8. Performing Organiz.a tion Re rt No. SRS-398

10. Project/Task/Work Unit No.

11. Contract/Gram No.

~--------------------------------------------------------------4---------------------~ 12. Sponsoring Organization Name and Address

Department of Civil Engineering 'University of Illinois

13. Type of Report & Period Covered

2209 Civil Engineering Building Urbana, Illinois 61801

15. Supplementary Notes '

16. Abstracts

Final Report 14.

Studies of ribbed slabs subjected to single and multiple concentrated loads were carried out on both one-span and two-span continuous structures. The ribbed slab was decomposed into a series of beam and slab elements. Joint forces of the decomposed structure were determined under the condition that forces and displacements of both elements at each joint. be equal. A solution for the structure under the applied loads was obtained by substituting the calculated joint forces into the equilibrium equations of the individual elements.

The results obtained were internal moments and forces in the ribs due to the applied loads. Also calculated were the equivalent uniformly distributed loads on the ribbed slabs subjected to multiple loads such as might be found in parking garages.

17. Key Words and Document AnalY$is. 170. Descriptors

Structural Analysis, Loadings, Ribbed Slabs, Concrete Structures, One-way joists, Garages (parking), Bending Moment, Shear, Torsion,

17b. Identifiers/Open-Ended Terms

17c. COSATI Field/Group 13/13 18. Availability Statement

Release Unlimited

FORM NTIS-35 (REV. 3-72)

19 .. Security Class (This Report) .

UNCLASSIFIED 20. Security Class (This

Page UNCLASSIFIED

TH [S FORM MAY BE REPRODUCED

21. No. of Pages

165 22. Price

USCOMM-DC 14952-P72

iii

ACKNOWLEDGMENT

The author wishes to express his appreciation and sincere thanks

to his advisor, Dr. William L. Gamble, Associate Professor of Civil Engi

neering, for his invaluable suggestions and supervision throughout this

study.

The author also wishes to thank Dr .. Chester P. Siess, Professor

of Civil Engineering, for his helpful suggestions at the beginning of this

study.

Chapter

1

2

3

4

iv

TABLE OF CONTENTS

INTRODUCTION .

1 . 1 General. 1.2 Previous Studies 1.3 Object and Scope. . 1.4 The Typical Ribbed Slab. 1.5 Selected Parameters. 1.6 Notation.

METHOD OF ANALYSIS.

2. 1 Genera 1. . 2.2 Assumptions. . . . 2.3 One-Span Ribbed Slab. 2.4 Two-Span Ribbed Slab. 2.5 Multiple Loads. . . . 2.6 Acturacy of the Analysis

BEHAVIOR OF RIBBED SLABS UNDER A SINGLE CONCENTRATED LOAD .

3. 1 General. .. .• 3.2 Moment, Shear, and Torsion Diagrams. 3.3 Moment Envelopes and Influence Lines

for Moment at Midspan and Influence Lines for Shear and Torsion at Support.

3.4 Moment Distribution and Moment Envelope Across the Sp~n '. . .

3.5 Effects of Torsional Stiffness on Moment, Shear, and Torsion. .

3.6 Effects of Hand b/a on Moment, Shear, and Torsion • . . . .

3.7 Effects of Hand b/a on Influence Lines for Torsional Moment at the Support, and the Maximum Torsion.

RIBBED SLABS UNDER MULTIPLE LOADS.

Page

1 .

1 2 4 5 6 8

12

12 12 13 16 18 20

23

23 24

29

35

37

38

46

48

4.1 General. . . . . . . . . . . .. 48 4.2 Moment, 'Shear, and Torsional Moment Diagrams. 49 4.3 Multiple Loads for Maximum Moment and 'Shear • 51 4.4 Multiple Loads for Maximum Torsional Moment. 60 4.5 Variations of Maximum Moment, Shear,

and Torsional Moment,' with Hand b/a 65 4.6 Effects of a Stiffer Edge Rib. 68

Chapter

5

6

v

DISCUSSION AND RECOMMENDATIONS .

5.1 General. . 5.2 Discussion and Comparison. 5.3 General Remarks 5.4 Recommendations

SUMMARY AND CONCLUSIONS

6.1 General Outline of the Investigation 6.2 General Conclusions .

LIST OF REFERENCES

Page

72

72 72 77 78

79

79 80

84

Table

3.1

3.2

4.1

4.2

4.3

5.1

5.2

5.3

5.4

5.5

5.6

5.7

5.8

5.9

vi

LIST OF TABLES

DIMENSIONS AND PARAMETERS OF THE VARIOUS RIBBED SLABS STUDIED. . . . .

PARAMETERS AND DIMENSIONS OF THE RIBBED SLABS FOR THE STUDY OF THE EFFECTS OF TORSIONAL STIFFNESS ON MOMENT, SHEAR AND TORS ION . ...

MAXIMUM SIMPLE BEAM MOMENT AND SHEAR VS. SPAN.

MAXIMUM DEFLECTIONS OF ONE-SPAN RIBBED SLABS DUE TO MULTIPLE LOADS FOR MAXIMUM MOMENT

MAXIMUM DEFLECTIONS OF TWO-SPAN RIBBED SLABS DUE TO MULTIPLE LOADS FOR MAXIMUM POSITIVE MOMENT

EQUIVALENT UNIFORMLY DISTRIBUTED LOADS FOR ONE-SPAN RIBBED SLABS .

EQUIVALENT UNIFORMLY DISTRIBUTED LOADS ON -TWO-SPAN RIBBED SLABS. .

MAXIMUM TORSIONS AT THE SUPPORT OF ONE-SPAN RIBBED SLABS. ...

SHEARS ACCOMPANYING THE MAXIMUM TORSIONS AT THE SUPPORT OF ONE-SPAN RIBBED SLABS. .

MAXIMUM TORSIONS AT THE EXTERIOR SUPPORT OF TWO-SPAN RIBBED SLABS. ...

SHEARS ACCOMPANYING THE MAXIMUM TORSIONS AT THE EXTERIOR SUPPORT OF TWO-SPAN RIBBED SLABS.

MAXIMUM TORSIONS AT THE 0.95 SPAN LOCATION IN TWO-SPAN RIBBED SLABS. ., .

SHEAR_ACCOMPANYING THE MAXIMUM TORSIONS AT THE 0.95 SPAN LOCATION IN TWO-SPAN RIBBED SLABS

NEGATIVE BENDING MOMENTS ACCOMPANYING THE MAX1MUM TORSIONS AT THE 0.95 SPAN LOCATION IN TWO-SPAN RIBBED SLABS. . . . .

Page

86

87

88

89

89

90

91

92

92

93

93

94

94

·95

Table

5.10

5.11

vii

TORSIONS ACCOMPANYING THE MAXIMUM SHEARS AT THE EXTERIOR SUPPORT IN TWO-SPAN RIBBED SLABS .

TORSIONS ACCOMPANYING THE MAXIMUM SHEARS AT THE 0.95 SPAN LOCATION IN TWO-SPAN RIBBED SLABS .

Page

95

95

Figure

1 • 1

2. 1

3.1

3.2

3.3

3.4

3.5 .

3.6

3.7

viii

LIST OF FIGURES

TYPICAL RIBBED SLABS AND CROSS SECTIONS .

INTERNAL FORCES AT THE JUNCTION OF BEAM AND SLAB ELEMENTS, AND ON A SMALL ELEMENT OF BEAM .

MOMENT, SHEAR, AND TORSION DIAGRAMS' FOR A ONE-SPAN RIBBED SLAB DUE TO A SINGLE LOAD AT MIDSPAN ON THE EDGE AND CENTER RIBS.

, MOMENT, SHEAR, AND TORSION DIAGRAMS FOR A TWO-SPAN RIBBED SLAB DUE TO A SINGLE LOAD AT ONE MIDSPAN ON THE EDGE AND CENTER RIBS .

THE INTERPRETATION OF THE LOADING EFFECTS ON THE RIBS. . · .

THE INTERPRETATION OF REACTIONS AND COUPLES AT THE INTERIOR SUPPORT. . .

INFLUENCE LINES FOR MOMENT AT MIDSPAN AND MOMENT ENVELOPES FOR A ONE-SPAN RIBBED SLAB DUE TO A LOAD MOVING ON THE EDGE AND CENTER RIBS .

INFLUENCE LINES FOR MOMENT AT MIDSPAN, AND MOMENT ENVELOPES FOR A TWO-SPAN RIBBED SLAB DUE TO A LOAD MOVING ON THE EDGE AND CENTER RIBS .

INFLUENCE LINES FOR SHEAR AT THE SUPPORT OF A ONE-SPAN RIBBED SLAB DUE TO A LOAD MOVING ON THE EDGE AND CENTER RIBS.

3.8 INFLUENCE LINES FOR SHEAR AT THE EXTERIOR SUPPORT AND AT THE 0.95 SPAN LOCATION OF 'A TWO-SPAN RIBBED SLAB DUE TO A LOAD MOVING ON THE EDGE AND CENTER RIBS .

3.9

3.10

INFLUENCE LINES FOR TORSION AT THE SUPPORT OF A ONE-SPAN RIBBED SLAB DUE TO A LOAD MOVING ON THE EDGE AND CENTER RIBS.

INFLUENCE LINES FOR TORSION AT THE EXTERIOR SUPPORT AND AT THE 0.95 SPAN LOCATION OF A TWO-SPAN RIBBED SLAB DUE TO A LOAD MOVING ON THE EDGE AND CENTER RIBS . .

Page

96

97

98

99

101

102

103

104

105

, f"\C IUU

107

108

ix

Figure

3.11 MOMENT ENVELOPES AND MOMENT DISTRIBUTIONS AT MIDSPAN OF VARIOUS RIBS OF A ONE-SPAN RIBBED SLAB WITH THE REGULAR AND STIFFENED EDGE RIBS

Page

DUE TO A LOAD MOVING ALONG MIDSPAN. 109

3.12 POSITIVE AND NEGATIVE MOMENT ENVELOPES AND MOMENT DISTRIBUTIONS AT MIDSPAN AND AT THE INTERIOR SUPPORT OF VARIOUS RIBS OF A TWOSPAN RIBBED SLAB DUE TO A LOAD MOVING ALONG MIDSPAN. 110

3.13 EFFECTS OF TORSIONAL STIFFNESS ON MOMENT, SHEAR AND TORSION. . 111

3.14 MOMENT DISTRIBUTIONS AT MIDSPAN OF ONE-SPAN RIBBED SLABS DUE TO SINGLE LOADS AT MIDSPAN ON THE EDGE AND CENTER RIBS. . . . . 112

3.15 POSITIVE MOMENT DISTRIBUTIONS AT MIDSPAN OF TWO-SPAN RIBBED SLABS DUE TO SINGLE LOADS AT MIDSPAN OF THE EDGE AND CENTER RIBS.- 113

3.16 NEGATIVE MOMENT DISTRIBUTIONS AT THE INTERIOR SUPPORT OF TWO-SPAN RIBBED SLABS DUE TO SINGLE LOADS AT MIDSPAN ON THE EDGE AND CENTER RIBS. 114

3.17 SHEAR DISTRIBUTIONS AT THE SUPPORT OF ONESPAN RIBBED SLABS DUE TO SINGLE LOADS AT MIDSPAN ON THE EDGE AND CENTER RIBS. 115

3.18 SHEAR DISTRIBUTIONS AT THE EXTERIOR SUPPORT OF THE LOADED SPAN OF TWO-SPAN-RIBBED SLABS DUE TO SINGLE LOADS AT MIDSPAN ON THE EDGE AND CENTER RIBS .. 116

3.19 SHEAR DISTRIBUTIONS AT THE 0.95 SPAN LOCATION OF THE LOADED SPAN OF TWO-SPAN RIBBED SLABS DUE TO SINGLE LOADS AT MIDSPAN ON THE EDGE AND CENTER RIBS . . 117

3.20 TORSIONAL MOMENT DISTRIBUTIONS AT THE SUPPORT OF ONE-SPAN RIBBED SLABS DUE TO SINGLE LOADS AT MIDSPAN ON THE EDGE AND CENTER RIBS. . . 118

·3.21 TORSIONAL MOMENT DISTRIBUTIONS AT THE EXTERIOR SUPPORT OF THE LOADED SPAN OF TWO-RIBBED SLABS DUE TO SINGLE LOADS AT MIDSPAN ON THE EDGE AND CENTER RIBS.. ... . . . 119

Fi gure

3.22

3.23

3.24

3.25

3.26

3.27

3.28

3.29

3.30

4.1

4.2

x

TORSIONAL MOMENT DISTRIBUTIONS AT THE 0.95 SPAN LOCATION OF THE LOADED SPAN OF TWO-SPAN RIBBED SLABS DUE TO SINGLE LOADS AT MIDSPAN ON THE EDGE AND CENTER RIBS. .. .,

VARIATIONS OF MAXIMUM MOMENT WITH HAND b!a FOR ONE-SPAN RIBBED SLABS UNDER SINGLE LOADS AT MIDSPAN ON THE EDGE AND CENTER RIBS. .

VARIATIONS OF MAXIMUM POSITIVE AND NEGATIVE MOMENTS WITH HAND b/a· FOR TWO-SPAN RIBBED SLABS UNDER SINGLE LOADS AT MIDSPAN ON THE EDGE AND CENTER RIBS . .....

VARIATIONS OF MAXIMUM SHEAR AT THE SUPPORT WITH HAND b/a FOR bNE-SPAN RIBBED SLABS UNDER SINGLE LOADS AT MIDSPAN ON THE EDGE AND CENTER RIBS.. ..... .

VARIATIONS OF MAXIMUM SHEAR AT THE EXTERIOR SUPPORT AND 0.95 SPAN LOCATION WITH HAND b/a FOR TWO-SPAN RIBBED SLABS UNDER SINGLE LOADS AT MIDSPAN ON THE EDGE AND CENTER RIBS. .

VARIATIONS OF MAXIMUM TORSION AT THE SUPPORT WITH HAND b/a FOR ONE-SPAN RIBBED SLABS UNDER SINGLE LOADS AT MIDSPAN ON THE EDGE AND CENTER RIBS. . .. . . .

VARIATIONS OF MAXIMUM TORSION AT THE EXTERIOR SUPPORT AND 0.95 SPAN LOCATION WITH HAND b/a FOR TWO-SPAN RIBBED SLABS UNDER SINGLE LOADS AT MIDSPAN ON THE EDGE AND CENTER RIBS. .

INFLUENCE LINES FOR TORSION AT THE SUPPORT OF VARIOUS ONE-SPAN RIBBED SLABS DUE TO A LOAD MOVING ON THE EDGE AND CENTER RIBS. .

VARIATIONS OF MAXIMUM TORSION AT THE SUPPORT WITH HAND b/a FOR ONE-SPAN RIBBED SLABS UNDER SINGLE LOADS ON THE EDGE AND CENTER RIBS ..

LOADING SYSTEMS FOR MAXIMUM BENDING MOMENTS, SHEARS, AND TORSIONAL MOMENTS . .

BENDING MOMENT, SHEAR, AND TORSIONAL MOMENT DIAGRAMS OF A ONE-SPAN RIBBED SLAB DUE TO MULTIPLE LOADS. . . .. .

Page

120

121

122

123

124

125

126

127

128

129

133

Fi gure

4.3

4.4

4.5

4.6

4.7

4.8

4.9

4.10

4. 11

4. 12

4. 13

xi

BENDING MOMENT, SHEAR, AND TORSIONAL MOMENT DIAGRAMS OF A TWO-SPAN RIBBED SLAB DUE TO MULTIPLE LOADS .

MOMENT DISTRIBUTIONS AT MIDSPAN OF ONESPAN RIBBED SLABS DUE TO MULTIPLE LOADS FOR MAXIMUM MOMENT.

SHEAR DISTRIBUTIONS AT THE SUPPORT OF ONE-SPAN RIBBED SLABS DUE TO MULTIPLE LOADS FOR MAXIMUM MOMENT.

TORSIONAL MOMENT DISTRIBUTIONS AT THE SUPPORT OF ONE-SPAN RIBBED SLABS DUE TO MULTIPLE LOADS FOR MAXIMUM MOMENT.

POSITIVE MOMENT DISTRIBUTIONS AT THE 0.44 SPAN LOCATION OF TWO-RIBBED SLABS DUE TO MULTIPLE LOADS FOR MAXIMUM POSITIVE MOMENT.

NEGATIVE MOMENT DISTRIBUTIONS AT THE INTERIOR SUPPORT OF TWO-SPAN RIBBED SLABS DUE TO MULTIPLE LOADS FOR MAXIMUM NEGATIVE MOMENT .

SHEAR DISTRIBUTIONS AT THE EXTERIOR SUPPORT OF TWO-SPAN RIBBED SLABS DUE TO MULTIPLE LOADS FOR MAXIMUM POSITIVE MOMENT.

TORSIONAL MOMENT DISTRIBUTIONS AT THE EXTERIOR SUPPORT OF TWO-SPAN RIBBED SLABS DUE TO MULTIPLE LOADS FOR MAXIMUM POSITIVE MOMENT. ..

SHEAR DISTRIBUTIONS AT THE 0.95 SPAN LOCATION OF TWO-SPAN RIBBED SLABS DUE TO MULTIPLE LOADS FOR MAXIMUM POSITIVE MOMENT.

TORSIONAL MOMENT DISTRIBUTIONS AT THE 0.95 SPAN LOCATIONS OF TWO-SPAN RIBBED SLABS DUE TO MULTIPLE LOADS FOR MAXIMUM POSITIVE MOMENT.

SHEAR DISTRIBUTIONS AT THE 0.95 SPAN LOCATION OF TWO-SPAN RIBBED SLABS DUE TO MULTIPLE LOADS FOR MAXIMUM NEGATIVE MOMENT .

Page

134

136

137

138

139

140

141

142

143

144

145

Figure

4.14

4. 15

4.16

4.17

4.18

4. 19

4.20

4.21

4.22

4.23

4.24

xii

TORSIONAL MOMENT DISTRIBUTIONS AT THE 0.95 SPAN LOCATION OF TWO-SPAN RIBBED SLABS DUE TO MULTIPLE LOADS FOR MAXIMUM NEGATIVE MOMENT

RELATIONSHIPS OF THE MAXIMUM DEFLECTION COEFFICIENT WITH HAND bja ...

TORSIONAL MOMENT DISTRIBUTIONS AT THE SUPPORT OF ONE-SPAN RIBBED SLABS DUE TO MULTIPLE LOADS FOR MAXIMUM TORSION.

SHEAR DISTRIBUTIONS AT THE SUPPORT OF ONE-SPAN RIBBED SLABS DUE TO MULTIPLE LOADS FOR MAXIMUM TORSION. . . . . .

TORSIONAL MOMENT DISTRIBUTIONS AT THE . EXTERIOR SUPPORT OF TWO-SPAN RIBBED SLABS DUE TO MULTIPLE LOADS FOR MAXIMUM TORSION

TORSIONAL MOMENT DISTRIBUTIONS AT THE 0.95 SPAN LOCATION OF TWO-SPAN RIBBED SLABS DUE TO MULTIPLE LOADS FOR MAXIMUM TORSION.

SHEAR DISTRIBUTIONS AT THE EXTERIOR SUPPORT OF TWO-SPAN RIBBED SLABS DUE TO MULTIPLE LOADS FOR MAXIMUM TORSION. .. ... ..

SHEAR DISTRIBUTIONS AT THE 0.95 SPAN LOCATION OF TWO-SPAN RIBBED SLABS DUE TO MULTIPLE LOADS FOR MAXIMUM TORSION. . . . · . · . .

NEGATIVE MOMENT DISTRIBUTIONS AT THE INTERIOR SUPPORT OF TWO-SPAN RIBBED SLABS DUE TO MULTIPLE LOADS FOR MAXIMUM TORSION. ...

RELATIONSHIPS OF MAXIMUM MOMENT, MAXIMUM SHEAR AND THE ACCOMPANYING TORSION, WITH HAND bja FOR ONE-SPAN RIBBED SLABS UNDER MULTIPLE LOADS FOR MAXIMUM MOMENT. . .

RELATIONS OF MAXIMUM TORSIONAL MOMENT AND THE ACCOMPANYING SHEAR WITH HAND bja FOR ONE-SPAN RIBBED SLABS UNDER MULTIPLE LOADS FOR MAXIMUM TORSION. ..

Page

146

147

148

149

150

151

152

153

154

155

156

Figure

4.25

4.26

4.27

xiii

RELATIONSHIPS OF MAXIMUM POSITIVE MOMENT AT THE 0.44 SPAN LOCATION, MAXIMUM SHEAR AND THE ACCOMPANYING TORSION AT THE EXTERIOR SUPPORT WITH HAND b/a FOR TWO-SPAN RIBBED SLABS UNDER MULTIPLE LOADS FOR MAXIMUM POSITIVE MOMENT .

RELATIONSHIPS OF MAXIMUM SHEAR AND THE ACCOMPANYING TORSION AT THE 0.95 SPAN LOCATION AND NEGATIVE MOMENT AT THE INTERIOR SUPPORT WITH HAND b/a FOR TWO~SPAN RIBBED SLABS UNDER MULTIPLE LOADS FOR MAXIMUM POSITIVE MOMENT . ...

RELATIONSHIPS OF MAXIMUM NEGATIVE MOMENT AT THE INTERIOR SUPPORT, MAXIMUM SHEAR AND THE ACCOMPANYING TORSION OF THE 0.95 SPAN LOCATION WITH HAND b/a FOR TWO-SPAN RIBBED SLABS UNDER MULTIPLE LOADS FOR MAXIMUM TORSION. .

4.28 RELATIONSHIPS OF MAXIMUM TORSIONAL MOMENT AND THE ACCOMPANYING SHEAR AT THE EXTERIOR SUPPORT WITH HAND b/a FOR TWO-SPAN RIBBED

Page

157

158

159

SLABS UNDER MULTIPLE LOADS FOR MAXIMUM TORSION. 160

4.29

4.30

4.31

4.32

RELATIONSHIPS OF MAXIMUM TORSIONAL MOMENT AND THE ACCOMPANYING SHEAR AT THE 0.95 SPAN LOCATION AND NEGATIVE MOMENT AT THE INTERIOR SUPPORT WITH HAND b/a FOR TWO-SPAN RIBBED SLABS UNDER MULTIPLE LOADS FOR MAXIMUM TORSION.

EFFECTS OF THE STIFFENED EDGE RIB ON MOMENT, SHEAR AND TORSIONAL MOMENT DUE TO MULTIPLE LOADS .

EFFECTS OF THE STIFFENED EDGE RIB ON MOMENT, SHEAR, AND TORSIONAL MOMENT DUE TO A LINE LOAD ON THE EDGE RIB

MOMENTS, SHEARS, AND TORSIONS DUE TO A MIDSPAN LINE LOAD. . , .

161

162

163

164

CHAPTER 1

INTRODUCTION

1.1 General

Reinforced concrete ribbed slabs, or one-way joist floors, are mono

lithic combinations of regularly spaced reinforced concrete ribs and slabs

cast-in-place to form integral units with supporting beams or walls. Span

lengths of this floor type ,can be longer than the others. The ribbed slab

. behaves like a one-way structure or slab, as its geometry indicates, under a

uniformly distributed load. However, for the case of concentrated load, in

which the problem is more complicated, the direct design method for one-way

slabs is no longer applicable.

Ribbed slabs are one of the·more suitable floor types for a parking

garage, since they can be built with relatively long spans. Many such parking

garages have already been constructed, but some of them have had serviceabil

ity problems, such as severe cracking and punching shear failures of the slab,

and some have even collapsed (1). It might be claimed that such problems re

sulted from not using proper design criteria, since parking garages are sub

jected to wheel loads from vehicles which are concentrated, while most such

structures are designed for distributed loads only.

Current design methods for ribbed slabs are not specific but are

left· to the judgement of engineers. The Building Code Requirements for Re

inforced Concrete (ACI-3l8-71) (2), Section 8.8, has provided some limita-.

tions on the geometry of the cross section of one-way joist floor systems

and the requirements for. reinforcement, but it is very brief and specifies

2

nothi ng about concentra ted loads. ·A des; gn handbook· by Reese (3) has some

examples of design calculations and many tables for various span lengths

and loads. However, these tables are restricted to the calculation for uni

formly distributed loads only. For concentrated loads (i.e., in parking

garages) some building codes, such as the BOCA Basic Building Code (4), have

specified using an equivalent uniformly distributed load (i.e., 50 psf for

open parking structures) in lieu of a more accurate solution. The BOCA Code

also requires consideration of a 2000 Ib load distributed over an area 2.5

ft ~q, but this requirement seldom governs and ,apparently is often ignored.

l~2 Previous Studies'

Prev i ou s. stud i es wh i ch are re 1 evant to one-way j 0 is t floor .sys terns

have been concerned mainly with highway bridge structures such as orthotr~pic

steel plate deck with stiffeners, composite steel girders with slabs, and

composite precast prestressed concrete girders with slabs. Few studies exist

about ribbed slabs and nothing has been found concerning concentrated loads.

However, the ribbed slab structure is structurally similar to the girder-slab

highway bri~ge structure, which has been studi~d extensively. Numerous analyt

ical and experimental, studies of girder-slab bridges have been done at the

Engineering Experiment Station, University of Illinois. Most of these inves

tigations were concentrated on simple span bridges with five or six girders.

The greatest number of girders tested, by Hondros and Marsh (5) in their studies

of load distribution in composite, girder~sl~b systems, was ten.

In analysis, different approaches and assumptions have been used'in

solutions of integrated girder-plate type structures. Some investigators

3

looked upon the slab as being a series of members laid perpendicular to the

girders and the resulting equivalent grillage structure was analyzed. On

the other hand, some investigators considered the girders as stiffeners of

the slab and the girder-slab structure was replaced by an orthotropic plate

of equivalent stiffness. A numerical method was applied by Newmark and

Siess (6), who considered the girder-slab structure as a plate supported on

a series of girders' with no interface shear forces. The girders and slab

were then treated so that they deflected together with no separation. Com

posite action of the slab with the girders was taken into account by using

a beam stiffness equivalent to the composite section stiffness of the girder

and a portion of the slab. Then the assumed structure was analyzed by

Newmark1s moment distribution procedure (7). In-plane forces in the slab and

axial forces in the girder cannot be taken into account in this procedure.

However, this analysis is among the references listed in the present design

manual of highway bridges (8).

Various numerical methods constitute very important and powerful

approaches,especially when high-speed computers 'are available. Several

numerical techniques, such as the finite difference and finite element

methods, have become.important in the study of girder-slab bridges. Chen,

Siess, and Newmark (9) used the finite difference method to solve ske~ed

girder-slab bridge problems. Gustafson and Wright (10) used a finite ele

ment procedure to solve similar problems. The results of both methods are

in general agreement~ ~ut the latter procedure gave better solutions for the

same mesh size. The finite element method is also applicable to the study

of ribbed slabs. However, because of the geometry of the ribbed slab, a huge

4

number of elements are required to obtain an accurate solution. Therefore,

it might be an uneconomical procedure for this type of structure.

Sithichaikasem and Gamble (11), and Van Horn and Mortajemi (12),

using the Goldberg-Leve folded plate theory (13), considered the girder-slab

bridge deck as a series of slab and girder elements. At the joints between

elements, there are unknown forces, which take into account the in-plane

forces in the slab, and the biaxial bending and torsional moments in the

girders, in addition to the forces considered in the analysis by Newmark and

Siess. Solutions of the individual slab and girder elements were then ob

tained, with both bending and membrane theories being used for the slab ele

ments. For the girder elements, solutions were obtained by integration of

the equilibrium equations of a small element. Then compatibility at the

joints between slab and girder elements was restored to obtain solutions

for the bridge structure, in which T-beam action of the girder and slab

was directly taken into account rather than being represented by some approxi

mation.

The above studies were done on simple span bridges only. Wong and

Gamble (14) extended the investigation to continuous bridges, using the 'same

procedure. This method of analysis is one of the more rigorous procedures

and so was selected as the basis of the analysis for the study of ribbed

slabs subjected to concentrated loads.

1.3 'Object and Scope

This study examines the general behavior of ribbed slabs subjected

to a single concentrated load, and then investigates their characteristics

5

under groups of concentrated loads. Since problems concerning concentrated

loads are complicated, the characteristics of ribbed slabs under a single

concentrated load will be studied first. After obtaining vital information

about single loads, the ribbed slabs under multiple loads (i.e., as in parking

garages) will be examined.

The ribbed slab is gener'ally considered a flexure type structure;

hence, bending moment is the most important internal force. Besides bending,

shear and torsion may also be significant forces for this type of structure

under concentrated loads. The investigation will be performed using loadings

that result in maximum internal forces. Maximum·deflections corresponding

to the maximum moment loadings will also be presented. The relationships be

tween the internal forces due to the applied load and the important parameters

will be examined and reported. The various cross sections of ribbed slabs

to be" analyzed will be chosen such that the magnitudes of the parameters

cover the reasonable range of current practice. Both simple-span and two-span

continuous structures a0e to be investigated in this study.

The study considers the forces in the ribs due to various loadings~

but does not take into account local stresses within the slabs which are

caused by concentrated loads.

The information obtained from this study will lead to some recommen

dations for the design procedures of the ribbed slab under concentrated loads.

1.4 The Typical Ribbed Slab

The typical ribbed slab will be considered to consist of identical

parallel ribs at equal spacings and slabs of uniform thickness. For this

6

study, seventeen ribs will be used for the analysis. The ribs have a constant

thickness rectangular cross section, whereas in practice they usually have a

tapered cross section, which makes construction somewhat easier. The average

width of the tapered rib might be used as the rib width in design calculations.

The rib width is assumed constant along all spans, although the width is some

times increased near the supports in current construction practice.

The rib spacing, b, is the distance between the centers of any pair

of adjacent ribs, and slab thickness, t, is the total depth of the slab .. Span

length, a, is measured from center to center of the supports, and for the case

of two-span ribbed slab, two equal spans is the typical case. The ribbed slabs

are analyzed as being simply supported, while in practice the ribbed slab is

usually ~ast monolithically with the supports. In the analysis, the interior

support of the two-span ribbed slab is provided on the ribs only, while at

the exterior ends of spans, both ribs and slabs are simply supported.

Figure 1.1 shows typical ribbed slabs, both one- and two-span, and

also the cross sections of the tapered and equivalent rectangular ribs.

1.5 Selected Parameters

Parameters that influence the load carrying behavior of the ribbed

slabs include both geometry of the structure and material properties. However,

if the same materials are always used in both ribs and slabs, the remaining

variables depend only upon the geometry of the ribbed slab. The important

geometric variables are the span length, rib spacing, cross section of the

ribs, and slab thickness.

The magnitude of bending moment is a direct function of span length

7

in any flexure member. For the ribbed slab~ not only the span length but

also the rib spacing and slab stiffness, which is proportional to its thick

ness, affect the magnitude of bending in the ribs. So far, ribbed slabs gen

erally have been designed to resist bending and shear only. In this study,

torsional moments are found to be important, because of both the geometry

of the structure and the nature of the loading. The torsional moment in a

rib depends upon the torsional stiffness of the cross section as well as the

flexure stiffness.

It can be concluded that the major parameters which may affect the

load carrying characteristics of the ribbed slab are:

1.· Span length, a;

2. Rib" spacing, b ;

3. Flexural stiffness of slab, 0;

4. Flexural stiffnesS of the rib, EI;

5. Torsional stiffness of the rib, GJ.

Material properties, as mentioned above, might also affect the load

carying behavior of the ribbed slab, but throughout this study material pro

perties are kept constant. For concrete, Poisson1s ratio in the elastic range

is almost constant, and in this analysis the value of 0.15 is used. The modulus

of elasticity of concrete is considered to be 3,600,000 psi. The modulus of

elasticity does not affect the magnitude of the internal forces due to applied

load, but does affect the displacements. The effects of material properties

on the behavior of ribbed slabs under applied loads consequently are not in

cluded in this stydy.

The above parameters can be reduced to three and made more general

8

by converting to relative measures. The rib spacing can be considered rela

tive to the span length, since it is reasonable that the width of the spacing

should be compared to its span length. The flexural stiffness of the rib

could reasonably be considered relative to the stiffness of the slab. And

the torsional stiffness of the rib can be taken in terms of its flexural

stiffness. Therefore, three nondimensional parameters are introduced as

following:

1. b/a is rib spacing relative to span length (i.e., aspect ratio),

2. H = !~ is flexural stiffness of the rib relative to that of

the slab, and

3. T = ~i is torsional stiffness of the rib relative to its

flexural stiffness.

These three dimensionless parameters and their relationships to the

load carrying characteristics of the ribbed slab will be the focus of this

study.

1.6 Notation

The letter symbols throughout this study are defined as following:

a

b

h

Span length of the ribbed slab, measured from center to

center of supports

Rib spacing, measured from center to center of ribs

Width of the rib

Total height of the rib cross section (including the slab)

n

t

We

Et3 o =

12(1 - J2)

E

FG, FS

* F

G

H = .rr aD

I

J

9

Number of loads on a ribbed slab in the span direction for

the case of multiple loading

Thickness of slab

Equivalent uniformly distributed' load for ribbed slabs

Flexural stiffness of slab per unit width

Modulus of elasticity of material of the ribbed slab

Flexibility matrices for beam and slab elements, respec

tively

Flexibility matrix for the ribbed slab structure

Absolute flexibility matrix at the interior support for

the two-span ribbed slab

Shear modulus of material of the ribbed slab

Flexural stiffness of the rib relative to that of the slab

Moment of inertia of the composite section of the rib

and slab

Torsional constant of the cross section of the rib

Deflection coefficient

Displacement vectors at the left and right edges, re

spectively, of k element due to the applied load on that

element

* L

M

M. J

p

* p

10

Displacement vector of the structure due to the applied

load

Displacement vector at the edge x of i element due to the

applied multiple load on that element

Bending moment of the composite section of the rib and

slab

Maximum bending moment of a simple beam with the same span

length and load spacing as the ribbed slab

torsional moment of the rib

Moment per unit length around the immaginary joint, j, be-

tween the beam and slab elements

In-plane normal forces per unit length on joint j, on

the left and right edges of the element

The applied concentrated load

Internal force vectors at the left and right edges of

k element

Internal force vector of the. ribbed slab structure

Vertical reactions per unit length on joint, j, on the

left and right edges of element

T = GJ E1

v

11

Redundant reactive force vector at the interior support

of the two-span ribbed slab

In-plane shearing forces per unit length on joint, .j,

on the left and right edges of element

Torsiona'l stiffness of the rib relative to its flexural

stiffness

Shear force of the rib

Maximum shear of the beam loaded as for MB

Absolute displacements at the interior support of the

two-span ribbed slab due to the applied load, when the

interior support is ignored

Poisson's ratio of concrete

Absolute value of the maximum deflection of a ribbed slab

2.1 Genera 1

12

CHAPTER 2

METHOD OF ANALYSIS

The analysis of reinforced concrete ribbed slabs subjected to a

single or multiple concentrated load is based' on the method of analysis which

has been used in the studies of the effects of diaphragms in bridges with pre-

stressed concrete I-beam girders by Sithichaikasem and Gamble (11). This

method is possibly even better suited to the monolithic ribbed slab structure

than to the composite bridge structure, since the method was derived assuming

monolithic structures without joints.

In practice, a ribbed slab is usually cast in' plac'e monolithically

with its supports, which are then partially fixed. Hehce, the supporting

conditions deviate somewhat from the restrictions of the method of analjsis,

which is limited to simply supported structures only. This limitation results

from using of Fourier series in the analysis; in which the displacement func

tion, internal forces, and external loads are put in terms of a sine series

which satisfies boundary conditions of a simple support. However, the par

tially restrained supporting condition of the usual ribbed slab structure

would influence the results of the analysis toward the conservative side, so

far as the forces in the ribbed slab are concerned.

2.2 Assumptions

The assumptions in this analysis are concerned mainly with solving

for the'force-displacement relationships of plate and beam elements. Therefore,

13

the basic assumptions for flexure theory of a medium-thick plate, membrane

theory for a thin plate, and theory of biaxial bending combined with axial

force and torsion of a beam will hold for this analysis. Since the ribbed

slab is a reinforced concrete structure, the additional assumptions for

material properties are as follows:

1. Concrete is homogeneous, isotropic~ and elastic;

2. Poisson's ratio for concrete is equal to 0.15.

All supports are assumed to be unyielding.

2.3 One-Span Ribbed Slab

The method of analysis is initiated by decomposing the ribbed slab

into a series of plate and beam elements. At joints between the elements,

there are four unknown internal forces: in-plane normal force N, in-plane

shearing force S, vertical reactive force R, and moment acting perpendicular

to the joint Mj (see Fig. 2.1). Then solutions for the individual plate

and beam elements under the conditions of the applied load and the internal

forces along the joint are carried out separately.

The exact solution of the plate element is rather complicated, since

it is the problem of bending of a plate under the combined action of the lateral

load, edge moments, and forces in the middle plane. For this study, the bend-

ing theory and the plane stress theory for thin plates are applied independentl~

It is not extremely accurate, since bending moments' in the plate are affected

by the in-plane forces. However, this effect is negligible for plates with

small deflections. Another source of error is found in solutions of the mem-

brane theory since equations in the form of Fourier series do not sati'sfy the

14

end support condition ~hat membrane shear should be zero, but the effect is

also negligible, as shown by Sa vern (15).

The flexibility-matrix and displacement vectors at the left and

right-edges of the plate element are obtained by combining solutions of bend

ing and membrane theories. The flexibility matrix and displacement vectors

at the left and right edges of the beam element are determined by integration

of the equilibrium equations of a small element of the beam (see Fig. 2.1),'

and then using the force-displacement relationships for solutions. The de

tailed equations for solutions for both plate and beam elements are discussed

and reported in Ref. 11.

The internal forces can be determined by the compatibility conditions

at the joints of the deco~posed structure. In other words, the decomposed

structure is assembled under the condition that displacements of the plate

and beam elements at the,same joint are equal.

By this argument the solutions can be obtained as follows:

FG is the flexibility matrix of the beam element;

FS is the flexibility matrix of the plate element;

Lkl , Lkr are the displacement vectors at the left and right

edges, respectively, of element k ~ue to the applied

external load on that element;

Pkl , Pkr are the vectors of internal forces at the left and

ri'ght edges of element k

Both the plate and beam flexibility ~atrices have size of 8 by 8, since ,each'

15

element is subjected to eight internal forces (see Fig. 2.1). The flexibil-

ity matrix of each element can be partitioned into four submatrices, as far

as internal f6rces are concerned. For example, for an element k

Fkll 1 Fklr F I f krl: krr

At any joint N, the right edge of the beam element i is connected

to the left edge of the plate element j, or vice-versa. The displacement of

the right edge (i.e., r edge) of the beam element i is:

FGrl P'l + FG P. + L, 1 rr lr lr

and the displacement of the left edge of the plate element + 1 = j is:

For compatibility of joint N, the displacement at the right edge of element

i must equal the displacement at the left edge of element j, and the force

vectors of both elements at the edges of the common joint N must be the same,

tha tis:

and

p, . lr

where PN is the internal force vector at joint N. Then we can write

L'1 - L. J 1 r

16

By applying the above conditions for all the joints, then equations for the.

assembled structure are obtained as follows:

* * * F P = L

where

* F is the assembled flexibility matrix of the structure

* p is the force vector at all joints

* L is the applied load displacement vector

The internal forces at every joint are, therefore, obtained by

solving the above equations. Internal forces in any slab and "beam can be

determined by substituting the joint forces into the equilibrium equations

of the individual element. As far as the geometry of the structure is con-

cerned, it is reasonable that the rib and slab interact somewhat as aT-beam,

but there is no direct way to determine the effective width of the flange of

each rib in this analysis. In order to find the effective T-beam moment, the

T-beam action is evaluated for the condition that the sum of axial forces in

the effective composite T=beam section is zero under pure bending. Then the

composite bending moment of a rib including the interaction with the slabs is

calculated.

2.4 Two-Span Ribbed Slab

Solutions of a two-span ribbed slab are determined by the unit 'load

method, in which there are three steps in the calculations. First, the inte~

rior support is removed, and the structure becomes a simple span ribbed slab

under the applied load. Second, to determine the redundant reactions (vertical

17

reactions and twisting moments), unit loads and. couples are applied to each

rib at the interior support. Third, the solutions of the first and second

steps are combined to yield the solution of the two-span ribbed slab by re-

storing the interior support points to zero deflection and zero rotation about

the rib axes.

Solutibn of the first step is like that ofasimple span ribbed

slab subjected to externally applied load, so the calculation is similar to

that described in Section 2.3. Besides internal forces, solution of this

step also involves absolute displacements (deflections and rotations) at the

interior support. In the second step, the simple span ribbed slab is sub-

jected to unit loads and couples acting on every rib at the interior support

location. Displacements are found using the procedure in Section 2.3, re

sulting in the absolute displacements at the interior support, as well as

internal forces. Then, with the compatibility condition that there are no

vertical displacements or rotation about the rib axis at the interior support

of the structure, equations are formed as follows:

where

* FI is the absolute flexibility matrix of the structure at the

interior support;

RI is the redundant reactive force vector at the interior support;

6 is the absolute displacement vector at the interior support

loc~tion due to the externally applied l~ad, with the interior

support removed.

18

By solving the above equations, the redundant reactions at the interior sup

port line are determined.

Third step, solution of the two-span ribbed slab is obtained by

combining the solution of the first step due to the externally applied load

and that of the second step, using the calculated reactions instead of unit

loads and couples. Then we have:

where

* S = S 1

* S is the solution of a two-span ribbed slab subjected to the

applied load;

Sl is the solution of the first step, considering the applied

load;

S2 is the solution of the second step, considering the applied

unit loads and couples.

This technique of applying redundant reactions instead of the inte

rior support is not exactly correct, since the real support normally supports

not only the ribs but' also the slab. However, the inaccuracy affects only

the portions of slabs close to the support and is negligible for the structure

as a whole.

2.5 Multiple Loads

The analysis of highway bridges (1T,14) was made with single loads

only, and the superposition method was then used for the solution of truck

loadings· or any other combination of "loads. 'For this study, multipl~ loads

19

are considered (i.e., wheel loads of vehicles in parking garages). The super-

position method is not practical, because of the great number of wheel loads

possible in parking garages. Therefore, a direct solution for multiple loads

is necessary.

In the analysis, all vehicles are assumed to have the same wheel

spacings, to weight the same, and have the weight divided evenly among the

four wheels, as is further discussed in Chapter 4. Starting with the method

of analysis described in Sections 2.3 and 2.4, the only things to be changed

for multiple loads are the displacement vectors at the edges of the elements

due to the applied loads. Since the displacement vectors due to single loads

are available, the displacement vectors for multiple loads equal the sum of

displacement vectors at the corresponding edges due to all the externally

applied loads on an element. For example, if there are'M loads on an element

;, the displacement vector at the edge x will be as follows:

where

M I L. N

N=l 1 x

* L: is the displacement vector at the edge x of the element lX

due to M loads

L. N is the displacement vector at the edge x of the element i lX

due to a single load N

Solutions of ribbed slabs due to multiple loads can then be obtained

by the method of analysis in Sections 2.3 and 2.4, with the displacement vec

tors calculated by the above procedure.

20

2.6 Accuracy of the Analysis

The computer program was modified from the program developed by

Sithichaikasem and Gamble (11). The accuracy of the computation depen~s

largely on the number of terms of harmonics used in the calculation for the

Fourier series type equations. It has been shown by Wong and Gamble (14)

that the rate: of convergence toward the ,cor.rect ~olution for a simple-span

bridge is slow after 5 harmonics. At 35 harmonics, 97 percent computational

accuracy was obtained, comparing. total composite moment of all beams with

the static moment as calculated from the elementary beam theory_ The computing

work was carried out on th~ .IBM 360/75 computer, using double precision

arithmetic.

In this study, the computational accuracy of the solution for

shear is also examined, comparing total shear at all supports in all ribs

with the applied load. A simple span ribbed slab with H = 4.5, b/a = 0.052,

and T = 0.100, and a two-span ribbed slab with parameters of the same magni

tudes were taken as examples for showing the computational acturacy of solu

tions. Twenty harmonics were used for the simple span ribbed slab and 35

for the two-spane

The results of two loading cases, a single load P at midspan of

center and edge ribs, are as follows:

Simple-Span Ribbed Slab

Load P at midspan of the center rib:

Total moment at midspan of 17 ribs = 0.2427Pa, 97.1 percent.

convergenc'e

Total shear at supports of 17 ribs = 0.9588P, 95.9 percent

convergence

21

Load P at midspan of the edge rib:

Total moment at midspan of 17 ribs = 0.2443Pa, 97.7 percent

convergence

Total shear at supports of 17 ribs = 0.9652P, 96.5 percent

convergence

Two-Span Ribbed Slab

Load P at midspan of the center rib:

Total moment at midspan plus one-half total moment at interior

support of 17 ribs = 0.2408pa, 96.3 percent convergence

Total shear at all supports of 17 ribs =0.9864P, 98.6 percent

convergence

Load P at midspan of the edge rib:

Total moment at midspan plus one-half total moment at interior

support of 17 ribs = 0.2426Pa, 97.0 percent convergence

Total shear at all supports of 17 ribs = 0.9889P, 98.9 percent

convergence

The static moment is 0.25Pa.

For the simple span ribbed slab, the example above shows that the

computational accuracy for the solution of shear is slightly less than that

of the solution for moment. A reason for this is that some shear forces are

carried directly to the supports by the slab. The composite action between

the ribs and slab is taken directly into account in the solution of moments.

Some of the moment which is unaccounted for after 20 harmonics is resisted by

bending in the slab elements, and most of the rest by the loaded rib.

For the two-span ribbed slab, results' opposite-those of the

22

simple span are obtained. The computational accuracy of solution for: moment

is obtained by comparing total moment at midspan plus one-half of total moment

at the interior support of all ribs (i.e., 17 ribs) with the static moment.

Because of the additional dimension involved in this structure, the moments

at midspan of the unloaded ribs are not all maximum Values. The comparison

to the simple beam moment is' correct only if the total moments in all 17 ribs

are considered. A better result in the computational accuracy for the solu

tion for shear is obtained because only the ribs are supported at the interior

support, where about 70 percent of th~ applied load is carried. Therefore,

the amount of shear carried by the slabs is minimized, and a higher computa-

tional accuracy is obtained.

Metz Reference Room Civil Enginesring Department BlD6 C. E. Buil~ing University of Illinois Urbana, Illinois 61801

23 '

CHAPTER 3

BEHAVIOR OF RIBBED SLABS UNDER A SINGLE CONCENTRATED LOAD

3.1 General

Thi s study has consi dered both one-,span and two-span conti nuous

,ribbed slabs. A single concentrated load is the basic loading case for

studying the general load distributio~ behavior of ribbed ~labs. Various

load locations were chosen to cover the possible range of interest, such as

at midspan of center and edge ribs.

The structural behavior of ribbed slabs subjected to a concentrated

load might be characterized by slab and rib actions. An interpretation of

these actions might be made in two directions, relative to geometry of the

structure. First, in the transverse direction (direction perpendicular t6

the rib), slab action is likely to take the form of an elastic support of

the ribs, especially of the loaded rib. The ribs also act as elastic sup

ports of a continuous one-way slab. Second, in the longitudinal direction

(direction along the rib), the rib is somewhat like a beam on an elastic

f6undation as far as the transverse slab action is concerned. The slab also

acts as a flange of a T-beam which has the rib as its web.

This investigation of the ribbed slab under concentrated loads is

mostly concerned with the study of moment, shear, and torsion (such as their

d'i stri buti ons, as ; nfl uenced by vari ous parameters). The important parameters

in this study are H, b/a, and T, as discussed in Section 1.5. To examine the

influence of these' ~arameters on the load-carrying characteristics of both

simple span and two-span continuous ribbed slabs, various sections were chosen

24

for the analysis. The ribbed slabs analyzed have magnitudes of H varying

from 2.0 to 9.0, b/a from 0.035 to 0.10, and T kept about constant at 0.100

(see Table 3.1). For generality and to help comparisions, the presentations

of moment, shear, and torsion are in dimensionless forms as M/MB3 V/VB

, and

Mt/MB respectively.

3.2 Moment, Shear, and Torsion Diagrams

The study of moment, shear, and torsion diagrams due to single

loads was carried out on both simple span and two-span continuous ribbed

slabs with H = 4.5, b/a = 0.052, and T = 0.100. Two loading cases, for

loads at midspan on the center and edge ribs, will be presented in this sec

tion.

3.2.1 Moment Diagrams

a. One-Span Ribbed Slab

Figure 3.1 shows moment diagrams of a few ribs due to the load at

midspan on the center and edge ribs. The moment diagrams of all ribs, except

the loaded rib and the first adjacent rib, are more or less parabolic curves

which indicates that those ribs were subjected to a distributed load of some

kind. These loads are transmitted from the loaded to the unloaded ribs by

bending and shear forces in the slab. The slab also acts as an elastic support

for the loaded rib," causing a portion of the moment diagram for the loaded

rib between the support and the load to be concave upward. For the load at

midspan on the center" rib, the moment diagra~ of the first adjacent rib is

also concave upward between the support and midspan, indicating greater load

25

distribution occurr~d around midspan (which is close to the applied load).

Instead of being concave, the moment diagram for the first adjacent rib due

to the edge rib ioading is more or less a straight line,_ indicating that

this rib effectively is loaded only near midspan. As a resul·t of the ~se

of the Fourier series, the moment diagrams of the loaded ribs have a rounqed

curve instead of a sharp break at the point of maximum value.

The magnitudes of moments due to the edge rib loading are about

twice those due to the center rib loading. The reason for this is that the

load on the center rib is distributed to adjacent rib on both sides, which

is impossible on the edge rib. For the ribbed slab ~nalyzed in this section,

the maximum moments of the loaded center and edge ribs are O.056Pa and O.125Pa,

respectively, as compared to the simple beam moment of O.25Pa.

b. Two-Span Ribbed ~lab

The moment diagrams of the two-span ribbed slab are presented for

two loading cases--lbads at~midspan on the center and edge ribs of-one span.

Midspan loading location is not for maximum moment, but it is the simplest

location for studying general behavior of the two-span ribbed slab under a

concentrated load and. for making comparisons.

Figure 3.2 shows the moment diagrams of a two-span ribbed slab due

to the above loadings. The general characteristics of these diagrams are

similar to those of the simple span ribbed slab. For this particular ribbed

slab, absolute values of the maximum positive and negative moments of the

loaded edge rib ar~ O.094Pa and O.042Pa respectively, and those of the loaded'

center rib are O.051Pa and O.017Pa. :In. comparison, the maximum positive and·

negative moments of a correspondingly loaded two-span beam are O.203Pa and

26

O.094Pa respectively. It is clear that the maximum moments of the loaded

ribs are significantly smaller than those of the beam, and for the same rea

sons as in the simple span ribbed slabs.

3.2.2 Shear Diagrams


Shear ,diagrams of the unloaded ribs are more or less parabolic

curves between the support and midspan, with maximum values at the supports,

except that of the first adjacent rib which has high shear values on the

portions close to midspan before it curves down to zero there. For the

loaded rib, a distinctive high shear value was obtained on the portion close

to the load, as would be expected. The shear diagram of this particular

ribbed slab shows that shear is as high as O.4P and O.47P in the loaded center

and edge ribs, respectively, whereas simple beam shear is constant at O.5P

all the way to the support. However, shear diagram of the loaded ribs is

not like that of the simple beam, since for the edge rib loading, the maxi

mum shears decrease sharply to a value about the same as those of the un

loaded ribs; for the center rib loading, maximum shears decrease to a value

somewhat less than those of the first few adjacent ribs, indicating good

shear distribution. The maximum positive and negative shears of the loaded

ribs are found a little distance away from midspan, and the portion of shear

diagram connecting these two maximum shears is almost a straight line, which

is a result of using Fourier series.

The shear diagrams also show some idea about how various ribs are

affected when the load is applied at midspan on the center or edge rib. The

27

maximum shear of the loaded rib decreases sharply, indicating that the rib

is subjected to a significant downward load over the portion between the

maximum positive and negative shear points, and to a relatively great up-.- - - --

ward reaction from the slab over the high shear portion. the upward-roaa- ---

decreases rapidly from the point of maximum shear and continues as a small

distributed load to the support, since shear increases slightly from the

support to the high shear portion. Through slab action, the first adjacent

.rib is loaded downward by a load of the same magnitude as the upward load

on the loaded rib (or by one-half of that if the loaded rib is the center

one), although the force is distributed over a greater length than in the

loaded rib. The rib is also subjected to some distributed upward·load with

a more or less p~rabolic shape with the maximum value at mid~pan. Loads on

the other ribs also result from slab action and can be explained in the same

way as above. Examples of these loads are sketched and shown in Fig. 3.3.

b. Two-Span Ribbed Slab

Shear diagrams of the two-span ribbed slab due to the load at mid

span of the first span on the center and edge ribs are. shown in Fig. 3.2.

The general characteristics of the ~iagrams are similar to those of the simple

span, as discussed above. In addition, the decreasing shear at the interior

suppo~t results from the nature of the Fourier series, since the interior

support was replaced by sets of vertical reactions and twisting moments on

the ribs. Sketches sh~wing an interpretation of the shear and torsional mo

ment diagrams near the interior support are shown in Fig. 3.4. For the re

sults of the particular ribbed slab.analyzed in this section, the maximum

positive and negative shears of the loaded center rib are O.39P and O.41P

28

respectively, and those of the loaded edge rib are O.44P and O.50P (whereas

the maximum positive and negative shears of a two-span prismatic beam are

O.41P and O.59P). The difference between the maximum positive and negative

shears of the loaded ribs is much smaller than that of a beam--especially in

the loaded center rib. The effect might be attributed to the transverse

shear distribution that reduces the shear at the support of the loaded rib.

Slab action in the two-span ribbed slab is somewhat the same as that in the

simple span, as far as shear diagrams are concerned.

3.2.3 Torsional Moment Diagrams


Torsion diagrams due to the load at midspan on the center and edge

ribs are shown in Fig. 3.1. This is not the loading case for maximum torsion,

but rather is for maximum bending moment. The location of load for maximum

torsion is somewhere between the support and midspan, as shown by the influence

line for torsion at the support (see Fig. 3.9).

When the loading is at midspan on the center rib, there is no torsion

. on the loaded rib, because of symmetry, and maximum torsion is obtained at the

support of the second adjacent rib. For the first adjacent rib, torsion in

creases slightly from the support toward midspan, until close to the quarter

point of span. There the rate of increase is a little greater and torsion

reaches a maximum value somewhere between the quarter point and midspan,

whereas the other ribs have maximum values at the support. The torsion dia

grams of all ribs except the first adjacent rib are more or less of parabolic

shape between the support and midspan.

29

Load at midspan on,the edge rib results in maximum torsion at the

support of the loaded rib. It is different from the case of the load on the

center rib, sinc~ the edge rib has a free edge, and therefore is freer to

deflect and rotate than the others. The maximum torsion at the support of

the loaded edge ribis more than three times the max,imum torsion caused b,y

the center rib loading. All torsion diagrams for the edge rib loading are

also nearly parabolic for the portion between the support and midspan.


Figure 3.2 shows the torsion diagrams of the ribs of a two-span

ribbed slab due to a' load at midspan on the center and edge ribs of one span.

At the exterior support,of the loaded span, the maximum torsions of both

loading cases occur in the same manner as those in the simple span ribbed

slab.- At the interior support, the torsion diagrams are also affected by

the Fourier series, as are shear diagrams (see Fig. 3.4 for a graphical in

terpretation). The general characteristics of the torsion diagrams fdr the

two-span ribbed slab are similar to those of the simple span ribbed slab.

3:3 Moment Envelopes and Influence Lines for Moment at Midspan and Influence Lines for Shear and Torsion at Support

The studies of the moment envelopes and influence lines for moment

'at midspan and influence lines for shear and torsion,at support were carried

out for a 'ribbed slab with H = 4.5, b/a = 0.052, and T = 0.100, for both one-

and two-span structur~s~ Two loading cases were considered--one load at mid-

span of the center"and edge ribs. The moment envelopes and influence lines

of any ribbed slab will be similar iri general form. Therefore, the study in

30

this section is intended to show an example of the general configuration

and characteristics for a particular ribbed slab, which could be taken as the

reference;for any ribbed slab.

3.3.1 Moment Envelopes and Influence Lines for Moments at Midspan

a. One-Span Ri bbed Sl ab

Figure 3.5 shows influence lines for moments at midspan of various

ribs due to a lo?d moving on the center and edge ribs. These influence lines

have exactly the same shape and magnitude as the moment diagrams (see Fig.

3.1) due to a load at midspan on the corresponding center and edge ribs.

This phenomena .is explained by the fact that the simple span ribbed slab

obeys the Reciprocal Theorem just as well as a simple beam does. However,

the slab action causes both the shape and magnitude of the influence lines

to be different from those of a simple beam.

Moment envelopes of the loaded ribs and a few adjacent ribs are

also shown in Fig. 3.5. The ~oment envelope for the loaded rib is iimilar

to that of a simple beam, but the magnitude is considerably smaller, as dis

cussed above for the influence lines.

b. Two-Span Ribbed Slab.

For two-span ribbed slabs, the influence lines for moments at midspan

of one span and at the in~erior support due to a load moving on the center

and edge ribs are illustrated in Fig. 3.6. Moment envelopes of the center

and edge ribs and ~ few adjacent ribs are also shown in Fig. 3.6. The general

characteristics of the influence lines and moment envelopes of the loaded ribs

31

are similar to those of a two-span prismatic beam. Furthermore, moment en

velopes of the loaded ribs indicate that the load location for maximum posi

tive moment is approximately 0.4 times the span length from the exterior

support; for maximum negative moment at the interior support, the load- loca

tions for center and edge ribs are about 0.7 and 0.8 times the span length

respectively. These locations for maximum positive and negative moments

compare with 0.43 and 0.57 times span length -in the case of a two-span pris

matic beam. It is quite clear that load location for maximum positive mo

ment of rib and beam are about the same. For maximum negative moment, -the

load location on a rib (espetially a center rib)- is-much closer to the in

terior support than -is true on a beam. The cause of the difference could

be slab action--i~e., load distribution among ribs is better when the load

is far from the suppbrt. Note that better load distribution also means better

momen.t distribution and reduced magnitude of the maximum moment.

For the unloaded rib~, the influence lines for moment at midspan

and moment envelopes are more or less parabolic curves, especially for the

ribs beyond the first adjacent rib (see Fig. 3.6).

3.3.2 Influence Lines for Shear at Support

a. One-Span -Ribbed Slab

Influence lines for shear at the support show the distribution of

reaction shea~s in various ribs due to a load moving on the center and edge

ribs. Figure 3.7 shows:-that shear distribution at the support due to load

at any section on the center rib is better than that due load at the corres

ponding section on the edge rib. Wh~n the load is a certain distance f~om

32

the support, the reaction shear of the loaded center rib is smaller than

those of the first few adjacent ribs (see Fig. 3.7). It could be inter

preted that she~r can distribute transversely to adjacent ribs better than

longitudinally to the support of the loaded center rib. For the edge rib,

shear can distribute transversely only to one side, and it must transmit

the shear longitudinally to the support. These characteristics result from

the geometry of the structure.

Influence lines for shear at the support of the loaded ribs approxi

mate hyperbolic curves. This results from the two dimensional effect rif the

structure (slab action). The difference between the hyperbolic curve and a

straight line is the amount of shear that distributes transversely to adja

cent ribs by slab action. Also, as the load moves closer to the support;

transverse shear distribution is limited and a high reaction shear on the

loaded rib is obtained. For unloaded ~ibs, the influence lines are the re

sult of distributed shears.

Reaction shear is not the maximum shear under a single load, except

for the loaded rib with the load very close to, or at, the support. However,

reaction shear is important, because in case of multipie loading, the maximum

shear is .always the reaction shear.


Influence lines for shear at the exterior support and at the 0.95

span point (near b~t not at the interior support) due to a load moving on the

center and edge ribs are shown in Fig. 3.8. The general characteristics of

the influence lines for shear at the exterior support are similar to those of

33

the simple span. For shears at 0.95 span, a portion of the influence line

very close to the interior support of the loaded rib is affected by usage

of the Fourier series. In addition, shear distributions among the ribs,

both at the exterior support and at 0.95 span points, due to loads at Vari-

,ous sections on the center and edge ribs are different.from those in the sim

ple span ribbed slab in that shears in the loaded ribs are usually the maxi

mum shears. This could mean that shear distribution in the simple span ribbed

slab is better. From Fig. 3.8, the load location for maximum shear at 0.95

span would be somewhere between 0.90 and 0.95 of the span for both loading

cases. For the unloaded ribs, the characteris~ics of the influence lines

for shear at 0.95 span are also similar to those of the'simple span. There

fore, the affect tif the Fourier series on shear around the interior support

is confined to the loaded ribs only.

3~3.3 Influence Lines for Torsion at Support


Influence lines for torsion at the support of the ribs due to a

load moving on the center and edge ribs are shown in Fig. 3.9. For the ,load

on the edge rib, the greatest maximum torsion is obtained in the loaded rib,

and maximum torsions of the other unloaded ribs are smaller in proportion to

the, distance from the loaded one. The load location for maximum support tor

sionin the loaded rib is nearest to the support, and for unloaded ribs the

location is successively farther from the support, in proportion to the dis

tance from the ,loaded rib.

, When the load moves on the center rib, there is no torsion in the

34

loaded rib because of symmetry, as mentioned in Section 3.2.3, and the great

est maximum torsion is found on the first adjacent rib. In the portion of

the influence line from the support to the load location for maximum torsion

in the first adjacent rib, the rate of increase of torsion is great in the

first adjacent rib; but for the other portion of the span, the rate of de

crease of torsion is also great, resulting in smaller torsion on the first

adjacent rib in this ~ortion of the span (see Fig. 3.9). The characteristics

of load locations on the center rib for maximum torsion in the unloaded ribs

are similar to those of the edge rib loading, as discussed above.

The difference in load locations for maximum torsion at the support

of various ribs result from a shear distribution phenomena--i.e., sheaf dis

tribution is limited if the load is close to the support, as mentioned in

Section 3.3.2. Therefore, for the unloaded ribs far from the loaded one,

the load has to be some distance out on the span before the unloaded rib re

ceives greatest effect of distributed shears causing maximum torsion~


Influence lines for torsion at the exterior support and at the 0.95

span point due to a load moving on the center and edge ribs are shown in Fig.

3.10. The general characteristics of the influence lines for torsion at both

the exterior support and at 0.95 span are similar to those of the simple span,

as discussed above. But the magnitude of torsion at 0.95 span is somewhat

smaller than at the exterior SUpport. Torsions at the exterior -support of

various ribs are ~ery small when the load is on the other span.

35

3.4 Moment Distribution and Moment Envelope Across the Span

a. ,One-Span, Ribbed Slab

When a load moves along the midspan section of a ribbed slab~ maji

mum moments at midspan of all the loaded ribs are about the same, except that

the first few near the edge rib have greater moments. Figure 3.11 shows the

moment envelope at midspan of a ribbed slab with H = 4.5, b/a = 0.052, and

T = 0.100. For this particular ribbed slab, moments in the edge, second,

and third ribs are 1.89,1.29, and 1.10 of that of the center rib, respectively.

Moments in the second and third ribs can be reduted 'if the edge rib is made

stiffer (i.e., a strong ~dge beam is provided)., In Fig. 3.11, two cases of

moment 'enve lopes of the ri bbed, slab wi th stiffer edge ri bs are a 1 so ill us

trated. First, the width rif the edge rib is double that of a regular one,

and the results of moments in the edge, second, and third ribs afe 2.28 5

1.07, and 1.02 of that of the center rib, respectively. Second, the width

of the edge rib is triple that of a regular one, and result of moments in

the edge, second, and third ri bs are 2.44, 0.99, and 0.9'8 of that of the

center rib. Hence, moments in all ribs can be evened out by providing a very

strong edge beam, whi~h has to carry a greater bending moment.

Figure 3.11 also shows moment distributions of various ribs due to

a single load at their midspan. Since the Reciprocal Theorem holds for the

simple span ribbed slab, the moment distribution curves also represent the

influence lines for mom~nts at midspan of the loaded ribs due to a load

moving along midsp~n. For any ribbed slab, moment distributions due toa

load at the sam~ location will have similar shapes, but the ~agnitude of the

36

distributed moments depends upon the important parameters discussed in Sec

tion 1.5. Therefore, the study in this section should be considered as a

reference for a general ribbed slab. The affects of those parameters on

moments will be discussed later.


Figure 3.12 shows moment envelopes and moment distributions at mid

span of the first span of a two-span ribbed slab due to a load moving along

the midspan, and also those of the coresponding negative moment at the interior

support. The general behavior of moment envelope and moment distributions at

midspan is similar to the simple span ribbed slab; for example, moments under

the load at midspan of the edge, second, and third ribs are 1.84, 1.24, 'and

1.08 of that of the center rib, respectively. The mom~nt envelope at the

interior support due to the load at midspan is similar to those of the posi

tive moment at the midspan. But the negative moments of the edge and second

ribs, as compared t6 that of the center rib, are much greater than i~ the case

of the positive moments. The results in Fig. 3.12 indicate that negative

moments at the interior support of the edge, second, and third rib are 2.47,

1.53, and 1.06 of that of the center rib, respectively. Negative moment dis

tributions of various ribs at the interior support due to midspan loads are

also shown in Fig. 3.12. The distributed negative moment in the first ad

jacent rib relative to that of the loaded rib is considerably greater than

the relative distributed positive moment of the same rib at midspan. Other

characteristics of negative moment distributions in general are similar to

those of the positive moments at midspan.

37 .

. .. . .

The affects of a stiffened edge rib or edge beam are not presented

for the two-span' ribbed slab. However, results similar to those of the simple

span ribbed slab should be expected.

3.5 Effects .of Torsional Stiffness on Moment, Shear, and Torsion

The affects of the parameter T on moment have been studied and re

ported tn Ref. 5 arid 11. Reference 11 show~d that the moment of the loaded

girder decreases as magnitude of T increases, but a significant affect was

found ohly in the range of T fro~ 0.0 to 0.20. In Ref. 5, test results from

steel I-beam girder-slab bridges indicated that the affect of variation in

torsional stiffness of steel I-beam is negligible.

For this study, the influence of the parameter T on moment, shear,

and torsion are presented. Theinvestigati·on was performed 'on simple sp~a~ ribbed

slabs with constant magnitudes ,of H = 2.0 and 4.0, b/a = 0.05 and 0.10, and

with T varied from 0.063 to 0.161 (see Table 3.2). This is in the ~ange of

significant affects of T on moment as reported in Ref. 11. The study 'con~

sidered two loading cases (load at midspan on the center and edge ribs), and

results of the analysis are shown in .Fig. 3.13. The influence of T on the

moments at midspan, a~d shear at the support of the loaded ribs are similar-

moment and shear.decrease as T increases. But the changes are not very sig

nificant in the range of T investigated, which is considered the practical

range for conventional ribbed slabs. For example, the. results for ribbed

slabs w;'th H = 4 .. 0 and b/a = 0.05 show that the maximum moments of the center

and edge ribs decreased 8.5 and 13.7 percent, respectively, as T increased'

98.8 percent.

38 :

The influence of T on ~maximum torsional moment at the support due

to the above loadings is opposite the affects on moment and sheara As r'

increases, the maximum torsion also increases quite sJgnificantly. The re

sults of the same ribbed slabs with H = 400, bla = 0.05 show that the mix

mum torsion due to a load at midspan on the center and edge ribs increased

59.6 and 42.8 percent, respectively, as Tincreased 98.8 percent (see Fig.

3.13). However, in this study, the magnitudes' of T used are approximately

constant, because the influence of T on bending moment and shearare'very

small,· as mentioned above. The influence of T.on the torsion is significant,

but the incr.eased torsional strength accompanying increased torsional stiff

ness would be enough to comp~nsate for the amount of the increased torsion.

Furthermore, torsion is considered of secondary importance in the structure.

In addition, the magnitude of T selected for this analysis (about 0.100) are

considered common for general practice.

3.6 Effects of Hand bla on Moment, Shear, and Torsion

From the results of the studies of behavior of ribbed slabs under

a concentrated load in previous sections, it might be concluded that the

characteristics of moment" shear!! and torsion distributions are somewhat

changed according' to different load locations on different ribs. In order

to make the study in this section simple and yet reveal enough 'detail, solu

tions for two· loading cases are ·considered--l oad at midspan on the center

and edge ribs. These load locations give the lower and upper bound values of

moment for any ribbed'slab under a concentrated 10ad~ For shear and torsibn,

the center and edge ribs might be considered as being representative of the

39

interior and exterior ribs. And as shown in Sections 3.3.2 and 3.3.3, the

characteristics of the shear and torsion distributions also depend upon

locations along the ~pan; therefore, midspan is considered to be a suitable

load location for studying the general behavior.

The study of the influence of·H and bla on' moment distribution is

confined to the midspan section, where maximum moments are obtained. Shear

and torsion are studied at the support. The results 6f th~ ribbed slabs

analyzed with parameters of various magnitudes, as shown in Table 3.1, ~re

presented in this section.

3.6.1 Effects of H andbla on Moment Distribution Across Span and Maximum Moment of the Loaded Rib


The moment distributions among ribs of simple span ribbed slabs due

to a load at mirlspa~ on the center and edge ribs depend upon the values of

the parameters Hand b/a. The results of the analysis show that the moment

distribution is better for the ribbed slab with smaller ~.and bjavaloes, as

shown in Fig. 3.14. Comparing graphs of constant bla and various H values,

and those of constant H and various bla ratios, it is evident that the effect

of bla is somewhat greater than that of H. For a ribbed slab, the moment dis~

tribution due to the load on the center rib is better than that due ·to the

load on the edgerih. Examples of two simple span ribbed slabs with H = 2.0,

bla = 0.035, and H = 9.9, bla = 0.10 are as follows:.

H = 2.0, bla = 0.035

Load at midspan on the edge rib:

40

Maximum moment of the loaded rib = 0.0761Pa = 0.3044MB Moment of the eighth adjacent rib (center rib) = 0.0041Pa = 0.0164MB

Load at midspan on the center rib:

Maximum moment of the loaded rib = 0.0366Pa = 0.1464MB Moment of the eighth adjacent rib (edge rib) = 0.0043pa = 0.0172MB

H = 9.0, b/a = 0.10


Maximum moment of the loaded rib = 0.1507Pa = 0.6028MB Moment of the third adjacent rib (rib no. 4) = 0.0066Pa = 0.0264MB


Maximum moment of the loaded rib = 0.0940Pa = 0.3760MB Moment of the third adjacent rib (rib no. 6) = 0.0073Pa = 0.0292MB

From the above examples we can see that for the ribbed slab with H = 2.0,

b/a = 0.035, the moment of the eight rib from the loaded edge rib (i.e., the

center rib) is about 5.4 percent of that of the loaded rib, and the moment of

the eighth rib from the loaded center rib (i.e., the edge rib) is about 11.8

percent of that of the loaded rib. For the ribbed slab with H = 9.0, b/a = 0.10,

the moment of the third adjacent rib from the loaded edge rib (i.e., rib no. 4)

reduced to about 4.4 percent of that of the loaded rib, and the moment of the

third rib from the loaded center rib (i.e., rib no. 6) reduced to about 7.8

percent of that nf the loaded rib. It is quite obvious that the moment dis

tribution of the former ribbed slab is much better than that of the latter one,

and the maximum moments of the corresponding loaded ribs of the second ribbed

slab are much greater.

41

As a result of good moment distribution, maximum moment of the loaded

rib is smaller than if moment distribution is not good. It may be concluded

that the magnitude of the maximum moment of the loaded rib is a 'direct func

tion of Hand bla, which control the moment distributions amring ribs. The

relationships of the maximum moments of the loaded ribs with Hand bla are

shown in Fig. 3.23. As shown in Fig. 3.23, the effects of Hand bla on the

maximum moment are not in the same order. Graphs of the maximum moment versus

bla, for several constant H values, are closer together and have steeper

slopes than graphs of the maximum moment versus H. Therefore, it is evident

that bla has a greater effect on the maximum moment of the loaded ribs than

H., Furthermore, as a smaller maximum moment of the loaded rib alsQ means

better moment distribution and vice versa, it is evident that bla is a more

important parameter in controlling moment distribution than H.


The study of moment distributions among ribs of two-span ribbed

slabs concentrated at the sections at midspan of the first span and at the

interior support. Positive and negative moment distributions due to a load

at midspan of one span on the center and edge ribs are illustrated in Fig.

3.15 and 3.16. The results show that the general characteristics of moment

distributions at midspan are similar to those of the simple span. For nega

tive moment at the interior support, moment distribution characteristics are

similar to those of ,the-positive moment when load is on the edge rib. For'a

load on the center-rib, the negative moment in the first adjacent rib is

comparabJe to that of the loaded rib; as discussed in Section 3.4. The

42

effects of Hand b/a on moment distributions, both at midspan and the interior

support, are similar to those noticed in the simple span ribbed slab--i.e., a

better moment distribution is obtained in a ribbed slab with smaller Hand

b/a values.

Relationships of the maximum positive and negative moments of the

loaded ribs versus Hand b/a are also similar to those of the maximum moment

of the simple span--i.e., maximum moment of the loaded ribs increases at a

diminishing rate as Hand b/a increase (see Fig," 3.24).

3.6.2 Effects of Hand b/a on Shear Distribution Across Span and Maximum Support Shear


Shear distributions at the support due to a load at midspan on the

center and edge ribs of various ribbed slabs are illustrated in Fig. 3.17.

The distribution diagrams show that maximum support shear did not occur in the

loaded rib except in the case of a loaded edge rib with rather stiff ribs and

larger b/a ratios. For the center rib loading, the maximum shear can be in

any of the first few adjacent ribs, depending upon Hand bfa, but never in

the center rib. If- it is an edge rib loading, the maximum shear is either

in the loaded or first adjacent rib. The effects of Hand b/a on shear dis-

tributions are similar to those on moment distribution--i.e., the smaller H

and b/a values result in better shear distribution.

In this analysis, the smallest Hand b/a are 2.0 and 0.035, and

the greatest are 9.0 and 0.10, respectively. Comparisons of shear distributions

at support due to a load at midspan on the center and ribs of these two

43

ribbed slabs are as follows:

H = 2.0, b/a = 0~035


Shear of the loaded rib = O.0703P = O,.1406VS Maximum shear in the first adjacent rib (rib no. 2) = 0.0820P =, 0.1640Vs Shear of the eighth adjacent rib (center rib) = 0.0129P = 0.0258V

~ S


Shear of the loaded rib = 0.0166P= 0.0332Vs . Maximum shear in the third adjacent rib (rib no. 6) = 0.0369P = 0.0738VS Shear of the' eighth adjacent rib (edge rib)

H = 9.0, b/a = 0.10


Shear of the loaded rib (maximum shear)

Shear of the fourth adjacent rib (rib no. 5)


Shear of the loaded rib

= 0.0130P = 0.0260Vs

= 0.2200P = D.4400Vs = 0.0017P= 0.0034VB

= 0.0750P = 0.1500Vs Maximum shear in the first adjacent rib (rib no. 8) = 0.1097P =0.2194VB Shear of th~ fourth adjacent rib (rib no. 5) = 0.0050P = O.OlOOVs

Shear distributions in the above example are quite different; the former rib

bed slab with load at midspan on the edge rib, shear in the eighth adjacent

rib (i.e., center rib) is about 15.7 percent of the maximum shear, and with

load at midspan 'on·the center rib, shear in, the eighth adjacent rib (edge

rib) is as high as 35.2 percent of the maximum shear. In the second ribbed

44 .

slab with a load at midspan on the edge rib, shear in the fourth adjacent

rib (i.e., rib no 5) is reduced to about 0.8 percent of the maximum shear,

and with load at midspan on the center rib, shear in the fourth adjacent

rib (i.e., rib no 5) reduced to about 4.6 percent of th~ maximum shear." It

is clear that shear distributions of the former ribbed slab are much better

than that of the latter one.

The relationships of the maximum shear with changes in Hand b/a are

shown in Fig. 3.25. The graphs indicate that the maximum support shear due to

a load at midspan on the center and edge ribs increases more or less linearly

as b/a increases. The effect of H on the maximum shear is somewhat similar

to that of bfa, but is less significant.


Shear distributions at the exterior support and at the 0.95 span lo

cation due to a load at one midspan on the center and edge ribs are shown in

Fig. 3.18 and 3.19, respectively. Note that according to the sign convention

'used in this analysis (see Fig. 2.1), shears at 0.95 span are negative, but

for convenience, the absolute values are used in Fig. 3.19~ The character

istics of shear distributions, at both the exterior support and the 0.95

span position, a~e somewhat different from those of the simple span ribbed

slabs, in that the loaded ribs for both loading cases carry the maximum sup

port shears for nearly all values of the parameters Hand b/a. The maximum

shears near the interior support are generally about twice the shear forces

at the exterior supports for the midspan loading cases considered.

The effects of Hand b/a on the characteristics of the shear

45

distributions are similar to those in the simple span ribbed slabs, i.e.,

smaller Hand b/a values result in better shear distribution. The relation-

ships of the maximum reaction shear with Hand bfa, as can be seen in Fig. 3.26,

are also similar to those in the simple span ribbed slabs, i.e., the maximum

reaction shear increases as Hand b/a increase.

3.6.3 Effects of Hand b/a on Torsional ,Moment distribution, and Maximum Torsion


Torsional moment di~tributions at the support due to the same

loadings used in the'studies of moment and shear distributions are,considered

in this section. 'These loadings are not for maximum torsion, but they are

simply for general study and comparison. The torsion distributions of se

lected ribbed slabs with various values of Hand b/a are shown in Fig. 3.20.

Note that the torsions due to the load at midspan on the edge rib are negative

according to the sign convention, but the, absolute values are used in fig.

3.20. The effects of Hand b/a on torsion distributions are such that the

distribution is better for the smaller H and- b/a. Maximum torsion due to

the load on the edge rib is much greater than that due to the load on the

center rib, because of the geometry of the structure.

The relationships of maximum torsions for both loadings as, functions

of Hand b/a are shown in Fig. 3.27. The maximum torsion increases at a dimin

ishing rate as H and ~/a increase, in a manner similar to that described in

Section 3.6.1, but the effect of H on the maximum torsional moment is some-

what greater than that on the maximum bending moment.

46


Torsion distributions at the exterior support and at the 0.95 span

point due to a load at midspan of one span on the center and edge ribs are

shown in Fig. 3.21 and 3.22. Note that torsions at the exterior support due

to the load on the edge fib, and torsions at 0.95 span due to the load on

the center rib are negative according t6the sign convention used, but the

absolute values are used in Figs. 3.21 and 3.22. The general characteristics

of the torsion distributions, both at the exterior support and at 0.95 span,

are similar to those in the simple span ribbed slabs. The effects of Hand

bja on the torsion distribution are also similar to those in the simple span

ribbed slabs, i.e., the smaller Hand bja values result in better torsion

distribution.

The relationships of the maximum torsion with Hand bja are shown

in Fig. 3.28. The maximum torsion increases as Hand bja increase in the

manner similar to that in the simple span ribbed slabs.

3.7 Effects of Hand bja on Influence Lines for Torsional Moment at the Support, and the Maximum Torsion

We have learned that the load location for maximum torsion is un-

certain for different ribbed slabs. Therefore, this section will show the

influence lines for torsion at the support of various ribbed slabs. Since

the general characteristics of the influence lines for torsion at the supports

of one-and twa-span are similar, as shown in Figs. 3.9 and 3.10, the presenta

tion in this section will be made for the simple span ribbed slabs only.

Two loading cases, single loads moving on the center and edge ribs, are

47

considered. The influence lines of the ribs with the maximum torsion for

each loading case will be compared for various values of Hand bfa, i.e.,

the influence lines of the loaded edge rib, and the first adjacent rib (rib

no. 8) due to the load on the center rib.

Figure 3.29 shows the influence lines of the ribs mentioned above

for various ribbed slabs analyzed.. The load locations for the maximum tor

sion at the support for each ribbed slab are a little different, for the

two loading cases. This location is nearer to the support for the smaller

Hand b/a values. For the case of loading on the edge rib, load locations

for the maximum torsion at th~ support of the ribbed slabs analyzed are be

tween 0.15 and 0.35 span from the support; and for the case of loading on the

center rib, these-locations are in the range of 0.1 to 0.3 of the span. The

maximum torsion due to the former loading is much greater than that due to

the later loading.

The relationships of the maximum torsion with Hand b/a are shown

in Fig. 3.30, and they are similar to those due to a load at midspan on the

center and edge ribs, as.described in Section 3.6.3.

48

CHAPTER 4

RIBBEQ SLABS ,UNDER MULTIPLE LOADS

4.1 General

The study of the general characteristics of ribbed slabs subjected

to multiple loads, especially in parking garage floors, is the purpose of

this chapter, and both one- and two-span structures are investigated. Multi

ple loads are considered to be the wheel loads of several parked cars of

equal weight, and each car having the weight divided evenly among the four

wheels. The arrangements of the loads will be made in such way that worst

conditions are obtained, even though cars might not be parked like that in

general paractice. In fact, cars are movabl~ loads, which might cause_im

pact and repeated load effects. However, in this study, static load is con

sidered most important, because car speed in a parking garage is very slow

and not all cars move or brake at the same time.

In this study, it is assumed that the cars are parked perpendicular

to the ribs, so that the maximum bending moment, shear, and torsional moment

in the ribs are obtained. The length of wheel base and spread between wheels

(or tread width) are 'taken as ten and five feet, respectively. These dimen

sions are based on the information by Burrage and Morgen (16), and Baker (17).

The car spacing is based on a suggested minimum width of the stall in a parking

garage of eight feet. Distance between rows of cars from wheel base to wheel

base is taken as six feet (i.e., the cars are parked bumper-to-bumper).

The characteristics of ribbed slabs under multiple loads will be

studied ,under two situations; first, the perimeter wheel loads are on the

edge rib and second, the perimeter wheel loads are on the second rib. The

49

system of loading considered in this chapter is shown in Fig. 4.1. The cross

sections and span lengths of the ribbed slabs in Table 3.1 will be used for

the analysis. The presentation of results will be in dimensionless forms in

order to make the study more general, since some of the ribbed.slabs analyzed

have different span lengths and were subjected to different numbers of loads.

The dimensionless quantities are M/MS' V/VS' and Mt/MS' where M, V, and Mt are bending moment, shear, and torsional moment of a rib, respectively, and

MS and'VS are the maximum moment and shear of a simple span beam which has

the same span length and is subjected to loads with the spacing as the ribbed

slab. Values of various MB and Vs .are listed in Table 4.1 for ease of compar

ison with the results of ribbed slabs discussed in this chapter.

4.2 Moment, Shear, and Torsional Moment Diagrams

The studies of moment, shear, and torsional moment diagrams of the

ribs of one- and two-span ribbed slabs are carried out on a ribbed slab which

has parameters H = 4.5, b/a = 0.052, and T = 0.100. Loadings for maximum mo

ment, shear, and torsion are considered, in which the loadings for maximum mo

ment and shear are coincident. For two-span structure, loadings for maximum

positive and negative moments also produce maximum shears at the exterior and

interior supports, resp~ctively.

The moment, shear, and torsional moment diagrams, for both one- and

two-span ribbed slabs, due to loadings mentioned above, are illustrated in

Fig. 4~2 and 4.3. The general configurations of the moment diagrams are similar

to those of isolated beams subjected to a uniformly distributed load. This

phenomena might be explained that the loaded ribs or slabs were subjected to

50

groups of concentrated loads with rather closed spacings (i.e., three and five

feet alternately). Furthermore, these concentrated loads were replaced by a

Fourier series approximation, which tends to distribute the loads at the load

locations. The slab ~lso tends to spread out the loads as they are distributed

from rib to rib. Therefore, the influence of a group of concentrated loads be

comes similar to that of a distributed line load.

Shear diagrams for the ribs subjected directly to the loads have

some sharp slope discontinuities in the diagrams.at the load locations because

of the concentrated load effects. For the ribs which are not directly under

the loads the shear diagrams are more or less parabolic curves. Fo~· two-span

structure, a small portion of the shear diagram near the interior support is

affected by using Fourier series approximation; the maximum shear is found at

a very short distance from the interior support (i.e., at the 0.95 span point)

instead of the support.

Torsional moment diagrams for one- and two-span ribbed slabs are

also shown in Figs. 4.2 ·and 4.3, respectively. The general ch~racteristics

of these diagrams are similar to those of shear diagrams. for the ribs which

are not directly under the applied loads, as mentioned above. Torsional mo

ment diagrams for the two-span ribbed slabs due t6 loads on one span (i.e.,

loadings for maximum positive moment) or two-span symmetry (i.e., loading for

maximum negative moment at the interior support) are also affected by use of

the Fourier series at the interior support. But for two span asymmetric load

ing (i.e., loading for maximum torsional moment at the interior support of the

center rib), the torsional moment diagram is unique as shown in Fig. 4.3. Be~

cau~e torques are applied to the rib in the opposite di~ection at the interior

support by such a loading, the effect of using Fourier series is eliminated.

51

4.3 Multiple Loads for Maximum Moment and Shear

The study of moments of ribbed slabs subjected to multiple loads

is focused on the maximum moments of the ribs, and the distributions of mo

ments. The characteristics of moment distributions among ribs depend upon

the properties of the ribbed slabs and the loading conditions. In this study,

the results of the analysis for the ribbed slabs listed in Table 3.1 are con-

sidered to cover the normal range of the important parameters for ribbed

slabs. The loadings are again arranged such that the perimeter loads are

Note that the load patterns for maximum positive and negative moments are

coincident with those for maximum ~hears for dne- and two-span ribbed slabs,

respectively.

4.3.1 One-Span Ribbed Slab

For the one-span ribbed slab, bending moment is important at mid

span, whereas the combination of shear and torsional moment are most impor

tant at the, support. Loading for maximum moment ~lso results in maximum shear,

but torsional moment is small. However, the effect of torsional moment in

combination with shear should be considered. Therefore, moment at midspan,

and shear and torsional moment at the support due to the loadings for maximum

moment are presented.

a. Moments at Midspan

Maximum moments at midspan of various ribs due to both loading cases

are shown in Fig. 4.4 for comparison. For the loading -with the perimeter

52

loads on the edge rib, the maximum moment always occurred on the edge rib, and

its magnitude was considerably greater than the moments in the other ribs.

For the loading with the perimeter loads on the second rib, the maximum moment

might be in the edge or an interior rib, depending upon properties of the

ribbed slab, in which b/a has the major effect (i.e., the maximum moment is

in an interior rib for all ·the ribbed slabs analyzed with bja greater than

0.035). The maximum moment due to this loading is considerably smaller than

that of the former loading. The results in Fig. 4.4 show that moment distri

butions are quite uniform, especially for ribbed slabs with small Hand b/a

values, subjected to loads of the latter case.

b. Shear and Torsional Moment at the Support

For a flexure member, maximum shear is usually another force to be

considered; shear is as important as bending moment, especially for a member

without web reinforcement. For ribbed slabs s~bjected to concentrated loads,

not only shear but also .torsion exists at the support, as mentioned above.

As far as the combined shears and torsions are concerned, the most significant

torsion due to these loadings is not the maximum totsion, but the torsion in

the rib with the maximum shear. However, torsion'in various ribs is shown for

better understanding. Shear and torsion at the support due to both loading

cases for maximum moment are shown in Figs. 4.5 and 4.6, respectively. The

shear distributions are less uniform than moment distributions for both loading

cases. The toriion distributions are different and difficult to compare with

moment or shear distributions, because the sign for torsion depends upon the

direction of rotation of the ribs about their longitudinal axes. Therefore,.

53

torsion in the ribs on the opposite sides of a load would have a tendency to

have different signs.

The general characteristics of the effects of loading conditions,

H, and b/a on both shear and torsion are similar to those on moments. "For

the loading with the perimeter loads on the edge rib, both maximum shear and

torsion oc~urred in the edge rib. For the loading with the perimeter loads

on the second rib, the torsion corresponding to the maximum "shear is very

small.

For example, maximum moment, shear and the corresponding torsion

of two ribbed slabs with H = 2.0, b/a = 0.035, and" H = 9.0, b/a = 0.10 are

shown as follows:

H = 2.0, b/a = 0.035

Perimeter loads on the edge rib:

Maximum moment (in edge rib)

Maximum shear (in edge. rib)

Corresponding torsion (in edge rib)

Perimeter loads on the second rib:


Maximum shear (in second rib)

= =

0.3185MB 0.4000VB

= -0.0226MB

= =

0.2579MB 0.3016VB

Corresponding torsion (in second rib) = -0.0052MB

H = 9.0; bja =0.10



Maximum shear (in edge rib)

=

=

0.5685MB 0.5737VB

54

Corresponding torsion (in edge rib) = -0.0567MB


Maximum moment (in rib no. 6) = 0.4255MB Maximum shear (in rib no. 7) = 0.4l58VB Corresponding torsion (in rib no. 7) = 0.0031MB

These are the results of the smallest and greatest maximum moments for each

loading case among the ribbed slabs analyzed.

4.3.2 Two-Span Ribbed Slab

In the positive moment region only the bending moment is important,

whereas at the exterior support the combination of shear and torsion are im

portant. In the negative moment region, or at the interior support, bending

moment, shear, and torsion can exist simultaneously in many combinations,

depending on both the loading and the properties of the structure. Themost

important combinations could be considered as follows: First, the case of

maximum moment plus maximum shear, and torsion, and second, the case of maxi

mum torsion plus shear and moment. The first case will be discussed in this

section, ahd the second case will be discussedlat~r.

a. Moment at 0.44 Span and at the Interior Support

For a two-span ribbed slab, the maximum positive mDment due to the

applied load does not occur at midspan, but rather at a point close to midspan,

as shown by the moment diagrams in Fig. 4.3. For this study, the moment at

0.44 of the span from the exterior support is considered to be the maximum

positive· moment. If the maximum moment occurs at any other section, the

55

value of moment obtained at 0.44 span would close enough to the maximum moment

for any practical purpose, because the moment diagrams show that the rate of

change of moment is very small in that portion of the span (see Fig. 4.3),

The positive moment distribution at 0.44 span and the negative mo

ment distributinn at the interiot support of various two-span ribbed slabs

due to the loading cases for maximum positive and negative moments are shown

in Figs 4.7 and 4.8, respectively. The general characteristics of the solu

tions for the positive moment are similar to those of the moment at midspan

of the one~span ribbed slabs, as discussed previously. The effetts of Hand

b/a on the negative moment at the interior support are similar to those on

the positive moment at 0.44 span or the moment at midspan of the one-span

ribbed slabs, in that a better moment distribution is obtained for the smaller

Hand b/a values. But the negative moment distributions at the interior sup

port are a little less uniform than those of the positive moments. One reason

for this might be that the negative moment is most affected by loads which

are relatively close to the interior support, whereas the positive moment is

affected by most of the loads in the span, as shown ~y the influence line for

positive and negative moments at midspan and the interior support, respectively

(see Fig. 3.6). Note that the'distribution of loads near midspan is better

than that of loads near the support .

. The magnitudes of the maximum negative moments are greater than

those, of the maximum positive moments, and for a similar loading case, the

maximum positive and negative moments usually occurred in the same rib. In

addition, the negative moments at the interior support due to loadings for

maximum positive moment are exactly one-half the maximum negative moments

shown in Fig. 4.8.

56

b. Shear and Torsional Moment at the Exterior Support and at 0.95 Span

Shear and torsion distributions at the exterior support and at the

0.95 span position due to two loading cases for both maximum positive and nega

tive moments are shown in Figs. 4.9 to 4.14. The values at the 0.95 span point

are presented instead of at the interior support, because of the effects of

Fourier series, as mentioned before. The general characteristics of the shear

and torsion distributions, as shown in the figures mentioned above, are simi

lar to those of the one-span ribb~d slabs. For the loadings with the perimeter

loads on the second rib, torsion is relatively small as compared to that due

to the loading with the perimeter loadsbn the edge rib. The important tor

sions due to these loadjngs are those of the ribs with maximum shear at the

exterior support, and with maximum shear and moment at the interior suppo~t.

The shear and torsion distributions at the exterior support due to

loadings for maximum negative moment are similar to those due to loadings for

maximum positive moment, except that the magnitudes are smaller. They are

also very similar to the values for one-span structures discussed previously.

Therefore, such shear and torsion distributions at the exterior support are

not presented. At the.0.95 span location (or at ~he interior support for

moment), the combination of negative moment, shear, and torsion, makes the

problem more complicated than at the other end of the span. Shears at the

0.95 span point due to both loadings for maximum negative moment are consider

ably greater than those due to loadings for maximum positive moment. Negative

moment at the interior support due to the loadings for maximum negative mo

ment is twice that due to the loadings for maximum positive moment. The

combined shear, torsion, and bending moment at the interior support due to

57

loadings for maximum negative moment is the most important set of forces

governing the design of ribbed slab structures.

For example, solutions of two ribbed slabs with H = 2.0, b/a = 0.035,

and H = 9.0, b/a = 0.10, are shown as follows:

H = 2.0, b/a = 0.035

Loadings for maximum positive moment at 0.44 span


Maximum positive moment (in edge rib)

Maximum shear at the exterior support (in edge rib)

Torsion at the exterior support (in edge rib)

Maximum shear at 0.95 span (in edge rib)

Torsion at 0.95" span (in edge rib)

N~gative moment at the interior support (in edge ri~)


Maximum positive moment at 0.44 span (in edge rib)

Maximum shear at the exterior support (in rib no. 2)

Torsion at the exterior support (in rib no. 2)

Maximum shear at 0.95 span (in ri~ no. 2)

Torsion at 0.95 span (in rib no. 2)

= 0.2563MB = 0.36llVB = -0.0205MB = -0.4718VB = 0.0158MB = -O.1873MB

= 0.1992MB = 0.2667VB = -0.0046MB = -0.3099VB = 0.0034MB

Negative moment at the interior support (in rib no. 2) = -0.1266MB

Loadings for maximum negative moment at the interior support


Maximum negative:moment (in edge rib)


= -0.3746MB = -0.6390VB

58

Torsion at 0.95 span (in edge rib)




Maximum negative moment (in rib no. 2)

Maximum shear at 0.95 span (in rib no. 2)




H = 9.0, b/a = 0.10

Loadings for maximum positive moment at 0.44 span


Maximum positive moment (in edge rib)




Torsion at 0.95 span (in edge rib)

Negative moment at the interior support (in edge rib)

'Perimeter loads on. the second ri b:

0.01Q2MB 0.3314VB

= -0.0185MB

=

=

= -0.2531MB = _n L1.nnl\l

V •• VVI 'B

= 0.0018MB = O.2394VB = -0.0040MB

= 0.4579MB

= 0.5317VB = -0.0521MB = -0.6502VB = 0.0423MB = -0.3158MB

Maximum positive moment (in rib no. 6) = O.3353MB Maximum shear at the exterior support (in rib no. 7)' = 0.3809VB Torsion at the exterior support (in rib no. 7) = O.0042MB,


,Torsion at 0.95 span (in rib no. 7)

= -O.4608VB = -0.0027MB

Negative moment at the interior support (in rib no. 7) = -0.2234MB

59

Loadings for maximum negative moment at the interior support


Maximum negative moment (in edge rib)


To~sion at 0.95 span (in edge rib)




Maximum negative moment (in rib no. 7)





4.3.3 Maximum Deflections

=

=

=

=

=

=

=

=

=

=

-0.'6316MB -0.8673V

B

0.0290MB 0.4880VB

-0.0475MB

-0.4468MB -0.5994VB -0.0068MB

0.3449VB 0.0053MB

The maximum deflections df ribbed slabs, both one- and two-span sub

jected to multiple loads, resulted from loading for maximum positive moment.

The absolute values of maximum deflection (i.e., in inches) of the ribbed

slabs analyzed are shown in Tables 4.2 and 4.3.' Since the ribbed slabs ana~

lyzed have different span lengths and cross section properties, the deflection

coeffi ci ents in terms of the app 1 i ed load, span 1 ength, and the .fl exura 1

stiffness of the composite section are considered for ,the study of the gener

al relationships involved. The deflection coefficient is as follows:

where

60 "

K = deflection coefficient

o = absolute deflection

n = number of the applied loads in the span direction, con~idering

only those loads located more than the rib depth from the sup

port"

For example, the relationships of the maximum deflection coefficients

for one-span ribbed slabs under the loading with the perimeter loads on the

edge rib with Hand b/a are shown in Fig. 4.15. The coefficient K increases

slightly as H increases. The relationships between the coefficient K and b/a

are more or less linear with "the values of K considerably increases as b/a in

creases.

4.4 Multiple Loads for Maximum Torsional Moment

Since torsional mom~nts are significant, this section will focus on

maximum torsions of various ribbed slabs due to the loadings arranged as shown

in Fig. 4~1. Load patterns that maximize torsional moment in two~span ribbed

slabs depends largely upon the absolute rib spacing, and slightly upon Hand

b/a. Therefore, for simplification, load patterns on two-span ribbed slabs

with the same magnitude of rib spacing were taken to be the same,(i.e., on" those

with bja = 0.07, and 0.10). Besides torsional moment, the corre~ponding

values of shear and bending moment (if ahy) will ~lso be presented, since the

combined torsion, shear, and bending moment in the rib is the important ph~nom

ena, as mentioned previously.

61


a. Torsional Moment at the Support

The transverse torsional moment distributions at the support of

various simple span ribbed slabs are shown in Fig. 4.16. Note that the ab

solute values of torsions are used in Fig. 4.16. The effects of Hand b/a

on the distributions are similar to those in the case of the moment distri

butions (i.e., the transverse torsion distribution is better for small H

and b/a values). The analysis of the ribbed slabs showed that the parameter

b/a has a greater effect on the torsion distribution than H.

For loading with the perimeter loads on the edge rib, the maximum _

torsion is always found in the edge rib, as was the maximum moment. For the

other loading with the perimeter loads on the second rib, the maximum torsio~

is always in an interior rib. -The magnitude of the maximum torsion in th~

latter loading case is significantly smaller than that of the former loadings.

In comparison with the maximum moments at midspan of one-span ribbed slabs in

Section 4.3.1 (a), the maximum torsional moments are about 9 to 11 percent of

the maximum bending moments for the former loading case, and about 6 to 10

percent for the latter loading case.

b. Shear at the Support

The important shear at the- support due to loadings in this section

is not the maximum shear, but rather the value of shear corresponding to the

maximum torsion in the rib. However, for better understanding, the transverse

shear distribution to various ribs at the support are shown in Fig. 4.16. For

62

the loading with perimeter loads on the edge rib, both maximum shear and tot

sion occur in the edge rib. Their magnitudes are not much different from the

shear and torsion for the maximum moment loading case considered in Section

4.3.1. For the loading case with the perimeter loads on the second rib,

shear is somewhat smaller but torsion is much greater than that due to loading

for maximum moment.

For example, the maximum torsion and the torresponding shear at the

support of two one-span ribbed slabs with H = 2.0, b/a = 0.035, and H = 9.0,

b/a = 0.10, are shown as follows:

H = 2.0, b/a = 0.035


Maximum torsional moment at the support (in edge rib)

Corresponding shear at the support (in edge rib)


= -0.0331MB = O.3778VB

Maximum torsional moment at the support (in rib no. 8) = -0.0208MB Corresponding shear at the support (in rib no. 8)"

H = 9.0, b/a = 0.10

Perimeter loads qn the edge rib:

Maximum torsional moment at the support (in edge rib)

Corresponding shear at the support (in edge rib)


O.1404VB

= -0.0600MB = O.5694V

B

Maximum torsional moment at the support (in rib no. 6) = -0.0369MB

Corresponding shear at the support (in rib no. 6) = 0.2290VB"

63


a. Torsional Moment at the Exterior Support and at the 0.95 Span Point

Torsion at the exterior support and at the 0.95 span section'of

variou~ two-span ribbed slabs due to both cases of loading for maximum tor

sion are shown in Figs. 4.18 and 4.19, respectively. The magnitudes of maxi-

mum torsions at 0.95 span are somewhat smaller than thbse at the exterior

supprirt. The general characteristics of the transverse torsional moment

distribution among ribs at the exterior support and at 0.95 span are similar

to those in the simple span ribbed slabs, but the magnitudes of maximum tor

sions in the two-span slabs are slightly smaller than those in the one-span

case. The effects of the loading conditions, Hand bfa, on the characteristics

of the transverse torsion distributions are similar to those in the simple'

span ribbed slabs, as discussed in Section 4.4.l(a).

'b~ Shear at the Exterior Support and the 0.95 Span Point, and Bending Moment at the Interior Support

Shear at the exterior support due to multiple loads for maximum

torsional moment is shown in Fig. 4.20; shear and negative moment at the in-

terior support are shown in Figs. 4.21 and 4.22, respectively. The transverse

negative moment distributions in various ribs are fairly'uniform, whereas

shear and torsion distributions are not because of the loadings. The magni

tudes of the negative moments are about the same as those due to the maximum

positive moment loadings discussed in Section 4.3.1 (b).

The general characteristics of the transverse distributions of shear,

both at the exterior support and at 0.95 span, are similar to those in the

64

simple span ribbed slabs, as discussed above. The magnitudes of maximum

shear due to loadings in this section are not much smaller than the maximum

shear due to multiple loads for maximum positive moment. However, the impor

tant shear for loadings in this section is riot the maximum shear, but rather

the shear corresponding to the maximum torsional moment. Unfortunately for.

the structure, the maximum shear, both at the exterior support and at 0.95

span, is the shear accompanying the maximum torsional moment in the case where

the perimeter loads are on the edge rib. Hence,. for this loading case, the

combined torsional moment, shear, and bending moment at (or near) the interior

support could be very important. For the loading with the perimeter loads

on the second rib, the shear corresponding to the maximum torsion is consider

ably smaller than. the maximum shear. The maximum torsion due to this loading

is much smaller than that due to the former loading. In order to minill1i:ze the

effects of the combined torsional moment, shear, and bending moment, it may

be worthwhile to consider restricting the loading so that the perimeter loads

cannot go outside of the second rib.

For ~xa~ple, maximum torsional moment, and the corresponding shear

and bending moment of two two-span ribbed slabs with H = 2.0, b/a = 0.035,

and H = 9.0, b/a = 0 . .10, are shown as follows:

H = 2.0, b/a = 0.035

Perimeter loads on the edge rib (forces in edge rib):

Maximum torsion at the exterior support

Shear at the exterior support

Maximum torsion at 0.95 span

= -0.0308MB

=

=

0.3379VB 0.0262MB

65

Shear at 0.95 span

Bending moment at the interior support

= -0.4473VB = -0.1897M

B'

Perimeter loads on the second rib (forces in rib no. 9):




Shear at 0.95 span


H = 9.0, b/a = 0.10

Perimeter loads on the edge rib (forces in edge rib):




Shear at 0.95 span


Perimeter loads on the second rib (forces in




Shear at 0.95 span


rib no. 6) :

= -0.0207MB

= 0.1134VB = 0.0200MB = -0.1789VB = -0.1056MB

= -0.0567MB

= 0'. 5254V B

= O.0473MB

= -0.6415VB = -0.3142MB

= -0.0371MB

= 0.1908VB = 0.0364MB

= -0.3324MB = -0.2420MB

4.5 Variations of Maximum Moment, Shear, and T9rsional Moment, with Hand b/a

It has been observed that good moment, shear"and torsion distribu-

tions are the charactistics nf a ribbed'slab with small Hand b/a; hence, as'

66

far as statics is concerned, decreased maximum moment, shear, and torsion

could be expected in such a ribbed slab. One might also conclude that the

maximum moment, shear, and torsion of a ribbed slab due to any multiple

loading case are direct functions of the parameters Hand b/a. The relation-

ships of the maximum moment, shear, and torsion with Hand bla are presented

in this section, and also the corresponding values of shear, torsion, and

moment (if any).


The relationships of maximum bending moment at midspan, maximum

shear and the corresponding torsion at the support due to both cases of loadings

for maximum moment with Hand bla are shown in Fig. 4.23. The maximum moments

increase at a diminishing rate as bla increases, and the effect of bla is

greater for the larger values of H. The maximum moments increase slightly

as H increases, with the larger changes occurring when bla is larger. For

the ribbed slabs analyzed, the relationship between the maximum shear (at the

support, and for both loading cases) and bla is approximately a parabolic curve

with the maximum value somewhere between bla = 0.07 and 0.10. In the other

words, the maximum sh~ar increases as bla increas~s until bla = 0.07, and at

bla = 0.10 the maximum normalized shear is slightly smaller than at bla = 0.07.

The maximum shear also increases at a very small rate as H increases. The tor

sional moments corresponding to the maximum shear due to the loading with the

p~rimeter loads on the second rib are negligible. For the loading with the

perimeter loads on the edge rib, the torsional moments corresponding to the

maximum shear are significant, and they increase as both Hand bla increase.

Metz Reference Room Civil Engineering Department BI06 C. E. Bui~=ing University of Illinois Urbana~ Illinois 61801

67

The effect of b/a on the torsional moment is greater for greater H values, and

the effect of H is also greater for greater b/a values.

Figure 4.24 shows the relationships of the maximum torsion~l moments

and corresponding shears at the support with Hand bfa, for -both maximum tor

sional moment loading cases. For loading with the perimeter loads on the

edge rib, the characteristics of the maximum torsions and corresponding shear

are similar to those of the maximum shear and corresponding torsions due to

the same loading case. For loading with the perimeter loads on the second rib,

the general characteristics of the maximum torsions are similar to those of

the maximum torsions due to the former loading case, except that the maximum

torsions at b/a = 0.10 are slightly smaller than those at b/a = 0.07, as also

happened for the maximum shear values. The relationships of shear corresponding

to maximum torsions in the latter loading case with Hand b/a are more ambigu- .

ous. However, for a general description, it might be said that the magnitude

of this shear is about 50 to 60 percent of the shear due to maximum moment

loading, as discussed abbve.


For both cases of loadings fo~ maximum positive moment at 0.44 span,

the relationships of the maximum moment, maximum shear and the corresponding

torsion at the exterior support with Hand b/a are shown in Fig. 4.25; the

relationships of maximum shear at 0.95 span, and the corresponding negative

moment at the interior support and torsion at 0.95 span with Hand b/a are

shown in Fig. 4.26. For the two maximum negative moment loading cases at

the interior support, the relationships of the maximum moment and the corres

ponding shear and torsion at 0.95 span with Hand b/a are shown in Fig. 4.27.

68

For both maximum torsion loading cases at the supports, the relationships of

the maximum torsion and corresponding shear at the exterior support with H

and b/a are shown in Fig. 4.28; the relationships of the maximum torsion and

corresponding shear at 0.95 span and negativ~ moment at the interior support

with Hand b/a are shown in Fig. 4.29.

All these relationships of the moment, shear, and torsion with H

and b/a are similar to those for the one-span ribbed slabs. But at 0.95

span, the maximum shear decreased as b/a changed .from 0.07 to 0.10 with the

response somewhat greater than that extent at the exterior support, or in

the one-span ribbed slabs.

4.6 Effects of a Stiffer Edge Rib

The study of the effects of the stiffer edge ribs were considering

the simple span ribbed slab with H = 4.5, b/a = 0.052, T = 0.100. The width

of the edge rib was twice the regular width. The absolute flexural stiffness

of the edge rib was then double of that of the regular edge rib, but the rela-

. tive flexural stiffness H of the edge rib, composite with the slab, was 50

percent greater than that of the regular edge rib. For the torsional stiff

ness, both the absolu~e value and the relative vaiue rif T were 5.7 times. that

of the regular edge·rib. Two types of multiple loads are considered; first,

multiple loads for maximum moment, shear, and torsion as shown in Fig. 4.1

and second, a line load on the edge rib. For the second type of loading, a

simulated line load (i.e., concentrated loads at every 2.5.·ft) was used in

stead of the actual line load, since the method of analysis was prepared for

concentrated loads only.

69

4.6.1 Multiple Loads for Maximum Moment, Shear, and Torsion

Two loading cases, perimeter loads on the edge and second ribs as

shown in Fig. 4.1, are again considered. The results of the analysis are

shown in Fig. 4.30 for both cases of loadings. The results of the regular

ribbed slab are also shown in Fig. 4.30 for comparison.

For the case of loading with the perimeter loads on the edge rib,

the maximum bending moment in the stiffened edge rib was about 40 percent

more than that of the regular edge "rib. The stiffer edge rib caused some

reduction of the maximum moment in the adjacent ribs. The effect of the

stiffer edge rib on the maximum shear is ~imilar to that on the maximum mo

ment (i.e., the maximum shear in the stiffer edge rib was about 23 percent

more than that in the regular edge rib). The effect of the stiffer edge rib

on the maximum torsion is greater than on the maximum moment and shear (i.e.,

the maximum torsion in the stiffer edge rib was about 100 percent more than

that"in the regular edge rib) because the torsional stiffness T had increased

5.7 times, whereas the flexural stiffness H increased 1.5 times, as mentioned

above.

For the case of loading with the perimeter lqads on the second rib,

the maximum moment also occurred in the edge rib; and the maximum moment in

the ~econd and near adjacent ribs were reduced when the edge rib was stiffened.

The effect of the stiffer edge rib on the maximum shear is similar to that on

the maximum moment (i.e., the maximum shear in the stiffer edge rib increased

and maximum shear in the adjacent ribs decreased). The maximum torsion for

this loading is not affected by the stiffer edge rib, sjnce it occurs in an

interior rib. For the stiffer edge rib, torsion changed sign which means that

70

the rotation in the stiffer edge rib occurred in the opposite direction to

that of the regular edge rib for this loading (i.e., a ball placed over the

first rib tends to roll toward the edge of the structure for the case of the

regular rib and toward the interior with the stiffened rib).

4.6.2 Line Load on the Edge Rib and Across the Span

The study of line load on the edge rib in this section is intended

to show the effects of possible line loads (such as a wall or dead load of

the widened edge rib) on moment, shear, and torsion in the ribs. The results

for both the stiffened edge rib and the reg~laredge rib are shown in Fig.

4.31. The line load produced significant values of moment and shear (i.e.,

greater than 10 percent of the maximum value) to rib no. 5, but the effect

on torsion is significant to rib no. 8 ..

The effects of the stiffer edge rib are similar to those due to the

loadings described in the previous section. The maximum moment, shear, and ~

torsion in the edge rib were about 30, 20, and 120 percent, respectively, more

than those of the regular edge rib. The reductions in the maximum moment,

shear, and torsion in the second rib were about 20, 15, and 57 percent, re

spectively, of the maximum values in the regular. edge rib. For the same rea

son as described in the previous section, the change of torsional moment is

significantly greater than of bending moment or shear. The effects of the

stiffer edge rib exist in only a few ribs near the loaded edge rib (i.e.,

to rib no 4 for bending and shear for the particular ribbed slab analyzed).

For the case of a line load across the span (parallel to the sup

ports), an approximation was made by applying single concentrated loads on

71

e~ery rib (i.e., single loads at mldspan on all ribs). This loading shows the

effects of the regular edge rib and of the stiffer edge rib, as illustrated

in Fig. 4.32. For the regular edge rib, the moment, shear, and torsion in an

outer rib are slightly greater than in an inner rib due to the e"dge effects.

In fact, moment and shear due to this loading should.be.about the same as the

static values in a simple beam, but they are slightly smaller because of the

computational inaccuracy, as discus~ed in Section 2.6. The effect of a

stiffer edge rib is somewhat similar to the effect of a line load on the edge

rib. Moment and" shear are greater in the stiffer edge rib and smaller in the

adjacent ribs, as compared to the regular edge rib case. Torsion increases in

all ribs, with a significant value in the stiffer edge rib.

72 '

CHAPTER 5

DISCUSSION AND RECOMMENDATIONS

5.1 General

The results of the investigatio~ (bending moment, shear, and tor

sional moment) were presented relative to the static moment and shear in

simple beams. Since static torsion does not e~ist in the ~imple beam" tor

sional moment was reported relative to the simple beam bending moment.

The results are presented as functions of Hand b/a. However, the

different absolute values of b and a also have some influence on the results.

For the case of single loads, this effect is ~egligible; for the case of

multiple loads, small effects exist because of the different number of loads

and loading locations relative to the rib (e.g., compare the maximum mome~ts

shown in Fig. 3.1 'and 4.2 with the graphs of maximum moment in ~Fig. 3.23

and 4.23 respectively). HoweVer, the riq spacing b for this type of struc

ture is in a very narrow range of about 2 to 3 feet, and for this reason the

effect of the different absolute values of b and a for a particular Hand

b/a would be limited.

5.2 ,Di'scussion ~nd Comparison

The discussion of the results of the investigation in this section

wi 11 ,concern the resul ts produced by mUl'ti p l'e loads as descri bed' in Chapter

4. Since these multi~le loads were considered to be wheel loads of vehicles,

in parking garages~ the discussion will compare the results obtained from the

analysis with the current design provisions for such structures; for example,

73

the equivalent uniformly distributed load for the design of an open parking

structure is specified to be 50 psf by Ref. 4.

For the purpose of comparison, the equivalent uniformly distributed

loads for the results of this analysis will be evaluated. By considering the

rib as an equivalent beam, then the equivalent uniformly distributed loads

can be determined by'the formulae as follows:

For One-Span Ribbed Slab:

Equivalent load based on the maximum moment:

w . = e

8M max -2 ab

Equivalent load based on the maximum shea~:

2V, max ab

For Two-Span Ribbed Slab:

Equivalent load based on the maximum positive moment:

. w = e

Equivalent load based on the maximum shear at the exterior support:

l6V max 7ab

Equivalent load based on the maximum negative moment:

.+.

74

Equivalent load based on the maximum shear at 0.95 span:

w = e

40V max 23ab

where we is the equivalent uniformly distributed load per unit area.

- The magnitudes of the concentrated wheel loads P are taken as 1000

lb for the calculation of the equivalent loads w in this section. _ . . e

For both one- and two-span ribbed slabs, the case of loading with

the perimeter loads on the edge rib resulted in significantly greater maxi-

mum moment, shear, and torsion in the edge rib than in the interior ribs,

especially at the interior support of a two-span structure. However, the

maximum forces in interior ribs due to this loading are nearly the same as

the maximum values produced in interior ribs when the perimeter loads are

on the second rib. Therefore, it might be. said that the results from the

investigation for the latter loading case could be considered as the reference

for all the interior ribs; and the results due to the former loading case

could-be considered as the reference for the edge rib when loading is possi-

ble to load it directly.

a. Perimeter Lo~ds on the Second Rib

The following discussion is concerned mainly with the results pro

duced by the loading with_the perimeter loads on the second rib, for both-

the one- and two-span structures described in Chapter 4. The equivalent loads

we for these results calculated by the above formulae are approximately in the

range of 30 to 40 psf, except that the equivalent- loads based on the maximum

75

shear at 0.95 span for the two-span structure are somewhat larger, at 40

to 50 psf. The equivalent loads calculated are listed in Tables 5.1 and 5.2.

These equivalent loads suggest that the current specified design load of 50

psf for an open parking structure is adequate as long as significant area

load reductions, as permited by Ref. 4, are not allowed. However, none of

the building codes has mentioned any possibility of the effects of torsional

moment in combination with shear, or shear and bending moment. The results

of this investigation indicates that significant torsion exists simultaneously

with shear and bending moment.

For one-span ribbed slabs, maximum torsion at the support (as shown

in Table 5.3) is about 6 to 10 percent of the maximum bending moment. (Maxi

mum torsion and maximum moment are caused by different loading patterns, as

shown in Fig. 4.1.) The shear corresponding to the maximum torsion is gener

ally not the maximum shear but is approximately 40 to 60 percent of the maxi

mum shear, as mentioned in Chapter 4 (see Table 5.4).

For two-span ribbed slabs~ maximum torsion at the exterior support

is about 6 to 10 percent of the maximum negative moment at the interior support

(see Table 5.5). The shear corresponding to this maximum torsion is usually

in the range of 40 to 50 percent of the maximum shear at the exterior support

(see Table 5.6). Maximum torsion near the interior support (i.e., at 0.95

span) is about 6 to 9 percent of the maximum negative moment at the interior

support (see Table 5.7). The shear corresponding to this maximum torsion is

about 40 to 60 percent of the maximum shear at the 0.95 span location (see

Table 5.8); and the corresponding negative moment is about 40 to 55 percent

of the maximum negative moment. at the interior support (see Table 5.9).

76

Note that torsion accompanying the maximum shears and bending moments

are generally negligible.

b. Perimeter Loads on the Edge Rib

The equivalent load~ based on maximum moment and shear values cal

culated by the above formulae, for both one- and two-span, are mostly in the

range of 40 to 50 psf. However, the equivalent loads based on the maximum

shear at the 0.95 span location of the two-span structure is greater and

varies from about 55 to 80 psf. These equivalent loads are somewhat greater

than the often specified design load of 50 psf. According to these results,

it might be said that if loading directly on the edge rib is possible, a

stronger edge rib or edge beam is needed. Furthermore, considerable torsion

occurred simultaneously with maximum shear at the exterior support (i.e., about

7 to 11 percent of the maximum positive moment, see Table 5~10); and some tor

sion occurred with maximum shear and bending moment at or near the interior

support (i.e., about 3 to 5 percent of the maximum negaiive moment, s~e Table

5.ll)~

The other important combination of forces at the support is due to

loading for maximum torsion. For one-span ribbed slabs, the maximum torsion

is about 9 to 11 percent of the maximum bending moment (see Table 5.3). The

shear corresponding to this maximum torsion is about-the same as the maximum

shear (i.e., about 95 to 100 percent, see Table 5.4). For two-span structures,

maximum torsion at the exterior support is about 7 to 9 percent of the maximum

negative moment a~ the interior support, and shear corresponding to this maxi

mum torsion is about the same as the maximum shear at the exterior support

77

(i.e., about 95 to 100 percent). Maximum torsion at the 0.95 span location

is about 6 to 7 percent of the maximum negative moment at the interior sup

port (see Table 5.7); shear corresponding to this maximum torsion is about

70 to 75 percent of the maximum shear at 0.95 span, and the corresponding

negative moment is about 50 percent of the maximum negative moment (see

Tables 5.8 and 5.9). Note that the above combinations of the results are

also shown graphically in Chapter 4 for both cases of loading. According

to the results discussed above, if loading on the edge rib is possible, the

effects of the above combinati6ns of shear and torsion, or shear, torsion

and bending moment ought to be taken into consideration.

5.3 General Remarks

On the basis of the results of the investigation as discussed in

the previous sections, most of the problems which have occurred in in-service

parking garage ribbed slabs (i.e., Ref. 1) probably did not result from use

of an inadequate design live load (if 50 psf or more was used). The trouble

may have resulted from various other causes which have not been included in

this study, for example:

1. Restrained shrinkage effects in concrete (i.e., tensile stress

reduces shear strength of reinfdrced concrete).

2. Repeated load effects on the fatigue strength of the concrete,

especially in the plate elements, in the lower stories of a

ramp parking structure.

3. Temperature change effects, especially when the effects of

temperature and shrinkage are additive.

78

5.4 Recommendations

The following recommendation for the design of ribbed slabs sub

jected to concentrated loads (as in parking garages) are based on the results

of this investigation:

1. When possible, loading on the edge rib should be avoided so as

to eliminate the large-bending moment, shear, and torsional

moment in that edge rib.

2. If the edge rib is to be loaded, a stiffened edge beam should

be provided (i.e., double the width and reinforcement).

3. In so far as the calculated moments and shears are concerned,

the equivalent design load of 50 psf is

ribs, unless a very heavy line load is applied to the edge rib.

4. The increase in the allowable shear stress for ribs allowed by

the ACI Building Code (2) should not be permitted, unless the

effects of potential torsional moments are considered.

79

CHAPTER 6

sur~MARY AND CONCLUS IONS

6.1 General Outline of the Investigation

The analytical study presented in this report is concerned with the

bending moments; shears, and torsional moments in the ribs of one-span and

two-span continuous ribbed slabs subjected to concentrated loads.

All ribbed slabs analyzed consi~ted of seventeen idehtical ribs

spaced uniformly with the cross section of the rib constant along the span.

The range of the parameters considered is 0.035 to·D.10 forb/a, and 2.0 to

9.0 forH~ and a nearly constant T of·D.100. The supports were considered

to be nondeflecting and to provide complete restraint against torsional ro

tation of elements about their longitudinal axes, but to provide zero flexural

restraint.

Solutions were obtained by a numerical proce~ure based on the

Goldberg-Leve folded plate theory (13). The studies considered two cases:

a .. Solutions involving single concentrated loads, mainly with

loadings at midspan on the tenter ~nd edge ribs, and'

biMultiple loadings as in parking garages.

The loading systems were set, as shown in Fig. 4~1, to obtain maximum bending

moment, shear, and torsional moment in the ribs. In addition,the effects of·

various loads (including a line load} on the stiffened edge rib were also

examined. The presentations were made using dimensionl~ss quantities (i.e.,

M/MB' V/VB, and Mt/MB) for most of the results.

80

6.2 General Conclusions

The following conclusions for this inve~tigation are believed to be

applicable to all simply supported one-span and two-span continuous ribbed

slabs which have at least 17 ribs and values of parameters H, bfa, and T in

the range considered in this study.

For Single Loads

1. The, moment', shear, and torsion distributions are better for

small Hand b/a valuesi

2. The maximum moment, shear, and torsion increase as Hand b/a

increase'" but in the range b/a :::: 0.07 to 0.10 the maximum shear

decreases slightly, and b/a has somewhat greater effett than H.

,3. The' maximum moments at midspan in th~ center and edge ribs 'of

one-span ribbed slabs are about 0.15MB to 0.38MB, and 0~30MBto

0.60 MS' respectively.

4. The maximum shear occurs near the applied~load, and the magnitude

can be as great as a simple beam shear (i.e., O.~P).

5. A stiffer edge rib reduces the maximum moments in the ribs near

the edge rib to about the same as in the other interior ribs,

but increases the maximum moment in the edge rib.

6. The load locations for maximum torsional moment at the support

of one-span ribbed slabs are about 0.15 to 0.35 of the span

length from the support for the edge rib lo~ding, and 0.1 to

0.3 of the span length for the center rib, loading~

81

For Multiple Loads:

1. The maximum moment at midspan of the simple span ribbed slabs

is about a.32MB to a.57MB for the case of the perimeter loads

on the edge rib, and O.26MB to O.38MB when the perimeter loads

are on the second rib.

2. The ~aximum shear at the support of the simple span ribbed slabs

is between a.40VB and O.59VB for the case of the perimeter loads

on the edge rib, and between a.30VB and 0.45VB for the case of

the perimeter loads on the second rib.

3. The maximum torsional moment at the support of the simple span

. ribbed sl~bs varies from about 0.033MB to 0.060MB for. the case

of the perimeter loads on the edge rib, and 0.021MB to O.037MB for the case of the p~rimeter loads on the second rib.

4. The maximum positive moment at the 0.44 span location of two-span

ribbed slabs is from 0.26MB to 0~46MB for the case of the perim

eter loads on the edge rib~ and from a.20MB to 0.33MB for the

case of perimeter loads on the second rib.

5. The maximum negative moment at the interior support varies from

0.38MB to a.63MB for the case of the perimeter loads on the edge

rib, and from a.25MB to a.45MB for the case of the perimeter

loads on the second rib.

6. The'maximum·shear at the exterior support is about 0.36VB to

0.59VS for the case of the perimeter loads on the edge rib, and

from"a.27VS to O.38VS for the case of the perimeter loads on the

second rib.

82

7. The maximum shear at the 0.95 span location of the two-span

structure is about 0.64VB to 1.00VB for the case of the perim

eter loads on the edge rib, and fromO.40VB to 0.70VB for the

case of the perimeter loads on th~ second rib.

8. The maximum torsion at the exterior support is about 0.031MB to 0~057MBfor the case of the perimeter loads on the edge rib,

and from 0.021MB to 0.037MB for the case of the perimeter loads

on the second rib.

9. The maximum torsion at the 0.95 span location varies between

0.026MB and 0.047MB, and from 0.020MS to 0.036MB for the case

of the perimeter loads on the edge and second ribs, respectively

10. The maximum moment, shear, and torsion increase as Hand b/a

increase, but in the range of b/a = 0.07 to 0.10 the maximum

shear decreases somewhat.

11. The influence of b/a on the maximum moment and shear is some

what greater than that of H; they have about the same influence

on the maximum torsion.

12. The maximum deflection coefficient K increases more or less

linearly with b/a; the coefficient K is nearly independent of H.

13. The effects of a line load on the edge rib are significantly

transmitted to only a few adjacent ribs.

14. A stiffer edge rib (double the width) carries more internal

forces, and reduces those in a few adjacent ribs, as compared

to t~e regular edge rib.

15. When the perimeter loads are limited to the second rib, the.

83

equivalent uniformly distributed loads calculated based on the

maximum moment and shear are between 30 and 50 psf for both

one- and two-span ribbed slabs.

16. If the perimeter loads are on the edge rib, the equivalent loads

based on the maximum moment and shear in the edge rib are about

40 to 55 psf for one-span ribbed slabs, and 40 to 80 psf for

two span. The equivalent loads for interior ribs would be in

the same range as those due to the perimeter loads on the second

rib.

84

LIST OF REFERENCES

1. Engineering News Record, April 23, 1970, p. 15.

2. "Building Code Requirements for Reinforced Concrete (ACI 318-71)," American Concrete Institute (ACI),1971.

3. Reese, R. C., "Floor Systems by Ultimate Strength Design," Concrete Reinforcing Steel Institute (CRSI), Chicago, 1968.

4. "BOCA Basic Building Code,1l Building Officials and Code Administrators, Inc., 1970.

5. Hondros, G. and J. G. Marsh, "Load Distribution in Composite GirderSlab Systems,1I Journal of the Structural Division, Vol. 86, No. STll, November 1960, pp. 79-109.

6. Newmark, N. M. and C. P. Siess, "Moments in I-Beam Bridges,1I University of Illinois, Engineering Experiment Station Bulletin Series No. 336, 1942.

7. Newmark, N. M., IIA Distribution Procedure for the Analysis of Slabs Continuous Over Flexible Beams,1I University of Illinois, Engineering Experiment Station Bulletin Series No. 304, 1938.

8. IIStandard Specifications for Highway Bridges," American Association of State Highway Officials (AASHO), 1969.

9. Chen,T. Y., C. P. Siess, and N., M. Newmark, "Moments in Simply Supported Skew I-Beam Bridges,1I University of Illinois, Engineering Experiment Station Bulletin Series No. 439, 1957.

10. Gustafson, W. C. and R. N. Wright, IlAnalysis of Skewed Composite Girder Bridges,1l Journal of the Structural Division, Vol. 94, No. ST4,ApriJ 1968, pp. 919-41.

Ii. Sithichaikasem, S. and W. L. Gamble, iiEffects of Diiaphragms in Bridges with Prestressed toncrete I-Section Girders," Civil Engineering Studies, Structural Research Series No. 383, Department of Civil Engineering, University of Illinois, Urbana, February 1972.

12. VanHorn, D. A.' and D. Mortajemi, IITheoretical Analysis of Load Distribution in Prestressed Concrete Box-Beam Bridges,1I Lehigh University, Fritz Engineering Laboratory, Report No. 315.9, 1969.

13. Goldberg, J. E. and H. L. Leve, IITheory of Pri sma ti c Fo 1 ded P1 a te Structures, II IABSE, Zurich Switzerland, No. 87, 1957, pp. 59-86.

85

14. Wong, A.Y.C. and W. L. Gamble, IIEffects of Diaphragms on Continuous Slab and Girder Highw~y Bridges,1J Civil Engineering Studies, Structural Research Series No. 391, Department of Civil Engineering, University of Illinois, Urbana.

15. Savern, R. T., liThe Deforma ti on of a Rectangul ar Sl ab Sti ffened by Beams Under Transverse Loads,1J Magazine of Concrete Research, Vol. 14, No. 41, July 1962, pp. 73-7.

16. Burrage, R. H. and E. G. Morgen, IIParking," The Eno Foundation for Highway Traffic Control, Saugatuck, Connecticut, 1957.

17. Barker, G. and B. Funaro, IJParking," Reinhold Publishing Corporation, New York, 1958.

b/a ·H

0.035 2.0

4.0

6.0

9.0

0.05 2.0

4.0 6.0 9.0

0.052 4.5

0.07 2.0

4.0 6.0

9. O·

0.10 2.0

4.0

6.0

9.0

86 .

Table 3.1

DIMENSIONS AND PARAMETERS OF THE VARIOUS RIBBED SLABS STUDIED

T t br h

( in. ) (i n ~ ) (i n. )

0.102 4.'0 7.0 .19.8

0.100 3.5 7.4 21.8

0.098 3.0 7.2 21 .7

0.097 2.5 6.9 21 . 1

0.100 4.0 6.8 18.5

0.102 3.0 . 6.6 17.8

0.104 : 2.5 6.4 17.2 0.104 2.5 7.0 19.4

0.100 . 3.0 6.3 17.7

0.098 4.0 6.6 17.3

0.103 3.5 7.2 18.8

0.100 3.0 7.0 18.7

0.102 2.5 6.8 18. 1

0.099 4.0 6.1 15.7

0 .. 102 3.5 6.6 17.7

0.098 3.0 6.4 16.9

0.099 2.5 6.2 16.3

b a (i n. ) (ft)

24.0 57.0

24~0 57.0

24.0 57.0

24.0 57.0 .

30.0 50.0

30.0 50.0

30.0 50.0 30.0 50.0

25.0 40.0

36.0 42'.8 36.0 42.8 36.0 42.8.

36.0 42.8

36.0 30.0 36.0 30.0

36.0 30.0

36.0 30.0

87

Table 3.2

PARAMETERS AND DIMENSIONS OF THE RIBBED SLABS FOR THE STUDY OF THE EFFECTS OF TORSIONAL STIFFNESS

ON MOMENT, SHEAR, AND TORSION

b/a H T t br h b a

(i n. ) ( in. ) (i n. ) (i n. ) (ft)

0.05 2.0 0.075 4.0 6.0 18.8 30.0 50.0

0.100 4.0 6.8 18.5 30.0 50.0

0.146 4.0 8.0 17.8 30.0 50.0

0.05 4.0 0.081 3.0 6.0 18.2 30.0 50.0

0.102 3.0 6.6 17.8 30.0 50.0

0.126 3.5 8.0 19.8 30.0 50.0

o. 143 3.0 7.6 17.2 30.0 50.0

0.161 3.0 8.0 17.0 30.0 50.0

0.10 4.0 0.063 3.5 5.4 17.8 36.0 30.0

0.102 3.5 6.6 17.7 36.0 30.0

o. 117 3.5 7.0 16.8 36.0 30.0

88

Table 4.1

MAXIMUM SIMPLE BEAM MOMENT AND SHEAR VS. SPAN

Span Length MB VB (ft) (K-ft) ( K)

40.0 49.5 5.0

57.0 101 .0 7.0

50.0 77.5 6.0

42.8 56.5 5.0

30.0 28.0 4.0

89

Table 4.2

MAXIMUM DEFLECTIONS OF ONE-SPAN RIBBED SLABS DUE TO MULTIPLE LOADS FOR MAXIMUM MOMENT

<5 (in.) Loading H b/a = 0.035 b/a= 0.05 b/a == 0.07

Perimeter Loads 2.0 0.7343 0.6152 0.4570

on the Edge 4.0 0.5557 0.7341 0.3474 Rib 6.0 0.5926 0.8549 0.3707

9.0 0.6947 ·0.5792 0.4298

Perimeter Loads 2.0 0.5856 0.4712 0.3540 on the Second 4.0 0.4370 0.5586 0.2675 Rib 6.0 0.4624 0.6496 0.2845

9.0 0.5391 0.4400 0.3304

Table 4.3

MAXIMUM DEFLECTIONS OF TWO-SPAN RIBBED SLABS DUE TO MULTIPLE LOADS FOR MAXIMUM POSITIVE MOMENT

<5 (in.)

b/a=O.10

0.1686 0.1310 0.1411 0.1655

0.1235

0.0956 0.1026 0.1210

Loading H b/a = 0.035 b/a = 0.05 b/a = 0.07 b/a = 0.10

Perimeter Loads 2.0 0.5272 0.4409 0.3283 0.1226

on the Edge 4.0 0.4002 0.5283 0.2504 0.0958

Rib 9.0 0.5034 0.4191 0.3118 0.1224

Perimeter Loads 2.0 0.4124· 0.3320 0.2495 0.0878

on the Second 9.0 0.3813 0.3119 0.2352 0.0886

Rib

Loading

Perimeter Loads on the Edge Rib

Ca 1 culati on Based On

Mmax

Vmax

Perimeter M L6adson max the Second Rib V max

* .

Table 5.1

EQUIVALENT UNIFORMLY DISTRIBUTED LOADS FOR ONE-SPAN RIBBED SLABS

* we (psf)

H = 2.0 H = 9.0

b/a= 0.035 b/a = 0.05 b/a = D.07 b/a = 0.10 b/a = 0.035 b/a = 0.05 b/a = 0.07 b/a = 0.10

40 40 38 41 42 43 42 47

49 46 42 46 54 50 46 51

32 31 30 31 33" 37 33 35

37 35 35 33 40 38 32 37

The equivalent load we based on the magnitude of the concentrated wheel loads P = 1000 lb.

1..0 C>

Table 5.2

EQUIVALENT UNIFORMLY DISTRIBUTED LOADS ON TWO-SPAN RIBBED SLABS

* (psf) we Calculation Loading. Based On H == 2.0 H == 9.0

b/a == 0.035 b/a == 0.05 b/a == 0.07 b/a == 0.10 b/a == 0.035 b/a == 0.05 b/a == 0.07 b/a= 0.10

Perimeter Mm'ax 41 40 40 43 46 44 44 50 (at 0.44 Span)

Loads Vmax 44 48 45 48 56 54 49 54

. on the (at Exterior Support)

Ed,ge Rib . Mmax 47 44 42 45 54 51 48 52 (at Interior

U)

Support) .......

Vmax 68 64 60 56 81 73 68 67 (at 0.95 Span)

Perimeter Mmax 33 31 31 31 ... 33 33 33 36 (at 0.44 Span)

Loads V . max 38 37 33 34 41 40 37 39

on the (a t Exteri or Support) .

Second Mmax 32 31 31 32 34 34 35 37 (at Interior

Rib Support) Vmax 43 41 42 40 '48 47 48 47

(at 0.95 Span)

* The equivalent load we based on the magnitude of the concentrated wheel loads P = 1000 lb.

92

Table 5.3

MAXIMUM TORSIONS AT THE SUPPORT OF ONE-SPAN RIBBED SLABS

Mt (% Mmax) Loading H max

b/a = 0.035 b/a = 0.05 b/a = 0.07

, Perimeter Loads 2.0 10 9 9 on the Edge Rib 9.0 11 11 10

Perimeter Loads 2.0 8 7 7 on the Second Rib 9.0 10 10 9

Table 5.4

SHEARS ACCOMPANYING THE MAXIMUM TORSIONS AT THE SUPPORT OF ONE-SPAN RIBBED SLABS

V (% Vmax ) Loading H b/a = 0.035 bfa' = 0.05 bfa' = 0.07

'Perimeter Loads 2.0 94 97 98 on' the Edge Rib 9.0 97 97 98

Perimeter Loads 2.0 47 45 57 "on the Second

Rib 9.0 46 43 58

b/a=0.10

9

11

6

9

b/a == 0.10

100

100

53

53

93

Table 5.5

MAXIMUM TORSIONS AT THE EXTERIOR SUPPORT OF TWO-SPAN RIBBED SLABS

Loading H b/a = 0.035 b/a = 0.05 b/a = 0.07 b/a = 0.10

Perimeter Loads 2.0 8 8 7 on The· Edge Rib 9.0 9 9 9

Perimeter Loads 2.0 8 7 6 on the Second Ri-b . 9.0 10 9 8

Table 5.6

SHEARS ACCOMPANYING THE MAXIMUM TORSIONS AT THE EXTERIOR SUPPORT OF TWO-SPAN RIBBED SLABS

8

9

6

8

V (% Vmax at Exterior Support) Loading H b/a = 0.035 b/a = 0.05 b/a = 0.07 b/a =0.10

Perimeter Loads 2.0 94 96 96 98 on the Edge Ribs 9.0 96 97 97 99

Perimeter Loads 2.0 41 .39 49 48 on the Second Rib 9.0 41 37 50 50

Loading


Perimeter Loads on the Second Rib

94

Table 5.7

MAXIMUM TORSIONS AT THE 0.95 SPAN LOCATION IN TWO-SPAN RIBBED SLABS

M t max (% Neg. Mmax)

H b/a = 0.035 b/a .= 0.05 b/a = 0.07

2.0 7 7 6

9.0 7 7 7

2.0 8 6 6

9.0 9 9 8

Table 5.8

SHEAR ACCOMPANYING THE MAXIMUM TORSIONS AT THE 0.95 SPAN LOCATION IN· TWO-SPAN RIBBED SLABS

V (% Vmax at 0.95 Span) Loading H b/a = 0.035 b/a = 0.05 b/a = 0.07

Perimeter Loads 2.0 70 73 74 on the Edge' Rib 9.0 69 72 74

Perimeter Loads 2.0 45 51 54 on the Second Rib 9.0 42 48 53

b/a=O.lO

6·

7

6

8

b/a= 0.10

74

74

57

55

95

Table 5.9

NEGATIVE BENDING MOMENTS ACCOMPANYING THE MAXIMUM TORSIONS AT THE 0.95 SPAN LOCATION IN TWO-SPAN RIBBED SLABS

: ..

M (% Neg. Mmax) Loading H b/a = 0.035 b/a = 0.05 b/a = 0.07

Perimeter Loads 2.0 51 50 51 on the Edge Rib 9.0 50 50 50

Perimeter Loads 2.0 40 52 r~

0-:)

on the Second Rib 9.0 41 53 54

Table 5.10

TORSIONS ACCO~1PANYING THE MAXIMUM SHEARS AT THE EXTERIOR SUPPORT IN TWO-SPAN RIBBED SLABS

Mt · (% POSe Mmax) Loading H b/a = 0.035 b/a = 0.05 b/a = 0.07

Perimeter Loads 2.0 8 7 7 on the Edge Rib r\ r\ , , , , '1"\ ::1. V I I I I IV

Table 5.11

TORSIONS ACCOMPANYING THE r,1AXIMUM SHEARS AT THE 0.95 SPAN LOCATION IN TWO-SPAN RIBBED SLABS

Mt (% Neg. Mmax)

Loading H b/a = 0.035 b/a = 0.05 b/a = 0.07

Perimeter Loads 2.0 3 3 3 on the Edge Rib 9.0 4 4 · 4

b/a = 0.10

50

49

rr 00

54

b/a = 0.10

8

, , I I

b/a=O.lO

3

5

96

A Center Lines of the Ribs

A.J I .. a

(a) One-Span Ribbed Slab

A.J J ...

a

(b) Two-Span Ribbed Slab

b

Simply Supported

Section A-A

B..J .. I ..

u

u

(/')

..0

Simply Supported

a

Section B-B

.1

u ......... u .c CE;

(/')

.c 'r->-

,......

o u \lr Equivalent br .5-'

Tapered Cross Section FIG. 1.1 TYPICAL RIBBED SLABS AND CROSS SECTIONS

Beam Element

Z

97

Slab Element

Forces at the Junction of Beam and Slab Elements

aM M + -zdx z ax aM

M + ~ dx x ax aM

II-+-~ My + at- dx

F Z

p

Z

aF v _ F + -" dx x ax

aF F + ~dx y oX

aF F + _z dx

Z ax

Internal Moments and Joint Forces Internal Forces and the Applied Load

Small Element of Beam

FIG. 2 .. 1 INTERNAL FORCES AT THE JUNCTION OF BEAM AND SLAB ELEMENTS, AND,ON A SMALL ELEMENT OF BEAM

98

0.50----~----~----~----~--~

Moment Diagrams Moment Diagrams 0.40

Ri b No.:

0.20

0.10

o~~~----~----~----~--~

1.00~--~----~----~---.----~

0.80 Shear Diagrams Shear Diagrams

>.r:tJ 0.60 Ri b No.: ..........

::>

0.40

0.20/--_____ _

O~ __ ~ ____ ~ ____ ~ __ ~ __ ~_

0.40, I I T .... ""'c:';n.., n;~I"1""~rn ... I VI ..::I I VII Lli U~:p UIII":;'

0. 30r-_

0.10

. Rib No.:

o~--~----~----~--~-----

Rib No.:

6

I I Torsion Diagrams

Ri b No.:

o o. 1 0 O. 20 O. 30 0 . 40 0 .:s a . 0 O. 1 0 O. 20 0 . 30 0 . 40 O. 50

Distance from Support/a Load at midspan on the edge rib

Distance from Support/a. Load at midspan on the center rib

FIG. 3.1 MOMENT, SHEAR, AND TORSION DIAGRAMS FOR A ONE-SPAN RIBBED SLAB DUE TO A SINGLE LOAD AT MIDSPAN ON THE EDGE AND CENTER RIBS

99 0.40

0.30 Moment Diagrams

0.20

0.10 co

::a: -.. ::a: 0

-0.10

-0.20

1 .00 ~

~ 0.80

0.60 Rib No.: Shear Diagrams

0.40

0.20 2

co 0 4 > -.. >

-0.2

0.3~~--~~--~~--r-.--.--.--.--r--r~~'--.--.--r--r--r~

0.2 Ri·b No.: Torsion Diagrams

0.10 co

~ 0~----------~~------------4-------~~~--------------~ -!-l :::E

-0.1

Distance from End Support/a

(a) Load at Midspan on the Edge Rib

FIG. 3.2 MOMENT, SHEAR,AND TORSION DIAGRAMS FOR A TWO-SPAN RIBBED SLAB DUE TO A SINGLE LOAD AT ONE MIDSPAN ON THE EDGE AND CENTER RIBS

co > ......... >

Moment Di agrams,

Rib No.: Shear Diagrams

-0.6

-0.8

Torsion Diagrams

o. 1

o~--~~~--~--~~~~~~~==~~

-0.10

-0.20

-0.30 L-~~ __ ~~ __ L-~~ __ ~~ __ ~~~ __ ~~ __ ~~~ __ ~~~

o 0.20 0.60 0.80 0.80 0.60 0.40 Distance from End Support/a·

(b) 'Load at Midspan on the Center Rib

FIG. 3.2 (CONT.)

o

Rib No.9

Rib No.8

Rib No. 7

RihNo.:

101

Reaction Transmitted by Slab

Shear Diagrams

-----....;"""""IIIIIIa!I!I~fttttHfHttiHtitttHfl~~----- Fi rs t Adj acen t - Rib

FIG. 3.3 THE INTERPRETATION OF THE LOADING EFFECTS ON THE RIBS

A Portion of Shear Diagram

102

Shear

Interior Support

Vertical Reaction

(a) A Portion ofa Shear Diagram and Its Interpretation near the Interior Support

A Portion of. Torsion Diagram

Torsion

Interior Support

Span

(b) A Portion of a Torsion Diagram and Its Interpretation near the Interior Support

FIG. 3.4 THE INTERPRETATION OF REACTIONS AND COUPLES AT THE INTERIOR SUPPORT

0.50

0.40

0.30

';2:.C:C 0.20 ......... ';2:.

0.10

0

0 .. 40

0.30

';2:.C:C 0.20 ......... ::E:

0.10

0 0

Influence Lines Due to Load on the Edge Rib

Rib No.:

Influence Lines Due to Load on the Center Rib

0.20

Rib No.:

0.40 Distance from Support/a

6

0.50 0

Moment Envelopes Due to Load on the Edge Rib

Rib No.:

Moment Envelopes Due to Load on the Center Rib

Rib No.: 9

0.10 0.20 0.30 0.40

Di~tanc~ from Support/a

FIG. 3.5 INFLUENCE LINES FOR MOMENT AT MIDSPAN AND MOMENT ENVELOPES FOR A ONE-SPAN RIBBED SLAB DUE TO A LOAD MOVING ON THE EDGE AND CENTER RIBS

0.50

o w

co ::'E ......... ::'E

co ::E ......... ::'E

0.50

0.40

0.30

0.20

0.10

0

-0.10

-0.20

-0.30

0.30

0.20

0.10

0

-0.10

104

Influence Line for Moment at Midspan

Rib No.:

Influence Line for Moment at Interior Support

Influence Line for Moment at Midspan

Rib No.:

Influence Line for Moment at Interior Support

Moment

Ri b No.:

Influence Line for Moment at the Other Midspan

Load on the Edge Rib

Moment Rib No.:

Influence Line for Moment at the Other Midspan

Load on the Center Rib -0.20 L-__ -L ____ ~ __ ~~ __ ~ ______ ~ __ ~ ____ ~----~--~ ____ ~

o 0.20 0.40 0.60 0.80 1.00 0.80 0.60 0.40 0.20


FIG. 3.6 INFLUENCE LINES FOR MOMENT AT MIDSPAN, AND MOMENT ENVELOPES FOR A TWO-SPAN RIBBED SLAB DUE TO A LOAD MOVING ON THE EDGE

. AND CENTER RIBS ~

o

1 .00

0.90

0.80

0.70

0.60

:> 0.50

0.·40

0.30

0.20·

0.10

Unit Load on the Edge Rib Unit Load on the Center Rib

Rib No.: Rib No.:

0.20 0.40 0.60 Distance from Support/a Distance from Support/a

FIG. 3.7 INFLUENCE LINES FOR SHEAR AT THE SUPPORT OF A ONE-SPAN RIBBED SLAB DUE TO A LOAD MOVING ON THE EDGE AND CENTER RIBS

0.80 1 .00

o Ul

>

0.4

0.2

-0.2

-0.4

-0.6

-0.8

0.20

0

-0.20

-0.40

-0.60

-0.80

-1.00 0

106

Influence Lines for r Shear at the Exterior Support

Influence Lines for Shear at the 0.95 Span Location

Rib No.:

Influence Lines for r- Shear at the Exterior

Support

Influence Lines for Shear at the 0.95 Span Location

Unit Load on Edge Rib

Unit Load on Center Rib

0.20 0~40' 0.60 0.80 1.00 0.80 0.60 0.40 0.20 0


FIG. 3.8 INFLUENCE LINES FOR SHEAR AT THE EXTERIOR SUPPORT AND AT THE 0.95 SPAN LOCATION OF A TWO-SPAN RIBBED SLAB DUE TO A LOAD MOVING ON THE EDGE AND CENTER RIBS

107

0.05 ~--~--~----'---II---.--~r----r--~----'---I

Rib No.: Unit Load on the Edge Rib

0.04

0.03

0.01

OL---~--~ ____ ~ __ ~ __ ~ ____ ~ __ ~ __ ~ ____ ~ __ ~

0.03 ~--~--~----~--~---'----'---'----'----'---I

Unit Load on the Center Rib 0.02 Rib No.:

1.00 Distance from Support/a

FIG. 3.9 INFLUENCE LINES FOR TORSION AT THE SUPPORT OF A ONESPAN RIBBED SLAB DUE TO A LOAD MOVING ON THE EDGE AND CENTER RIBS

ca ~ '-..

+l :E:

0.05

0.04

0.03

0.02

. 0.01

0

-0.01

-0.02

-0.03

-0.04

0.03

0.02

0.01

-0.01

108

Ri b No.: Influence Lines for Torsion at the Exterior Support

Influence Lines for Torsion at the 0.95 Span Location

Rib No.: Influence Lines for Torsion at the ·Exterior Support

Unit Load on the Edge Rib

Unit Load on the Center Rib

Influence Lines for Torsion at the 0.95 Span Location

-0.02L-__ ~~ __ ~ ____ -L ____ ~ ____ ~----~--~~--~----~----~ o 0.20 0.40 0.60 1.00 0.80 0.60 0.40 0.20


FIG. 3.10 INFLUENCE LINES FOR TORSION AT THE EXTERIOR SUPPORT AND AT THE 0~95 SPAN LOCATION OF A TWO-SPAN RIBBED SLAB DUE TO A LOAD MOVING ON THE EDGE AND CENTER RIBS

o

0.70

0,60

0.50

0.4.0

co :E 0.30 ........... :E

0.20

0.10

0

0,70

0,60

0.50

0.4.0

co :E 0.30 ........... :::E'

0.20

O. to

a

0.70

0.60

0.50

D.ltO co

:::E' ........... ::E: 0.30

0.20

0.10

a

" , "-"-

1 2 3

109

Regular Edge Rib

Double-Width Edge Rib

Moment Envelope

Moment Distributions

Triple-Wi~th Edge Rib

Moment Envelope

Moment Disiributions

li 5 6

Rib Number 7 B s

FIG. 3.11 MOMENT ENVELOPES AND MOMENT DISTRIBUTIONS AT MIDSPAN OF VARIOUS RIBS OF A ONE-SPAN RIBBED SLAB WITH THE REGULAR AND STIFFENED EDGE RIBS DUE TO A LoAD MOVING'ALONG MIDSPAN

110

FIG. 3.12 POSITIVE AND NEGATIVE MOMENT ENVELOPES AND MOMENT DISTRIBUTIONS AT MIDSPAN AND AT THE INTERIOR SUPPORT OF VARIOUS RIBS OF A TWO-SPAN RIBBED SLAB DUE TO A LOAD MOVING ALONG MIDSPAN

co :E: ..........

>< rt! E

::E

co >-..........

>< rt! E

>-

co ::E ..........

>< rt! E

+-> ::E

111 .

0.60 H ~ 4 0 b ..... ---..:.-., Ia..:::. .10 --.......... _-...

0.50 .

0.40

0.30

G-____ !~ 4.0, b/a ::: .05 .... -- ~ = 2 -0 b ..... 7 -----· - - ...... - - --:. ~ -!::!.I.,S... = ...D.5-- - --e

• H :::; 4, bl a, ::: .~ 10 .' - ~ . 8 @

0.20 II H = 4. 0 ,. ~/ a, = . 0i H ::: 2.0, b/a, = .05

0.10

o b-__ ~ __ ~ __ ~ __ ~~~ __ ~~~~~~~~ __ ~

0.50 .... -l!.:.~. 0, b/a = .10

0.40 ----..... __ ...

0.30

0.20

0.10

0,

0.05

0.04

0.03

0.02

0.01

__ H = 4.0; b/a::: 05 e- __ --:..-=---&--. -_.........:.._....:._. : .. ----.=------ ~--- ........

•• uHr _::---A4-. riO ,~·"'Lb /'-a:--=-."""'=;l·~-·-.-· . ....... iT ;-~ 0-:-~ = . 05. ' ... 1 .... • H= 4~0, b/a =.05

@Ii " II -~ ~

-H = 2.0, b/a =.05 .

T

Load at Midspan on the Edge Rib Load at Midspan on the Center Rib

FIG. 3.13 EFFECTS OF TORSIONAL STIFFNESS ON MOMENT, SHEAR AND TORSION

co L -.... ::2:

co ~ -.... L

0.70

o.6ol bja = 0.035 bja = 0.05 Load on the Load on the

0.50 L Edge Rib H = 9.0

Edge Rib H = 9.0

o.qOWH = 'I.U Load on the H = 4.0

H = 2.0 Center Rib ;; H = 2.0

0.30 . H = 9 0 . ----, -t ~ " '\." H = 4.0

0.20 I ~ H =.2.0

0'.10

0

0.70 I

Load on the 0.60 L Load of the bja = 0.07

Edge Rib Edge Rib bja=O.lO H = 9.0

0'.50 ~ / H = 9.0 H = 4.0 H = 4.0

Load on the H = 2.0

o.qOf,\ Center Rib H = 9.0 ~

0'.30 H = 4.0 H = 2.0

0'.20

0.10

0 2 3 It 5 6 7 8 9 2 3 It 5

Rib Number Rib Number

FIG. 3.14 MOMENT DISTRIBUTIONS AT MIDSPAN OF ONE-SPAN RIBBED SLABS DUE TO SINGLE LOADS AT MIDSPAN ON THE EDGE AND CENTER RIBS

6

Load on the Center Rib

H =

I N


H = 9.0 H = 4.0

7 B 9

0.10

0.60 _ b/a = 0.035 b/a = 0.05

0.50 L Load on the Load on the Edge Rib

0.40 L Edge Rib H = 9.0

f 0.30Ur H ~ 2.0

0.20 r , 0.10

0

0.10

0.60 l Load on the

0.50 L Edge Rib H = 9.0 o.qOW H ~ 2.0

o::::l :::E .......... L 0.30

0.20

0.10

0 "2 3

H = 9.0 Load on the H = 2.0 Center Rib H = 9.0 H = 2.0 ="

b/a = 0.07 Load on the b/a = O. 10 Edge Rib

,-H = 9.0

Load on the

J KH ~ ~.U Center Rib

H : 9.0 ~ H - 2.0 --.

4 5 6 7 8 9 2 3 4 5 6

Rib Number Rib Number

FIG. 3.15 POSITIVE MOMENT DISTRIBUTIONS AT MIDSPAN OF TWO-SPAN RIBBED SLABS DUE TO SINGLE LOADS AT MIDSPAN OF THE EDGE AND CENTER RIBS


H = 9.0 H = 2.0

--I

w


H = 9.0 H = 2.0

7 8 9

0.70~1 ----~------~----~----~----~----~._----._----I

co :::E::

0.60

o.so

0.40

~ 0.30 :::E::

0.20

0.10

o


H= 9.0 H == 2.0

b/a == 0.035


H·= 9.0 H = 2.0

0.70~1 ----~------~----~----~----~----~~----~----.

0.60 ~.

0.50 ~.

0.40 co

:a:: ~ :::E:: 0.30

0.20

0.10

o . 1


H = 9.0 H = 2.0

2 3

b/a = 0.07

4 5 5

Rib Number


H = 9.0 H= 2.0

7 B 9


H = 9.0

H = 2.n


H = 9.0 H = 2.0

2 3

b/a = D.n5

b/a=0.10

4 5

Rib Number 6


H = 9.0 H = 2.0


7 B

FIG. 3.16 NEGATIVE MOMENT DISTRIBUTIONS AT THE INTERIOR SUPPORT OF TWO-SPAN RIBBED SLABS DUE TO SINGLE LOADS AT MIDSPAN ON THE EDGE AND CENTER RIBS'

..j::::.

9

O.70~, ----~----~----~~----~----~----._----_.----I

0.60

0.50

ro 0.4.0 :> ........... :>

0.30


H = 9.0 H = 4.0 H = 2.0

0.20 t- I ,

b/a = 0.035

Load on the Center Rib H = 9.0 H = 4.0 H = 2.0

o.~o~ I =======-~ O.70~, ----~--~----~--~~--_.----._----._--I

ro :>

0.60

Load 0.50 LEdge

0.4.0

........... 0.30 :>

0.20

0.10

o 2

= 9.0 4.0

= 2.0

3

b/a = 0.07


H = 9.0 H = 4.0 H = 2.0

4. 5 6 7

Rib Number B 9


H = 9.0 H = 4.0 H = 2.0

Load on the· Edge Rib

2

H = 9.0 H = 4.0 H = 2.0

'3

b/a = 0.05

Load on the Center Rib H = 9.0 H = 4.0 H = 2.0

b/a = 0.10


H = 9.0

4. 5 6 7

Rib Number

FIG. 3.17 SHEAR DISTRIBUTIONS AT THE SUPPORT OF ONE-SPAN RIBBED SLABS DUE TO SINGLE LOADS AT MIDSPAN ON THE EDGE AND CENTER RIBS

(Jl

B 9

0.70 I . I . I

0.60

o.so

ro > 0.40 ........... >

0.30

0.20

0.10

o


H = 9.0 H = 2.0

b/a = 0.035


H 9.0 H = 2.0

0.70-1------~--------__ ~----~~----~----~----~._--~

ro > ........... >

0.60

0.50

0.20

0.10

o


2

H = 9.0 H =2.0

3

b/a = 0.07

456

Rib Number


H= 9.0 H = 2.0

7 8 9


H = 9.0 H = 2.0


2

H= 9.0 H = 2.0

3

b/a =0.05


H = 9.0 H = 2.0

b/a = 0.1n


H = 9.0

5 IS 7 B

Number

FIG. 3.18 SHEAR DISTRIBUTIONS AT THE EXTERIOR SUPPORT OF THE LOADED SPAN OF TWO-SPAN RIBBED SLABS DUE TO SINGLE LOADS AT MIDSPAN oN THE EDGE AND CENTER RIBS

O'l

9

PIG. 3.19 SHEAR DISTRIBUTIONS AT THE 0.95 SPAN LOCATION OF THE LOADED SPAN OF TWO-SPAN RIBBED SLABS DUE TO SINGLE LOADS AT MIDSPAN ON THE EDGE AND CENTER RIBS

O.07~1 ____ ~-L--~----._----._~--._----._----._--~

0.06

0.05

O.Ol!


kiH =9.0 H = 4.0

.. ~iH = 2.0 co 0.03 F ' ( ( ::z:: J J

............ +->

::E: 0.02 I

b/a = 0.035

Load on the' Center Rib H = 9. 0 --'----. H = 4.0 H = 2.0

a·OI h ~

o

O.07~1 ----~--~----._--_.----II----._--~---: Load on the Edge Rib 0.06

0.05

co :E 0.0l! ............

+-> ::z:: O. 03 r:-------L-

0.02

0.01

o 2

H = 9.0 H = 4.0 H =2.0

3 I! 5

b/a = 0.07


H = 9.0 H = 4.0 H = 2.0

6 7 '8

Rib Number 9


H = 9.0 H= 4.0

H = 2.0

I I


H= 9.0

2 3 I! 5

b/a = 0.05


H = 9.0 H = 4.0 H = 2.0

b/a = n.10


H = 9.0 H = 4.0 H =

6 7 8

Rib Number

FIG. 3.20 TORSIONAL MOMENT DISTRI~UTIONS AT THE SUPPORT OF ONE-SPAN RIBBED SLABS DUE ' TO SINGLE LOADS AT MIDSPAN ON THE EDGE AND CENTER RIBS

co

9

0.07

0.06

0.05

0.04 co

:E ............

+> 0.03 :E

0.02

0.0l

0

0.07

0.06

0.05

0.04

,co :E 0.03 ............

+> :E

0.02

0.01

0 1


H = 9.0 H =2.0


2

H = 9.0 H = 2.0

3

bla = 0.035

II


H = 9.0

H = 2.0

b/a = 0.07

5

Load on the Center Rib H = 9.0 H = 2.0

6 7 8

Rib Number

9

bla = n.05

Load on the Edqe Rib

H = 9.0

H = 2.0 Load on the Center Rib H = 9.0 H = 2.0

bla = 0.10

Load on the

2

Edge Rib H = 9.0 H = 2.0

3 II 5


H = 9.0

H = 2.0

6 7

Rib Number'

8

FIG. 3.21 TORSIONAL MOMENT DISTRIBUTIONS AT THE EXTERIOR SUPPORT OF THE LOADED SPAN OF TWO-RIBBED SLABS DUE TO SINGLE LOADS AT MIDSPAN ON THE EDGE AND CENTER RIBS

I.D

9

0.07

0.06

0.'05

0.04

CO L 0.03 ..........

+-> L

0.02

0.01

0

0.07

0.06

0.05

0.04

CO 0.03 L ..........

+-> L 0.02

0.01

0

b/a = 0.035


H = 9.0

H = 2.0


H = 9.0

H = 2.0

2 :3

I·

b/a = 0.07

II 5 6

Rib Number


H = 9.0

H'= 2.0


H = 9.0 H = 2.0

7 B 9

b/a = 0.05


H = 9.0

H = 2.0


H = 9.0


H = 9.0 H = 2.0

b/a = O.ln


2 :3 II 5 6 7 B

Rib Number

FIG. 3.22 TORSIONAL MOMENT DISTRIBUTIONS AT THE 0.95 SPAN LOCATION OF THE LOADED SPAN OF TWO-SPAN RIBBED SLABS DUE TO SINGLE LOAnS AT MIDSPAN ON THE EDGE AND CENTER RIBS

9

N o

121 0.70 .

0.60

0.50

co 0.40 :,'E ...........

>< I'tS 0.30 :,'EE

0.20

0.10

o ~~~ ____ ~ ______ -L ______ ~~ ____ ~ ____ ~ ____ ~~

0.035 0.05 0.07 0.10

b/a

0.70 ----.,.....---r-----r----,.------r----..---~

0.60 b/ a==O.lO _----__e__---...e------------- b/a == 0.07 -+-. __ -- ----. -----.......----0.50

_ b~-=-Ch.02- --+- - --- - _._----.....--- 35 . b/a == 0.10

b/a == ~_ ---0.40

co :,'E

'x 0.30 _ ...... ---

b/a == 0.07 n::s E

:,'E

L:==========~------_=============Jb~/~a~=~OJ.0~3~5==~ 0.20 E

b/a = 0.05

o. 1 0,:

o ~ ____ ~ ______ ~ ____ ~ ______ ~ ____ ~ ____________ ~ 2.n 4.0 6.0 9.0

H

- - Load at Mi dspan on the Edge Ri b ---- Load at Midspan on the Center Rib

FIG. 3.23 VARIATIONS OF .MAXIMUM MOMENT WITHH AND b/a FOR ONE-SPAN RIBBED SLABS UNDER SINGLE LOADS AT MIDSPAN ON THE EDGE AND CENTER RIBS

co ;:E:

'-. x' rt:S

;:E:E

122

0.70

0.60 Positive Moment at Midspan

0.50

0.40

0.30

U ::: 9.0 _-n _---_--11'0 --._..._- U ::: "t. _-....- n ---- . ---0 _-- _---u ::: Z. ---__ __..e-- n_---

......-- --- ........ ---.-...'---------- - - --

-""'--- --- ----- --- ----- --- -----0.20

0.10

0

-0.10 H = 2.0

-0.20

-0.30

-==::::-_ H = 9.0 _ -- =:t----_ ---.:::J::'------- ........ - H = 20 ------. - ....... -----~------- ....-.~-- ----. Negative Moment at - - - -H = 9 0 - -=--=-= the Interior Support .

-0.40 __ ~~ ____ ~ ______ ~ __ ~ __ ~ ____ ~ ______ -L ____ ~

0.035 0.05 0.07 0.10

b/a

--Load at Midspan on the Edge Rib -- Load at Mi dspan on the Second Ri b

FIG. 3.24 VARIATIONS OF MAXIMUM POSITIVE AND NEGATIVE MOMENTS WITH HAND b/a FOR TWO-SPAN RIBBED SLABS UNDER SINGLE LOADS AT MIDSPAN ON THE EDGE AND CENTER RIBS

123

O. 60 ---rl----~I----rl----,...-I---,-'---Ilr----,

0.50 ~ -H=9.0~ -4 ..-- ..... ---::.....c H = 6.0 --::..-- - _....:4 --- -- .......... --- ..,.,. -- --- ::. -=-...... --- ,.".,.... --- """0

0.40 i-

,..,.. -,:::t::.: - --4(.\-\ = 4. --~ ~ -~-~--- ---- -2" O. 30 f- _ - ::: - _ - _ - - "'tu = '2.0

o -:;:::-::::-- ____ -- n - _.....ft:"'-_-_-rt:S -_::::::IF- --- u - 9 0 .....01

_E o.20t~-:-:-:-~~::-~~~:-~-~~~~~~~~~~f\~-~.~~~~~ - ..... ~:::,::..---- --~ -- ~ ~- . I

.--::: i'

0.10 . - H _ 6.01 LH ~ 2.0 -H = 4.0

co >-..........

>< I'd E

>-

o I 1 0.05

I I I

0.035 0.07 0.10 b/a

0.60 I I I I I I

0.50 f0- -

0.40

0.30

b/a = 0.10 .. . __ ---~ .....-.- ------ . , .... -------- .------- 07 . _----4 1,...--- b/a =Jl.:..----e---- ----~---- -f- . __

0.20 ~------- b/a = _O...:..O~-+--- ra.-::-rrTn- --~ -- --e------ 0 a - . ...... ---- 035 ~ t- b/ a = O. -& - bl - 0 07

0.10

-- ...... -,...::::.-- ..... a - . t;:":--":::'=- ~ h/a = 0.05 Ii-

0 2.0

-

I I I I

4.0 6.0

H


b/a =0.035 I I

9.0

FIG. 3.25 VARIATIONS OF MAXIMUM SHEAR AT THE SUPPORT WITH HAND b/a FOR ONE-SPAN RIBBED SLABS UNDER SINGLE LOADS AT MIDSPAN ON THE EDGE AND CENTER RIBS

co >-"-..

x f'1j

E >-

co >-"-.. x

f'1j

E >-

124

0.70 ~----'----~I-------~I-----"-------'l-------r-,-----,

0.60 f- Maximum Shear at the Exterior Support -

O. 50 f- -

0.40

0.30

0.20

0.10

0

0.80

0.70

0.60

0.50

0.40

0.30

0.20

0.10

\-\ :: 9.0 _---~ ---- --------__ - U :: 2.0 _--~ _-- f\ _---

_____ - _ --0-- - - --- ----- ---10-0- .....-I" __ ,....--- H - 9.0

-- H :: 2.0 -

I I j ~ I 1

Maximum Shear at the 0.95 Span Location

--U :: 9.0 _----n _----------- -__ -- \J :::: 2. 0 __ ----__ --. n_-.,....--- ------- ----------- -----. ---------,.....-

----"" -""----""

H :: 2.0

-

o 0.035 0.07 0.10

b/a

Single Load at One Midspan on the Edge Rib Single Load at One Midspan on the Center Rib

FIG. 3.26 VARIATIONS OF MAXIMUM SHEAR AT THE EXTERIOR SUPPORT AND 0.95 SPAN LOCATION WITH HAND b/a FOR TWO-SPAN RIBBED SLABS UNDER SrNGLE LOADS AT MIDSPAN ON THE EDGE AND CENTER RIBS

co :E '-- x

rcr E

+-l :E

0.06

0.05

0.Q4

0.03

0.02

125

~ : ~:~ ) H = 4.0 _------ ---- ------- ------- ...-.-- --- ---- ---- --- ----.--- --- ---- ---- ---

--...- ----- - ----...-- --....,..-- ...- - -.-...- 0 ...- ......---- ----- rH ::: s.:...---

-" ......... ...- -- --1---...-..,;".. .......... ........-- .. ----...... -- .-- -~ - ......... --- ------ _-0----- H = 9.0

::: 6.0

o ~~~ ____ ~ ______ ~ ______ ~~ ____ ~ ____ ~ __ ~ __ ~ 0.035 0.07 0.10

b/a

0.06 __ ----~------~----~------~----~------~----~

b/a::: 0.10_---..------........-----------__ ........... ---- bh/a = 0.07 ___ ~

0.05

....,...,. ~---__ ..... __ ._ ..... _ .... __ .... __ ......... ____ .'-__ n._OL .-:-----~-~.~.-.---~-----------~~-= :::!"'--=-=..--.-o-no-.- _ ............. - ... -....... -- _ ...... _- .-.--.-... -

--- -- bil::: .------.---- --~-- --co ___ --- ---e------~ x 0.03 _-...----- _---o---QL~~Q.:..035 ___ -

rcr --4iB'-" - -- . . ..

+-lE ---L 0.02

o ~ ____ -L ______ L_ ____ ~ ______ ~ ____ ~ ______ ~ ____ ~

2.0 4.0 6.0 9.0 H

---- Load at Midspan on the Edge Rib --.-- Load at Midspan on the Center Rib

FIG. 3.27 VARIATIONS OF MAXIMUM TORSION AT THE SUPPORT WITH HAND b/a FOR ONE-SPAN RIBBED SLABS UNDER SINGLE LOADS AT MID- . SPAN ON THE EDGE AND CENTER RIBS

0.06

0.05

126

Maximum Torsion at the Exterior Support H = 9.0

-e-- - - H =--zr. D.:- _ -------- .....--.....-- --- --- ----- ....--.-- -- --- -:- -

0.04 ------ -- 2 0 ___ --- H = . _------ -- ----------- ---- ........ --....... ---- ------....... ..e----------H = 9.0 H = 4.0

~~::====~====== ____ --~------~H~=~2~.~0------~ 0.01t

O~ __ ~ ____ ~ ______ ~ ______ ~ ______ ~ ______ ~ ____ ~

0.06 I I I I I I

Maximum Torsion at the 0.95 Span Location 0.05~ -

O. 04~ -H = 9.0 _---4

--- -=-- - ---_e__---------..JIll-- -0~03~ -

----......... --.,~--H = 2.2-_---~ --e-------0.02 - __ -e--- ------ --- -- -

~....--- H = 9.0 o. 01 ~ ...

.- H - 2.0 - -

o I 1 1 I I I

0.035 0.05 0.07 0.10 b/a


FIG. 3.28 VARIATIONS OF MAXIMUM TORSION AT THE EXTERIOR SUPPORT AND 0.95 SPAN LOCATION WITH HAND b/a FOR TWO-SPAN RIBBED SLABS UNDER SINGLE LOADS AT MIDSPAN ON THE EDGE AND CENTER RIBS

co :E ..........

x n:s E

+l ::E

0.06

0.05

0.04

0.03

0.02

0.01

0 0.06

0.05

0.04

0.03

0.02

0.01

b/a = 0.035

b/a = 0.07 --.- H = 9 0 ~--......................... .

'?_-- .. " , Ii 1''' '.::: ........ " H = 6.0

/ ----'-... '~\~ H = 4.0 1// '.........'" H = 2.0

1/ " ' , II H = 9. 0................ " " , I H - 6 0 '" I - -. ,,,~,,

" ~~, I ~H = 4.c), "'. ~ I H = 2.0 ',~, I ~ -............... '''~~' I. '~

~ ~

0.2 0.4 0.6 0.8 1.0 0

b/a = 0.05 H = 9.0 .---- /i;. H::: 6.0 .1.::--- ....... "

/~.-- ....................... " ................. (f£ H = 4.0 ~/ ' ....... ', ........ , ----. , ....... % H = 2.0 - -- , // ......................::--...

/ ....... ,,' I 6 0 ,,,~~ ~f=.9.0 , ">'~ , . H =. , ,,~~

H - 4 0 ....... '::", ,

. -., """ 2 0 ......... ":-..:

I H =. '<:'\, I ~ '~ " ".

~ __ -:: ......... , b/a = 0.10 !/ -,"

/; <:::'<', (/;H = 9.0 ~-- " " '/.;' -- ......... , ,~"" H = 6.0

/1 '''''~H = 4 0 // ........,' . ,1/ H : 9.0'" 'AH = 2.0 I H - 6. 0 ",'\ ,,'\ I , ~~ J , ~~ I " ,,~ I ' ~~ , ~ I ,~

,~ ,~ ,,~

~.\.

0.2 0.4 0.6 0.8 1 .0 Distance from Support/a

-- Load on the Edge Ri b ---- Load on the Center Rib

FIG. 3.29 INFLUENCE LINES FOR TORSION AT THE SUPPORT OF VARIOUS ONE-SPAN RIBBED SLABS DUE TO A LOAD MOVING ON THE EDGE AND CENTER RIBS

--I

N -......J

128

0.07

0.06

H = 9.0 H = 6.0~ __ H = 4.0 __ -.-:-

--- ---- ---- ------- --- --- --- ~ ~ ---- ----------~ ---- ---- ------------- - ------- ------ -- ----..-.-~-------- ---~~~-- --T .,.".-::;;~:::""- -e-----H= 2.0 -::=:---:::...---"'- -----...- ----------

0.05

0.03 H = 9:0

0.02 H :: 4.0

'2§:::====::====~------+-------~HI=~2~.Oa-----~ 0.01 t;

O~~ ______ ~~ ____ ~ ______ ~ ____ ~~ ____ ~ ____ ~ 0.035 0.10

b/a

Load on the Edge Ri~ Load on the Center Rib

FIG. 3.30 VARIATIONS OF MAXIMUM TORSION AT THE SUPPORT WITH H AND b/a FOR ONE-SPAN RIBBED SLABS UNDER SINGLE LOADS ON THE EDGE AND CENTER RIBS

129

1 .5 1 51 3 I Center Line of Center Line of

11 I r· the Edge Rib the Second Rib

- -......-.--e--+- .......... a 1/ ,....

U • . " II • " . .. . II ......... • • fa • • • • • • • U

1.0 'I • • " . •• •• • l..() - • • • • • • • • • • N a

®

'I •• •• •• • • • t/')

lID lID "

«II • " " • • • -c .r-c::::

I. 40' .I I. 40' ./ f"-

/. 57 1 I. 57 1 I "'"I

Center Line of Center Line of the Edge Rib the Second Rib

- ........ ~ .......... -................... -a r-- 11

• •• • • • • •• •• • u - • • • •• • • •• • • • ......... 1.:0 U

• •• • • • • •• • • • - • • • •• • • •• •• • a a M

®

• •• • • • • • • •• CD t/')

- • CD • e • • • fa • •• e -c \.0 e " . • • • • " . • • • .r-

c:::: • • • " II •• • • II • • f"-

l. 501 m I I. 50' .j

Perimeter Loads on the Edge Rib Perimeter Loads on the Second Rib

• Concentrated Wheel Load P (a) Multiple loads for Maximum Moment on One-Span Ribbed Slabs.; for Two

Span, These are the Load Patterns on One~Span for Maximum Positive Moment, and on Both Spans the Patterns are for Maximum Negative Moment

FIG. 4.1 LOADING SYSTEMS FOR MAXIMUM BENDING MOMENTS, SHEARS, AND TORSIONAL MOMENTS

130

-Cen~EJr tine of Center line of 2.71 5' 3 1

the Edge Rib the Second Rib t 1 I r

-0 ...... \0

"" ., • • ee • e •• • LO

... • · '" -- e • @ @ • II \.0 • • • -. • • •• • u

CD • • • • • • • • • .......... - u 0 I"""""

• • • CD • • • •• • \0 ("I") • • • • e • • •• • 1..0 • • • •• • • • • CEJ • • • • • • • • •• • VI - ...c

0 .,.... ...... 0::::

• • • ../ • • •• • "" .. • • • • •• •• • ...... I I I I~

I. 50' ./ I~ 50' '~ I'~'" ,"'"

0.5' 5 1 3' Center Line of Center line of

1/- I -I the Edge 'Rib the Second Rib

-0 ...... = •• • • •• \.0 - •• f) • •• ,......

1..0 LO

•• • • •• II - • • •• •• 0 u ...... ..........

•• • • u •• -\.0 • • •• •• \.0 •• • • •• ("I")

• • •• •• ~ - jg' 0 ...... .-•• • • ee 0::::

•• •• o • ,...... ......

I. 30' ~l 1. 30 1

Perimeter Loads on the Perimeter loads on the Edge Ri b Second Rib

FIG. 4. 1 ( CONT .. )

131

Center Line of the Edge Rib

of

Center Line of Rib No. 9

the Second Ri b ~---

2 J 5 J 3' I I I I

40'


Center Line of the Second Rib '~'~'-G-e-.~ .

• ED ED •• ED • 0 •• 00 00 00 00 00

57' .i.

... CD ......

o 00 00 00 00

40' ./

Center Line of Rib No.8 Center Line of Rib No. ,11

• •• o • •• • • •• o .f!!. CD a

0 00 o 0 00 o 0 o 0 o 0 0

57' ./

2. 5' 5' 3'

Center Line of the Ed'ge Hi b

Center Line of the Second Rib

Center Line of Rib No.7 Center Line of Ri b No .. ' 8

~[ I·t '··1

I.

. -e-e-. -e--e-.-e-e-.

•• •• •• •• 00000000

50' .1.

. ...........,. ............ ~.~. 3

.~.~ ............ ~.- ~

• ., Iii ., ••••••• 'W

o 00 00 00 00 00 0

50' .1

• ,Perimeter Loads, on the, Edge Ri b o Perimeter Loads on the Second Rib

(b) Multiple Loads for Maximum Torsion on Two-Span Ribbed Slabs, and the Left Span Taken Separately Shows the Load Patterns for Maximum Torsion on One-Span Str~ctures

FIG. 4. 1 (CONT.)

2.7' 5' 3'

'" "I I' "' ~[ ~


Center Line ·of

o

the Second Ri b --+-------'

I. 42.8'

0.5' 5' 3' lie ",

" { -0 r-- --Center ::L i ne of

00

the Edge Rib

Center Line of the Second Rib

3~'

132"

. ~~ 00 0

.-e-e-.

--o 0

./.

FIG. 4. 1 ( CONT . )

Center Line of Rib No.5 Center Line of Rib No.7

a 000000000.0

42.8'

Center Line of Rib No. Center Line of Rib No.

aa s "

B 0 r--

00 o 0

3D'

5 7

co > . ......... >

co ::E .........

4-> :::E

133

0.60~--~----~----~----~--~

0.50

0.40 Rib No.:

Moment Diagrams Due to Multiple Loads for Maximum Moment

Rib No.:

o ~ __ ~ ____ ~ ____ ~ ____ ~ __ ~ 0.70~--~----~----~----~--~

0.60 Shear Diagrams Due to Multiple Loads for Maximum Shear

Rib No.:

Or06~--~----~----~----~--~

0.05

0.03

0.02

0.01

0

Rib No.:

4

Torsion Diagrams Due to Multiple Loads for Maximum Torsion

Rib No.:

1

0 0.10 0.20' 0.30 0.40 0.50 0 0.10 0.20 0.30 0.40 0.50 Distance from Support/a Distance from Support/a


FIG. 4.2 BENDING MOMENT, SHEAR, AND TORSIONAL MOMENT DIAGRAMS OFA ONESPAN RIBBED SLAB DUE TO MULTIPLE LOADS

a:J ::E ........... ~

a:J > ........... >

a:J :E: ...........

+J ::E

0.40

0.30

0.20

0.10

0

-0.10

-0.20

-0.30

0.60

0.40

0.20

0

-0.20

-0.40

-0.60

0.03

0.02

0.01

0

-0.01

-0.02

-0.03

-0.04

0

Rib No.:

Rib No.:

Rib No.:

134

Moment Diagrams Due to Multiple Loads for Maximum Moment

Shear Diagrams Due to Multiple Loads for Maximum Shear

Ri b No.:

Rib No.:

Torsion Diagrams Due to Multiple Loads for Maximum Torsion

9 8 7

2

0.20 0.40 0.60, 0.80 1.0 0 0.20 0.40 0.60 0.80 1.0 Distance from End Support/a Distance from End Support/a


FIG. 4.3 BENDING MOMENT, SHEAR, AND TORSIONAL MOMENT DIAGRAMS OF A TWO-SPAN RIBBED SLAB DUE TO MULTIPLE LOADS

135

0.05~--~----~----~----~----~----~--~~--~----~----~

0.04

0.03

0.02

0.01

co 0 :E ~

+.J

:E -0.01

-0.02

. -0.03

-0.04

Torsion Diagrams Due to Multiple Loads for Maximum Torsion at the Interior Support of the Center Rib (Rib No.9)

Rib No.:

Rib No.:

Load Pattern

-O~·05i O~--~----~----~----~--~l~.O----~----~----~--~~--~O


FIG. 4.3 (CONT . )

co :::E: ............. :::E:

co :::E: ............. :::E:

0.70

0.60

0.5'0

0.4.0

0.30

0.20

0.10

0

0.70

0.60

0.50

0.4.0

0.30

0.20

0.10

0

b/a = 0.035


H = 2.0 H = 4.0 H = 9.0

H = 2.0 H =-4.0 H = 9.0


b/a = 0.07 Perimeter Loads on the Edge Rib

H = 2.0 H = 4.0 H = 9.0

H = 9.0 H = 4.0 H = 2.0


2 3 q 5 6 7

Rib Number B 9

I

b/a = 0.05


H = 2.0 H = 4.0 H = 9.0

H = 2.0 H = 4.0 H = 9.0



H = 2.0 H = 4.0 H = 9.0


2 3 q 5 6 7

Rib Number B 9

FIG. 4.4 MOMENT DISTRIBUTIONS AT MIDSPAN OF ONE-SPAN RIBBED SLABS DUE TO MULTIPLE LOADS FOR MAXIMUM MOMENT

---I

W CJ)

0.70

0.60

0.50

0.1l0

co >- 0.30 .......... >-

0.20

0.10

a

0.70

0.60

0.50

0.1l0 co

>-.......... >- 0.30

0.20

0.10

0

b/a = 0.035


b/a = 0.07

H = 2.0 H = 4.0 H = 9.0

Petimeter Loads on the Second Rib

I


H = 9.0 H = 4.0 H = 2.0


2 3 456

Rib Number 7' B 9

b/a = 0.05 Perimeter loads on the Edge Rib

H = 2.0 H = 4.0 H = 9.0

H = 9.0 H = 4.0 H = 2.0


b/a=O.lO Perimeter Loads on the Edge Rib

H = 2.6 H = 4.0

,H = 9.0

H = 9.0 H = 4.0 H = 2.0

Perimeter Loads on the Second Rib 2 3 456

Rib Number 7

FIG. 4.5 SHEAR DISTRIBUTIONS AT THE SUPPORT OF ONE-SPAN RIBBED SLABS DUE TO MULTIPLE LOADS FOR MAXIMUM MOMENT

B 9

W ""-....I

('

0.03 ~----~----~----~------~~--.------.-----.----~

0.02

0.01


H = 2.0 H = 4.0 H= 9.0

b/a = 0.035

CQ 01 :E III ......... } } }

+l :E -0.01

-0.02

H= 2.0 H = 4.0 H = 9.0


-O.Oij~ ____ ~ ____ ~ ______ ~ ____ ~ ____ ~ ______ ~ ____ ~ ____ ~

CQ :::E

'+.>

0.031~--~----~----~--~----~----~----~--~

0.02

0.01

Perimeter Loads on the Edge Rib H = 9.0 H = .4.0 H = 2.0

/\ b/a = 0.07

o I ~//?C>. ~0, :It:= ----- ~'- ~ .j

:E -0.01

-0.02

H = 2.0· Perimeter Loads on H = 4.0 the Second Rib H = 9.0

2 3 4 5 6 7 8

Rib Number 9

H = 2.0 "'-H = 4.0

H = 9.0

b/a = 0.10

on b/a = 0.05


J J J

Pe~imeter Loads on the Edge Rib

H = 9.0 H = 4.0 H = 2.0

Perimeter Loads the Second Rib

0-H : 2.0 ,"""-H-4.0

3

H = 9.0

456

Rib Number 7 8

FIG. 4.6 TORSIONAL MOMENT DISTRIBUTIONS AT THE SUPPORT OF ONE-SPAN RIBBED SLABS DUE TO MULTIPLE LOADS FOR MAXIMUM MOMENT

9

w OJ

0.70

0.60

0.50

0.40

(Q

:E 0.30 .......... :E

0.20

0.10

0

0.70

0.60

0.50

0.40

(Q

:E 0.30 ..........

:E

0.20

0.10

0

b/a = 0.035


H = 2.0 H = 9.0

H = 2.0 H· =9.0


I

b/a = 0.07


H = 2.0 H = 9.0


2 345

Rib Number 6 7 8 9

b/a = 0.05


H = 2.0 H = 9.0

H = 2.0 H =9.0


b/a=O.lO


H = 2.0 H = 9.0

. Perimeter Loads on the Second Rib 2 3 4 5 6 7

Rib Number

FIG. 4.7 POSITIVE MOMENT DISTRIBUTIONS AT THE 0.44 SPAN LOCATION OF TWO-RIBBED SLABS DUE TO MULTIPLE LOADS FOR MAXIMUM POSITIVE MOMENT

8 9

w

""

0.70

0.60

0.·50

0.40

co ~ 0.30 .......... ~

0.20

0.10

0

0.70

0.60

0.50

n.lI co ~ .......... ~

U • .:JU

0.20

0.10

0

b/a = 0.035


H = 2.0 H = 9.0


b/a ~ 0.07

Perimeter Loads on the Edge Rib H = 9.0 H = 2.0


2 345 Rib Number

6 7 B 9

b/a = 0.05

Perimeter Loads on the Edge Rib H= 2.0 H = 9.0


2

b/a= 0.10 on the Edge Rib

Perimeter Loads on the Second Rib 3 456

Rib Number 7

FIG. 4.8 NEGATIVE MOMENT DISTRIBUTIONS AT THE INTERIOR SUPPORT OF TWO-SPAN RIBBED SLABS DUE TO MULTIPLE LOADS FOR MAXIMUM NEGATIVE MOMENT

6 9

+::o

0.10

0.60

0.50

co 0.40 :>-'-... :>-

0.30

.· •• 1 \ 0.20 t-:".: i--j ~ .. ,

0.10

o·

0.10

0.60

0.50

co 0.40 :>-'-... :>-

0.30

0.20

0.10

0

b/a = 0.035

Perimeter Loads on the . Edge Hi b

H = 2.0 H = 9.0

Perimeter Loads ori the Second Rib

"

b/a = 0.07

H = 2.0 H = 9.0



2 345 Rib Number

15 1

b/a = 0.05

Perimeter Loads on the Edge Rib H = 9.0 H = 2.0-


b/a = 0.10


Perimeter Loads on the Second Rib 8 9 2 3 4 5 6 7

Rib Number

FIG. 4.9 SHEAR DISTRIBUTIONS AT THE EXTERIOR SUPPORT OF TWO-SPAN RIBBED SLABS DUE TO MULTIPLE LOADS FOR MAXIMUM POSITIVE MOMENT

+:0

8 9

,;£00

O.031~------------------~----~----~----~----~----1

0.02

0,01


H = 2.0 H = 9.0

o L I I :;0- -====s-,.. ~ ~ ~ 7 I oc::::::::.. ""-

...........

,;£+-3-0.01

-0.02

-0.03

Perimeter Loads on the Second Rib -0.04 LI __ -L __ --L-~_-l-__ __L_ __ _L_ __ ...l__ __ ..J__ _ ____'

0.03 I I I

0.02

0.01


H = 2.0 H = 9.0

b/a = 0.07

o I "'~ r;fyz=:--.... ...... \( :;;>"~~" :;tL AI q OJ Cl il~oo ::1 f-1 f-J •. (~ -0 01 f-J. 0 -G c+""" • <l (J") ...... ~ CD f-1 I-i (") !:l:I -0' 02 m' I?:;I CD • f-Jo ~ H~ IT t:?:l O'q (j) '<l. 1-'. I-~

i-:i (D ob::1 m t:l [-b~Q)O

r:-s. I-:l (D H 1-;-1 !...Ao !--I 1=1J h! !:d I~ I-!' ~ 0 J-Io i-:i 0 ::1 oq CJ S

2

H = 2.0 H = 9.0


'345

Rib Number 6 7 6 9

b/a. = on the Edge Rib


b/a = 0.10


345

Rib Number 6 7 8

o m (-I. Id rn P'

~ c+ E3 CD ::1 c+

FIG. 4.10 TORSIONAL MOMENT DISTRIBUTIONS AT THE EXTERIOR SUPPORT OF TWO-SPAN RIBBED SLABS DUE TO MULTIPLE LOADS FOR MAXIMUM POSITIVE MOMENT

9

--' +:a N

0.70

0.60

0.50

0.40 co

:::> '-...

0.30 :::>

0.20

0.10

0

0.70

.0.60

0.50

co 0.40 :::>' '-... :::>

0.30

0.10

o

b/a = 0.035


H =2.0 , H = 9.0



2 3 'lA 5 6 7

Rib Number 8 9

I.

b/a = 0.05



b/a=0.10


Perimeter Loads on the Second Rib 2 3 lA 5 6 7

Rib Number

FIG.4.l1 SHEAR DISTRIBUTIONS AT THE 0.95 SPAN LOCATION OF TWO-SPAN RIBBED SLABS DUE TO MULTIPLE LOADS FOR MAXIMUM POSITIVE MOMENT

8 9

-J:::::. W

O.04~. ----~----~----~----~----~----_.----_.--~_.

O.OJ

0.02

0.01


H = 2.0 H = 9.0

b/a = 0,,035

co :E ...........

+l :E

o r== \ \ ~"5 ========--:;:;v-z" =1 \ '"

-0.01

-0.02 Perimeter Loads on the Second Rib

-0.031~ ____ ~ ____ ~ ____ ~ ____ ~~ ____ ~ ____ ~ _____ ~ ____ ~

O.04 r• ----.---~.-----~----~--~----~-----------~

0.03

0.02

0.01

co :E

~ :E -0.01

-0.02


H = 2.0 H = 9.0

H =2.0 H = 9.0

rre:::7

(

b/a = O.Ol

Perimeter Loads on the Second Rib -0.03 I I I I I I I I

1 2 3 ij 567 8 9

Rib Number


2.0 H = 9.0

b/a = 0.05



H = 2.0 H = 9.0

bja = 0.10


2 3 ij 5 6 7 8

Rib Number

FIG. 4.12 TORSIONAL MOMENT DISTRIBUTIONS AT THE 0.95 SPAN LOCATIONS OF TWO-SPAN RIBBED SLABS DUE TO MULTIPLE LOADS FOR MAXIMUM POSITIVE MOMENT

9

--' +::a +::a

1.40 , '-,

, 1.20

co > ..........

1.00

0.80

> 0.60

0.40

0.20

o

b/a = 0.035


~ ----' on the Second Rib

1.40 i '---,

o

b/a = 0.07


Perimeter Loads on the Second Rib 2 3 It 5 6

Rib Number 7 8 9


H = 2,.0 H = 9.0


b/a = 0.10


H= 2.D H = 9.0

I.


2 3 It 5 6 7

Rib Number

FIG. 4.13 SHEAR DISTRIBUTIONS AT THE 0.95 SPAN LOCATION OF TWO~SPAN RIBBED SLABS DUE TO MULTIPLE LOADS FOR MAXIMUM NEGATIVE MOMENT

6 9

,..(::::0 Ul

c::t:I :::E

" +-> ~

c::t:I :::E

" +-> ::E

O.Oll

O.OJ b/a == 0.035

Perimeter Loads on the Edge Rib 0.02 H = 2.0

H = 9.0 0.01

0

-0.01

-0,02 Perimeter Loads on the Second Rib

-0.0'3

O.Olt.~---------------------------------------------------~

0.03 b/a == 0.07 Perimeter Loads on the Edge Rib

H = 2.0 0.02 H = 9.0

0.01

01 ~"'""""" ::::>~:::>7..?-""":--s.""",,,"~

-0.0.1

-0,02 L Peri meter Loads on the Second Ri b

-0.03 I I I I

1 2 3 ij 567 B 9

Rib Number

b/a == 0.05

Perimeter Loads on the Edge Rib H == 2.0 H = 9.0


b/a = 0.10 Perimeter Loads on the Edge Rib H == 2.0 H == 9.0


2 ~3 ij 5 . 6 7

Rib Number

FIG. "4.14 TORSIONAL MOMENT DISTRIBUTIONS AT THE 0.95 SPAN LOCATION OF TWO-SPAN RIBBED SLABS DUE TO MULTIPLE LOADS FOR MAXIMUM NEGATIVE MOMENT

8 9

-J:::. m

147

12 I I I I I I

10 - -b/a = 0.10 - -8 t- -

b/a = 0.07 ... -(V) 0

6 ,..- I- b/a = 0.05 .... --'=!!

x b/a = 0.035 - .... ~ 4 - -I-

2

() v

r- 1 I I I , I I I I I

2.0 4.0 6.0 9.0

H

12

10

8 \-\ ::: 9.0

M 0 6 \-\ ::: 2. 0 ,..-

x ~ 4

2

o I I I I I 0.035 0.05 0.07 0.10

b/a

FIG. 4.15 RELATIONSHIPS OF THE MAXIMUM DEFLECTION COEFFICIENT WITH HAND b/a

0.07

0.06 Perimeter Loads on b/a = 0.035 the Edge Rib

2.0 0.05 4.0 Perimeter Loads on

= 9.0 the Second Rib 0.04.

co :E .......... :E+J 0.03

0.02

0.01

0

0.07. I I I

Perimeter Loads on the Edge Rib 0.06 L H = 2.0 b/a = 0.07

H = 4.0 Perimeter Loads on 0.05~ II II _ n n the Second Rib

H:::9.0~

0.04.~~. H = 4.0

~ H=2.0'1' ~

co :E 0.03 ..........

+J :E

0.02

0.01

a ./ /

-I

I I


H = 2.0 H = 4.0 H = 9.0

H = 2.0 H = 4.0 H = 9.0

Perimeter Load~ on the Edge Rib ..

H = 2.0 irH = 4.0 rH = 9.0

I- ........ II \ \

b/a = 0.05


b/a=O.lO


H =

:3 4. 5 6 7 B 9 /J~

j

~/ Rib ~umber7 8

Rib Number

FIG. 4.16 TORSIONAL MOMENT DISTRIBUTIONS AT THE SUPPORT OF ONE-SPAN RIBBED SLABS DUE TO MULTIPLE LOADS FOR MAXIMUM TORSION

+::0 co

9

0.70

0.60

0.50

0.40

o::l > 0.:10 ............ >

0.20

0.10

a

0.70

0.60

0.50

~ 0.40 ............ >

0.30

0.20

0.10


H = 2.0 H = 4.0 H = 9.0

b/a = 0.035


I I I I

Perimeter Loads on the Edge Rib H ::: 2.0 H = 4.0 H = 9.0

b/a = 0.07 Perimeter Loads on the Second Rib

H = 2.0 H = 4.0 H = 9.0

I I I


H = 2.0 H = 4.0 H = 9.0

I ,_


H = 2.0 H = 4.0 H = 9.0

I

b/a = 0.05


= 2.0 = 4.0 = 9.0

b/a = 0.10


o =---:::;:I 2 ~----~----~--_--L-- ' ~ 345

Rib Number 5· 7 B 9 2 3 4 5 6 7 8 9

Rib Number

FIG. 4.17 SHEAR DISTRIBUTIONS AT THE SUPPORT OF ONE-SPAN RIBBED SLABS DUE TO MULTIPLE LOADS FOR MAXIMUM TORSION

+::\..0

0.07 _ ·1

0.06 L Perimeter Loads on the Edge Rib

0.05 L -H = 2.0 H = 9.0


H = 2.0 H = 9.0

b/a = 0.05

Perimeter Loads on ~ 0.04L II the Second Rib

Perimater Loads on the Second Rib

.......... +->

:::E

ro :::E ..........

+-> :::E

0.03~ "-... H = 2.0 H = 9.0

0.02

0.01

0

0.07 I

b/a = 0.07 Perimeter Loads on 0.06 L the Edge Rib

H = 2.0 O.05~ H = 9.0


H = 2.0 H = 9.0 0.04

0.03

0.02

0.01

0 '345

Rib Number 5 7 B

H = H =


H = 2.0 H = 9.0

Rib

b/a = 0.10

Perimeter Loads on the Second Rib H = 2.0 H = 9.0

6 7

FIG. 4.18 TORSIONAL MOMENT DISTRIBUTIONS AT THE EXTERIOR SUPPORT OF TWO-SPAN RIBBED SLABS DUE TO MULTIPLE LOADS FOR MAXIMUM TORSION

U1 o

5

Rib Number

b/a = 0.035


H = 2.0 H = 9.0

b/a = 0.07


(5

H = 2.0 H = 9.0


H = 2.0 H = 9.0


H = 2.0 H = 9.0

3

b/a =: 0.05


= 2.0

b/a == 0.10


f5 7

= 2.0 9.0

Rib Number

FIG. 4.19 TORSIONAL MOMENT DISTRIBUTIONS AT THE 0.95 SPAN LOCATION OF TWO-SPAN RIBBED SLABS DUE TO MULTIPLE LOADS FOR MAXIMUM TORSION

U1 ........


H :: 2.0 H = 9.0

I I I


H = 2.0 H = 9.0

b/a = 0.05


b/a = 0.10


2 3 456

Rib Number

FIG. 4~20 SHEAR DISTRIBUTIONS AT THE EXTERIOR SUPPORT OF TWO-SPAN RIBBED SLABS DUE TO MULTIPLE LOADS FOR MAXIMUM TORSION

--I

U1 N

co > .......... >

co > .......... >

O.70~, ----~----~----~~----~----._----._----_.----~

Perimeter Loads on 0.60 L the Edge Ri b

H 2.0 0.50 1\ / / H = 9.0

0.40

0.30

0.20

0.10 H = 2.0 H = 9.0

b/a = 0.035

Perimeter Loads on the Second Rib 0

0.70

on b/a = 0.07 0.60 Perimeter Loads on

the Second Rib 0.50

0.1!0

0.30

0.20

0.10

0 2 '3 456

Rib Number

H = 2.0 H = 9.0

7 8 9

Per~mete~ Loads' on the Edge Rib

H = 2.0 H = 9.0

H = 2.0 H = 9.0

b/a = 0.05


, on b/a = 0.10


2 3 456

Rlb Number

H = 2.0 H = 9.0

7 8

FIG. 4.21 SHEAR DISTRIBUTIONS AT THE 0.95 SPAN LOCATION OF TWO-SPAN RIBBED SLABS DUE TO MULTIPLE LOADS FOR MAXIMUM TORSION

9

--I

<.n w

b/a = 0.05


b/a = 0.10


2

H = 2.0 H = 9.0

3 l! ,5


H = 2.0 H = 9.0


6 7 8

Rib Number

FIG. 4.22 NEGATIVE MOMENT DISTRIBUTIONS AT THE INTERIOR SUPPORT OF TWO-SPAN RIBBED SLABS DUE TO MULTIPLE LOADS FOR MAXIMUM TORSION

9

Ul ..p,.

0.70

0.60

0.50

o:::l :::E 0.40 -.........

x I'd E 0.30 :::E

0.20

0.10

0

0.70

0.60

0.50

o:::l 0.40 >---...:..

>< rtS E 0.30 >-

0.20

0.1: r ,

0.07

0.06

0.05

0.04 co

:E -... +-'

:E 0.03

0.02

0.01

0 0.035

155

Maximum r~oment

Accompanying Torsion

L H = 9.0 H = 6.0

O ---H -::: 9.:-_---- 0 --_--u -::: 6. _--_--e-- r\_~-O ..._.--- --u - £\.. ---_--- .-...-- fI""':_------- --- --_- _-.e---'-- _--e---

..."....... ."."......,.".,.-.."""- ~--------------- ----- H = 2.0 ----...-.------_--e---------------- H = 9.0

0.05

H = 6.0 H = 4.0

H = 2.0

b/a 0.07


0.1:0

FIG. 4.23 RELATIONSHIPS OF MAXIMUM MOMENT, MAXIMUM SHEAR AND THE ACCOMPANYING TORSION, WITH HAND b/a FOR 'ONE-SPAN RIBBED SLABS UNDER MULTIPLE LOADS FOR MAXIMUM MOMENT

0.07

0.06

0.05

0.04 . c::c

::E --.......

156

Maximum TOrsion H ::: 9.0 ------fi==6...JL ---.......------ ---------.----- ------- --- --- --- -- ---___ --- -- ---- -- ___ ~ --------__ ---------11= 4. 0

~ --- --- --- ----- ----- -- --- ---- -- -- 2 0 ---....--e-- _--- . H == . ---- --...-.-- ----- -- -- - ---.------H-- 9 0 ,.,."...-::::.,..,.."...."""""- ------------ - . -- --e------- H == 4.0

~ 0.03 H == 2.0 E +>

::E

co ::> --....... >

0.02

0.01

0

0.70

0.60 Accompanying Shear

0.50

0.40

0.30

'---;#~-::::::~:~H=r9 a Wii[! .....-:::-:::::::::=:::::::-.-.---- • --:::::: ::;::::..;,..--...- H == 6 0 ...---:::;::::::' ..- .

;,..--;:...-"'- H ::: 4.0 H = 2. r;H == 9.0

H ::: 4.0 ::: 2.0

0.20

0.10

O~~ ______ ~ ______ ~ ______ ~ __ ~ __ ~ ____ ~~ ____ ~ 0.035 0.05 0.07

- - Perimeter Loads on the Edge Ri b -- Perimeter Loads on the Second Ri b

FIG. 4.24 RELATIONS OF MAXIMUM TORSIONAL MOMENT AND THE .. ACCOMPANYING SHEAR WITH HAND b/a FOR ONE-SPAN

RIBBED SLABS UNDER MULTIPLE LOADS FOR MAXIMUM TORSION

0.10

co :::E ............

>< ro E

:::E

co :::E ............

-I-l :::E

157 0.60

0.50 Maximum Positive Moment H = 9.0

0.40 ---___ -------H=-2.o

-----=-~-----H-::-9-0----- -- --~ --- . -- ---------- ----- H = 2.0 0.2

o ~~ ______ -L ______ ~ ____ ~~ ____ ~~ ____ ~ ____ ~

0.60----------~------------~~----~------~----~

0

0.06

0.05

0.04

0.03 .

0.02

0.01

0 0.035

H = 9.0 _-e----------__ --- H-20--Maximum Shear

_-- ___e_-- -. ............. --------- ----------.""..... ---------- .-.--- H == 9.0

H == 2.0

Accompanying Torsion U :: 9.0 _----n _-

----------------.-

-------------H :: 2.0 -----------------~-__ ---e-----

-------- ----- --- --- -- -----

o .05 b/a

0.07

H = 9.0 H = 2.0


n.10

FIG. 4.25 RELATIONSHIPS OF MAXIMUM POSITIVE MOMENT AT THE 0.44 SPAN LOCATION, MAXIMUM SHEAR AND THE ACCOMPANYING TORSION AT THE EXTERIOR SUPPORT WITH HAND b/a FOR TWO-SPAN RIBBED SLABS UNDER MULTIPLE LOADS FOR MAXIMUM POSITIVE MOMENT

co >--....

x n::s E

>-

158 0.80

0.70

0.60

,,------ H Maximum Shear ~-- ---:::: 9 0 ,,-"'- -- .

.,.-"'- ---~ ,,-~- H --.........- .,.-"'- --- :::: 2 0 .,.----- ~~.,.- --~-

~ -~ ----- ----------.,.-.-"

0.3

0.20

0.10

0

0.50 J J • J J I I

0.40 r- Accompanying Moment -

H = 9.0 O 30 _--------4P . - _-----+-----Tr = 2.0 --...---- -----------------~~ O 20~---=-==-.- ----- H = 9.0

. ii=---- - -----=::::::::1= H 2 . 0

0.10~

o

0.05

0.041-

r

I

-

I I

I I

Accompanying Torsion

-

I I I

I I I

gO _---4 \-\ ::: -:..----- -

....... -----....... ---~ 0.03r- _- ....... ---- -

. --~~- -H- = 2.0 ---~ ---0.02_ _--------e-------- -~~------.-

0.01 - -r H = 9.0 Ir H - 2•O

ot;;;;:::::::~::~Jt::dI~;;~~~~==~====~:~I~==~ 0.035 0.05 0.07 0.10

b/a - - Perimeter Loads on the Edge Ri b -- Perimeter Loads on the Second Ri b

FIG. 4.26 RELATIONSHIPS OF MAXIMUM SHEAR AND THE ACCOMPANYING TORSION AT THE 0.95 SPAN LOCATION AND NEGATIVE MOMENT AT THE INTERIOR SUPPORT WITH HAND bla FOR TWO-SPAN RIBBED SLABS UNDER MULTIPLE LOADS FOR MAXIMUM POSITIVE MOMENT

o:::l L ............

X ~ E

L

o:::l > ............

x ~ E

>-

0.70

0.60

0.50

0.40

0.30

0.20

O~ 1 0

0 1 .10

1 .00

0.90

0.80

0.70

0.60

0.50

0.40

0.30 0.04

O. 03 ~

159

Maximum Negative Moment H = 9.0 _----- -- --- -- --.-8------ 0 _---- H~3~-----_~- _--0------

----- ---- H = 9 0 ...- ~- . --__ ---~ H = 2.0

Maximum Shear ...-......e-.~_ 1-1

...-"'- --L.L...,:::: 00 ...- -~ ...- --~~ --~ ~.. --"..,....~ ~ ---- H :::: 2 0 ,.,..,.....- ~...-. ---~

-...-~...- ----~...- ---~-- ... --,.,..,../ // ,.,..,..

I I I I I I

H = 9.0 _-----..:aI ....e--------Accompanying Torsion

...._e___-.....:....------o. 02 F-~---- -

H = 2.0 ___ u . . _._------- - ---------

-0.01 ~-----...--- H = 9.0

0~~====~~-~~~;;;;;;;;;:~~~::::~H:=~2~.0~:J II"""

0.035 0.05 0.07 b/a


0.10

FIG. 4.27 RELATIONSHIPS OF MAXIMUM NEGATIVE MOMENT AT THE INTERIOR SUPPORT, MAXIMUM SHEAR AND THE ACCOMPANYING TORSION OF THE 0.95 SPAN LOCATION WITH HAND b/a FOR TWO-SPAN RIBBED SLABS UNDER MULTIPLE LOADS FOR MAXIMUM TORSION

0.07

0.06

0.05 CQ

::E ........ >< 0.04

rt:S E

+-> 0.03 ::E

0.02

0.01

0

0.70

0.60

0.50

0.40 CQ

> ........ > 0.30

0.20

0.10

0 0.035

160

Maximum Torsion at the Exterior Support

H ::: 9.0 ---...---------------....... -- --- --- --- ----------.---------- H - 2.0 __ ;... ...... ----------

-------------- - 9 0 ..... ---..------....... - .

H = 2.0

Accompanying Shear H = 9.0

-~------------------ H = 2.0 ------- ---------------------- -----

------ -------- --------

0.05 0.07

b/a


H = 9.0

H = 2.0

0.10

FIG. 4.28 RELATIONSHIPS OF MAXIMUM TORSIONAL MOMENT AND THE ACCOMPANYING SHEAR AT THE EXTERIOR SUPPORT WITH HAND b/a FOR TWO-SPAN RIBBED SLABS UNDER MULTIPLE LOADS FOR MAXIMUM TORSION ,

c:::c ~ ...........

x ro E

+J ~

c:::c :> ........... :>

c:::c :E: ........... ~

0.06

0.05

0.04

0.03

0.02

0.01

O.

0.80

0.70

0.60

161

Maximum Torsion at the 0.95 Span Location

------------- ...... ---H = 9.0 __ _ ----------------H = 9.0

_~ __ ~r---------~-----H~2~---

H = 2.0

Accompanying Shear ----- H - 9 ............. -------- - 0 ."".......-......- --------:...... ,........-- ----

,.... ------ H -.......- -- --- =: 2 0 ~ -,.... -- .

~~,....-- ----~ -- --~ ,....- -

~ ~

0.50

0.40

0.30

0.20

0.10

0

0.40

0.30

-...-,.-"'-

H = 9.0

H = 2.0

H = 9.0

AccompanYi: T:Si:._---.:::=_=__=_ H = 2.0 ~-=-=== 0.20 --- --- --- --- ------ - -- --- --~. -4------,--==:=1 --. _---e-.--

--- ____ --__ ------- / I --.

0.10

o 0.035 0.05

b/a 0.07

H = 9.0 H = 2.0


0.10

FIG. 4.29 RELATIONSHIPS OF MAXIMUM TORSIONAL MOMENT AND THE ACCOMPANYING SHEAR AT THE 0.95 SPAN LOCATION AND NEGATIVE MOMENT AT THE INTERIOR SUPPORT WITH HAND bia FOR TWO-SPAN RIBBED SLABS UNDER MULTIPLE LOADS FOR MAXIMUM TORSION

162.

Multiple Loads for Maximum Moment

Regular Edge· Rib

Stiffer Edge Rib

---..::::. :::: :::: :::: = = ..... - -----

Multiple Loads for Maximum Shear

Regular Edge Rib

tiffer Edge Rib

Multiple Loads for Maximum Torsion

Regular Edge Rib

Stiffer Edge Rib

0.02 ------ --- ---..."...,.. ........ ~.-- - -........-:~~,.... ,,-- .

o ~~~----------------------~--------------~ .,.,/

-0. 02 L--_---L __ -I...-__ ..l-_---' __ --L-__ -:-'---"'-------'

'1' 2 3 4 5 6 8 9 Rib Number


FIG. 4.30 EFFECTS OF THE STIFFENED EDGE RIB ON MOMENT, SHEAR AND TORSIONAL MOMENT.DUE TO MULTIPLE LOADS

163 O~60~--~----~----~----~----.----.-----r--~

0.50 Moment at Midspan

0.40

co 0.30

Regular Edge Rib Stiffer Edge Rib

::E --------------------------- --------- ------ - -----~------ ---------------- ----- ---------- --------

::E 0.20

co > ........... >

co ::E: ...........

0.10

oL-~~~ __ ~ __ ~ __ ~~~~~ __ ~ 0.70

-O.DO Shear at Support

0.5e)

0.40

0.30

0.20

0·'00 t O. 14

O. 12

0.10

0.08

Regular Edge Rib

Stiffer Edge Rib

Torsion at Support

j

';E+J 0.06

Regular Edge Rib

Stiffer Edge Rib

0.04

0.02.

o ~--~~--~----~--~~--~~--~----7_--~ 1 9

FIG. 4.31 EFFECTS OF THE STIFFENED EDGE RIB ON MOMENT, SHEAR, AND TORSIONAL MOMENT DUE TO A LINE LOAD ON THE EDGE RIB

co :::E: ......... :::E:

co >-......... >-

co :::E: .........

+-> :E:

FIG. 4.32

164 1.60~--~--~~----~----~----~--~-----.----~

1.20

1 .00

0.80

0.60

0.40

0.20

0

1.40

1.00

0.80

0.60

0.40

0.20

0

0.05

0.04

0.03

0.02

0.01

0 1

MOMENTS,

Moment at Midspan

Regul ar Edge Ri b Stiffer Edge Rib

.... ---........ --................... -.. . .... - .................. - ........................... - ...................... ·············.··-··.--·-······.··--.-1.-···.·.···.···-.- .-.- --•. ----... .

Shear at Support

Regular Edge Rib Stiffer Edge Rib

Torsionat Support

Regular Edge Rib

Stiffer Edge Rib

4 5 Rib Number

SHEARS, AND TORSIONS DUE TO A MIDSPAN LINE LOAD

9

RIBBE SL BS UN E C E TR TED l 5 - IDEALS

Documents

Transcript of RIBBE SL BS UN E C E TR TED l 5 - IDEALS