Model Study of the Proposed Precast-Prestressed Bridge …A model study of a particular...

MoDOT

TE 5092 .M8A3 no.68-9

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"MODEL STUDY OF THE PROPOSED

PRECAST-PRESTRESSED BRIDGE SYSTEM"

Prepared for

MISSOURI STATE HIGHWAY DEPARTMENT

by

JOHN R. SALMONS

and

SHAHROKH MOKHT ARI

DEPARTMENT OF CIVIL ENGINEERING

UNIVERSITY OF MISSOURI

COLUMBIA, MISSOURI

in cooperation w ith

U.S. DEPARTMENT OF TRANSPORTATION

FEDERAL HIGHWAY ADMINISTRATION

BUREAU OF PUBLIC ROADS

The opin ions, findings, and conclusions

expressed in this publication ore not necessarily

those of the Bureau of Public Roods .

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ABSTRACT

A model study of a particular prestressed-precast com

posite member is presented. The proposed composite mem

ber is composed of a prestressed concrete channel, an in

terior void form, and a top slab of cast-in- place con

crete. Theoretical model analysis is developed for the

particular member and possible deviations from theory are

discussed. A one-half scale model member is designed

to satisfy the 1961 AASHO Code structural requirements

and the similitude criteria are established.

To check the agreement between theoretical similitude

ratios and the experimental values, a series of half-scale

models were constructed. Results of flexural tests per

formed on three composite members are presented in this

report. Instrumentation of each test member was designed

to measure deflection, strain, slip between the two con

crete portions, and top slab separation. Model test re

sults were compared to test results from three full-scale

members tested previously.

In general, model members predicted the behavior of

full-scale members adequately and reasonable agreement was

observed between the predicted values and the actual full

scale test data.

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TABLE OF CONTENTS

CHAPTER PAGE

I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 General 1

1.2 Literature Survey ...............•........ 6

1.3 Scope .................................... 10

II. THEORY AND DESIGN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2

2.3

2.4

2.5

Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Practical Considerations . . . . . . . . . . . . . . . . . 16

Design 2. 4. 1 2.4.2 2.4.3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 General . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Design Assumptions ................ 19 Design Values . . . . . . . . . . . . . . . . . . . . . 19

Design Comparison of Model and Full-Scale Members ....................... 22

III. FABRICATION OF TEST SPECIMENS ................. 29

3 .l General .................................. 29

3.2 Channels ................................. 29 3. 2. 1 Formwork . . . . . . . . . . . . . . . . . . . . . . . . . . 2 9 3.2.2 Prestressing ...................... 29 3.2.3 Shear Reinforcement ............... 30 3.2.4 Casting and Curing ................ 32 3.2.5 Companion Specimen ................ 35

3.3 Top Slabs ................................ 35 3.3.1 Reinforcement ..................... 39 3.3.2 Casting and Curing ................ 39

IV. TESTING PROCEDURE • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 42

4.1 Flexural Test ...........•................ 42

4 . 2 Ins trumen ta tion • . . . . . . . . . . . . . . . . . . . • . . . . . 4 3

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CHAPTER

4.3 Testing Procedure ..................... .

V. RESULTS AND COMPARISONS .................... .

5 . 1 Genera 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.2 Experimental Results .................. . 5.2.1 Deflection History of the Pre-

stressed Channels .............. . 5.2.2 Load-Deflection Relationship ... . 5.2.3 Load-Strain Data ............... . 5.2.4 Slip and Slab Separation Data .. . 5.2.5 Ultimate Load Capacity ......... .

5.3 Comparison of Model and Full-Scale Test Results .......................... . 5.3.1 Deflection Similitude .......... . 5.3.2 Strain Similitude .............. . 5. 3. 3 Load-Slip Behavior ............. . 5.3.4 Ultimate Load and Mode of

Failure ........................ .

VI. SUMMARY AND CONCLUSIONS .................... .

6. 1 Summary ............................... .

PAGE

48

52

52

52

52 53 57 67 69

69 72 74 77

77

83

83

6.2 Conclusions . . . . . . . . . . . . .. . . . . . . . . . . . . . . 85

SELECTED REFERENCES ................................ . 88

APPENDIX A SIMILITUDE ANALYSIS OF STRUCTURAL MODELS · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · Al

A.l General A2

A.2 Dimensional Analysis .................. . A2

A. 3 Types of Models ....................... . A4

A. 4 Structural Models .................. · ... . AS

A. 5 The Applied Force, R AlO

A. 6 Dead Load Al2

APPENDIX B SUMMARY OF DESIGN COMPUTATIONS FOR A ONE-HALF SCALE MODEL ................. . Bl

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CHAPTER PAGE

B.l General • • • • . • • • • • • • • • • • • . • • • • • • • . . • • • . • Bl

B.2 Design Sununary • • • • • • • • • • • • • • • • • • • . • • • . • B2

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LIST OF FIGURES

FIGURE PAGE

1.1 Proposed Bridge System.......................... 4

1.2 Cross-Sectional View of Composite Section ....... 5

1.3 Transverse Distribution of Midspan Deflection 8

1.4 Variation with Size of the Crushing Strength of Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1 Stress-Strain Relationship for Model and Pro-totype Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Cross-Sectional Dimensions of the Full-Scale Member . . . . • . . . • . . . . . . • . . . • . . • . . . • . . . . . . . . . . . . . . . 2 0

2.3 Cross-Sectional Dimensions of the Half-Scale Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.4 Channel Reinforcement ........................... 23

2.5 Top Slab Reinforcement .........•................ 23

2.6 Bottom Fiber Stress vs. Span Length, Full-Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.7 Bottom Fiber Stress vs. Span Length, Model ...... 27

2.8 Distortion of Bottom Fiber Stress ............... 28

3.1 Formwork for Prestressed Channel ................ 31

3.2 Pretensioning Operation ......................... 31

3.3 Stirrup Arrangement ............•................ 33

3.4 Casting of the Channels ......................... 34

3. 5 Steam Curing Operation . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.6 Time-Temperature Recording Device ..•...•........ 36

3.7 Concrete Temperature During Steam Curing ........ 37

3.8 Welding of Reinforcement ........................ 40

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FIGURE PAGE

3. 9 Reinforcement Arrangement ..................... . 40

3.10 Cast-in-place Top Slab ........................ . 41

4.1 Loading System ................................ . 44

4.2 Suspended Support and a Partial View of the F r arne . . • . . . . . . . . . . . . • . . . . . . . . • . . . . • . . . . • . . . . . . . 44

4.3 Strain Meters and Slip Dial . .................. . 46

4.4 Slab Separation Dial ................. . ........ . 46

4.5 Location of the Strain Meters .............. . ... 47

4.6 Strain Measurement Instruments ................ . 49

4. 7 Loading System ................................ . 49

4.8 Flexural Test Instrumentation ................. . 50

5.1 Deflection Instrument ....•..................... 52

5.2 Channel No. l Deflection History . ... ........... 54

5.3 Channel No. 2 Deflection History . .............. 55

5.4 Channel No. 3 Deflection History ............... 56

5.5 Load-Deflection Similitude ............ ......... 58

5.6 Load-Deflection Similitude .................... . 59

5.7 Load-Deflection Similitude 60

5.8 Load-Deflection Similitude 61

5.9 Typical Strain Distribution of Midspan ········· 63

5.10 Load-Strain Similitude at Midspan ·············· 64

5.11 Load-Strain Similitude at Midspan ·············· 65

5.12 Load-Strain Similitude in Shear Span ......•.... 66

5.13 Slip Load Similitude ..... . .................... . 68

5.14 Ultimate Load Similitude ...................... . 71

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FIGURE PAGE

5.15 Model #1, Diagonal Tension Failure •............. 79

5.16 Model #1, Slip Surface Near Midspan ............. 79

5.17 Model #2, Slab Separation and Diagonal Tension Crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.18 Model #2, Shear Cracks and Secondary Compression Failure in the Channel .............. 81

5.19 Model #3, Moment Cracks and Secondary Compression Failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.20 Model #3, Diagonal Tension and "Negative Moment" Cracks at the Support ................... 82

B.l Midspan Stress Gradient ......................... B5

LIST OF TABLES

TABLE PAGE

I 2 .1 Scale Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 Section Properties of the Model Member ........ 21

2.3 Design Distortions 24

3.1 Concrete Information from the Companion

I Specimens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5.1 Ultimate Load Capacity ......•................. 70

B.l Predicted Deflections ......................... B4

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CHAPTER I

INTRODUCTION

1.1 GENERAL

Model testing as a tool of structural design and re

search has been developed over the past half century. Dur

ing the past decade, structural model testing has been ad

vanced toward becoming a tool of major importance in prac

tical design of specific concrete structures as well as in

research work. The need for model testing could arise from

any of these four possible situations:

(1) Theoretical background for the problem not avail

able or mathematical analysis is virtually impossible. This

situation is frequently encountered in various behavior of

concrete members such as bond, creep and shrinkage, effect

of prestressed forces, buckling phenomena, composite ac

tion and inelastic behavior.

(2) Theoretical analysis, though possible, is complex

and tedious. Often times, analytical solutions are so

idealized that they no longer adequately describe the

physical problem. Model testing gives more realistic

answers in such cases. Highly complex structures, such

as arch dams, shell roofs are examples of this category.

(3) Economy and available equipment are major lim

itations in some research work. Considerable savings in

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materials, space, fabrication cost, handling and testing

equipment required and the subsequent possibility of carry

ing out tests in controlled laboratory conditions make

model testing extremely advantageous in these respects.

(4) Importance of the problem is such that verifi

cation of the theoretical analysis by model test is war

ranted.

The objective of tests of a structural model may gen

erally be placed in one or all of the following four cat

egories:

(l) Stress analysis such as axial force, shear stress

es, bending moment and torsional stresses.

(2) Determination of stress distribution and load

transfer in the case of composite structures.

(3) Determination of the ultimate or buckling loads.

(4) Analysis of the characteristics of the stiffness

and the normal modes of vibration.

With the light of the above discussion, one can see

that model testing can successfully be applied to almost

any structural system, ln particular, concrete composite

members. Due to numerous structural and functional advan

tages, concrete composite members have become highly de

sirable in recent years. However, stress state, load

transfer, slip and deflection characteristics of such mem

bers are, often times, highly involved. Based on the ad

vantages sited above, model tests are favored in many cases.

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One particular example of a concrete composite mem-

ber is a prestressed-precast, composite bridge system

(Fig. 1.1) A schematic sectional view of this type of

member is shown in Fig. 1.2 The void space does not ex-

tend through the entire span length but is interrupted pri-

or to reaching a pier and at intermediate points between

h 0 t 'd 1 1 b'l't l* t e p1ers o prov1 e atera sta 1 1 y •

It can be seen that the behavior of this structural

system is quite complicated with respect to composite ac-

tion of the system, that is the nature of load trans-

fer from the cast-in-place slab to the precast channels.

Construction and testing of a full-scale bridge of this type

in a normal size laboratory is impractical, whereas the

testing of a scaled model would be much more feasible.

Using the model laws, test results of a scaled model can

be extrapolated to predict the behavior of the full-scale

bridge. However, the accuracy of the model laws and sim-

ilitude ratios might seem questionable when used to extrap-

olate for such a composite member.

The foregoing discussion initiated the course of study

which is presented in this paper. The object of the study

was to investigate the similitude relationship between the

model and the prototype through a model test program. Theo-

retical and experimental model relationships were investi-

gated with due consideration of the special features of

Superscripts refer to entries in the Bibliography.

- ----------

~ ../'

./" ..,.

~MPOSIT7 DECK

PRECAST CHANNEL-

Fig. l.l Proposed Bridge System

---

,_, ../'

~

-

-------------------

VOID FORM CAST-IN-PLACE

VOID

PRECAST-PRESTRESSED-UNIT

Fig. 1.2 Cross-Sectional View of Composite Section

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the proposed structural system.

1. 2 LITERATURE SURVEY

In an exploratory investigation by Z. Y. Alami2

anum

ber of reinforced concrete model beams were tested to fail

in diagonal tension, bond and flexural compression. In

each case ultimate stress, center deflection, compression

strain and moment cracks were compared with the test val

ues of prototype beams and hence the accuracy of model

theory applied to concrete beams was checked. These re

sults show:

(1) Models failed to predict the behavior of the pro

·totypes when bond was the primary or secondary reason for

failure. When flexure or shear failure was expected, mod

els with scales of 0.2 to 0.3 closely predicted the proto

type behavior.

(2) Load-deflection and load-strain curve, and thus

the overall response of the beams to load, was reasonably

predicted by testing models with scales as small as 0.334.

(3) Only approximate similitude for average distance

between moment cracks was obtained.

(4) The difference between the actual and predicted

stiffness of a model became more significant as the scale

was made smaller.

(5) Scaling the maximum size of aggregates seemed to

improve the accuracy of the models.

A continuous study of structural model testing is be-

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ing carried out at the Structural Laboratory of Portland

Cement Association. Alan H. Mattock3

cited a few exam-

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ples of structural concrete models tested in recent years:

(1) A half-scale model of a continuous precast-pre

stressed concrete bridge was tested. This type of bridge

consisted of precast-prestressed !-shaped girders spaced

at 6'-6" in accordance with AASHO-PCI requirements, with

a continuous situ-cast deck slab.

(2) A one twenty-fourth scale model of the same type

of structure was constructed and tested in plexiglas. Ex

cellent agreement was found between the behavior of the

half-scale and twenty fourth scale model when subjected to

service load. Distribution of deflection across the width

of the loaded span is tabulated in Fig. 1.3 accompanied

by sketches of the loadings. Deflection of each girder is

expressed as a percent of the total deflection of the five

girders. As can be seen there is very good agreement of the

results from the two models, despite their large difference

in size.

(3) A l/13th scale prestressed mortar model of a con-

crete prototype bridge was tested by Cement and Concrete

Association, London, England, to study the load distribution

characteristics of different components of the bridge and

the vibrational behavior of the slab. Results of this in-

vestigation were used to predict the behavior of the full

scale bridge utilizing the well-known model laws.

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(a) Load On Outer Girder

~ Scale Model 64.5 27.9 9.1 0.9 -2.4

1/24 Scale Model 64.8 28.4 8.1 0.8 -2.1

(b) Load On First Interior Girder

~ Scale Model 28.8 41.1 21.6 7.3 1.2

1/24 Scale Model 28.5 40.1 22.8 7.9 0.7

(c) Load On Center Girder

t ill ~ Scale Model 9.4 21.7 37.8 21.7 9.4

1/24 Scale Model 8.3 23.0 37.5 23.0 8.3

Fig. 1.3 Transverse Distribution of Midspan Deflection

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(4) A l/35th scale plexiglas model of a 28 story

apartment building was constructed and tested by the En-

gineers Collaborative, Des Plaines, Illinois. As a result

of the model test, it was found that columns were correct-

ly dimensioned, but that the reinforcing steel in the slab

could be reduced by 15 percent. The resultant saving

was approximately four times the cost of the model tests.

In all these cases, it was found that the small scale

models predicted the behavior of the prototypes very close

ly. The deflections, crack patterns, lateral distribution

of moment, mode of failure, and ultimate load were in close

agreement with predictions.

Guralnick and LaFraugh4

report the results of a model

study of a 45-foot square flat plate which was tested at

the University of Illinois. The objective of the study

was to re-evaluate and improve the design methods of flat

plates with application to ultimate strength theories. A

l/4th scale model slab was correlated to a 3/4th scale

model slab tested at PCA Structural Laboratory. Once again,

deflection, crack-patterns, distribution of moments due to

service load, and mode of failure were in close agreement

with results of the prototype.

5 In a study undertaken by Donald D. Magura at the PCA

Research Laboratories, 16 ordinary reinforced and 14 pre-

stressed mortar model beams were tested. Test results

showed that, in general, the behavior of small mortar beams

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was ~n accordance with performance of full-scale members.

However, it was seen that the measured ultimate loads were

generally greater than those predicted by the design, and

that in the case of shear failure inaccuracies were quite

large.

In a laboratory investigation undertaken by M. Amara

tunga6 miniature prestressed continuous beams and portal

frames were tested. Results showed the same order of

variation from the predicted design values as one can ex-

pect from testing normal size structures. Maximum v a ria-

tion of 19 percent was observed.

7 In the papers presented by E. H. Brown , R. F. Blanks

and C. c. McNamara 8 , effect of maximum aggregate size in con-

crete models was discussed. Test results presented in both

papers showed that when the concrete test cube or cylinder

was made smaller, failure stress became higher as can be

seen from Fig. 1.4. The same type of behavior should be

expected from small-scale structures. Results also showed

that this increase in strength can be compensated for by

reduction in the maximum aggregate size. The authors sug-

gested that, to obtain comparable results from model test,

aggregate size should be proportionally scaled down.

1.3 SCOPE

A particular type of prestressed-precast composite

member (Fig. 1.2) has been proposed for use in highway

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8 z ~ u p:: ~ P-l

::r: 8 l'J z ~ p:: 8 (/)

~ > H 8 ,::X: ...:l

~

110

105

100

95

90

85

10 CYLINDER 0

CUBE 0

~ STRENGTH RELATED TO 6

INCH SIZE AS STANDARD

1\ ~ \ ~ ~ .......__

80

6 12 18 24 30 36

DIAMETER OF CYLINDER OR CUBE-INCHES

Fig. 1.4 Variation with Size of the Crushing Strength of Concrete

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bridge construction on primary and secondary roadways. A

study of the structural behavior of three full-scale members

has been presented 1 .

The object of this study was to design a one half

scale concrete model of the member and to investigate the

similitude relationship between the model and the prototype

through a model test program. In addition to the theoret

ical approach, the report describes the fabrication of

three 18-foot long composite beams and five prestressed

channels to be used for further bridge study.

Also, results of flexural tests performed on the three

model beams are compared with the prototype test results

and similitude correlation is presented.

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CHAPTER II

THEORY AND DESIGN

2.1 GENERAL

In order to predict the behavior of a prototype from

the structural response of the model, there must exist some

mathematical expressions which relate the behavior of the

model to the full scale structure.

Based on the assumptions and the mathematical presenta

tion in Appendix A, a table of model scale factors for the

general properties of the composite beam is presented. In

addition, with the consideration of special features of this

study, possible deviations from theory are discussed. Fi

nally, design of a one-half scale model beam is presented.

2.2 ASSUMPTIONS

To idealize the problem the following assump~ions are

adopted:

(1) Material properties of the model and the proto

type members are identical. Or more specifically modulus

of elasticity, modulus of rigidity and consequently the

Poisson's ratio should be the same for the model and the

prototype in both elastic and inelastic ranges. This as

sumption would result in identical stresses and strains un

der similar conditions of testing and is an idealized case

of similitude of deformation. Fig. 2.1 demonstrates a

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e: p

1 = -o Cl. p

STRAIN

0 p

(a) General Case

STRAIN

STRAIN

(b) Case, S = 1

& A m

(c) Simplified Case, a = S = 1

0 p

Fig. 2.1 Stress-Strain Relationship for Model and Prototype Material

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more general scope of similitude of deformation. Case (a)

shows the stress-strain relationship for two materials, dif-

ferent in composition but similar in behavior. "A " and p

"A " are corresponding points, with respect to ultimate conm

dition, on the two curves. The ordinates of any point "A " m

can be related to the ordinates of the corresponding point

"A " on the upper curve utilizing the constants a and s. p

Case (b) represents the condition where one of the con-

stants, say S, is equal to unity, and thus imposing the

condition of identical strains. Case (c) is the simpli-

fied condition where both of the constants are equal to

unity and is the idealized condition assumed earlier.

(2) The model and the prototype are homologous i.e.,

complete geometric similarity exists between the model and

the prototype and no dimensional distortion is introduced.

(3) All assumptions of mechanics are applicable:

Plane sections before bending remain plane after bending,

concrete is completely homogeneous and isotropic, etc.

Based on these assumptions, the theoretical relation

between a prototype and a homologous model can be summar-

ized in the following statement:

If the linear dimensions of a prototype structure are

scaled by a factor (k) and the applied forces are scaled

by the factor (k2), then the applied moments are scaled by

a factor (k 3 ), the deflections will be scaled by the factor

(k) and the strains and stresses will remain unchanged.

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Scale factors for all the significant properties of con

crete members are shown in Table 2.1. A mathematical treat

ment of the similitude analysis is presented in Appendix A.

2.3 PRACTICAL CONSIDERATIONS

Due to some practical difficulties encountered in de

sign of reinforced concrete members, there exist some dis

crepancies between the theoretical and the experimental

similitude results.

(1) Effect of Dead Weight. As shown in Table 2.1 weight

of a concrete member is a function of its volume while ap

plied live load is a function of surface areas. Therefore,

dead load and resulting stresses are scaled down one degree

higher than the live load stresses. In other words, if the

live load stresses and strains are identical for the model

and the prototype, dead load stresses and strains are k

times smaller for the model than the prototype. This is in

compatible with the original assumptions.

In ordinary reinforced concrete model beams, this con

dition is eliminated by hanging additional weights along

the length of the member. In the case of this study, the

prestressing design was so modified to introduce addition

al stresses which when combined with dead load stresses

would produce stress profile similar to full-scale members.

(2) Reinforcement and Prestressing. Since the avail

able commercial sizes of reinforcing bars and prestressing

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Description of the parameter Scale factor

Live load strains (imposed condition)

Live load stresses (imposed condition)

Linear dimension

Surface area (concrete, steel, transformed)

Volume

Moment of Inertia

Concentrated live load: stress X area

Concentrated live load moment: load X length

Dead load per unit length (function of volume)

Dead load stresses

Dead load moment

Deflection

Table 2.1 Scale Factors

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strands are limited, it is generally not possible to scale

down the steel areas exactly according to theory. However,

a combination of sizes were selected which best approximated

the scale factors.

(3) Design Code Limitations. To comply with design

limitations on the minimum thicknesses and the minimum con

crete cover set by most design codes, locations of strand

and steel bars were distorted in few instances. But the

distortions were very small and apparently did not affect

the test results.

(4) Effect of Aggregate Size. Similitude analysis

does not include the aggregate size as a parameter influ

encing the concrete behavior. However, as discussed in

Chapter I, experiments have shown that scaling the maximum

aggregate size may improve the model test result. Hence,

this refinement was considered in the design of model beams

presented in the next section.

2.4 DESIGN

2.4.1 General. A one-half scale model was proposed

for the present study. The model design procedure was

roughly the same as the full-scale members presented in

Reference (l). As a preliminary design, all the features

of the full-scale member were scaled down using proper scale

factors. To satisfy the conditions of identical stresses

in the prestressed channels, it was necessary to modify the

prestressing pattern and strand sizes. In the following


pages, the design assumptions, final design features and

various properties of the member are presented.

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2.4.2 Design Assumptions. Design of this particular

composite member is based on the following assumptions:

(1) Variations of strain are linear through the entire

composite depth.

(2) There is no slippage between the precast and the

cast-in-place concrete. That is, composite action is

present up to failure load.

(3) No slab separation takes place.

(4) The loss of prestress force is assumed to be six

percent of the initial force at the time of transfer.

(5) Total loss of prestress is eighteen percent of

the initial force.

(6) No tension is allowed in the precast section un

der working load.

(7) AASHO specifications are adopted for both working

and the ultimate design.

2.4.3 Design Values. Design computations are similar

to common methods of prestress design and hence they will

not be presented. Critical values at different stages

of design, predicted deflection, and states of stress are

presented in Appendix B.

Cross-sectional dimensions of the full scale and the

model members are shown in Figs. 2.2 and 2.3 respectively.

Section properties of the model beam areshown in Table 2.2.

I I I

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I I I I I I

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4"

60"

4"

Fig. 2.2 Cross-sectional Dimensions of the Full-scale Member

1.--------------- 3 0"

2"

Fig. 2.3 Cross-sectional Dimensions of the Half-scale Model

20

•J

--- -- - - --------

Wt Area I st sb yb yt Type 2 4 3 3 plf in in in in in in

Channel 88.5 85 262 52.1 133.0 1. 97 5.03

Slab 80.5 77.3

Composite* 168.5 191.4 1912 412.0 438.5 4.36 4.64

*Transformed to 4000 psi concrete

Table 2.2 Section Properties of the Model Member

I

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I

Figs. 2.4 and 2.5 show the reinforcing arrangements for

the channel and the top slab, respectively.

2.5 DESIGN COMPARISON OF MODEL AND FULL-SCALE MEMBERS

22

As mentioned earlier in this chapter, due to physical

conditions it was not possible to design and construct a

true model. Table 2.3 shows the actual model relationship

between the full-scale and one half scale members. Design

values are not shown, instead, theoretical ratios are com

pared with those achieved in design.

As can be seen, stresses at transfer were not the same

for the model and the prototype; however, as shown in Table

2.3 similitude of stresses were partly satisfied after cast

ing of the top slab. As mentioned before it was not possi

ble to satisfy these two stages of stress state without

hanging additional weights to the beam, therefore, this dis

tortion was unavoidable.

Due to limited sizes of strands, a combination of 4-3/8"

strands and 11-1/4" strands was used in contrast to 26-7/16"

strands used in the prototype. Due to the dead load effect,

the eccentricity at midspan was reduced considerably and elim

inated the necessity of midspan strand depression as was the

case for the full-scale member. Hence, the stress distri

bution of the model and the prototype varied throughout the

span length due to the variation of the eccentricity of the

prestressing force.

To show the magnitude of this distortion, Figs. 2.6,

I I I I I I I I I

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I

#3 @ VARXlNG $PA. WlTH ~II X ~~~ X 1/8"

L ON FACE

.---4-3/8" <P STR.

#3 STIRRUP

~~~~.2~'~1• 11-~"<jJSTR. SPA. @ 2" CTR. ~ .,.__ __________ 30"

Fig. 2.4 Channel Reinforcement

#3 $PA. @ ABT.

--- _____. \

#3 @ ABT. 12" CTR.

CORRUGATED SHEET METAL

30"

Fig. 2.5 Top Slab Reinforcement

23

7"

8"

l"

.I

l"

4~"

--Parameter

Geometric Dimensions

-- --Theoretical scale factor

0.5

Design scale factor

0.5

Stresses due to prestressing

Midspan - bottom 1 1

Midspan - top 1 1

End - bottom 1 1. 08

End - top 1 0.645

Live load moment capacity 0.125 0.125

Ultimate moment capacity 0.125 0.115

Cracking moment 0.125 0.11

Stirrup Steel area 0.25 0.25

Horizontal shear stress 1 1

Deflection 0.5 0.5

Table 2.3 Design Distortions

---Distortion

possible

X

X

X

X

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2.7 and 2.8 are presented. In Figs. 2.q and 2.7, the

bottom fiber stresses, including prestressing stress ex

pressed as i + ~e , were superimposed and plotted versus b

25

the span length for the full scale and the model respect-

ively. The region indicated as excess compression is the

net compressive stress remaining in the member after appli-

cation of the design working load. In Fig. 2.8, the net

compressive stress was plotted for the model and the pro-

totype to show the magnitude of distortion. The broken-

line curve represents an alternate trial design of model,

where strand depression at midspan was considered.

Comparison of the two solid-line curves reveals that

the net bottom stress in the model member is distorted as

high as 25 percent around the ~ points of the span length.

The broken-line curve representing the alternate design

reduces the stress distortion at end regions but does not

change the high distortion existing at ~ points. Compar-

ison of the two design approaches shows that the distor-

tion is inherent in the shape of the dead load stress dis-

tribution which is different for the model and the proto-

type.

---

H U)

Pol

1500 U) U)

~ 8 U)

p:; lCOO ~

~ H ~

~ 0 8 8 0 ~ 500

0

---

0 5

.-------+- !+ A

10

---------

11'-4"

15 20 25 30 34

SPAN - FEET

Fig. 2.6 Bottom Fiber Stress vs. Span Length, Full-Scale N 0'\

------- ----------

2000

1500

1000

500

0 0 2.5

LIVE LOAD AND IDEAD LOAD

EXCES~ COMPRES~ION

~---+---~=LIVE LOAD ---1-------1--~

5.0

5 '- 8" ~--~-- ~~--~

7.5 10.0

SPAN - FEET

12.5 15.0

Fig. 2.7 Bottom Fiber Stress vs. Span Length, Model

17

----1600

H (/) 1200 P-t

6 H Ul (/)

~ P-t ::E: 0 ()

Ul (/)

800

tj 400 :><: !:4

0

- ---

MODEL, CONSTANT ECCENTRICITY

~--~~~~MODEL, VARIABLE ECCENTRICITY~------~~------~~~--~

0

0

FULL-SCALE

2.5

5

5

10

7.5

15

10

20

SPAN - FEET

12.5

25

Fig. 2.8 Distortion of Bottom Fiber Stress

15

30

17

34

N 00

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I

CHAPTER III

FABRICATION OF THE SPECIMENS

3.1 GENERAL

Eight prestressed concrete channels were constructed

in the Civil Engineering Laboratory at the University of

Missouri. All these channels were one-half scale model of

the full scale members constructed previously. The over

all channel length was 18 feet with a supported length of

17 feet. Cross-sectional dimensions of the channel and the

top slab for both the model and the prototype are shown in

Chapter II, Figs. 2.2 and 2.3.

Top slab was cast only on three of the channels for

the purpose of this model study. The other five channels

are to be used as precast elements for the model bridge

during a separate study in the near future.

3.2 CHANNELS

3.2.1 Formwork. A combination of plywood and steel

forms was used in the construction of the channels. Forms

were used for all vertical and inclined surfaces, however,

no forms were utilized for horizontal surfaces, with the

exception of the soffit. A maximum tolerance of ±1/32-inch

was maintained for formed dimensions. A general view of the

forrnwork is shown in Fig. 3.1.

3.2.2 Prestressing. Fifteen prestressing strands

I I I

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30

were used with an arrangement as shown in Chapter II, Fig.

2.4. All strands were of A$TM A-416 Grade and were pre-

tensioned according to current precasting procedures. A

schedule of strands properties as reported by ASTM is

shown as follows:

Minimum Number of Nominal Ultimate Tensioning Total

Strands Diameter Strength Load Load in lb lb lb

11 1/4 9000 6300 69300

4 3/8 20000 14000 56000

Tensioning load is 70 percent of the minimum speci-

fied ultimate strength. To account for the force lost

due to slippage of the strand vices and the deformations

of the prestress frame during the tensioning operation,

the cables were overtensioned to 80 percent of the speci-

fied ultimate strength. To check the actual force in the

cables, the strands were retensioned and the load was

checked with a test shim, if the load was less than that

required, metal shims were inserted between each strand

vice and the bearing plate on the prestressing frame.

The prestressing load was applied by means of a 30-ton

hydraulic jack (Fig. 3.2) and was measured by means of a

pressure cell connected to a strain indicator.

3.2.3 Shear Reinforcement. Two types of stirrups

were used in each channel. The two stirrup types and the

I I I I

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31

Fig. 3.1 Formwork for Prestressed Channel

Fig. 3.2 Pre-tensioning Operation

I I I I

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32 arrangement are shown in Fig. 3.3. All stirrups used were

No. 3 bars with minimum yield strength of 40 ksi. In chan

nel No. 3 a No. 3 bar was welded to the top of all the stir

rups in the longitudinal direction on each side of channel.

This extra reinforcement would prevent excessive cracking

of the channel under high live load and prevent spreading

of channel legs as was observed in the full-scale tests.

3.2.4 Casting and Curing. Since the prestressing

bed is 40 feet long, it was more economical and time saving

to cast two channels at the same time. A total of four

castings of two channels each was made in the laboratory.

Concrete for construction of these members was obtained

from a local ready-mix plant and the maximum aggregate

size specified was 3/8-inch. The concrete was cast and vi

brated thoroughly by shaft vibrators to prevent honeycomb

and to assure complete filling of the forms. The surface

of the concrete was levelled by means of a screeding angle.

Fig. 3.4 shows a typical casting and the crew at work.

To simulate the current practice of precasting, a

steam curing process was used. The curing cycle consist

ed of a 5-hour period of initial set at laboratory temper

ature followed by a 13-hour steam curing period. The pur

pose of the initial curing period was to prevent any dis

turbance of the chemical hydration of the concrete by too

early application of the steam. During the 13-hour period

of steam curing, the prestress bed was covered with insu-

-- --- ------

~-- -~ TYPE 'A' STIRRUP TY~E 'B' STIRRUP

D

~ r------ 13TYPE 'A I AT8" = 8'

I I· 9TYPE'B'AT8 11 = 5'-4"'-"-------1~

__, 4"1- L -1 8'' PLAN VIEW

Fig. 3.3 Stirrups Arrangement

---

w w

I I I


I I I

34

Fig. 3.4 Casting of the Channels

I I

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I

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I

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35 lating plastic and steam was controlled to maintain the

temperature rise and maximum curing temperature, Fig. 3.5.

Maximum rate of rise of temperature was 40°F. per hour and

maximum curing temperature was l75 ° F. Two continuous re

cording thermometers (Fig. 3.6) were used to record the con

crete temperature at different locations, which were approx

imately maximum and minimum conditions . . Concrete temperature

versus curing time is presented in Fig. 3.7.

3.2.5 Companion Specimens. Four creep and shrinkage

prisms and six 6-inch diameter cylinders were made during

each casting, and were cured in the same manner as the pre

stressed channels. The prisms were instrumented shortly

after curing and creep prisms were loaded to find the long

term creep coefficient of the prestressed concrete. Three

of the 6-inch diameter cylinders were tested prior to re

lease of the strands and the remaining three were tested

prior to the flexural tests to determine the compressive

strength and modulus of elasticity of the concrete at these

times. These data are given in Table 3.1.

3.3 TOP SLABS

Formwork used in casting of the top slab was relatively

simple. 3/4-inch plywood boards were aligned in position

and tightened against the vertical sides of the channel by

means of adjustable chain clamps. Corrogated metal sheets

bent into arches of the required curvature were used to form

the void space inside the composite beam.

I I I I I I I I I I I I I I I

I I I

36

Fig. 3.5 Steam Curing Operation

Fig. 3.6 Time-Temperature Recording Devi ce

--

!i.. 0

j:LI p:; ::J 8 ,::r:; p:; j:LI

~ .:. j:LI 8

j:LI

8

~ u ~ 0 u

-- -- -180

170

160

150

140

130

120

110

~ ....... / v~ -!...--\" [\

~ ~ K\ ~ ~Y\

,Vf; v \ \\ [\_ \~ ~ FIRST CASTING

/A v \ SECOND CASTING

~~ ~ THI RD CASTING 'I II j\._ FOURTH CASTING ~\~ 1/ \\

IJ \ if

100

90

80 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

TIME AFTER APPLICATION OF STEM~ - HOURS

Fig. 3.7 Concrete Temperature During Steam Curing

--

w -...)

--- -- ------ -----Release Modulus of

Specimen Strength Test f' Elasticity Creep Cement Slump (psi) (psi) c (ksi) Coefficient Factor* (in.)

Top slab 1 5675 4080 6 3~

Top slab 2 5940 4100 6 3

Top slab 3 5910 4200 6 3

Channel 1 2700 5000 4400 0.79 6 2

Channel 2 2700 5000 4400 0.79 6 2

Channel 3 3700 7900 4710 0.65 8 2~

# Channel 4 & 5 3580 8 2~

# Channel 6 & 7 2830 8 3~

# Channel 8 3700 8 2~

# No test results are reported i n this paper

* Bags of Cement per cubic yard

Table 3.1 Concrete Information From the Companion Specimens

w 00

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39

3.3.1 Reinforcement. To provide adequate resistance

to the horizontal shear stress developed at the interface

of the top slab and the channel No. 4 bars were bent to

approximately go o and welded to the angles imbedded in the

channel leg (Fig. 3.8). These angles were in turn welded

to No. 3 bars which extended for about 4-inches in the chan

nel legs and were installed during the casting of the chan

nels. Compression steel consisted of five No. 4 bars equal

ly spaced in the longitudinal direction. No. 3 bars spaced

at 12 inches were used as a nominal transverse reinforce

ment. Reinforcements in both directions were welded to

the shear connector at common points to help prevent spread

ing of the channel legs in the transverse direction under

high loads near the failure, (Fig. 3.8 and 3.9). It should

be noted that channel spreading is only observed in members

~ested individually, while in an actual bridge slab, this

spreading is resisted by the adjacent channels.

3.3.2 Casting and Curing. The top slab concrete was

cast in a manner similar to that of the channels. Those

portions of the channel exposed to fresh concrete were

coated with wet sand mortar to prevent honeycomb and pro

vide better bond. The concrete was vibrated thoroughly,

screeded, and given a dry brush finish (Fig. 3.10).

Fresh concrete was covered with plastic sheets for two

days, after which the top slab was covered by wet cloths

for at least one week.

I

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40

Fig. 3.8 Welding of Reinforcement

Fig. 3.9 Reinforcement Arrangement

I I I I I I I

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41

Fig. 3.10 Cast-in-place Top Slab

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I

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I

I

CHAPTER I V

TESTING PROCEDURE

4.1 FLEXURAL TEST

Three composite model members were tested in this

study. These tests were conducted to determine the flexural

characteristics of the members such as deflections, strains,

and mode of failure. The mechanism of testing was as sim

ilar as possible to the full scale members tested previous

ly so that comparable test data could be obtained.

A loading system consisting of two third-point loads

was used to simulate the prototype loading and to provide

a constant moment region for instrumentation. Each load

was distributed transversely by means of a spreader beam

6 inches wide and 30 inches long. Loads were applied to

the test specimen by means of two 20-ton hydraulic rams.

To maintain stability in the loading s y stem and to transfer

the load to the two loading points evenly, a longitudinal

load distributor beam was used. Fig . 4.1 shows the jacks

and test frame attachments, the distributor beam, spreader

beam, and the test specimen.

Each test specimen was supported on two box beams

suspended from the testing frame by means of one inch di

ameter rods. Both of the supports allowed freedom of

rotation and horizontal movement which would cause an un-

I I I I I I

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~I

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I

43

stable condition under applied horizontal loads. Since

the applied loads were only vertical, the stability re

quirements were satisfied, resulting in a simply supported

span length of 17 feet. Due to elongation of the rods un

der the applied load, a condition of support settlement

existed. Such a condition introduced some error in the

midspan deflection readings, consequently, a dial gage was

installed under each support to measure any possible settle

ment. Fig. 4.2 shows a test specimen on the suspended sup

ports, the dial gage under the support, and a partial view

of the test frame.

4.2 INSTRUMENTATION

The test specimen was instrumented to measure deflec

tions, strains, slip and separation at the interface of

the top slab and the channel.

Two 0.001-inch dial indicators were used to measure

the deflection at midspan. One dial was rigidly attached

to the laboratory floor beneath the specimen to measure

deflection at the bottom surface. Another dial was attached

securely to the test frame above the member to measure

deflection at the top surface. Fig. 4.1 shows the top

indicator in place. The top dial indicator readings were

corrected to account for the displacement in the test frame.

Concrete strains were measured using clip-on strain

meters. These 6-inch gage length strain meters consisted

of flexible aluminum brackets upon which four wire resist-

I I I

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I

Fig. 4.1 Loading System

Fig. 4.2 Suspended Support and a Partial View of the Frame

44

I

I

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I

I

45

ance electric strain gages were mounted in the form of a

full-bridge circuit. A detailed discussion on the mechanism

of these strain meters can be found in reference (9). Each

strain meter was calibrated prior to the flexural tests.

Three strain distributions were measured along the

depth of the specimen. Two series of strain meters were

located at midspan, one on each side of the specimen. The

third series was mounted at 1/6 of the span length distance

from the support, but only on one side of the member. Fig.

4.3 shows a typical series of strain meters, while Fig. 4.5

shows the dimensional location of the strain meter.

A power supply, switching units, and digital volt

meter were utilized to record strain data. All the read

ings taken from the digital voltmeter were later converted

to strains, using the calibration factors of the strain

meters obtained prior to testing. Fig. 4.6 shows the set

up of these instruments.

Horizontal slip at the interface of the top slab and

the channel was measured by means of ten .001-inch dial

gages on each side of the member. Slab separation was meas

ured by four .0001-inch dial gages. Fig. 4.3 shows a slip

dial mounted beside the strain meters, while Fig. 4.4 is a

close-up view of a slab separation dial.

The applied loads to the specimen were measured by

means of a pressure cell containing strain gages which in

turn was connected to a strain indicator device. The

I I I I

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46

Fig. 4.3 Strain Meters and Slip Dial

Fig. 4.4 Slab Separation Dial

-------------------

-r-3"

+-21<" _L

1" f

Fig. 4.5 Location of the Strain Meters

I I I I I I I I I I I I I I I I I I I

48

jacks, pressure cell, and strain indicator were calibrated

prior to testing with a mechanical screw-driven testing

machine. Fig. 4.7 shows the loading set-up used in these

tests.

Fig. 4.8 shows the location and arrangement of deflec

tion dials, strain meter, horizontal slip dials, and slab

separation dials for specimen No. 1. From results of test

No. 1, it was found that a few of the slip dials could be

omitted without reducing the useable data obtained from

the test. Consequently, those dial gages marked by (*) in

Fig. 4.8 were omitted during Tests No. 2 and No. 3.

4.3 TESTING PROCEDURE

In order to measure permanent deformation and inelastic

slip between the channel and the top slab, the test speci

mens were loaded cyclically, with the maximum load in each

cycle higher than that of the preceding cycle. A loading

increment of one kip was used for the loading range from

zero to 10 kips which was 2 kips higher than the theoreti

cal cracking load. Since all members were cracked notice

ably at 10 kips, a loading increment of 0.5 kips was used

for the range from 10 kips to failure. Four kip increments

were used during unloading.

I I I I I I I I I I

I I I I I I I I

49

Fig. 4.6 Strain Measurement Instruments

Fig. 4.7 Loading System

-------------------

*

2'-10"

5'-8" 1'-6"1'-6" 1'-6" 1'-6" 1'-6"

8'-6"

c STRAIN METER * Omitted in Tests No. 2 and No. 3.

§ SLIP DIAL

aD SEPARATION DIAL

Fig. 4.8 Flexural Test Instrumentation

*

Vl 0


5.1 GENERAL

CHAPTER V

RESULTS AND COMPARISONS

In the first part of this chapter an attempt is made

to present the data, collected during the construction and

testing periods, in the most informative way. Next, these

data are utilized to examine the similitude relationships

established earlier in this study. This is achieved by

extrapolating the model test results and comparing them with

the available test data from three full-scale members. How

ever, due to the limited scope of this investigation, the com

parisons do not include such relationships as moment-cur

vature, shear capacity of the members or crack spacing, etc.

Finally, ultimate capacity and mode of failure of the test

ed members are discussed.

5.2 EXPERIMENTAL RESULTS

5.2.1 Deflection History of the Prestressed Channels.

Time-deflection data were collected during the curing per

iod for all the half-scale channels. Deflection readings

were taken from the time of the release of the prestressing

force until the time of preparation for the flexural test.

The deflection instrument consisted of three parts as

shown in Fig. 5.1. At the time of forming the top slab,

the deflection device was removed and further readings were


Fig. 5.1 Deflection Instrument

52

I I I I

I I I I I I I I I I I I I I

53

taken by means of .001" dial gage mounted beneath the chan

nel at midspan.

Figs. 5.2, 5.3, and 5.4 show the predicted and actual

time-deflection of the three model members tested. Deflec

tion is given in inches and time measured as days of elapsed

time after casting of the prestressed channel.

The initial portion of the predicted deflection curve

is the elastic deflection immediately following release of

the prestressed force. Actual concrete strength, as found

from 6-inch diameter cylinder tests prior to transfer of

prestressing force, was used in this prediction. A creep co

efficient was computed for every channel from the creep meas

urements of the companion specimens. These coefficients are

listed in Chapter III, Table 3.1. Predicted deflection was

increased for the first 30 days after release of the pre

stress force utilizing these creep factors. The creep coef

ficients were assumed to stay constant after 30 days, resulting

in the horizontal portion of the prediction line prior to

casting of the top slab. Deflections due to placing of forms

and casting of the top slab were computed by conventional

elastic methods.

5.2.2 Load-Deflection Relationship. Deflection was

measured with two dial gages at the midspan as shown in var

ious figures in Chapter IV. The top dial readings were cor

rected for the elastic deflection of the test frame caused

by the applied load. These corrected values were in turn

-------------------

0.3

0.2 Ul

•• •••• • • • • • • ••• • •

... --.. 14 ::r: • ~ u 0.1 z H.

,.-

l ....._ ....-

z 0.0 • .. 0 H 8 • EXPERIMENTAL u -0.1 14 ....:! PREDICTED li-t 14 Cl -0.2

,.__ PLACING OF FORMS

¢:=1 CASTING OF TOP SLAB

-0.3

0 10 20 30 40 50 60 70

TIME - DAYS

Fig. 5.2 Channel No. 1 Deflection History

-------------------

0.3

0.2 (I)

I'Ll ::r:

• • • • u 0.1 z H

1'""

v z 0.0 0 H • E-t u -0.1 I'Ll --....::1 ~ --I'Ll Q -0.2

~

-0.3

0 10

• • •• •••

• ••• -1--

. ~

EXPERIMENTAL • • • • PREDICTED

PLACING OF FORMS

CASTING OF TOP SLAB

20 30 40 50

TIME - DAYS


60 70

U1 U1

-------- ----------

0.3

0.2 Ul ~ tt: • • • • u 0.1 z H

1--- • • • • •

• ... """ =

z 0.0 0 H E-t • EXPERIMENTAL • • • u -0.1 ~ H

-- PREDICTED lil. ~ - PLACING OF FORMS c::l

-0.2 ¢::=:1 CASTING OF TOP SLAB

-0.3

0 10 20 30 40 50 60 70

TIME - DAYS



57

corrected for support settlement. As was mentioned in the

previous Chapter, two dial gages were installed under the

supports and the average of these two readings was consid

ered to be the support settlement.

Subtracting the support settlement from the deflection

dial reading would give the net deflection due to the ap

plied load.

The load deflection relationships from the three flex

ural tests are shown in Figs. 5.5, 5.6, and 5.7. In these

figures, the curve designated as (M) is the load deflec

tion curve for the model member and the solid circles

represent the experimental results. It should be noted

that the curve line is not a theoretical deflection pre

diction and is simply connecting the solid circles to dem

onstrate the smooth flexural behavior of the model members.

In Fig. 5.8 all three tests were combined together to pro

duce the curve band designated as (M). In all load-de

flection diagrams, the curves marked (P) are the prototype

predictions from the models and the curves denoted by (F)

are measured prototype deflections.

5.2.3 Load-Strain Data. As discussed in Chapter IV,

three strain distributions were measured on each test spec

imen. Two series were located on each side of the member

at midspan and the strain measurements of these two distri

butions were averaged wherever used in this chapter. The

third distribution was located in the shear span at L/6 from

-------------------80

70

60

:::.:; u o::t: IJ 50 ~ ~ p.

U) 40 P-c H :::.:;

Cl 30 o::t:: 0 H

20

10

0

p -----~ ---___::::::. ~ F-3

:::--~

~ ~

~

/ v

~ ;) MODEL TEST NO. 1

F: FULL-SCALE

7 M: MODEL

P: PREDICTED

7 -~ ~ M

--v 0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0

DEFLECTION AT MIDSPAN - INCHES

Fig. 5.5 Load-Deflection Similitude

V1 00

-------------------

~ u l'l! t-)

~ r:4 p..

Ul Pi H ~

Q ~ 0 H

80

70

l?_ --------;:; 60 ~

~/ ~ 50 ............: ~

40

30

20

~

/ ;/

) ~

( I .---------......

r 10

0 0 1.0 2.0

HODEL TEST NO.

F: FULL-SCALE

H: MODEL

P. PREDICTED

M

3.0 4.0 5.0 6.0 7.0 8.0



2

9.0

-- ---------- -- --80

----~---p

60

~ u ~ ~ 50 p:; J:il P-t

Ul 40 P-t

H ~

Q 30 ~ 0 ....:!

20

---------~3 -------~ __.... =----

~ ~

/

/ ~ u

I / MODEL TEST NO. 3

F: FULL-SCALE

I M: HODEL

P: PREDICTED

/_ ~ .. H

~

r

70

10

0 0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0



--------- -- -----80

~

..,.,., ~"""

!'1fl+ ~ 70

60 F-3 l?

.... ~ u F:t:

50 IJ

~ ll:l 1=4

(/). 40 1=4 H ~

n 30 F:t: 0 H

~

~ ~ \

""" F-2 F-1

A .t-1: MODEL

'f P: PREDICTED

F: FULL-SCALE

20

J ~ ~ M

v } 10

0 0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0




the support, where L was the simply supported length of

the member.

A typical linear distribution of strains along the

depth of the member is shown in Fig. 5.9. This figure

shows strain measured in the test specimen at given load

level in micro-inches per inch, versus gage location in

inches.

62

The major portion of the strain data 1s presented in a

series of load-strain similitude diagrams shown in Figs.

5.10, 5.11, and 5.12. These diagrams compare the strain

of the model and the prototype at a fixed location on the

member. In all three diagrams the curve band denoted as

(M) represents the experimental results obtained from the

three model tests, while the band marked (P) is the prototype

prediction from the model and the solid curves are the

measured prototype strains. Compression strain curves

were based on data collected at a distance of 0.5 inches

from the extreme top fiber. On the other hand, tension

strains presented here were measured at a distance of 3.5

inches from the extreme bottom fiber. Tension strains were

plotted only up to the load which caused the cracks to

reach the measured location. The appearance of cracks at

the measured location in each test is marked by an open

circle as seen in Fig. 5.11.

To present the data in a realistic fashion, the ex

perimental relationships are shown as curve bands. It was

-------------------:-r- • LOAD = 4 KIPS ,

' ~/ / N

I 0 LOAD = 8 KIPS

// v o::l a LOAD 12 KIPS J ~ = H U) IY v - ~ I LOAD = 16 KIPS

-/ ~ v

// / :: / I [' / _7 -~ I

- _/ H

7 7 7 ~ i<i: / 0 / / I

I /o I -'-- / of 4

2700 2100 1500 900 300 0 300 900 1500

STR~IN - 10-6 INCHES PER INCH

Fig. 5.9 Typical Strain Distribution at Midspan

-------------------80

70

60

• MODEL TEST 1 ~ u • MODEL TEST 2 .:t: 50 1-:1 0 MODEL TEST 3 p::; rLI M: MODEL 1=4

Ul 40 F: FULL-SCALE 1=4 H P: ~ PREDICTED

30 Q

.:t: 0 ....:!

20

10

0 0 250 500 750 1000 1250 1500

COMPRESSION STRAIN - 10-6 INCHES PER INCH

Fig. 5.10 Load-Strain Similitude at Midspan

--------- --- ----

::.:: u .::!! ~

Pi rz:l ~

(/)

~ H ::.::

0 .::!! 0 H

50 ~------------~-------------.r-------------,-------------~--------------,

40

• MODEL TEST 1

• MODEL TEST 2

0 MODEL TEST 3 30

M: MODEL

F: FULL-SCALE

P: PREDICTED

20

10

0 0 200 400 600 800

TENSION STRAIN - 10-6 INCHES PER INCH

Fig. 5.11 Load-Strain Similitude at Midspan

m tJl

- - --- - - --80~--------------~--------------r---------------r-------------~~-------------,

60

:::,::; • MODEL TEST 1 u ,.:( • MODEL TEST 2 F-). 50 ~ 14 0 MODEL TEST 3 P<

Ul 40 M: MODEL

fll H F: FULL-SCALE :::,::;

P: PREDICTED

Q 30 ~ 0 H

20

0 100 200 300 400

COMPRESSION STRAIN - 10-6 INCHES PER INCH

Fig. 5.12 Load-Strain Similitude in Shear Span


67

assumed that precision o~ the strain measurements were ± 10

micro-inches per inch. Therefore, to pass a curve line

through the points obtained from each test would have been

unrealistic. At best, one can confine the scattered data

by a curve band which contains all the points. Also,

different load-strain scales were used in these diagrams

for a clearer presentation.

5.2.4 . Slip and Slab Separation Data. A number of

dial gages were installed on each test specimen to measure

slip and slab separation at the interface of the two con

crete portions. Consistent load-slip data would provide

the means to estimate the shear force at the interface and

that carried by the shear connectors. However, test re

sults showed that no appreciable slip occurred in any of the

test members up to the vicinity of the failure load. Hence,

the load-slip data were insufficient to predict the hori

zontal shear force transferred into the shear connector.

Nevertheless, at load levels close to failure small

magnitudes of slip, in the order of one-thousandth of an

inch,were recorded. The magnitude of the load at the on

set of these small slips is of some significance and may

be termed as ''slip load", these load levels are listed be

low and are shown as (x) in Fig. 5.13.

Test specimen No. 1 15 . 0 kips per jack

Test specimen No. 2 13 . 5 kips per jack

Test specimen No . 3 1 7 . 0 kips per jack

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70

60

50

0 0

MODEL TEST

FULL-SCALE

PREDICTED

SIMILITUDE CURVE BAND PREDICTED FROM MODEL TESTS

0.25 0.50

)<.

3 • •

0.75

MODEL SCALE FACTOR

1.0

Fig. 5.13 Slip Load Similitude

68


69

With the exception of specimen No. 2, none of the test

specimens showed any slab separation up to the failure load.

Test data of member No. 2 showed slab separation at 14 kips

per jack. At 15.5 kips per jack (just before failure) a

slab separation of 0.021 inches was measured.

5.2.5 Ultimate Load Capacity. The failure load was

considered to be the load at which the test specimen was

not capable of carrying an increase of load and collapsed

within a short span of time. The failure loads of the three

specimens tested are listed in Table 5.1 and are shown as

(x) in Fig. 5.14.

5.3 COMPARISON OF MODEL AND FULL-SCALE TEST RESULTS

Three full-scale members were tested in a previous

1 study and the results have been reported . These tests

showed that at high loads close to failure, behavior of the

test specimens was greatly influenced by the efficiency of

the shear connectors. Loss of composite action was encour

aged as a result of spreading of the channel legs and the

excessive slab separation. To remedy these effects, arrange-

ments of the top slab .reinforcement and the shear connector

were modified in the third full-scale member and all the

model specimens. Therefore, only full-scale specimen No. 3

can be considered as the true prototype for the model mem-

bers with regard to the ultimate load capacity.

Nevertheless, to broaden the scope of this comparison

------------- -----

Test Specimen Failure Load Predicted Full-Scale Percent of Kips per Jack Failure Load-Kips* Error

Model No. 1 18.00 72.00 +4.35

Model No. 2 16.00 64.00 -7.25

Model No. 3 19.00 76.00 +10.1

Full-scale No. 3 69.00 69.00 o.oo

*Predicted failure load for full-scale members based on model failure load

Table 5.1 Ultimate Load Capacity

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~ u ~ t-:l

p:; I'Ll ~

U)

~ H ~

Cl ~ 0 H

71

75

70

MODEL TEST X

FULL-SCALE 3 • PREDICTED •

60

50

SIMILITUDE CURVE

BAND PREDICTED

FROM MODEL TESTS

40

30

0 ~~------~--------~--------~------~~---0 0.25 0.5 0.75 1.0

MODEL SCALE FACTOR

Fig. 5.14 Ultimate Load Similitude

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72

in the discussion to follow, with the exception of the slip

behavior and ultimate capacity, test results of three model

members are compared with experimental results of ~11 the

prototype members tested previously. The following nota-

tions are used for the prototype members:

Full scale specimen No. 1 F-1



5.3.1 Deflection Similitude. As was mentioned pre

viously, the dead load of the model channel introduced dis-

tortions in prestressing design which in turn distorted

the time-deflection data. Consequently, deflection his-

tory of the model channels can not be correlated to the

prototype time-deflection data. However, the measured

deflections were in reasonable agreement with the computed

curves based on conventional elastic methods, as shown in

Figs. 5.2, 5.3, and 5.4.

Applied load-deflection similitude of the composite

members is illustrated in Figs. 5.5, 5.6, 5.7, and 5.8.

Predicted curves were obtained using the following rela-

tionships: 2

Predicted load = 1/k (model load)

Predicted deflection = 1/k (model deflection)

Where k = ~; scale factor

In Figs. 5.5, 5.6, and 5.7, the model members

are considered separately and the corresponding pre-

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73

dieted curves are compared with the full-scale test No. 3.

In Fig. 5.8, the above similitude ratios were utilized to

obtain the predicted curve band from the model curve band.

In all cases, close agreement was observed between the pre

dicted load-deflections and the experimental results ob

tained from the three full-scale tests.

The nonlinear portions of the predicted curves have

slightly steeper slopes than the experimental curves from

the full-scale tests. This phenomenon has been observed in

some experimental work on models,and some investigators, 2 ' 7

have related this to the "scale effects" inherent in re

duced-scale models. In this particular study, in spite of

the utmost care taken to produce a true geometric model,

some slight distortions were introduced in the model mem

bers. The stiffness of the models was slightly increased

by the presence of the additional #3 bars used in stirrup

cages (Fig. 2.4).

Also, better composite action was observed during the

flexural tests on the model members. Improved composite

action would mean slower propagation of cracks at high load

which in turn would result in a stiffer section. This in-

creased stiffness was particularly observed in the third model

member (Fig. 5.7) where an additional #3 bar had been

placed in the channel legs in the longitudinal direction

at l-inch from the top. The extra reinforcement prevented

the cracks from propagating to the interface of the two com-

I

74

ponents, and thus i~proved the co~posite action.

It can be noted that predicted deflections and full

scale test results are in much better agreement in the elas

tic range than in the nonlinear region. Divergence from

the similitude theory could be due to many effects such as

cracking pattern, mode of failure, and the nonlinear be

havior of concrete. Nevertheless, the maximum error ob

served in all three tests was approximately 10 percent, and

this is often considered to be acceptable for test results

involving concrete in nonlinear range.

5.3.2 Strain Similitude. The typical strain profile

shown in Fig. 5.9 indicates that composite action existed

up to about the failure load. The location of the neutral

axis was calculated as 4.36 inches from the bottom of the

composite member, while the strain distribution shows the

neutral axis to be between 4.4 and 4.6 inches for strains

in the elastic range.

Load-strain similitude is illustrated in Figs. 5.10,

5.11, and 5.12 by comparison of the predicted curve bands

and the various load-strain curves obtained from the full

scale tests. Based on an ideal material, it was assumed

in Chapter II that stresses and strains are identical for

both prototype and the model regardless of the scale factor.

The relationships shown in Fig. 2.l.c were assumed as the

basis for the model design. However, such a condition is

unrealistic and almost impossible in the case of concrete,

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75

even for two samples from the same batch. Quite often, Fig.

2.l.a represents the actual condition, where corresponding

stresses and strains in model and the prototype are related

by constants different than unity. Stress-strain curves

for the model and the prototype must be established from

compression tests on concrete cylinders.

In this study, stress-strain data in the elastic

range and the f'c obtained from the concrete cylinder tests

revealed that the compressive strength of concrete in the

models and in the full-scale members was of the same order

of magnitude. Hence, the assumption of identical stresses

in the models and the prototypes was adopted. Thus, the

following relationships exist:

therefore;

Based on the foregoing discussion, the predicted bands

were obtained using the following similitude relationships:

Predicted load = l/k2

(model load)

Predicted strain E

= m (model strain) Ef

where k = 1/2 is the scale factor

Utilizing the above factors, experimental data from

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76

each model test resulted in a si~sle prediction curve.

The predicted band is a combination of the three predic

tion curves. Since prototype No. 3 was the only one which

had similar top slab steel arrangement to the models,

the moduli of Elasticity (E) for this member was used as Ef

in the above correction. It should be noted that for com

pression strain correction, E of the top slab was used,

while E of the channels was used for tension strains. Of

importance is the fact that such a correction is true in

the elastic range but would be crude in the nonlinear re

gion of the load-strain diagram.

Predicted loads versus compression strains at midspan

show good agreement with the actual full-scale tests. The

closest agreement was observed for full-scale specimen No. 3

where the discrepancies were least in the elastic range and

greatest in the nonlinear region. Near failure a differ

ence of 4% to 7% was found between the predicted band and

F-3. Good agreement was also observed in the tension strain

data at midspan. Once again, full-scale test No. 3 showed

better agreement than tests No . 1 and No. 2. For the third

test, the load-strain curve coincided with the upper boun

dary of the predicted band. Load-strain data at the L/6

location (Fig. 5.12) varies considerably for both the model

and the prototype test series. Consequently, the pre

dicted band is rather broad at high loads . It should be

noted that L/6 l ocation was in the shear span, and the

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77

strain measurements were atfected by shearing distortions.

5. 3. 3· Load-s lip B'ehavior. As mentioned before, load

slip data were not conclusive as far as the shearing capac

ity of the shear connector is concerned. The load at which

measurable slip initiated was termed "slip load" and was

indicated in Fig. 5 .13. This 'slip load similitude" diagram

is introduced by the author to illustrate an interesting

feature of similitude laws. For any geometrically true

model, load capacity, whether slip load or ultimate load,

of the prototype can be obtained by multiplying the cor

responding load capacity of the model by the square of the

scale factor. As a result, the load capacity of a member

can be expressed in the following form:

P = C(K) 2

where P = the predicted load,

K = the model scale factor, and

C = model load determined by the experimental results.

In this model study, the scale factor was constant, hence

three prediction functions were obtained, which differed

only by the constant C. Fig. 5.13 shows the band gen

erated by these three functions plotted for various scale

factors K. The slip load of prototype No. 3, with the

scale factor of unity, falls within the band and is shown

in Fig. 5.13.

5.3.4 Ultimate Load and Mode of Failure. In general,

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78

failure of the models in each test resulted from a combina

tion of loss of composite action, slight slab separation,

and diagonal tension cracking. However, the ultimate load

capacity of each member was different due to varied influ

ence of these factors.

In test specimen No. 1, moment cracks appeared, as

expected, at approximately 8 kips per ram. At a loading

of 16 kips per ram, diagonal tension cracks developed 1n

the shear span in the vicinity of the support. At 18 kips

per ram, this diagonal tension cracking was accompanied by

some measurable slip, and complete loss of composite action

which allowed tension cracks to propagate into the top slab

which in turn resulted in complete failure of the member

(Fig. 5.15). Maximum slip was measured in the vicinity

of the loading point, at a distance of 3 feet from the mid

span. Fig. 5.16 shows the slip surface.

Although specimen No. 2 was cast at the same time and

from the same concrete batch as member No. 1, its ultimate

load was lower and its mode of failure was slightly dif

ferent. Horizontal slip started at a load of 13.5 kips

per ram and increased rapidly with increase of the applied

load. In addition to horizontal slip, excessive diagonal

tension cracks were observed at the support. A large gap be

tween the top slab and the channel accompanied by a widen

ing tension crack at the support initiated the failure at

16 kips per ram (Fig. 5.17). At the same time, due to loss

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79

Fig. 5.15 Model #1, Diagonal Tension Failure

Fig. 5.16 Model #1, Slip Surface Near Midspan

----------------------------------

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of composite action, the top of the channel was under ex

cessive compression strains which resulted in a secondary

compression failure of the channel as shown in Fig. 5.18.

Failure load of member No. 3 was larger than those

80

of the previous two tests, and complete composite action

was maintained prior to reaching the failure load of 19.0

kips per ram. The additional #3 bar which was mentioned

earlier was very probably responsible for this high

horizontal shear resistance. It has been observed that loss

of composite action is encouraged by the cracks which prop

agate into the top slab. The additional reinforcing bar

curbed these cracks as shown in Figs. 5.19 and 5.20. It

can be observed from these figures that the diagonal cracks

become horizontal near the composite interface. Failure

resulted from excessive diagonal tension cracks, loss of

composite action, and "negative moment" 1 over the support.

An ultimate load similitude diagram is presented in

Fig. 5.14 which was obtained in a similar fashion to that

of the slip load similitude diagram discussed previously.

Once again only results from prototype No. 3 were compared

with the model tests due to the reinforcement similarity

of this specimen to the models.

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Fig. 5.17 Model #2, Slab Separation and Diagonal Tension Crack

Fig. 5.18 Model #2, Shear Cracks and Secondary Compression Failure in the Channel

81


Fig. 5.19 Model #3, Moment Cracks and Secondary Compression Failure

82

Fig. 5.20 Model #3, Diagonal Tension and ''Negative Moment" Cracks at the Support


CHAPTER VI

SUMMARY AND CONCLUSIONS

6.1 SUMMARY

Similitude characteristics of a proposed composite

member consisting of a prestressed precast channel, an in

terior void form, and a top slab of cast-in-place concrete

were studied. The study was based on laboratory investiga

tion of three one-half-scale composite members together

with theoretical model analysis developed for the particu

lar member.

A mathematical similitude analysis for structural mod

els was developed in general terms and is presented in

Appendix A. Based on the model ratios established, sig

nificant design values for a one-half scale model are

presented in Appendix B. In addition, a discussion on the

unavoidable model distortions inherent in concrete struc

tural models is presented.

In order to check the agreement of the theoretical

similitude ratios with the experimental determined ratios,

a series of one-half-scale models was constructed and

tested. Eight prestressed channels 18 feet long, 30 inches

wide, and 7 inches deep were fabricated. Top slabs were

cast on three of these channels and the remaining five

were retained f or further testi ng. The results of flexural


84

tests performed on the three composite members are present

ed. All the specimens were tested as simply-supported mem

bers with concentrated loads applied at the one-third points.

Instrumentation of each test specimen was designed to meas

ure deflection, strain distributions, slip and separation

between the two components of the composite member. Stan

dard dial gages were used in measuring deflection, slip,

and separation. Strains were measured by means of clip-on

strain meters with a gage length of 6 inches.

Hydraulic rams were used to apply point loads to the

member. The magnitude of the applied load was measured by

means of a pressure cell containing electric strain gages.

Cyclic loading was used with every cycle having a higher

maximum value than the previous cycle. This procedure

was continued until failure of the test specimen.

During the curing period, deflection history of the

prestressed channels was maintained by means of a vibrating

reed instrument. Experimental results from three prototype

members tested previously were compared with the model

test results. Similitude relationships were examined in

the cases of load-deflection, various load-strain measure

ments, slip-load, and the ultimate load. Generally, good

agreement between the theoretical model ratios and the ex

perimental data was observed. Model members were found to

be slightly stiffer than the prototype members and propor

tionately higher failure loads were observed for the models

I I I I I I I I I I I I I I I·

I

than for the prototypes.

6.2 CONCLUSIONS

Based on the results of this experimental study, the

following conclusions may be drawn:

85

1. The overall structural response of one-half-scale

models was in close agreement with the design calculations.

The experimental elastic deflection, cracking load, ul

timate load, and the mode of failure were similar to the

design computations listed in Appendix B. This indicates

that the common design practices can be applied to small

scale members without encountering any major discrepancy.

2, Utilizing the theoretical similitude ratios, the

one-half-scale model of the proposed composite member pre

dicted the structural behavior of the prototype member

satisfactorily. Experimental results of the prototype

tests agreed closely with these predictions.

3. Load-deflection of the prototype can be predicted

by one-half-scale model tests with negligible amount of

error. However, the model members are slightly stiffer than

the prototype members and the resulting predictions might

be slightly less conservative.

4. Load-strain relationship of prototype members can

be predicted by flexural tests on one-half-scale model mem

bers with sufficient accuracy. Results of this investi

gation showed a maximum error of 16 percent. Much of this

error was due to the difference in the material behavior of

I I I I

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86

the concrete used in model and the prototype members

tested. In order to obtain more dependable results, the

author suggests that complete stress-strain curves be ob

tained for both model and prototype members through com

pression tests of 6-inch diameter cylinders. Later, these

curves could be used to correct the experimental load-strain

diagram more precisely. Also, better agreement may be ob

tained by casting both the model and the prototype members

at the same time and from the same concrete batch.

5. Ultimate load capacity of full-scale members can

be predicted quite accurately by testing one-half-scale

models. A maximum discrepancy of 10 percent was observed

in this study, which was primarily due to more complete com

posite behavior of the model members.

6. Shear capacity of the shear connector, and the hor

izontal shearing force exerted on this connector could not

be determined by the instrumentations used in this series

of tests. Push-off tests should be designed and conducted

to determine the load-slip characteristics of the type of

shear connector used in this study. It is suggested that

an independent shear connector study be conducted to eval

uate the load transfer characteristics of the proposed con

nector.

7. Composite action of the model members under applied

load can be improved by the addition of longitudinal rein

forcement in the top of the prestressed channels. This re-

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inforcement helps prevent the flexural cracks from propa

gating to the composite interface.

87

8. In general, the structural behavior of the proto

type members can be satisfactorily predicted by tests on

model members. Predictions are at least as reliable as the

actual prototype test results.

SELECTED REFERENCES

1. Salmons, J.R., and Poepsel, J.r., "Investigation of a Prestressed-Precast Bridge System," Missouri Cooperative Highway Research Program Report 68-8, University of Mi ssouri-Columbla, Department of Civil Engineering, September 1968.

2. Alami, Z.Y., "Accuracy of Models Used in Research on Reinforced Concrete," Ph.D. Dissertation, The University of Texas, Austin, June 1962.

3. Mattock, Alan H., "Structural Model Testing - Theory and Applications," Journal of the PCA Research and Development Laboratories, 4. No. 3, September 1962, pp. 12-23.

4. Guralnick, S.A. and LaFraugh, R.W., "Laboratory Study of a 45-Foot Square Flat Plate Structure," Journal of the American Concrete Institute, 60, No. 9, September 1963, pp. 1107-1185.

5. Magura, D.D., "Structural Model Testing- Reinforced and Prestressed Mortar Beams," Journal of the PCA Research and Develo~ment Laboratories, 9, No. 1, January 1967, pp. 2-25.

6. Amaratunga, M. "Tests on Models of Concrete Structures," Engineering (London), val. 194, No. 5021, July 1962, pp. 62-63.

7. Brown, E.H., "Size Effect in Models of Structures," Engineering (London), val. 194, No. 5037, November 2, 1962, pp. 593-596.

8. Blanks, R.F., and McNamara , C.C., Journal of American Concrete Institute, vol. 31, 1935, pp. 280-320.

9. "Instrumen.tation," Technical Report No . 1, Cooperative Research Project, "Structural and Economic Study of Precase Units," University of Missouri , Department of Civil Engineering, 1957.

10. Buckingham , E. "On Physically Similar Systems; Illustrations of the Use of Dimensional Equations," Physical Review, vol. IV, No . 4, 1914.

I I I

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I I I I I I I I I I

ll. Murphy, G., "Similitude in Engineering," The Ronald Press Company, 1950.

89

12. Langhaar, "Dimensional Analysis and Theory of Models," Wiley, 1951.

13. Streeter, V. C., "Fluid Mechanics," Third Edition, McGraw-Hill, 1962.

I I I I APPENDIX A

I SIMILI TUDE ANALYSIS OF STRUCTURAL MODELS


- --

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NOTATlONS

= any linear dimension of the member

E = Modulus of Elasticity

G = Modulus of Rigidity

k = length scale factor

L = length of the member

M = any applied moment

p = any concentrated load

q = uniform load per unit length

R = resultant force acting at any cross-section

v = load per unit area

w = load per unit volume, density

o = any deflection

e = deformation per unit length, strain

cr = flexural stress

A = a notation to represent any d1

, d2

, •..

n = any dimensionless product

~ = Poisson's Ratio

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A2

A.l GENERAL

To design a suitable model and to interpret the re-

sults of a model testing, one must have a basic knowledge

of the underlying theory which predicts the behavior of

such models. The similitude laws governing the design

of models are based on the theories of dimensional anal-

ysis. The objective of this appendix is to provide ad-

equate terminology and to present the few simple laws

which are needed to understand and appreciate the use of

models in structural engineering. A brief discussion on

basic theories of dimensional analysis is followed by

formulation of structural model similitude relationships.

Finally, these relationships are solved to obtain the mod-

el ratios used in design.

A.2 DIMENSIONAL ANALYSIS

The concept of dimensional analysis was developed

by Buckingham10 , based on the following theorems:

1. Any mathematical equation expressing a physical

2.

property must be in a dimensionally homogenous

form. That is, the relationship must hold, re-

gardless of the system of units chosen.

Consequently, any dimensionally homogeneous

equation of the form

, A ) = 0 n

(Al)

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A3

can be expressed as

(A2)

where the n terms are independent and dimension-

less products of variables A1

, A2

, A3

, . , (~

is the number of variables, and (m) is the num-

ber of basic dimensions involved. The second

theorem is known as the Pi theorem.

The only restriction which has been placed on the Pi

terms is that they be dimensionless and independent. While

there are several procedures available for the determina

tion of a suitable set of Pi terms11

'12

, the method sug

gested by V. L. Streeter13

is presented in this discus-

sion.

Select (m) of the A-quantities, with different dimen-

sions, that contain among them the (m) number of basic

dimensions, and to use them as repeating variables togeth-

er with one of the other A-quantities for each n. For

example, let m = 3, and let A1 , A2 , A3 contain all the

three dimensions, not necessarily in each one, but collec

tively. Then the Pi parameters are made up as


A4

xl Y1 zl TI = A1 A~ A3 A4 1

x2 Y2 z2 TI = Al A2 A3 AS 2

(A3)

TI

n-m

In these equations the exponents are to be determined so

that each n term is dimensionless. The A-quantities are

substituted by their dimensional form and the exponents

of the dimensions are set equal to zero respectively. The

Examples in the following pages clarify the use of this

method.

A.3 TYPES OF MODELS

A model is a device which is so related to a physical

system that observations of the model may be used to pre-

diet accurately the performance of the physical system in

the desired respect. The physical system for which the

predictions are to be made is called the prototype. Four

types of models exist:

(a) True models are models in which all significant

characteristics of the prototype are faithfully reproduced

to scale. The model must satisfy geometric similarity,

and design restrictions of the prototype. Most model

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AS

studies are based on the use of true models.

(b) Adequate models are models from which accurate

predictions of one characteristic of the prototype may

be made, but which will not necessarily yield accurate

predictions of other characteristics. For example, let

moment of ine~tia of the mOdel be similar to that of the

prototype but width and depth dissimilar. Such a model

would be expected to give accurate results for bending de

flection, but will not predict shearing deflection or tor

sional resistance.

(c) A distorted model is the one in which some de

sign condition is violated sufficiently to require correc

tion of the predictions. If the length and depth scale

is different from the width scale for a given model, then

a distorted condition is obtained.

(d) Dissimilar models are models which bear no

apparent resemblance to the prototype but which through

suitable analogies give accurate predictions of behavior

of the prototype. A soap bubble is an analogous model to

a shaft under torsional stresses.

A.4 STRUCTURAL MODELS

The primary purpose of constructing structural models

is to obtain information concerning the behavior of the

structure under load. The behavior is normally evaluated

in terms of the induced stresses and deformations. The

I I I I I I I I I I

I I I I I I I I

A6

nature and the magnitude of such stresses and defor

mations depend on geometric configuration and material

properties. At any section in any member a resultant force

(R) is developed, and at another section a short distance

away another resultant force is developed. The two re

sultant forces will induce stresses and deformation in the

short element of the member between the section.

Structural problems of static nature usually involve

the two independent dimensions of length (L) and force (F).

The variables entering in such a problem are geometric

dimensions, resultant of all the forces applied, Modulus

of Elasticity, Modulus of Rigidity, deflection, stresses,

and distortion which is represented by Poisson's ratio.

To perform a dimensional analysis of such a problem, one

has to determine all the dimensionless n terms. Let us

consider the flexural stresses developed in a concrete

member under certain resultant load and moment. The

general expression would be

a= f(R, L, M, E, G, d 1 , d 2 , · .. , ~) (A4)

including a in the bracket, and expressing all d 1 , d 2 ...

by A, (A4) becomes

F(R, L, M, a, E, G, A, ~) = 0

Following the method described in the previous section,

the first three dimensionless Pi terms can be expressed

(AS)


A7

as X y

Til ;::; (R) ). (L) - 1 (M)

1T2 ;:; (R)x2(L)y2(a) (A6)

X y 1T3 = (R) 3 (L) 3 (E)

Substituting in the corresponding dimensions of all the

variables

X y Til ;:; (F) 1 (L) 1 (FL)

X y 2 1T2 = (F) 2 (L) 2 (FL-) (A7)

X y 2 1T3 ;:; (F) 3 (L) 3 ( FL- )

x's andy's should be determined in such a way that

the above products become dimensionless as required by

the Pi theorem. Since n 1 , n 2 , and n 3 are dimensionless,

they can be expressed as n = (F) 0 (L) 0 therefore:

(F) 0 (L) 0 X y

:: (F) 1 (L) 1 (FL)

(AS)

(F) 0 (L) 0 :: (F)x 3 (L)y 3 (FL- 2 )

equating the exponents of (F) and (L) on both sides

I I I I I


A8

(A9)

y3 - 2 = 0

Solving these first degree equations and substituting

x 's and y 's in original n terms in equation (A6)

(R)-l(L)-l(M) M Til = = RL

TI 2 (R)-l(L) 2 ( cr ) aL 2 = = """""R

{R) -l (L) 2 (E) EL 2 TI 3 = = """""R

the rest of the n terms can b e found in the same manner or by

simple inspection. Hence, a dimensionless expression for

flexural stress of the member will be

(AlO)

Let subscript (m) denote the model parameters, then equa-

' tion (All) would express the flexural stress of the model

member.

M m g(~,

m m

2 a L m m R m

E L2

m m R m

, . . . ) = 0

For a true model, the dimensionless products should be

equal to the corresponding terms of the prototype. If

(All)

all the geometric dimensions of the prototype are reduced


I I I I

A9

by factor (k) ' then:

L EL

2 E L2

m mm L = k,

~ = R m

M ~L; = RL R L

m m

crL2 cr L2 m m

-R- R m

Assuming E = E and substituting L /L = k in the above m m

equations

R k2 m (Al2) R =

om 1 (Al3) ~ =

M k3 m

(.A.l4) M =

From equation (Al3) one can deduce similitude of strains

(J E £ m m m 1 - = ~

= (J

or £m r- = 1 (AlS)

Also deflection similitude can be deduced from L /L = k, m

since deflection is one form of linear dimension. There-

fore; 0 m

T = k (Al6)


AlO

The above similitude ratios can be summarized in the

following: If all the linear dimensions of the prototype

are scaled by the factor (k) , and the applied forces are

scaled by the factor (k2), then the applied moment will be

scaled by (k3), the deflection by (k), and the strains

and stresses will remai n unchanged.

It should be noted that the above results completely

depend on the assumption of Em = E or the assumption of

identical stresses and strains for the model and the proto-

type. This assumption was elaborated on in Chapter II and

its scope broadened to more general cases. In a general

case where stresses and strains in the model are not equal

to those in the prototype, the above relationships should

be corrected for material distortion using the a and s co-

efficient of Fig. 2.1.

Another common distortion in model analysis is geo-

metric distortion of the prototype dimensions due to prac-

tical limitation encountered in laboratory conditions. A

detailed derivation of the similitude relationships under

various combinations of distortion factors is beyond the

scope of this paper and can be found in a number of texts

0 0 10 t d 1 0 12 on s1m1 1 u e ana ys1s .

A.S THE APPLIED FORCE, R

In the previous discussion, the symbol R was used to

indicate the magnitude of the resultant force developed at


All

any cross-section in the structure. The resultant force

in turn depends on the various forms of applied load,

geometry of the structure, and the material properties in

the case of statically indeterminate structures. Therefore:

R = h (p, q, v, w, L, A , E, G)

A non-dimensional form of this equation can be

H (E . E._ E._ E._ A !) = 0 R, qL, 2, 3, L, G

vL wL

A similar equation can be written for the model and through

a similar procedure as outlined in the previous section,

similitude ratios can be formulized as the following:

L m = k L

p R k2 m m = p =

R

qm (Al7)

- = k q

v m = l - v

wm l = k w

Hence with a linear scale factor of k, concentrated load

will be scaled by k 2 , load per unit length by k, load per

unit volume enlarged by 1/k, and load per unit area remains

unchanged.


Al2

A.6 DEAD LOAD

Dead load stress and distortions are developed by the

weight of the member which is a function of density or force

per unit volume. According to the previous section, force

per unit volume should be enlarged by factor 1/k for any

true model. In other words, density of model material

should be 1/k larger than the prototype material. This

creates certain difficulties from the practical standpoint.

Obviously, if the length scale is selected as unity, den

sity of model material will be the same as that of the pro

totype. But such a condition would result in distortion

of geometry since the length scale will not be the same as

the cross-section scale.

However, to solve this problem in true model design,

several practical expedients are available. A common

method is to add external force to supply the deficiency

in weight. This is usually done by hanging sand bags

at various locations along the length of the member.

Another alternative would be to increase the gravitation

al attraction by means of an elevator arrangement, a cen

trifuge, or creation of a magnetic or any other force

field.

I I I I I


APPENDIX B

SUMMARY OF DESIGN COMPUTATIONS FOR A ONE-HALF MODEL


I

Bl

B.l GENERAL

A one-half scale model was proposed and desi~ned as

a part of this investigation. The model member was designed

as a simply supported component of a highway bridge sys-

tem satisfying all the requirements of the 1961 AASHO Code.

Both lane and truck loadings were considered in the design

and both working stress and ultimate strength criteria were

utilized.

Based on the assumptions enumerated in Chapter II, the

following similitude criteria were utilized:

1) All geometric dimensions of the model member

shall be 1/2 of the corresponding dimensions

of the full-scale member.

2) All steel areas shall be 1/4 of the corresponding

steel areas in the full-scale member.

3) Total prestressing force shall be 1/4 of that

applied to the full-scale member.

4) Stress gradient at the mid-span section of the

prestressed channel after casting the top slab

shall be identical to that of the full-scale

member.

Based on these design criteria, design computations

of the model members were carried out in a similar procedure

as presented in Appendix A of "A Proposed Precast Prestressed

Bridge System" 1 . The significant design values are listed

in the following sections:


B.2 DESIGN SUMMARY

Minimum Compressive Strength of Concrete:

Prestressed Channel

Release Strength

Cast-in-place Top Slab

Maximum Aggregate Size in Concrete Mix

Minimum Yield Strength of Reinforcements:

Reinforcing Steel Bars

Sevel-wire Stress Relieved Strands

1/4" Diameter

3/8" Diameter

Allowable Concrete Stresses:

Allowable Compressive Stress at Release

Allowable Tensile Stress

Allowable Compressive Stress at Working Load

Strand Arrangement:

4-3/8" strands and 11-1/4" strands as shown in Fig. 2.4

Centroid of Channel Section

Centroid of Prestress Force

Constant Eccentricity

Prestress Force at Transfer

Final Prestress Force

Stresses at Transfer of Prestress:

Midspan - bottom

6000 ps1

3000 psi

4000 psi

3/8 in.

40 ksi

9 kips

20 kips

1800 psi

Zero

2400 psi

l. 97 in.

l. 60 in.

0.37 in.

115.9 kips

101.1 kips

1.395 ksi

B2


Midspan - top

End - bottom

End - top

Stress After Casting of Top Slab:

Midspan - bottom

Midspan - top

End - bottom

End - top

Top Slab Reinforcement: As shown in Fig. 2.5

Longitudinal Steel, 5-#4 bars equally spaced

B3

l. 276 ksi

1.684 ksi

0.539 ksi

0.895 ksi

l. 951 ksi

1.470 ksi

0.471 ksi

Transverse Steel, #3 bars at 12" spacing

Live Load Moment Capacity of Composite Section:

Working Stress Design

Ultimate Moment Capacity:

With ~ factor

Without ~ factor

Cracking Moment

Horizontal Shear at Design Load

Horizontal Shear Connector

#4 at 12" spacing in shear span

#4 at 24" spacing ~n constant moment region

Diagonal Tension Reinforcement

#3 stirrups at 6" spacing

265.5 in.-kips

940 in.-kips

1044 in.-kips

603.2 in.-kips

0.140 ksi

Graphical representations of midspan stress gradients

for various stages are shown in Fig. B.l. Deflections of


STAGE

Release o~ prestress force to channel

30 days after release

Casting of top slab

Design Live Load

DEFLECTlON

0.1 II t

0.06 11 t 0.14 11 t 0.15 11

~

NET DEFLECTION

0.1 II t

0.16 11 t 0.02 11 t

Table B.l Predicted Deflections

Allowable Deflection = L/800

= (17) (12)/800

= 0.25 11 > 0.15 11 OK

B4

the beam at different stages have been calculated elastic-

ally and, the results are shown in Table B.l.

------ -- -- ------STRESS-PSI

1276

1395

TRANSFER

1360

1395

1951 7

7

895

AFTER CASTING TOP SLAB

1205

895

Fig. B.1 Midspan Stress Gradient

580 l=::t420 2400

7

/ /

/ 7

'/

~ 153

LIVE LOAD

580 t==1420

f7 153

w Ul

AD LIBRARY

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