DOWEL ACTION IN REINFORCED CONCRETE CONSTRUCTION [PDF]

C I

DOWEL ACTION IN REINFORCED CONCRETE CONSTRUCTION (BEAM-COLUMN

CONNECTIONS)

by ELY E .

KAZAKOFF

B . A . S c , University of B r i t i s h Columbia, 1971

A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE

In the Department of CIVIL ENGINEERING

We accept t h i s thesis as conforming to required standard

THE UNIVERSITY OF BRITISH COLUMBIA A p r i l 197^

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D e p a r t m e n t o f C I V I L ENGINEERING The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, Canada 1974

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of this

thesis f o r

be a l l o w e d w i t h o u t my w r i t t e n

Ely

April

of B r i t i s h

s h a l l make i t f r e e l y

g r a n t e d by t h e Head o f my Department It

f u l f i l m e n t of the

f o r an advanced degree at t h e U n i v e r s i t y

Columbia, I agree that

extensive

this

». K a z a k o f f

permission.

i

ABSTRACT The

t r a n s f e r of shear i n a beam-column j o i n t by

a c t i o n a l o n e was

experimentally

and a n a l y t i c a l l y s t u d i e d .

l a b o r a t o r y work i n v o l v e d the shear c a p a c i t y d e t e r m i n a t i o n

dowel The of

i n d i v i d u a l r e i n f o r c i n g s t e e l dowels embedded i n c o n c r e t e .

Two

main s e r i e s o f e x p e r i m e n t a l

and

t e s t s were conducted on bottom

top dowels - component p a r t s of a beam-column j o i n t .

A l l experi-

mental r e s u l t s were compared t o a t h e o r e t i c a l a n a l y s i s . The

t h e o r e t i c a l a n a l y s i s c o n s i s t e d of c h o o s i n g a

r a t i o n a l p h y s i c a l model, i . e . , a mode of b e h a v i o u r f o r each of the two

component p a r t s of the j o i n t .

experimental

r e s u l t s was

t h a t the model p r o v i d e s of the j o i n t .

done.

No c u r v e - f i t t i n g t o

the

These r e s u l t s do show, however,

a s a f e lower bound on the shear c a p a c i t y

A l s o , the model p e r m i t s r e a s o n a b l e e x t r a p o l a t i o n

t o o t h e r d e s i g n problems where the c o n d i t i o n s of the problem are not e x a c t l y the same as those imposed d u r i n g the

experimental

tests. A design

example of p r e d i c t i n g the shear c a p a c i t y of a

beam-column j o i n t on t h e b a s i s of dowel a c t i o n of the r e i n f o r c i n g s t e e l i s p r e s e n t e d f o r any

c o m b i n a t i o n of top and bottom dowels.

TABLE OP CONTENTS Abstract

i

Table of Contents

ii

L i s t of Figures

iii

L i s t of Tables

iv

Acknowledgements

i Page V

Chapter 1

Introduction

1

Chapter 2

Laboratory Program

7

2.1

Material

7

2.2

Fabrication of Test Specimens

7

Chapter 3

Foundation Modulus of Concrete K

10

Chapter 4

Bottom Dowel Tests

19

4.1

Experimental Procedure

19

4.2

Analysis

20

4.3

Comparison of Results with Previous Work

35

Chapter 5

Top Dowel Tests

40

5.1

Laboratory Test Program

40

5.2

Analysis

45

Chapter 6

The J o i n t :

Chapter 7

Conclusions

Sum of Top and Bottom Dowels ..,. j has been assumed to be independent of

concrete strength and the stress l e v e l at the cracked interface. Since this may not actually be so, Mast l i m i t s the term pf

to

i

15% of the concrete cylinder strength f . c

He also recommends

that #6 bars (intermediate grade steel) be taken as an upper l i m i t i n shear f r i c t i o n design. Hofbeck, Ibrahim and Mattock^' investigated the shear transfer strength of r e i n f o r c i n g dowels (stirrups) crossing a shear plane.

Concrete specimens with and without i n i t i a l cracks

along shear planes were experimentally tested.

When the concrete

specimens had an i n i t i a l crack along the shear plane, there was considerable contribution to shear transfer strength by dowel action.

For uncracked specimens, the reinforcement i s put into

tension as a t r u s s - l i k e action develops, i . e . ,

a saw-tooth action

as one face t r i e s to s l i p r e l a t i v e to the other. The s h e a r - f r i c t i o n design concept, as proposed by Mast, has been successfully applied to several design situations of which the author i s f a m i l i a r .

In one instance, a precast load

bearing beam-panel was dowelled into a cast-in-place column.

Due

to shrinkage, i t was feared that the two concrete surfaces may separate and f r i c t i o n would not develop between the two surfaces. In the hope of preventing t h i s , the cast-in-place column was revibrated within 90 minutes of the i n i t i a l pour i n order to "squeeze-out"

the excess water and thus minimize shrinkage.

In such cases as described above, i t may be useful to consider the dowel action of the s t e e l bars and design the beamcolumn connection on that basis.

The additional work and expense

of revibration could be avoided.

Also, i n order to insure shear-

f r i c t i o n a c t i o n , stringent construction tolerances

necessitate

that the precast units be positioned snugly against the forms of the cast-in-place u n i t s . In certain s i t u a t i o n s ,

shear keys are provided i n

columns against which a beam i s l a t e r cast.

Some design

engineers

consider this a very s t i f f connection and an i d e a l area for stress concentrations.

On the other hand, the design of such beam-

column j o i n t s on the basis of dowel action provides for a d u c t i l e j o i n t as characterized by the shear-deflection behaviour of i n d i vidual dowels (Appendix 1 and 2). Many connections are subjected to forces a r i s i n g from settlement, creep, and shrinkage.

These forces are generally

unknown and therefore the connection must possess d u c t i l i t y i n order to accommodate the additional stresses imposed by these forces. The following chapters present an experimental and a n a l y t i c a l study of dowel action i n a beam-column j o i n t and the results of this work are intended to f a c i l i t a t e the design of such connections.

7. CHAPTER 2.

LABORATORY PROGRAM

The laboratory work of forming, casting and curing followed a standard procedure for each test s e r i e s .

This chapter

describes the methods involved. 2.1

MATERIAL A l l the concrete was delivered by truck from a l o c a l

ready-mix plant.

Type III (High Early) Portland Cement and 3/4"

maximum size aggregate was used i n the mix.

A slump of 3" was

specified for each mix. The deformed bar r e i n f o r c i n g s t e e l was of the type used on construction projects (40 and 60 grade) and was obtained from a l o c a l supplier - cut and bent to the required shape. Steel samples were tested i n tension to determine y i e l d stress f

and ultimate stress f . Three concrete cylinders were tested at the beginning

of each test series and three at the end.

The value of the

compressive strength f* which was used in the analysis of the test results was an average of the six t e s t s . The concrete and s t e e l properties for each test series are 2.2

tabulated i n Table 2.1. FABRICATION OF TEST SPECIMENS After the plywood forms were coated with o i l , the pre-

fabricated cages of reinforcement were positioned i n the forms. During pouring, the concrete was consolidated with a v i b r a t o r .

8. Six companion cylinders ( 4 " x 8") were poured with each test series./

_ < . Wet burlap sacks were placed over the poured specimens

and everything was covered with.a p l a s t i c sheet to prevent moisture l o s s .

The burlap sacks were repeatedly moistened everyday.

The forms were stripped two days after pouring, but moist curing continued for a t o t a l duration of 10 days, after which the p l a s t i c and burlap sacks were removed and the specimens l e f t dry cure on the laboratory f l o o r .

to

Table 2 . 1 TEST SERIES •

Concrete and Steel Properties fi

(Ksi)

f

(Ksi)

f

u

(Ksi)

6.33 for a l l specimens

K TESTS

BOTTOM DOWEL TESTS Bar Sizes

#3

54

79

#4

56

80

#5

66

101

71

103

73

110

#6 #7

4.2 for allspecimens • :.

#8

69

#9

69

112

#10

66.4

102

#11

66.4

93

79,5

97.5

STS TOP DOWEL TE Bar Sizes

#4

5.675

65

#5

3.13

70.5

110

#6

3.13

66.4

100

#7

5.675

69.5

110

#8

3.13

64

104

#9

5.675

62.7

109

#10

5,675

62.7

87.5

#11

6.0

62.7

87.5

CHAPTER 3.

FOUNDATION MODULUS OF CONCRETE K

As previously mentioned, the t h e o r e t i c a l analysis for the bottom dowels required the value for the foundation modulus of concrete K as a function of dowel s i z e . each dowel s i z e ,

To determine K for

i t was decided to test 4 dowel sizes and i n t e r -

polate for the others.

Three specimens were cast for each of

dowel sizes #4, #6, #8 and #11.

Pouring and curing of concrete

followed the standard procedure as described i n Chapter 2. A t y p i c a l specimen i s shown i n F i g . 3.1a. Only the bottom-half of the dowel was embedded i n concrete.

The specimens

were tested i n a Baldwin loading machine with load and deflection simultaneously recorded on a X-Y p l o t t e r .

F i g . 3-lb is a

schematic representation of the laboratory set-up.

F i g . 3-2 shows

a specimen in the Baldwin just before the beginning of a t e s t . The deflection of the s t e e l dowel was measured with l i n e a r transformers positioned at each end of the dowel. x-y p l o t t e r recorded the average of the two deflections

The and also

the load which was applied continuously at an average rate of 6 Kips per minute. There were no v i s i b l e signs of distress specimen u n t i l a substantial load was applied'.

i n the concrete

Crushing and

s p a l l i n g of the concrete immediately below the dowel were the f i r s t v i s i b l e signs of progressive f a i l u r e .

For bar sizes #4 and

#6, the extent of f a i l u r e was only crushing of the concrete below the dowel.

For the #8 and #11 dowels, the usual crushing and

s p a l l i n g occurred at the i n i t i a l stages of loading.

Also, a

1.1.

LMWM6 P L A T E

*3

BEfNFORCING

STEEL

, P i g . 3:la Foundation Modulus Test Specimen

T o JC-Y

PLOTTER

L I M E LOAD

LINEAR

TRANSFORMER

STEEL TO

Fig.

PLATES

GLUED

SPEC I MEM

3-lb Test Specimen In Baldwin

L c

*°$

^

Fig.

3.3 F a i l u r e

of Test

Specimen

1.3.

h a i r l i n e crack began to propagate v e r t i c a l l y downwards and at the completion of the t e s t , the crack had progressed to the base of the specimen. An "explosive" type of f a i l u r e was prevented by the horizontal #3 reinforcing bars ( F i g . 3 . 1 a ) .

F i g . 3 . 3 shows the

specimen at the end of the t e s t . The load-deflection graphs for the #8 dowel tests are presented i n F i g . 3 - 4 . This set of graphs is t y p i c a l of the other series.

In order to amplify the straight l i n e portion of the

graphs, the v e r t i c a l scale on tests 2 and 3 was doubled.

This

f a c i l i t a t e d i n establishing the value for the slope of the graph. The foundation modulus K i s calculated.by determining the slope of the s t r a i g h t - l i n e portion of the load-deflection graphs and dividing the value by the width of the specimen which was 8 inches. slope s

l

K

o

p

e

=

=

A6

^Lope

Kl£ in. Ksi

Therefore the constant K denotes the reaction per unit length of the beam (dowel) where the deflection i s equal to unity (Timoshenko

).

-

The results of a l l the tests are tabulated i n Table 3 . 1 . and F i g . 3 . 5 i s a plot of the average K value for each dowel s i z e . The graph was drawn by j o i n i n g the experimental points.and extrapolating to the #3 dowel s i z e .

At each averaged point is

a heavy dark l i n e which gives the range i n the experimental values.

DOWEL

SIZE

Table 3 . 1

Foundation Modulus Tests

TEST NO.

FOUNDATION MODULUS K Ksi

#4

#6

1

512

2

536

3

323

1

820

2

875

3

665

1

925

#8

2

•

1

CONCRETE:

f c 1

=

457

787

863

870

3

#11

795

AVERAGE K Ksi

1,010

2

986

3

1,020

6,330

psl

1,005

The graph i n F i g . 3 . 5 i s for a concrete strength f* of 6 , 3 3 0 p s i as determined from the standard cylinder t e s t s . This graph can be scaled for other values of concrete strengths by the following method. The modulus of e l a s t i c i t y of concrete E of^jfTas

g i v e n by t h e e m p i r i c a l

E

pcf)

c

i s a function

equation:

= 3 3 w ^ J f * " \ (w = u n i t w e i g h t o f h a r d e n e d concrete i n c c and the foundation modulus K varies d i r e c t l y with E 2

,

K

2

K

l

Therefore

;

=

The factor for s c a l i n g the graph of F i g . 3 - 5 to other concrete strengths

is

or

This has been done for several concrete strengths as shown in Fig.

3 . 6 . The foundation modulus K i s not too sensitive to

varying concrete strengths since the curves of F i g . 3 - 6 l i e i n a narrow band.

i M!

1

I'M I M i " L i n LLLLUJ-.i.:..-..j...i..

:

CHAPTER 4 . 4.1

BOTTOM DOWEL TESTS

EXPERIMENTAL PROCEDURE In order to determine the shear capacity of the bottom

dowels, 36 concrete specimens, as shown in P i g . 4 . 1 , were formed and cast.

The variable involved i n this study was the dowel size

Pour specimens were cast for each dowel size ranging from #3 to #11.

-N

PLAN #3

Z

TIES

•#S

BARS

a BOTTOM

DovvEL

ELEVATION

F i g . 4 . 1 Bottom Dowel Specimen

')

The method of pouring and curing of concrete was as described in Chapter 2. It was desired to load the protruding steel dowels i n shear only.

For this purpose, a wide flange beam was clamped to

20. the steel dowels and the load applied at the mid-point of the beam.

F i g . 4.2 and 4.3 show the positioning of the test

(two per test) and the method of load a p p l i c a t i o n . applied with an Amsler hydraulic jack.

specimens

The load was

The deflection of the

s t e e l dowel was measured at the column face

(positions

1 and 2,

F i g . 4.2).- Since the deflection probes from the transformers were positioned on the dowel i t s e l f , the s t e e l clamps were attached 1/4" away from the column face to provide the necessary space for the probes.

As a result of t h i s set-up, some bending

moment would be developed in the dowel at the column face. is considered i n the t h e o r e t i c a l analysis; were again used to measure the deflections

This

Linear transformers and both

deflections

and load were simultaneously recorded on punched paper tape on a D i g i t a l Data Acquisition u n i t .

A computer program converted

the paper tape data into the shear-deflection graphs which are presented i n Appendix 1.

Four curves were obtained for each

dowel s i z e . For simulating the actual column conditions, the concrete column specimens were compressively stressed to 1 Ksi with the tension rods ( F i g . 4.2).

The force i n each tension rod was de-

termined with a strainsert b o l t . 4.2

ANALYSIS The behaviour of the bottom dowel embedded i n the con-

crete column specimen was modelled as a beam-on-elastic foundation. F i g . 4.4b shows a semi-infinite beam on an e l a s t i c discussed i n Timoshenko

'.

foundation as

This model i s assumed to represent

the section shown i n F i g . 4.4a.

STEEL

CLAMP

BRONZE

SHIM

TRANSFORMER

1/

TEST

PROBE FOR M E A S U R I N G DEFLECTION A T COLUMN

FACE

[

SPECIMEN LOAD TENSION

P

ROD

i z

T7777T

/

/

/

/

/

/

/

Fig.

/

/

i i / / / /

STRAINS6KT LOAOING

BEAM

BOLT

I

4.2 Loading Apparatus For Bottom Dowels

/T7T

23.

CONCRETE

BOTTOM

COLUMN

DOWEL

RESIOM

A

Fig.

MODELLEP

AS

BEAM-OH-ELASTIC FoUMPATIOM

4.4a Bottom Dowel Specimen

BOTTOM

ELASTIC

Fig.

SPECIMEN

OOWEL

FOUNDATION"

4.4b Bottom Dowel Specimen as a Beam-on-elastic Foundation

The solution to the d i f f e r e n t i a l equation for a semii n f i n i t e beam on an e l a s t i c foundation as shown i n P i g . 4 . 4 b is y(x)

=

e"

where

(Pcosgx --pMc fcosgx-sinBx-]•) ( 4 - 1 )

ex

3 =

h

,^

K

=

Foundation Modulus

E

=

modulus of e l a s t i c i t y of the beam

I

=

moment of i n e r t i a of the beam

The values obtained for the foundation modulus i n Chapter 3 were used in c a l c u l a t i n g the g term.

Since the bottom dowel specimens

had a concrete strength of 4 , 2 0 0 p s i , the values for the foundation modulus were scaled by using a factor of

•^H§-

=

(Refer to page 17)

0.815.

The units of K are Ksi and the value for E in a l l the analysis was 2 9 , 0 0 0 Ksi (modulus of e l a s t i c i t y of the s t e e l dowels). To determine the deflection at the column face,

the

value x = 0 must be substituted into equation 4 - 1 . y (x=0) ;

=

2

^

(P- M ) P

o

(4-2)

or rearranging P

=

2g EIy 4. 6 M 3

(4-3)

O •

As previously mentioned, since some room had to be provided forL-positioning the deflection probes onto the dowels, s t e e l clamps were not snug against the column face.

the

Hence, the

bending moment that i s developed i n the dowel at the column face i s opposite i n sign to that shown i n F i g ; 4.4b.

With the change

i n s i g n , equation 4-3 becomes P When M

Q

=

(4-4)

2g EIy - g M . 3

Q

i s zero, equation 4-4 reduces to P

=

(4-5)

2g EIy 3

The two extreme values of "M • are zero and M - the ° P p l a s t i c moment of the s t e e l dowel.

Equations 4-4 and 4-5 were

superimposed on the shear-deflection curves of dowel sizes #4, #7 and #11,

as shown i n P i g . 4 . 5 a , 4.5b and 4 . 5 c . The experi-

mental shear-deflection curve is an average of the 4 curves as shown i n Appendix 1 for the corresponding dowel s i z e . l i s t s the variables involved i n t h i s

Table

4.1

analysis.

As can be noted from the graphs, the t h e o r e t i c a l curves are below the experimental curve for the #11 dowel up to a def l e c t i o n of 0.04".

As the dowel size i s reduced the two t h e o r e t i -

cal curves shift closer to the experimental curve u n t i l the upper l i n e (equation 4-5) begins to exceed the experimental results at a deflection of 0.02"

(#4 dowel s i z e ) .

P i g . 4.6 is a plot of the shear at 0.03" deflection for the range of bar sizes tested.

Equations 4-4 and 4-5 are also

plotted with the value of "y" equal to 0.03".

The majority of

the experimental points are bounded.by the two extreme equations. (Heavy dark v e r t i c a l lines show the range in the experimental results.) 4.6.

Table 4.2 l i s t s the values required i n p l o t t i n g F i g .

The 3 term was evaluated for a concrete strength of 4,200 p s i .

T a b l e 4 . 1 Bottom Dowel

Dowel Size

Diameter d foment o f I n e r t i a (in.) ' I (in. ) 4

Variables

F o u n d a t i o n Modulus K (Ksi) (for

fi

= 4200 p s i )

3 =

\f~P

V4EI (E = 2 9 0 0 0 /

( 1 ) (in.) Ksi)

P l a s t i c moment M = 0.l67f a p. y 3

(KIP-IN j

#3

0.375

0.00097

180

1.13

0.48

#4

0.5

0.00306

372

1.01

1.17

#5

0.625

0.0075

520

0.88

2.7

#6

0.75

0.0155

640

0.77

5.

#7

0.875

0.0286

670

0.67

8.15

#8

1.0

0 . 049

700

0.59

11.5

#9

1.12

0.0775

730

0.53

16. 2

#10

1.25

0.12

770

0 . 49

21.6

#11

1.38

0.178

815

0.45

29.2

Table 4.2 Shear at 0.03 Deflection f* = 4200 p s i Dowel Size

•

E = 29000 Ksi

Average Shear at 0 . 0 3 " Deflection (Experiment) (KIPS)

y = 0.03"

P = 2g EIy - BM (Kips) 3

p

P = 2$ EIy (Kips) 3

#3

2.5

1.9

2.4

#4

3.6

4.3

5.5

#5

4.9

6.5

8.9

#6

10. 4

8.6

12. 4

#7

10.4

9.5

15.

#8

13.6

10.9

17.7

#9

18.4

11.9

20.5

#10

27.4

13.3

23.8

#11

30.

14.4

27.4

32.

The "knee" of the shear-deflection curves occurs (in most cases) at around the 0 . 0 3" value of deflection with the concrete s t i l l in the e l a s t i c range.

At this deflection there were

no v i s i b l e signs of crushing or spalling' of the concrete around the dowel.

Thus, for t h i s reason the t h e o r e t i c a l beam-on-elastic

foundation equation was compared to the 0 . 0 3 " value.

Extrapolating

the equation to higher values of deflection would result i n overestimating the shear capacity, since the concrete under the dowel begins to crush and crack and the shear-deflection curves assume a shallower slope. Nevertheless,

the experimental and t h e o r e t i c a l values

are i n close agreement, at the 0.03" value for the entire range of dowel s i z e s , with some sizes experiencing more deviation than, others. P i g . 4 . 7 is a plot of equation 4 - 5 for varying values of concrete strengths.

As was shown i n P i g . 3 . 6 , the foundation

modulus K i s not very sensitive to differences i n concrete strength.

Hence the 3 term i s also rather insensitive to con-

crete strength, with the result that the two graphs (Pig. 4 . 7 ) do not have much v a r i a t i o n .

For a 50$ increase i n concrete strength

the maximum increase i n shear capacity (for a #8 bar) is about 17%. The ultimate shear for each dowel was taken to be the stage at which the concrete was crushing under the dowel and no increase i n load was possible.

The ultimate shear and the shear

at 0 . 0 3 " deflection i s plotted i n F i g . 4 . 8 . In most cases the ultimate shear i s double that at 0 . 0 3 " d e f l e c t i o n .

A design

based on the 0 . 0 3 " deflection curve would provide a safety factor of 2 in most cases.

3.5.

4.3.

COMPARISON OF RESULTS WITH PREVIOUS WORK

The results of these tests were compared to previous work which has been done with r e i n f o r c i n g s t e e l dowels and metal studs. 4 . 9 presents the experimental results and two

Fig.

expressions

from the ACI-ASCE C o m m i t t e e . T h e r e the allowable shear for reinforcing s t e e l dowels i s given by the

where

For

/(A f cose) where a longitudinal crack propagated from the top dowel out towards the beam sides and then horizontally along the beam.

The area over which direct tension

occurs i s a rectangle 6" (beam width) by 1" (distance to stirrup), i . e . ,

6 square inches.

first

With the t e n s i l e strength of

52.

Pig.

5.7

Crack

Propagation

i n Top

Dowel

Test

.5.3.

concrete taken as 7 . 5 A ^ ^ J the shear force required to crack the section can be calculated -directly as = .7.5AFT *6

V

(5-3)

The results of this c a l c u l a t i o n are l i s t e d in Table 5 . 3 and F i g . 5 - 9 shows a plot of equation 5 - 3 i n r e l a t i o n to other experimental values. The effect of the f i r s t s t i r r u p y i e l d i n g s h a l l be considered next.

F i g . 5 . 8 i l l u s t r a t e s the condition at the f i r s t

s t i r r u p where the shear force V i s resisted by the tension i n the s t i r r u p .

The values given i n Table 5 . 4 are plotted i n F i g .

5.9,as two discontinuous straight lines #4).

(stirrup sizes #3 and

These two lines agree reasonably well with the ultimate

values obtained from experiment. In this test s e r i e s ,

i t was d i f f i c u l t to compare the

experimental and model deflections.

Since the f i r s t s t i r r u p

w i l l s t r a i n and therefore extend under the application of load, the deflection that i s measured at positions 1 (or 2.) i s not i d e n t i c a l l y the same as the deflection of the top dowel v e r t i c a l l y above p o s i t i o n 1 (or 2 ) .

(Refer to F i g . 5 . 2 . )

As shown i n F i g : 5 . 9 , the ultimate shear i s considerably higher than that obtained at 0 . 0 3 " deflection.. f a i l e d i n shear at ultimate ( F i g . 5 . 1 0 ) . #8 dowels,

The #4 dowel

In the case of #7 and

the f i r s t s t i r r u p ruptured at ultimate. ( F i g . 5 . 1 1 ) .

The #9, #10 and #11 test specimens had #4 size stirrups and i n these three cases the concrete beam f a i l e d i n shear (at the end with the larger s t i r r u p spacing - F i g . 5 . 1 3 ) ,

Table 5-3 Direct Tensile Force

Dowel Size

f

r

=

7

' fc' 5 /

(

K

s

l

)

V = 7 . 5 A ^ * 6 (KIPS)

0.565

3.39

#5

0.42

2.52

#6

0.42

2.52

0.565

3.39

#8

0.42

2.52

#9

0.565

3.39

0.565

3.39

O.58O

3.48

#11

F i g . 5.8

Y i e l d i n g o f the F i r s t

T a b l e 5.4

Stirrup

T e n s i o n at S t i r r u p Y i e l d A

Dowel Size

Stirrup Size

s f Stirrup Area ( i n . ) 2

(Ksi)

2T = A f KIPS s

J

#4 #5 #6

#3

0.22

54

0. 40

60

11.88

#7

#8 #9 #10

#11

#4

24