SHEAR BEHAVIOR OF REINFORCED CONCRETE BEAMSWITHA SMALL AMOUNT OF WEB REINFORCEMENTSongkram Piyamahant

A dissertation submitted to
Kochi University of Technology
in partial fulfillment of the requirements for
The Degree of Master of Engineering

Supervisor
Professor Hiroshi Shima

Department of Infrastructure System Engineering
Kochi University of Technology
Kochi, Japan

January 2002

Abstract

ABSTRACTIn 1995, Hyogoken- nambu Earthquake destroyed many of Shinkansen viaductstructures that were a very large column section, approximately 900 mm., and a small
amount of web reinforcement. With respect to this evidence, the shear strength formula
was clarified in size effect but it still does not clarify the superposition method when asmall amount of web reinforcement is extremely employed. Following that, the aim of
this research is based on shear behavior and superposition method Vcr+Vs by intending
on a minimal amount of web reinforcement. Accordingly, 4 reinforced concrete
specimens with web reinforcement ratio equal 0.035%, 0.05%, 0.065%, and 0.08%
were conducted under monotonic loading. Moreover, two measuring systems for
aggregate interlocking and shear resisted by web reinforcement were designed to use ininvestigating shear resistance mechanism. Consequently, the experiment shows thatshear carrying capacity of 3 smallest amount of web reinforcement is the same and thesuperposition method is safe to predict the shear carrying capacity of reinforced
concrete beam with a small amount of web reinforcement as shear span ratio 3.0.
Similarly, other design codes also show the same degree of safety as the superposition
method is employed. Not only that, but also the result of aggregate interlocking shows
the same amount of stress transfer across the crack since the larger crack widthcorresponding to larger sliding of crack surface possesses the same stress transfer as thatof smaller crack width and sliding.

Shear Behavior of Reinforced Concrete Beams with A Small Amount of Web ReinforcementiAcknowledgements

ACKNOWLEDGEMENTS

I would like to express my deepest gratitude to my advisor Professor Hiroshi Shima forhis kind encouragement and valuable advice throughout the course of this study. I
express my profound gratitude to Professor Hajime Okamura for his guidance especiallyon the current problem of reinforced concrete and his encouragement during the courseof this study. My grateful appreciation is extended to Professor Masahiro Ouchi for his
suggestions. It has been an honor and a privilege to work with them for theiroutstanding examples of scientific dedication in their field.
Grateful acknowledgements are also extended to Professor Mikio Kadota for theirinterest and serving as members of examination committee.
Sincere words of gratitude are expressed to Mr. Masaru Ueno for his useful advice andsupport in experimental work.
I also would like to thank Mr.Supakit Swatekititham and Mr.Thammanoon
Dengpongpan, and all of graduate students for their help, support and patience towards
many of my problems during the course of the experimental work.

Shear Behavior of Reinforced Concrete Beams with A Small Amount of Web ReinforcementiiTable of Contents

TABLE OF CONTENTS

Abstract
Acknowledgements
Table of contents
1.

i
ii
iii

1.1
1.2
1.3
1.4
1.5

GENERAL INTRODUCTION
General and Problem Definition
Literature Review
Research Objective
Research Strategy and Content
References

1-3
4-6
7
7-8
9-10

2.1
2.2
2.3
2.4

EXPERIMENTAL DETAILSSpecime ns DetailMeasuring Systems
Experimental Procedures
References

11-13
13-20
20
21

3.1
3.2
3.3
3.4

EXPERIMENTAL RESULTS AND INVESTIGATION
Strength and Determination Comparison
Discussion on Experimental Results
Conclusion
References

22
22-30
30
31

4.1
4.2
4.3
4.4

MECHANICAL IN SHEAR RESISNTANCE
Mechanism of Stress Transfer
Mechanism of Web Reinforcement
Conclusion
References

32-36
37
38
39

2.

3.

4.

5.

CONCLUSION

406.

RECOMMENDATIONS OF FURTHER STUDY41

APPENDIX A
APPENDIX B

Crack Mapping
Crack Deformation at Shear Cracking of Each BeamShear Behavior of Reinforced Concrete Beams with A Small Amount of Web Reinforcementiii

CHAPTER 1
GENERAL INTRODUCTION• General and Problem Definition• Literature Review• Research Objective• Research Strategy and Content

General Introduction

Chapter 1

GENERAL INTRODUCTION

1.1 GENERAL AND PROBLEM DEFINITION
Amount of stirrup has a direct relation on the behavior of reinforced concrete
members of general structure, since the struc tures are possible to fail in brittle mannerwithout any warning sign if the shear stress rides over the shear carrying capacity.
Great example of shear failure is the collapse of super-structures during GreatHanshin Earthquake, in 1995. With respect to that evident, many of viaducts
structures constructed as a rigid-framed [1] were destructed as shown in Figure 1-1.
According to the mentioned structure, the amount of stirrup was lightly used.
Therefore, the following question has been asked by many researchers for such a longtime that did the estimation of shear strength of those was miscalculated?

Figure.1_1: Collapsed Structure.

One of possible and discovered reason is the size effect of the structure members,
which was introduced by Okamura [2]. With respect to this reason, the formuladerived from the specimen in experiments, which are very tiny compare to the real
structure, did not include some important factor. It can be illustrated that strength ofsmall specimens in experiments are affected in crack propagation by the reinforcing
bar, so called bond effect, which increased the fracture energy of concrete.
Nevertheless, in actual, the effect of bond from the reinforcement is very tiny since
the members cross-section is so large that crack propagation is not able to confine bybond effect of reinforcement. It used to clarify and accept this problem, size effect, bymany researchers and the consideration in its was added as size effect term in theshear strength design formula.
Another possible problem but it does not clarify yet is the shear carrying capacity’s
design concept since the concept is used as the strength of concrete at either failure or
shear crack load and sum up with shear resistance of web reinforcement. Why theshear strength design concept seems to be not enough, it is possible that a small
amount of web reinforcement cannot maintain the shear strength resist by concrete to
Shear Behavior of Reinforced Concrete Beams with A Small Amount of Web Reinforcement

1General Introduction

Chapter 1

be the same up to yielding point of stirrup itself. The remaining and possible reason isgoing to study and describe in this research thesis.
1.1.1 Current Use of Stirrup
As mentioned, the important factors drives shear failure in super-structure during
Kobe’s earthquake was explored. One factor originally based on the past design code
since it used to overestimate in size-effect. A related consideration is because Japan
Society of Civil Engineering’s design guide at the constructing period used allowablestress design concept, and it had required a suddenly increasing of stirrup if theallowable shear strength of concrete did not enough as Figure 1-2. Consequently,
escaping from suddenly increasing of stirrup, enlarge section of member to increase
the allowable shear stress in concrete was commonly done. In summary, large section
and small amount of stirrup are the consequence of designing that finally destructedby the earthquake.

Figure.1_2: Evolution of Stirrup in Design Code.

To solve the problem, the amount of stirrup was later renewed and published by JSCE
again in late year. Accordingly, the minimum requirement of stirrup was strictly
assigned to 0.2% for the beneficial in seismic performances. Although, the designed
structures follows the requirement have resulted in too much material used recently.
In the last, it is possibly that the good shear strength prediction method for whole
range of stirrup can reduce that of necessary cost.
1.1.2 Generality and Applicable of Superimposition Method
By considering simplicity, the several design guides stated that the shear carrying
capacity can be determined by adding the contribution of shear carrying capacity ofconcrete with that of stirrup. Whatever the shear strength of concrete are proposed inmany design guide, they are assumed that shear carrying capacity of concrete remains
the same up to the yielding of stirrups if the minimum requirement of stirrup isprovided. As it was claimed about small amount of web reinforcement, this is possible
Shear Behavior of Reinforced Concrete Beams with A Small Amount of Web Reinforcement

2General Introduction

Chapter 1that the shear carrying capacity of concrete can not remain constant until the yieldingof stirrup, and again this will result in ineffective calculation by Vc + Vs as shown in
Figure1-3.

VapplyVc+Vs(Test results)

Vc(prediction)Vs
ρv

ρw,min = 0.2%

Figure.1_3: Assumption on Load Carrying Capacity by Varying Stirrup.

Shear Strength Prediction in Varying Amount of
Web Reinforcement ratio

Vexperiment/V calculation

Vexp/(Vc+Vs)
Vexp/(Vcrk+Vs)
Vcrk/Vniwa

1.4

1

0.6
0.00

0.05

0.10

0.15

0.20

0.25

Web reinforcement ratio(%)
Figure.1_4: Prediction of shear strength by superposition method by varying web ratio.

To clarify and preliminary study, Figure.1_4 is the plot along the web reinforcement
ratio, x-axis, by comparing the shear strength calculated by Niwa’s equation [3], andthe assumption of 45 degrees crack to the member axis with the experimental. The
data used in the plotted graph are the large beams that tested in monotonic 3 and 4
point bending with depth higher than 500 mm., and contain quite small until verysmall amount of stirrup [4]-[9]. Beyond the web reinforcement ratio about 0.08%, it isclearly observed that the superposition method is safe enough to determine the shear
carrying capacity. Although it is seemingly that the estimation by superposition
method is not good enough. In summarize, the assumption and real experimental data,
even they were rarely discovered in research publications, are given us reasonableidea that the shear strength of reinforced concrete beam with a small amount of web
reinforcement can not simplicity determine by superposition method.
Shear Behavior of Reinforced Concrete Beams with A Small Amount of Web Reinforcement

3General Introduction

Chapter 1

1.2 LITERATURE REVIEW
1.2.1 Mechanical and Determining of Shear resistance
Generally, as in Figure.1_5, the shear resisting mechanism of reinforced concrete canbe qualitatively classified upon their behaviors as; 1.) Shear stress of concrete incompression zone; 2.) Aggregate interlocking along the crack plane; 3.) Dowel actionof longitudinal reinforcement; 4.) Stirrup. Although, it is difficult to quantify thequantitative model in estimating shear resistance of concrete part item by item.

Figure.1_5: General Mechanical of Shear Resistance.

By considering that difficulty, many researchers [3,10] proposed the model tocalculate the shear crack load of reinforced concrete without web reinforcement atshear cracking level by relying on the collected experimental results. Therefore, theshear strength of reinforced concrete beams with stirrup can be finally determined byadding that of concrete and stirrup based on 45-degree truss analogy. Those shear
crack load can be determined by the following equation;
13

Okamura’s Equation: Vcr = 0.20 f c'

(0.75 + 1.4 d a )(1 + β p + β d )

βp =

(

Where,

f c’
a
d
pw

:
:
:
:

pw − 1β d = d −1 4 − 1

Vcr = 0.2 p w f c'

Niwa’s Equation:

1-1) d (0.75 + 1.4 d a )
13

−1 4

1-2

compressive strength
shear span length
effective depth
tensile reinforcement ratio

In somewhat advance concept, both concrete and stirrup are recognized to possess
some interaction between them. Those were known as beneficial to the beam action[11] by effectively increasing in; 1.) The dowel action, because of the support offeredby stirrups to longitudinal bars; 2.) The strength of concrete tooth, due to an inclined
compression field associated with truss mechanism; 3.) The aggregate interlock
strength. With respect to this, it shown that stirrup effectiveness can be defined by‘stirrup effectiveness function’ [12] as 1-3 and 1-4, which is also function of moment
induced in beam action, Mb, as shown.
Ψ = kI b + p
1-3
Shear Behavior of Reinforced Concrete Beams with A Small Amount of Web Reinforcement

4General Introduction

Chapter 1
I b = M b M uc

Where,

k, p
Ib
Mb
Muc

:
:
:
:

1-4

constant
beam action index
moment contributed by beam action
ultimate moment

Conceptually, it can be said that the amount of stirrup has the direct relations to theshear resistance by increasing the effectiveness of beam action. Accordingly, in thisterminology, shear strength contributed by concrete was remained constant, no effectof stirrup effectiveness, but changed in term of shear resisted by stirrup for
simplifying. Shear strength of beams with web reinforcement can be determined byequation 1-5
Vu = Vc + ΨVs

1-5

According to the advance concerning, the shear carrying capacity of reinforced
concrete beam in concrete part after cracking should not be regard to be the same asjust crack [13] but it should be changed. However, in that experiment, the amount ofstirrup used as much more larger than the minimum requirement. Nevertheless, atleast, the behaviors of it after cracking like either load resistance or crack opening andcrack sliding, which control the load-deformation of itself, it mainly depends on theamount of stirrup.
In summary, the shear carrying capacity of reinforced concrete beam with web
reinforcement does not simply sum up that of concrete and stirrup together as Vc+Vs,
but it has to concern with the interaction between them, which actually depend on
amount of web reinforcement.
1.2.2 Provision of Minimum Stirrup in Several Design Codes
As aforementioned, the shear carrying capacity of reinforced concrete beam with web
reinforcement can be simply calculated by Vc+Vs, which the shear force resisted bystirrup has to provide almost minimum not less than the code specified below since it
is assumed that shear force resisted by concrete will remain the same. The following
requirement is the minimum requirement of stirrup according to many designing
codes.
JSCE 1986 [14]Seismic design code:ρs = 0.20%

ACI 318-83 [15]Normal strength concrete (Fc’ < 69 Mpa)
0.33bw sAv =:ρs = 0.08 – 0.13%
fy
ACI 318-89 [16]High strength concrete (Fc’ > 69 Mpa)

Shear Behavior of Reinforced Concrete Beams with A Small Amount of Web Reinforcement

5General Introduction

Chapter 1

f c' (0.33bw s ): ρs > 0.08
35fy
CSA 84 (Canada standard association) [17]0.35bw sAv =:ρs = 0.08 – 0.13%
fy
CSA 94 [18]b sAv = 0.06 f c' w
fy
AASHTO (LRFD Bridge design specification 1994) [19]b sAv = 0.083 f c' w
fy
Krauthammer [20]0.448bw sAv =:ρs = 0.1 – 0.16%
fy
Av =All of current codes same as that of ACI, CSA, and AASHTO, already provided theminimum requirement of stirrup for high strength concrete since the higher strength of
concrete affect in more brittle behavior and less shear transfer. Nevertheless, the shear
carrying capacity has some possibilities to be less than the calculation even theminimum stirrup was provided, this is because the shear carrying capacity of concrete
are not exactly predicted, which clearly observes in the case of ACI code due to either
size effect or lightly reinforcement [4].
1.2.3 Shear Strength of Lightly Web Reinforcment Used
In case that the shear strength has to examine corresponding to the less amount ofstirrup than minimum requirement that guidance is not provided in any design code,
Angelakos, Bentz, and Collins [4] suggested to estimate by using the interpolation
method. This method is to interpolate by tracing the straight line between shear
strength without web reinforcement and full AASHTO minimum amo unt of stirrups
using as shown in Figure.1_6.

Figure.1_6: Proposed method to determine shear strength of beams with less than
AASHTO specified minimum stirrups
Shear Behavior of Reinforced Concrete Beams with A Small Amount of Web Reinforcement

6General Introduction

Chapter 1

1.3 RESEARCH OBJECTIVE
Nowadays, in Japan, the existing reinforced concrete structures used the stirrup atleast equal to the minimum requirement specified in JSCE design guide; although, notheoretical insists that the amount of stirrup equals to 0.2% is appropriate, and how todetermine shear carrying capacity as a small amount of web reinforcement used.
Up to this point of concerning, first objective of this study is to clarify the applicableof the superposition method that used in predicting shear carrying capacity ofreinforced concrete beam with a small amount of web reinforcement at the shear span
ratio 3.0. Parallel to the first objective, second objective is to make clear that whatmechanisms take place when small amount of stirrup used. In summary, the great
understanding of reinforced concrete member with a small amount of stirrup can bepresented and hit it into shatter.

1.4 RESEARCH STRATEGY AND CONTENT
To perform the study in systematic and cover as much as details exist under several
constrains, the topic in each chapter was designed to study in different levels but
support to each other. Outline of research study is shown in Figure 1_7.
Chapter 2: This chapter is to illustrate experimental information, measuringsystems, and data interpretation methods, which they will be stated in details as
force transfer across shear crack and shear resisted by stirrup.
Chapter 2
Experimental Investigation

Assumption

Experimental work
(Beam test under monotonic loading)•••Specimen details.
Measuring systems.
Data interpretation methods

VapplyVy

Vc
ρ w,min- With shear span to depth
ratio equal to 3.
- Vary stirrup ratio at 0.03%,
0.05%, 0.06% and 0.08%.

Vs

Chapter 3
Experimental Results•••

Beams results.
Strength comparisons.
Wcomd Analysis results.

ρv

Chapter 4
Mechanism in Shear Resistance•Mechanism of stresstransfer

Figure.1_7: Research strategy.

Chapter 3: The experimental result corresponding with observation duringexperimental work will be noticed. The results of load carrying capacity
compared with theoretically calculating equation are shown with some degreeof difference.

Shear Behavior of Reinforced Concrete Beams with A Small Amount of Web Reinforcement

7

General Introduction

Chapter 1

Chapter 4: Leading to whole understanding in behavior of reinforced concrete
structure with a small amount of web reinforcement, this chapter is toinvestigate and explain the mechanical of shear resistance by the means ofstress transferring across the crack and stress contributed by stirrups.

Shear Behavior of Reinforced Concrete Beams with A Small Amount of Web Reinforcement

8

General Introduction

Chapter 1

1.5 REFERENCES
1.) JSCE, “Report on The Hanshin-Awaji Earthquake Disasater: Investigation of
causes of damage to Civil Engineering Structures, Bridge Structures”, JSCE, V.1,
1996.
2.) Okamura, H., Saeki, M., and Kanatu, T., “History of Seismic Design Code forConcrete Structure,” the scientific lecture for the great Hanshin-Awaji earthquake,
JSCE, Tokyo, 1996.
3.) Niwa, J., Yamada, K., Yokozawa, K., and Okamura, H., “Reevaluation of The
Equation for Shear Strength of Reinforced Concrete Beams without Web
Reinforcement,” Concrete Library International of JSCE, No.9, June 1987, pp.6584.
4.) Angelakos, D., Bentz, E. C., and Collins, M. P., “Effect of Concrete Strength andMinimum Stirrups on Shear Strength of Large Members,” ACI Structural Journal,
V.98, No.3, May-June 2001, pp.290-300.
5.) Yoon, Y. S., Cook, W. D., and Mitchell D., “Minimum Shear Reinforcement in
Normal, Medium, and High-Strength concrete Beams,” ACI Structural Journal,
V.93, No.5, Sep.-Oct. 1996, pp.1-9.
6.) Frosch, R. J., “Behavior of Large-Scale Reinforced Concrete Beams with
Minimum Shear Reinforcement,” ACI Structural Journal, V.97, No.6, Nov.-Dec.
2000, pp.814-820.
7.) Collin, M. P., and Kuchma, D. K., “How safe are Our Large, Lightly Reinforced
Beams, Slabs, and Footings?,” ACI Structural Journal, V.96, No.4, July-Aug.
1999, pp. 482-490.
8.) Bazant, Z. P., and Sun, H. H., “Size effect in Diagonal Shear Failure: Influrence ofAggregate Size and Stirrups,” ACI Material Journal, No.4, July- Aug. 1987,
pp.259-272.
9.) Vecchio, F. J., “Analysis of shear –Critical Reinforced Concrete Beams,” ACI
Structural Journal, V.97, No.1, Jan.-Feb. 2000, pp.102-110.
10.) Okamura, H., and Higai, T., “Proposed Design Equation for Shear Strength ofReinforced concrete Beams without Web Reinforcement,” Concrete Library
International of JSCE, V.1, July 1993, pp.96-106.
11.) Park, R., and Paulay, T., “Reinforced Concrete Structures,” John Wiley &Sons, New York, 1975.
12.) Russo, G., and Puleri, G., “Stirrup Effectiveness in Reinforced Concrete
Beams under Flexure and Shear,” ACI Structure Journal, V.94, No.3, May-June
1997, pp.227-238.
13.) Hassan, H. M., “ Shear Cracking Behavior and Shear Resisting Mechanism ofReinforced concrete Beams with Web Reinforcement,” Doctoral Dissertation,
University of Tokyo, March 1988.
14.) JSCE, Standard Specification for Design and Construction of concrete
Structure, Part Ι(Design), First edition, Tokyo, 1986.
15.)
ACI committee 318, “Building code Requirements for Structural Concrete
and Commentary (ACI 318-83),” American Concrete Institute, Detroit, 1983,
pp.111.
16.) ACI committee 318, “Building code Requirements for Structural Concrete
and Commentary (ACI 318M-89/ACI 318R-89),” American Concrete Institute,
Detroit, 1989, pp.353.
17.) Canadian Standard Association, “Design of Concrete Structures for
Buildings,” CAN3-A23.3-M84, Rexdale, Ontario, 1984, pp.281.
Shear Behavior of Reinforced Concrete Beams with A Small Amount of Web Reinforcement

9General Introduction

Chapter 118.) Canadian Standard Association, “Design of Concrete Structures,” CSA A23.394, Rexdale, Ontario, 1994, pp.199.
19.) “AASHTO LRFD Bridge Design Specifications and commentary,” Second
Edition, (1998) and 2000 update, American Association of State Highway
Transportation Officials, Washington D.C., 1998, 2000.
20.) Krauthammer, T., “Minimum Shear Reinforcement Based on Interface Shear
Transfer,” ACI Structural Journal, V.89, No.1, Jan.-Feb. 1992, pp.99-105.

Shear Behavior of Reinforced Concrete Beams with A Small Amount of Web Reinforcement

10CHAPTER 2
EXPERIMENTAL DETAILS• Specimens Detail• Measuring System• Experiment Procedures

Experimental Details

Chapter 2

EXPERIMENTAL DETAILS

2.1 SPECIMENS DETAIL
2.1.1 Material Properties
For designing the reinforced concrete specimens and collecting the necessary data, thematerial properties should be correctly known by conducting the material tests. Thus,
all principal materials used in constructing the reinforced concrete specimens, which
are steel bars and ordinary concrete, are tested and shown in following content.
Steel Reinforced Bars
The general properties of used steel bars, which all is deformed type but different ingrade between longitudinal and stirrup, are shown in Table.2_1. In the analysis, thestress-strain curves of steel bars are necessary used as constitutive equation; therefore,
D4 stress-strain relation determined by regression method are shown in Figure.2_1.
The constitutive equation is easily to obtain by regression method since the D4 bar
shown non-exist of yield plateau. Therefore, it is proper to determine yield strength of
D4 bar by the 0.002 off set.

Stress-Strain Relation of D4 bars

Stress (Mpa)500400
part1 0 - 0.002

300

part2 0.002 - 0.003

200

part3 0.003 - 0.05

100

0.002 off set

0
0

0.01

0.02

0.03

0.04

0.05

0.06

StrainFigure.2_1: Stress-strain curve and Constitutive equation of D4 bars.

Table.2_1: General properties of used deformed bars.
Steel Type

Nominal dia.
(mm.)

Yield Strength(Mpa)

Ultimate Strength(Mpa)

D4 (SD 295)

4.0

350

457

D10 (SD 345)

10.0

391

586

D22 (USD 685)

22.0

718

985

Shear Behavior of Reinforced Concrete Beams with A Small Amount of Web Reinforcement

11Experimental Details

Chapter 2

Ordinary Concrete
The casted concrete of specimens is ordinary concrete with the compressive strengthof 30 MPa at 28 days. Maximum aggregate size used as 20 mm. The curing method isdone by pouring water on the foam sheet that covered the specimens for first sevendays. Then, the beam specimens are cured by the air temperature in the laboratory.
2.1.2 Specimens Dimension and Parameters
According to the preliminary study in literature review, 4 reinforced concrete beams
are designed and used in experiment with the same dimension but varying amount of
web reinforcement to be less than 0.08%. As design is, the beams have unbalancing inshear span ratio, a/d, which they approximately equal to 3 in left span and 1.5 in rightspan. Respect to the target of the test, the right span has to highly reinforce with D10bars to gain its ultimate strength higher than that of left span side. In Figure.2_2 a
and b, the beams cross-section and scale down specimens layout are shown,respectively.
Clear cover 30
mm.
400 mm.

D4 @ 80, 100, 130, 180
mm.
8 D22

Clear cover 30

P

400 mm.
300 mm

D10@70mm.

400 mm

300 mm(a)

Vary number of stirrups

250 mmRa = P/3

1080
mm

450 mm
2700 mm.

540 mm

315mm

Rb = 2P/3(b)
Figure.2_2: Tested beams layout.

Regarding to the beams layout and cross-section, the shear carrying capacity of eachbeams were determined according to Niwa’s equation [1] parallel with superposition
method, (Vc+Vs ), and they also used the material properties from material
experiments. The predicted shear strength of tested beams is shown in Table.2_2.

Shear Behavior of Reinforced Concrete Beams with A Small Amount of Web Reinforcement

12Experimental Details

Chapter 2

Table.2_2: Shear carrying capacity of each tested beams.

1

Spacing of web
Reinforcement (mm.)
D4 @ 80

Vs
(kN)
40.58

Vc (Niwa)(kN)
160.81

Vu
(kN)
226.6

2

D4 @ 10032.47

163.35187.8

3

D4 @ 13024.97

161.06190.8

4D4 @ 180

18.04

166.6

187.5

Beam NO.

2.2 MEASURING SYSTEMS
According to the research objectives, the mechanical of shear resistance have to becorrectly investigated and considered in tested beams. Thus, the careful measurement
methods have been designed according to the basis of shear resistance as the stresstransfer across the crack and the stress of web reinforcement. Measuring systems havebeen desired on two microscopic models that can meaningfully explain local
behaviors in the specified parts. Finally, it is possibly that mechanical of shear
resistance in each two locations of beams can be accurately explained by thesemeasuring systems by employing two microscopic models.
2.2.1 Measurement of Stress Transfer across Crack Planea) Theoretical Background:
Buja Bujadham proposed the model to calculate the stress across cracks concrete, so
called Universal model for stress transfer across the crack [2,3], which it is actually
the generalization of Bi Li’s model, contact density model for stress transfer across
cracks concrete [4]. The stress transfer across cracks concrete can be imaged androughly illustrated in Figure.2_3. From figure, the opening and sliding of crack, ω andτ, create the stress at contact unit of θs, and summation of this stress at whole contact
unit results in the stress transfer across the crack

Figure.2_3: Stress transfer across the crack.

Contact density model concerns to the complex nature of crack surface, and thenrepresents it in mathematical ways by using two proposals and three basic
Shear Behavior of Reinforced Concrete Beams with A Small Amount of Web Reinforcement

13Experimental Details

Chapter 2

assumptions. Two basic proposals were crack geometry and contact stress directionthat shown in Eq.2-1 and 2-2, respectively.
Area of a contact unit :Contact stress direction :Where,

At
Ω(θ)
θs
θ

:
:
:
:

dAθ = At Ω(θ )dθθs = θ

2-1
2-2

whole surface area per unit crack plane
stochastic density function
resultant contact angle
contact angle before a unit deformed

Other three Eq.2-3, 2-4, and 2-5, were basic assumptions used in explain contact
density function, elasto-plastic model for contact stress, and effective ratio of contact
areas.
Contact density function:
Elasto plastic model:
Effective ratio of contact:
Where,

ωθ ’
ωθp ’

:
:

Ω (θ ) = 1 2 cos θσ c' = Rs (ωθ' − ωθ' p )K (ω ) = 1 − exp (1 − 0.5Gmax ω )

2-3
2-4
2-5

local deformation in normal direction of crack unit
local plasticity in normal direction of crack unit

Regarding to these equations, the stress crosses cracks are able to calculate insomewhat specified loading paths, but not for complex loading paths same as mixmode loading path, crack opening and sliding, of shear crack in beams. The
unrealistic of contact stress direction proposal is the exclusion of frictional force,
which is produced at the contact unit since it undergoes in mix mode. Therefore,
application of normality rules is invalid to use in this research; comprising with theactual, anisotropic behavior should be considered instead.
Beside that of concerning, the effect generated from mix- mode loading path, theanisotropic plasiticity of mortar beneath aggregate, which generated from either thenature of contact angle or the loading directions, and its fracture are accounted in
Universal model for stress transfer across cracks in concrete [3].
b) Measuring system
Contact extensometer and chips are designed to use in measuring the deformation ofshear crack opening and sliding of tested beams. By utilizing coordinatetransformations, the contact chips are located on the beam surface in both sides, northand south surfaced, at the longer span part, which has a/d = 3.0. Contact chips are
cemented with the distance about 100 mm. apart from each others with the angle 0, 90,
135 degrees and 0,-45,-90 for north and south surfaces, respectively, as shown in
Figure.2_4.

Shear Behavior of Reinforced Concrete Beams with A Small Amount of Web Reinforcement

14

Experimental Details

Chapter 2(a) Chips coordinate (north surface of beams). (b) Chips coordinate (south surface of beams).

(c). Chips on the beam surface.
Figure.2_4: Measuring system of stress across the cracks

By the proposed method, after crack taken place, the angle of crack at the origin pointof contact chips coordinates are measured and utilized coordinate transformation
equation, Eq.2-6, which has been well known used in strain rosette.

 x  1 + cos 2θ 1 1 − cos 2θ 1
  
 y  = 1 + cos 2θ 2 1 − cos 2θ 2
 xy 1 + cos 2θ 1 − cos 2θ
  
3
3
Where,

θi
ai
x
y
xy

:
:
:
:
:

sin 2θ 1  a1 2 
sin 2θ 2 a 2 2sin 2θ 3  a 3 

2-6angle of i coordinate to tangential coordinate of crack
deformation respect to i coordinate
crack deformation in parallel direction of crack planecrack deformation in vertical direction of crack planecrack deformation in tangential direction of crack plane

Shear Behavior of Reinforced Concrete Beams with A Small Amount of Web Reinforcement

15

Experimental Details

Chapter 2c) Data Interpretation
From the measurement, deformation in 0, 90, 135 and 0, -45, -90 degrees related withbeam axis, and crack angle related to the beams axis are used in interpreting the crack
displacement in lateral and horizontal direction, y and xy to the crack plane,
respectively, as shown in the above equation.
Then, the universal model of stress transfer across the cracks is utilized to obtain theamount of stress and force resisted by the aggregate interlocking mechanism. The
algorithm of model in calculating is shown in Figure.2_5. However, this outcome isstill in the crack plane coordination and it need to convert to the beam coordinate
again. By the coordinate transformation, the stress in Z direction, which is stresscontribution in shear resistance process of aggregate interlocking mechanism, can be
determined.
START

Input f c’, Gmax, Initial ω, δ,
N = 150, dθ = π/nGiven dδ, dω
τ = 0, σ = 0, δ = δ+dδ, ω = ω+dω
N = n+1, θ = −π/2 + n.dθdδθ , dω θRc’
τ = τ + Rc’sinθs.dθ
σ = σ + Rc’sinθs.dθ
N=n
τ, σ

Next input

END

Figure.2_5: Algorithm of The Universal stress transfer across cracks concrete. [3]2.2.2 Measurement of Strain of Web Reinforcement at Crack Planea) Theoretical Background: Bond-Slip-Strain Relationship of Steel bar
The tensile strain of reinforcement embedded in concrete has non-linear characteristic
depends on the embedment length. This is because the present of bond between thereinforcing bar and surrounding concrete. With respect to the bond stress, the strain of
Shear Behavior of Reinforced Concrete Beams with A Small Amount of Web Reinforcement

16Experimental Details

Chapter 2

steel bar in several cases is shown in Figure.2_6 (a), (b), and (c), which are bond oflong embedment length, short embedment length, and bond of axial tension,respectively.

(a)

(b)

(c)
Figure.2_6: Strain of Embedded Reinforcing Bar. [5]From the figures, the strain at specific point of reinforcing bar is nonlinearly reduced
respect to the distance from the crack section. Moreover, it is not necessary that thesame amount of slips have to be created by the same strain value, since the strain
distribution of embedded bar is affected by boundary conditions, which are theembedment length, compressive strength of concrete, steel grade, and steel diameter.
Without considering the effect of strain value, many researchers failed to achievemodel to determine bond stress. On the other hand, Shima [6], proposed a universal
bond-slip-strain model for reinforced concrete, which take the effect of strain of
embedded bar at the considered point. Bond-slip-strain model is shown in Eq.2-7.
τ (ε , s ) = τ o (s )g (ε )2-7Where, τ(ε,s) is local bond stress and τo (s) is intrinsic bond stress when strain is zerodenoted by,τ o (s ) = f c'k [ln (1 + 5s )]1
g (ε ) =1 + 105 εc

Shear Behavior of Reinforced Concrete Beams with A Small Amount of Web Reinforcement

2-8
2-9

17

Experimental Details
Where,

f c’
k
c
s
S
d
ε

Chapter 2
:
:
:
:
:
:
:

compressive strength of concrete
constant equal to 0.73
constant equal to 3,
non-dimensional slip equal to 1000S/d,
slip
diameter of steel bar
strain of steel bar

Nevertheless, this experiment does not intend to measure the bond stress but it intends
to measure stress at the crack position. Fortunately, the strain at the crack position canbe determined by utilizing bond stress equation.
b) Measuring method
By utilizing a set of equations corresponding with strain from two nearing positions
along the embedded bar, stirrup, the strain at crack position can be indirectly
calculated. The set of equations according to the local bond behavior are shown in
Figure.2_8, which are bond-stress-slip equation, slip-strain relation, constitutive ofsteel bar, and equilibrium equation, respectively.
According to this method, the strains at two points are located at 3 cm. apart fromeach other. Furthermore, the position of strain gauges should be mouthed in the range
25 times of diameter from the crack section to ensure that the strain before yielding
can be investigated. With respect to this method, the approximate crack positions
should be determined in first place and it may be properly computed by some finite
element program, so called Wcomd (2 dimensions, plate element). Therefore, theapproximate solution by Wcomd is employed to locate the position of strain gauges at
each stirrup.

Bond-slip-strain model:τ = τ o (s ). g (ε )Slip-strain relation:

S = ∑ (ε .∆x )Constitutive of steel bar:σ = σ (ε )Equilibrium equation:

τσ.As(σ+∆σ).Asτ∆X∆σ . As = τ .πd .∆x

Figure.2_8: Equation for calculating strain of stirrup at crack plane.

Each reinforced concrete beam that analyzed by Wcomd shows almost the same crack
position in tested span as the sample in Figure.2_9; therefore, the crack formation can
aspect to be the same for simplicity in locating the strain gauge. The strain gauges
positions in each stirrup leg of all beam are shown in Table.2_3, which the height aremeasured from the bottom of stirrup, the distances origin at the support, and the crackheights are estimated from 45-degree crack tracing from the loading point.

Shear Behavior of Reinforced Concrete Beams with A Small Amount of Web Reinforcement

18

Experimental Details

Chapter 2

Figure.2_9: Crack pattern of tested beams.

Table.2_3: Stirrup strain position at each stirrup legs.
Specimen No.1 spacing 80 mm.
Position (mm)Height1
Height2
1080
1000
346
316
920
266
236
840
206
236
760
126
156
680
66
96
600
66
96
520
66
96
440
66
96
360
66
96
280
66
96
200
66
96
120
66
96
40
66
96
Specimen No.3 spacing 130 mm.
Position (mm)Height1
Height2
1080
950
296
266
820
186
216
690
66
96
560
66
96
430
66
96
300
66
96
170
66
96
40
66
96

Height3
286
206
266
186
126
126

Height3
236
246
126
126
26
26
26
26

Specimen No.2 spacing 100 mm.
Position (mm)Height1
Height2
1080
980
326
296
880
246
276
780
146
176
680
66
96
580
66
96
480
66
96
380
66
96
280
66
96
180
66
96
80
66
96

Specimen No.4 spacing 180 mm.
Position (mm)Height1
Height2
1080
900
266
296
720
86
116
540
66
96
360
66
96
180
66
96
0

Height3
266
306
206
126
126

Height3
326
146
126

c) Data interpretation
From the measuring data, two strain values at two exact distances from the crack
section are input in the computer algorithm for iteratively determining the slip of thefarthest distance gauge. After the strain and slip of one exact location was exactly
known, the extrapolating of strain distribution can be done by assuming strain in next
position and then checking the equilibrium of stress between the bond and stress ofembed bar in each step. The algorithm of calculating the strain distribution, slip, andstress at crack position is shown in Figure.2_10.

Shear Behavior of Reinforced Concrete Beams with A Small Amount of Web Reinforcement

19Experimental Details

Chapter 2

Input ε1, ε2

σ1 = σ(ε1 ), σ2 = σ(ε2 )∆σ = σ1 − σ2Assume S1
τ1 = f(S1 ).g(ε1 )τ2 = f(S1 +∆S).g(ε2 )Assume ∆εεi+1 =εi + ∆ε
σ = σ(εi+1)
Si+1 = Si + ∆S
τi+1 = f(Si+1).g(εi+1)
Bond deterioration length

σi+1 = σi + (τ2 +τ1 ).π.db.∆X /As2∆σ / (τ2 +τ1 ).π.db.∆X ≤ 0.01
Xi+1 = Xi + ∆X
Set B.C (S1 ,ε1 ) at 1 st point
Xi+1 < L

END
Figure.2_10: Algorithm for calculating strain of web reinforcement at crack plane.

2.3 EXPERIMENT PROCEDURES
Specimens were set up in the loading frame with hand-pumping loading system. As
illustrated in Figure.2_11, the experiment set up before testing in each beam wasshown. The loading method was a load controlling and the measuring can be read at
the each loading steps equal to 30 kN. The load deflection can trace according to theload-measured values and deflection measured value from the load cell and CDP,
respectively.

Figure.2_11: Specimen set up in the tested frame.
Shear Behavior of Reinforced Concrete Beams with A Small Amount of Web Reinforcement

20

Experimental Details

Chapter 2

2.4 REFERENCES
1.) Niwa, J., Yamada, K., Yokozawa, K., and Okamura, H., “Reevaluation of The
Equation for Shear Strength of Reinforced Concrete Beams without Web
Reinforcement,” Concrete Library International of JSCE, No.9, June 1987, pp.6584.
2.) Bujadham, B., Mishima, T., and Maekawa, K., “Qualitative Studies on
Mechanisms of Stress Transfer across Cracks in Concrete,” Procurement of JSCE,
V.17, No.451, Aug. 1992, pp. 265-275.
3.) Bujadham, B., and Maekawa, K., “ The Universal Model for Stress Transfer
across Cracks in Concrete,” Procurement of JSCE, V.17, No.451, Aug. 1992,
pp.277-287.
4.) Li, B., Maekawa, K., and Okamura, H., “ Contact Density Model for StressTrnasfer across Cracks in Concrete,” Journal of the Faculty of Engineering, The
University of Tokyo (B), Vol.XL, No.1, 1989, pp.9-52.
5.) Okamura, H., and Maekawa, K., “Nonlinear Analysis and constitutive Models ofReinforced Concrete,” Gihodo-Shuppan, Tokyo, Japan, 1991.
6.) Shima, H., Chou, L. L., and Okamura, H., “ Micro and Macro Models for Bond in
Reinforced Concrete,” Journal of the Faculty of Engineering, The University ofTokyo (B), Vol.XXXIX, No.2, 1987, pp.133-193.

Shear Behavior of Reinforced Concrete Beams with A Small Amount of Web Reinforcement

21

CHAPTER 3
EXPERIMENTAL RESULTS AND INVESTIGATION• Strength and Determination Comparison• Discussion on Experimental Results

Experimental Results and Investigation

Chapter 3

EXPERIMENTAL RESULTS AND INVESTIGATION

3.1 STRENGTH AND DETERMINATION COMPARISON
According to the aim of this research, the shear carrying capacity of reinforced
concrete member with a small amount of web reinforcement is the most important inconsideratio n and outcome. Thus, shear capacity of beams are now determined andcompared with the experimental result as it shown in Table.3_1.
Table.3_1: Comparison table.

Categories
1.) Compressive strength (Mpa)
2.) Stirrup (%)3.) Vs (kN)4.) Shear crack, Vcr, (kN)5.) Shear crack, Vc, (kN)6.) Shear capacity, Vu, (kN)7.) Vcr / Vc
8.) Vcr + Vs
9.) Vc + Vs
10.) Vu / Vcr+Vs
11.) Vu / Vc+Vs
12.) Failure mode***

Beam 141.5
0.079%
40.0
170.2
160.81
226.6
1.06
210.2
200.81
1.08
1.12
S.C.*

Beam 243.5
0.063%
32.0
157.7
163.35
187.8
0.97
189.7
195.35
0.99
0.96
S.C.

Beam 341.7
0.048%
24.6
145.6
161.06
190.8
0.9
170.2
185.66
1.12
1.03
S.**

Beam 446.15
0.035%
17.8
156.3
166.6
187.5
0.94
174.1
184.4
1.08
1.02
S.C.

Shear compression failure
Shear failure

Refer to the table of comparison, the category number 11 and 12 show that the designconcept by superposition method, no matter what shear crack strength is taken fromeither experiment or calculation, is good enough to predict the shear carrying capacityof reinforced concrete members with a small amount of web reinforcement. Even
though the shear carrying capacity of each beam, except Beam 1, has mostly the samelevel, this is going to quote and investigate into the mechanisms in next chapter.

3.2 DISCUSSION ON EXPERIMENTAL RESULTS
3.2.1 General Behaviors
Beam 1: The strongest beamThis beam contained the highest amount of web reinforcement, about 0.08%, which isless than the design code mentioned in the first chapter. As observed during the test,
shear crack was taken placed at the shear load level equal to 170.2 kN. This beam did
not illustrate large and clear dropping of load deflection curve when the shear crack
was taken place. It can be seen from the load deflection curve, Figure.3_1 that theshear crack load level did not affect in any drop of plotting.

Shear Behavior of Reinforced Concrete Beams with A Small Amount of Web Reinforcement

22

Experimental Results and Investigation

Chapter 3

Shear load (tf)

Load Deflection Curve of Beam 1
24
20
16
12
8
4
0
0

5

10

15

Deflection (mm)Figure.3_1: Load deflection curve of Beam 1.

After shear cracking formed, the load was increased further up to the failure point andthe beam was fail as shown in Figure.3_2. Regarding to failure load, the compression
zone was compression failure and followed by the large opening of shear crack since
the shear resisting in compression zone suddenly dropped and aggregate interlocking
at shear crack surface can not substitute shear resisting in compression zone. Thisphenomenon simultaneously taken place with the cut-off of web reinforcement
illustrated in Figure.3_3.

Figure.3_2: Shear failure of Beam 1.

Shear Behavior of Reinforced Concrete Beams with A Small Amount of Web Reinforcement

23

Experimental Results and Investigation

Chapter 3

Figure.3_3: Cut-off of Stirrup

Beam 2: A second highest amount of web reinforcement
Shear crack was taken place at load level in shear span equal to 157.7 kN, which isslightly less than that of Beam 1. The crack formation of Beam 2 was generally
similar to Beam 1 without any special consideration. Shear crack formation, shear
failure, and load deflection curve are shown in Figure.3_4

Figure.3_4: Shear failure of Beam 2.

It is possible that the larger amount of web reinforcement ought to delay the time togenerate the perfect shear crack since the web reinforcement has highly possibility tointersect with crack and effectively activate at the same load level as shown in
Figure.3_5 a and b. Similarly, it is too difficult to judge when the shear crack became
complete path since it does not occur suddenly as the case of beam without web
reinforcement. Therefore, it has possibility that the excess shear crack strength in last
beam compare to calculation is routed by them.

Shear Behavior of Reinforced Concrete Beams with A Small Amount of Web Reinforcement24

Experimental Results and Investigation

Chapter 3(a)(b)
Figure.3_5: Possibility of crack intersect with stirrup

Load Deflection Curve of Beam 2

Shear Load (tf)

20
16
12
8
4
0
0

2

4

6

8

10

Deflection (mm)Figure.3_6: Load Deflection Curve of Beam 2.

Onset of shear crack started, as it can be observed, Figure.3_6, load deflection curveshows a little drop of graph after shear crack. It meant that the shear crack penetrated
to the compression quite fast, and then the load was able to increase further up to thefailure point. Similar to previous case, the compression zone firstly came up to thecompression failure, and then followed by the large shear crack opening and cut off of
web reinforcements in simultaneously.
Beam 3:
By the same manner with two previous beams but it is important to mention, shear
crack load of Beam 3 was observed at the shear load level equal to 145.6 kN that isquite low when compares to previous cases. Even though this beam possessed thesame compressive strength and geometry as Beam 1, the shear crack loads of these
two beams were observed at different level of shear load. With respect to thisphenomenon, it is believed to be the same reason as explained in Beam 2 but it seems
to be clear since both beam had the same compressive strength. Shear crack formation,
shear failure, and load deflection curve are also shown in Figure.3_7.

Shear Behavior of Reinforced Concrete Beams with A Small Amount of Web Reinforcement

25

Experimental Results and Investigation

Chapter 3

Load Deflection Curve of Beam 3

Shear Load (tf)

20
16
12
8
4
0
0

2

4

6

8

Deflection (mm)Figure.3_7: Load Deflection Curve of Beam 3.

Figure.3_8: Shear Failure of Beam 3.

From the load deflection curve, the load had a little drop as shear crack taken place
and then the load was increased until the failure occurred. Different mode of failure
was observed in this beam since the compression zone did not fail, but the shear crack
was the controller instead.
Beam 4: Minimum web reinforcement
The smallest amount of web reinforcement beam, Beam 4, was created shear crack atshear load level equal to 156.3 kN. Although the compressive strength of cylindrical
specimen was the highest in all beams, the shear crack load level was not the highest
among them. At the first place, this is believed that the initial state of crack did notconfine by the web reinforcement so that the stress at the crack front was propagate
very fast as aforementioned.

Shear Behavior of Reinforced Concrete Beams with A Small Amount of Web Reinforcement

26

Experimental Results and Investigation

Chapter 3

Figure.3_9:Geometry of crack propagation.

In more details to support previously claimed, it was discovered that bond between
the concrete and the main reinforcement is a major cause for the initiation of inclined
shear cracking in RC beams [1]. Since this bond are shown to have excellent highshear stress induce at the main reinforcement height. Its propagation believed to becontrolled by longitudinal cracking along the main bar, so-called horizontal crack [2],which actually comes after inclined shear cracking. Therefore, the less amount of web
reinforcement usually yields lower load level to resist horizontal crack will show
larger and faster horizontal crack, which finally results in faster perfect shear cracking.
The illustrative sketching can be shown as Figure.3_9. However, this is not dowel
action developing by web reinforcement since the dowel action is believed and shown
to be affected by the position of first stirrup from the crack [3].
Load Deflection Curve of Beam 4

Shear Load (tf)

20
16
12
8
4
0
0

2

4

6

8

10

Deflection (mm)Figure.3_10: Load Deflection Curve of Beam 4.

As shown in Figure.3_10 and 3_11, they illustrate about load deflection and shear
failure pattern. According to the load deflection curve, there is very large drop of it
since it was not enough amount of stirrup to confine the crack in both incline shear
crack and horizontal crack [3]. However, onset of shear cracking, the equilibrium
became satisfy again, and then the shear load can be increased further up to the failure
point. The failure mechanism and pattern in this beam is similar to Beam 1 and Beam2, which the compression zone firstly crushed, and followed up by large crack
opening and cut-off of stirrups.
Shear Behavior of Reinforced Concrete Beams with A Small Amount of Web Reinforcement27

Experimental Results and Investigation

Chapter 3

Figure.3_11: Shear Failure of Beam 4.

3.2.2 Shear Carrying Comparison among Beams
Beams behaviors under experiment were explained as aforementioned; currently,
shear carrying capacity of all beam are going to be compared. Since the compressive
strength in each beam is different; therefore, the comparisons have to be done bynormalizing compressive strength. However, dividing the shear load by compressive
strength seems no meaning. Fortunately, there are some equations for determining
shear crack load and those equations used the compressive strength to be one ofparameter [4,5].
By normalizing the effect of compressive strength term, the remaining terms areconcerned with the beam geometries, and all beams had used the same geometry
during this experiment. Therefore, the comparing result by normalizing thecompressive strength term in shear crack prediction formula is used, and then it
consequently shows in Figure.3_12.

ShearLoad
Normalized by fc'

Load Deflection (Normailized fc' )
35
30
25
20
15
10
Beam 1: Pw = 0.08%

5Beam 2: Pw = 0.065%

0
0

2

4

6

Deflection (mm)8 Beam 3: Pw10
= 0.05%
Beam 4: Pw = 0.035%

Figure.3_12: Shear carrying capacity comparison by normalizing compressive strength.
Shear Behavior of Reinforced Concrete Beams with A Small Amount of Web Reinforcement28

Experimental Results and Investigation

Chapter 3

As observed from the graph, the effect of compressive strength on the shear crackload was eliminated but the shear crack load and shear carrying capacity are still thesame, except Beam 1. It is meant that at a small amount of web reinforcement shear
carrying capacity cannot be exactly calculated by the concept of superposition;
however, the superposition method is still good enough to safely determine the shear
carrying capacity. As claimed, in actually, the bending members without web
reinforcement of the shear span ration, a/d, equal to 3.0 has some reserve strength
after the shear crack was taken place. It can be simply said that the shear strength of
the short span beam is not shear crack load.
3.2.3 Comparison between Several Calculation of Shear Strength
Several design codes, such as Eq.3-1, 3-2, 3-3, and 3-4, used superposition method byselecting the concrete contribution term equal to the shear crack load. Currently, it
should be know that “Can these codes satisfy the shear failure beams with a small
amount of web reinforcement?” The calculation results are shown in Table.3_2.
Vd 
ACI Committee 318-95 [6] Vcr =  0.16 f c' + 17.2 ρ s
3-1
b.d
M

220  'CSA Simplification [7]Vcr = 3-2
 f c b.d 1000 + d 
13
0.2  3d 
CEB-FIP [8]Vcr = 150.1 +(100. ρs )1 3 fc'1 3b.d 3-3d  a s 13

Vcr = 0.9 f c' (100 p s ) 1 3 (100 d )1 4 b.d

JSCE 1986 [9]3-4

Table.3_2: Comparison between Experiment Results and Calculation of Several Codes.
Specimens.
ACI
CSA
CEB-FIP
NewzealandJSCENo.
318-83
1994
NZS 3101
1986

Beam 1
Beam 2
Beam 3
Beam 4

1.15
0.98
1.05
1.03

1.19
1.01
1.09
1.07

1.42
1.22
1.32
1.32

1.10
0.93
1.00
0.97

1.17
1.19
1.19
1.21

Note: The values in table obtained from dividing the experimental with calculation.

Mostly, the design code of each country used the shear crack load in estimating shear
carrying capacity in superposition method. Therefore, the superposition method isseemingly applicable and good enough. With respect to safety, the comparisons tableshows that the testing values to calculating values ratio nearly to 1.0 andunderestimation in some codes, such as CEB-FIP and JSCE, since all of them were
provided some safety factor in the derivation processes. Especially, CEB-FIP shear
strength model was derived from fracture mechanic by assuming the dynamic modeof splitting crack. This mode of splitting crack in the horizontal cracking is actually
not dynamic but the quasi-statistic by considering with the energy used in opening
conical crack at the bond face [10].
However, it has to be reminded again that the tested beam had quite short span ratio
but it is scale down in dimensio n from Shinkansen viaduct structures. As it is short,the shear strength of the beam is not exactly the shear crack load but larger than that.
Shear Behavior of Reinforced Concrete Beams with A Small Amount of Web Reinforcement

29

Experimental Results and Investigation

Chapter 3

Implying of inefficiency stirrup can be highly possible, and it can be said in advancethat simply said that the superposition method is not safe for long shear span ratio.

3.3 CONCLUSION
1.) The shear carrying capacity of reinforced concrete beam with a small amount of
web reinforcement can be safely determined by the superposition method, Vc+Vs.

Shear Behavior of Reinforced Concrete Beams with A Small Amount of Web Reinforcement

30

Experimental Results and Investigation

Chapter 3

3.4 REFERENCES
1.) Kim, W., and White, R.N., “Hypothesis for Localized Horizontal Shearing Failure
Mechanism of Slender RC Beams,” Journal of Structural Engineering, ASCE,
V.125, No.10, Oct. 1999, pp. 1126-1135.
2.) Kim, W., and White, R. N., “ Shear-Critical Cracking in Slender Reinforced
Concrete Beams,” ACI Structural Journal, V.96, No.5, Sep.-Oct. 1999, pp.757765.
3.) Poli, S. D., Prisco, M. D., and Gambarova, P. G., “Cover and Stirrup Effects on
The Shear Response of Dowel Bar Embedded in Concrete,” ACI Structural
Journal, V.90, No.4, July-Aug. 1993, pp.441-450.
4.) Niwa, J., Yamada, K., Yokozawa, K., and Okamura, H., “Reevaluation of The
Equation for Shear Strength of Reinforced Concrete Beams without Web
Reinforcement,” Concrete Library International of JSCE, No.9, June 1987, pp.6584.
5.) Okamura, H., and Higai, T., “Proposed Design Equation for Shear Strength ofReinforced concrete Beams without Web Reinforcement,” Concrete Library
International of JSCE, V.1, July 1993, pp.96-106.
6.) ACI Committee 318, “Building Code Requirements for Reinforced concrete (ACI
318-95) and Commentary ACI 318-R95,” American Concrete Institute, Detroit,
1995.
7.) Canadian Standards Association, “Design of Concrete Structures for Buildings,”
CAN3-A23.3-M84, Rexdale, Ontario, 1984.
8.) Comite Euro-International Du Beton, “CEB-FIP Model Code 1990,” Thomas
Telford, London, 1993.
9.) JSCE, Standard Specification for Design and Construction of concrete Structure,
Part Ι(Design), First edition, Tokyo, 1986.
10.) Gastebled, O. J., and May, L. M., “Fracture Mechanics Model Applied to
Shear Failure of Reinforced Concrete Beams without Stirrups,” ACI Structural
Journal, V.98, No.2, Mar.-Apr. 2001, pp.184-190.

Shear Behavior of Reinforced Concrete Beams with A Small Amount of Web Reinforcement

31

CHAPTER 4
MECHANICAL IN SHEAR RESISTANCE• Mechanism of Stress Transfer• Mechanism of Web Reinforcement

Mechanical in Shear Resistance

Chapter 4

MECHANICAL IN SHEAR RESISTANCE
In generally, the shear resistance can be divided into concrete part and web
reinforcement part. According to the contribution of concrete part, the aggregateinterlocking, dowel action, and shear in compression zone will be raised up after shear
cracking. However, in this research, the measured component was only the aggregateinterlocking or shear crack deformation since there is a limitation in techniques andaccuracy. Finally, it can be said that the shear resistance mechanism of reinforced
concrete beam with a small amount of web reinforcement was obtained.

4.1 MECHANISM OF STRESS TRANSFER
4.1.1 Crack Deformation
Due to the geometry of shear crack, some of contact chip groups are properly selected
to use in determining shear stress across the crack. Fortunately, all beams mostly
yielded the same crack pattern so that the shear stress across the crack can be directlycompared to each other. Before going to compare the shear stress cross the crack, thelocation of measurement is shown by the circles in Figure 4_1.

C
B
A
Figure.4_1: Point of Measured Deformation.

At the considered location of crack deformation, the geometry of crack and beam
behavior have definitely correlation as Figure 4_2. Onset of shear crack, the crack
located in compression zone and the horizontal cracking, tension bars, mainly behavein opening and rarely exist in sliding since the crack angle is almost parallel to thebeam axis. In other hand, as the horizontal crack opening, the inclined crack is goingunder opening and sliding, simultaneously.
Compressive
zone
Inclined crack

Figure.4_2: Crack Geometry and its deformation.
Shear Behavior of Reinforced Concrete Beams with A Small Amount of Web Reinforcement32Mechanical in Shear Resistance

Chapter 4

The magnitude of crack opening and sliding can be considered by the length of thearrow vector. By this figure, the above-mentioned are brightly understood the total
different at the compression zone and the inclined crack location.
Point A: Horizontal crack
Crack Opening
Pw 0.08% : N

70Applied load (tf)

Pw 0.08% : S
65

Pw 0.065% : N
Pw 0.065% : S

60

Pw 0.05% : N
Pw 0.05% : S

55

Pw 0.035% : N
50

Pw 0.035% : S

45
40
0

0.5

1

1.5

Crack width (mm)Figure.4_3(a): Crack opening at the horizontal cracking.

In Figure.4_3 (a), beam specimen with web reinforcement ratio equal to 0.035%
shown the highest crack opening, whereas other beams with 0.065%, 0.05%, and
0.08% web reinforcement show smaller crack width at the same load level,
respectively. Even though the beam with web reinforcement ratio equals to 0.05%
should have crack width value larger than that of 0.065% but it is inversely perceived
from the measuring. However, beam with 0.05% web reinforcement ratio shown the
different in failure mode among others. Accordingly, it is believed that the reason
routed from different in failure mode. It can be concluded that the crack width ofbeam at horizontal cracking will become larger respect to the smaller amount of web
reinforcement used.
Crack Sliding
Pw 0.08% :N

70

Applied load (tf)

Pw 0.08% : S
65

Pw 0.065% : N
Pw 0.065% : S

60

Pw 0.05% : S
Pw 0.035% : N

55

Pw 0.035% : S
50
45
40
0

0.5

1

1.5

Crack slip (mm)Figure.4_3(b): Crack sliding at the horizontal cracking.

Definitely different from the crack opening at the horizontal cracking, the cracksliding shown in Figure.4_3 (b) does not clearly show the good relation between cracksliding with amount of stirrup used. However, it can be observed a little bit larger inShear Behavior of Reinforced Concrete Beams with A Small Amount of Web Reinforcement

33

Mechanical in Shear Resistance

Chapter 4

crack sliding in beam with 0.035% web reinforcement and almost same magnitude ofcrack sliding in beam with 0.05%, 0.065%, and 0.08%, of web reinforcement.
Point B: Bottom of inclined crack
Crack Opening
Pw 0.08% : N

Applied Shear (tf)70

Pw 0.08% : S
Pw 0.065% : N

65

Pw 0.065% : S
60

Pw 0.05% : N
Pw 0.05% : S

55

Pw 0.035% : N
Pw 0.035% : S

50
45
40
0

0.5

1

1.5

Crack width (mm)Figure.4_4(a): Crack opening at the bottom of shear cracking.

At the bottom of shear cracking, crack opening is largest when the web reinforcement
is smallest, and it became smaller when larger amount of web reinforcement is used
as observed in Figure.4_4(a). However, the beam with web reinforcement 0.05%
should have larger crack width than that of 0.065% but it is not actually since thefailure mode was different in beam that used 0.05%. These are also evidenced in
crack sliding as it shown in Figure.4_4(b).

Crack Sliding
Pw 0.08% : N

Applied load (tf)70

Pw 0.08% : S
Pw 0.065% : N

65

Pw 0.065% : S
60

Pw 0.05% : N
Pw 0.05% : S

55

Pw 0.035% : N
50

Pw 0.035% : S

45
40
0

0.5

1

1.5

2

2.5

3

Crack slip (mm)Figure.4_4(b): Crack sliding at the bottom of shear cracking.

Furthermore, it can be observed that beam with web reinforcement 0.035% has verylarge crack opening and sliding suddenly onset of shear crack initiated but it was not
the case for other beams.

Shear Behavior of Reinforced Concrete Beams with A Small Amount of Web Reinforcement

34

Mechanical in Shear Resistance

Chapter 4

Point C: Top of inclined crack
At the last location, tip of shear crack, the crack deformation, opening and sliding, isshown in Figure.4_5 (a) and (b). With respect to the measuring result in figures, thesame behavior with last two locations can be observed.
Crack Opening
Pw 0.08% : N
Pw 0.08% : S
Pw 0.065% : N
Pw 0.065% : S
Pw 0.05% : N
Pw 0.05% : S
Pw 0.035% : N
Pw 0.035% : S

Applied Load (tf)

70
65
60
55
50
45
40
0

0.5

1

1.5

Crack width (mm)Figure.4_5(a): Crack opening at top of shear cracking.

More deeply in detail of crack deformation, it can be observed that the crack width
was almost the same in three reference locations but it was highest in crack sliding at
the bottom of shear crack width. Finally, again, it is absolutely possible to said thatwhen the larger amount of web reinforcement were used, the smaller crack
deformation in both opening and sliding without regarding to the location of total
shear cracking path.
Crack Sliding
Pw 0.08% : N

70Applied Load (tf)

Pw 0.08% : S
65

Pw 0.065% : N
Pw 0.05% : N

60

Pw 0.05% : S
55

Pw 0.035% : N
Pw 0.035% : S

50
45
40
0

0.5

1

1.5

2

Crack Slip (mm)Figure.4_5(b): Crack sliding at top of shear cracking.

4.1.2 Crack Deformation Path
From Fig.4_6, the plots of crack sliding to opening ratio at the lower side of inclined
crack are illustrated. In this figure, the applied shear load of all beams was normalized by the
maximum applied shear load of beam with highest shear carrying capacity, which was thebeam with web reinforcement ratio 0.08%. Except the beam with web reinforcement ratio
Shear Behavior of Reinforced Concrete Beams with A Small Amount of Web Reinforcement

35Mechanical in Shear Resistance

Chapter 4

0.05% that failed in different manner compare with others, it can be observed that the plots ofcrack sliding to opening ratios were approximately regarded to be the same level.

Crack deformation path

Vapplied / Vmax

1.1
1
0.9
0.8
0.7
0.6

Pw = 0.035%
Pw = 0.05%

0.5
0

0.5

1

1.5

Sliding / Opening2
Pw = 0.065%

Pw = 0.08%

Fig. 4_6: Ratio of crack sliding to opening.

However, it can be also observed from this figure that the beams with smaller web
reinforcement had a large amount of crack sliding and crack opening just after the shear
cracking started. On the other hands, the beam with larger amount of web reinforcement
behaved only crack opening in the first state, and then gradually increased its sliding.

4.1.3 Shear Stress Cross Crack (Aggregate Interlocking)By summing up all measured point, the aggregate interlocking contributed to resist
the shear load in each beam is calculated. With respect to results, the characteristic instress transfer across the crack of each beam can be easily seen from the graph asshown in Figure.4_7. Start with Beam 2, 3, and 4, have similarly shear transfer, which
they are 100 kN up to 140 kN. Scattering is evidently observed but it is the limitationof method used to measure the crack deformation. Then, last specimen, Beam 1, ha s agreat different stress transfer across the crack from others, where it starts with smallvalue after the crack has initially propagated until a little bit higher load than theperfect shear crack taken place. After that, the shear stress across crack gradually riseup to the 160 kN level, which was the highest among 4 specimens, and then the shear
failure is consequence.
Related to the beams behavior during the experiments, the three smallest in web
reinforcement specimens evidently show the dropping of load resistance since theshear crack was suddenly opening and propagate as load deflection curves observed inprevious chapter. Thus, the suddenly jump to the maximum or near maximum ofstress transfer across the crack is the result of changing equilibrium, jump.
Nevertheless, it was not the matter for Beam 1 since the crack was not suddenly open
and propagate so fast that the load shown evidently dropping. Therefore, the shear
transfer did not rapidly increase but it delayed until the crack was widely open andwent under sliding. With respect to these considerations, the graph of stress transfer
across crack was shown in somewhat different patterns; however, the maximum shear
transfer can approximately the same with some degree of accuracy.

Shear Behavior of Reinforced Concrete Beams with A Small Amount of Web Reinforcement36Mechanical in Shear Resistance

Chapter 4

Shear Resisted by Aggregate Interlocking
180

Shear Resisted (kN)

160
140
120

Beam 1:Pw = 0.08%

100

Beam 2:Pw = 0.065%
Beam 3:Pw = 0.054%

80

Beam 4:Pw = 0.032%

60

Beam 1:Pw =0.08%

40

Beam 2:Pw =0.065%

20

Beam 3:Pw = 0.054%0140Beam 4:Pw = 0.032%
150

160

170

180

190

200

210

220

230

Shear Load (kN)
Figure.4_7: Stress transfer cross crack.

Finally, the most important item is the crack opening and sliding. Due to these, in
reinforced concrete beam with a small amount of web reinforcement, it is possiblythat the beam with larger crack width corresponding to the larger sliding wins to thesame level of aggregate interlocking or stress across crack as beam with smaller crack
width and sliding since it can be previously observed in shear transfer of beam
number 2,3, and 4 that used 0.065%, 0.05%, and 0.035% of web reinforcement,
respectively. However, beam with web reinforcement 0.08% is approximately
respected to have same shear transfer of beams even the first state yielded the low
shear transfer and then became large shear transfer.

4.2 MECHANISM OF WEB REINFORCEMENT
4.2.1 Yielding of Web Reinforcement
After the data was colle cted and carefully considered, it can be said that the collected
data from the strain gauges mounted along the web reinforcement cannot be employed.
Thus, it seems that the yielding of web reinforcement cannot be known from thedirect measurement. However, it is known that crack width equal to the amount ofpull out of steel bar, which is 2 times of slip. By considering this relation, it should
suit to approximate the slip from the crack width as Eq.4-1.
2*Slip = Crack Width4-1

Not only slip is calculated, but also strain at the crack should be found since theobjective is “Did the web reinforcement yield at the shear crack intersection? ” With
respect to this reason, long embedment length terminology is considered to employ inthis case since the web reinforcement is a tiny size compared with the height of thebeams. Therefore, 2 more equations of strain-slip relation [1] for long embedded bar
are tool for the analysis in this state, and those equation are specified in following;
Shear Behavior of Reinforced Concrete Beams with A Small Amount of Web Reinforcement37Mechanical in Shear Resistance

Chapter 4
s = ε s (6 + 3500ε s )4-2

2

Slip  f c'  3s=
D  20 
Where,

s
εs
D
f c’

:
:
:
:

4-3

normalized slip
strain of web reinforcement
diameter of web reinforcementcylindrical compressive strength of concrete

At this state, after all related equations was introduced, the yielding strain from thematerial test are going to be input in determining the normalized slip, and then slip
and crack width can be finally determined. Follow that describe, it can be finallycalculate that the crack width corresponding to the strain at yielding, which
determines by 0.002 offsets, is 0.42 mm, and then compare to the crack width of
every measured location along the shear crack, for example Figure.4_3, 4_4, and 4_5.
In conclusion, all web reinforcement intersected with the shear crack can beconsidered as yielding state before shear failure of the beams according to theprevious figures of crack opening and sliding.

4.2.2 Other Mechanisms Contributed by Web Reinforcement
As frequently referred to the strain of web reinforcement, the effect of unachievable inthis value influences on the analysis of other mechanisms such as shear resistance incompression zone and dowel action, which they cannot direct measure except adopt
from the indirect calculation. Therefore, it has to state in here that those mechanisms[2] have not been investigated for reinforced concrete beam with a small amount of
web reinforcement currently.

4.3 CONCLUSION
1.) The beam with smaller amount of web reinforcement show larger in crack
opening and sliding. Except beam with 0.05% of web reinforcement, the crack
opening and sliding show some different from the claimed relation since thefailure is absolutely different.
2.) Beams with a small amount of web reinforcement have the same stress transfer
across the crack even the crack opening of them are totally different. Since thelarger of crack opening shows larger of crack sliding, the stresses in those beams
are finally maintained to be the same.
3.) All web reinforcement intersected by shear crack is considered to reach yielding
state by relying on the strain-slip model of long embedded steel bar and therelation of crack width and slip.

Shear Behavior of Reinforced Concrete Beams with A Small Amount of Web Reinforcement

38Mechanical in Shear Resistance

Chapter 4

4.4 REFERENCES
1.) Okamura, H., and Maekawa, K., “Nonlinear Analysis and Constitutive Models ofReinforced Concrete,” Gihodo, Tokyo, 1991.
2.) Park, R., and Paulay, T., “Reinforced Concrete Structures,” John Wiley & Sons,
New York, 1975.

Shear Behavior of Reinforced Concrete Beams with A Small Amount of Web Reinforcement

39

CHAPTER 5
CONCLUSION

Conclusion

Chapter 5

CONCLUSION

Result of this research is very useful even the applicable length is very specific since theexperimental work was done only in the range of shear span equal to 3.0. However, it
can be said that most of viaducts and buildings are composed with the columns of shear
span ratio 3.0. According to the experimental results of four tested beams and theircorresponding measurement values of aggregate interlocking; therefore, it can be
confidently concluded as a unit as following:1.) The shear carrying capacity of reinforced concrete beam with a small amount of web
reinforcement as shear span ratio 3.0 can be safely determined by the superposition
method, Vcr+Vs .
2.) The beams with smaller amount of web reinforcement showed larger in crack opening
and sliding. Except beam with 0.05% of web reinforcement, the crack opening andsliding showed some different from the claimed relation since the failure is absolutely
different.
3.) All beams have the same stress transfer across the crack even the crack opening andsliding of them were totally different. Since the larger of crack opening showed thelarger of crack sliding; therefore, the stresses in those beams are finally maintained tobe the same.

Shear Behavior of Reinforced Concrete Beams with A Small Amount of Web Reinforcement40

CHAPTER 6RECOMMENDATIONS OF FURTHER STUDY

Recommendations of Further Study

Chapter 6

RECOMMENDATIONS OF FURTHER STUDY

Based on the experimental results according to the experimental plan and problems inanalyzing procedures, it is better to specify some recommendations for further study as
following:1.) Improving on measuring systems or data collecting method, for example stresstransfer across crack and strain of web reinforcement at the crack position, inorder to improve the accuracy and more detail understanding in the depth of
behavior.
2.) Extending the measuring systems or data collecting method to other locations or
items, for example the shear resistance of compression zone and that of dowel
action.
3.) More widely parametric studies are necessary to continue since this behavior
seemingly to different in longer shear span ratio, compressive strength, andabsolutely different in reversed cyclic loading.

Shear Behavior of Reinforced Concrete Beams with A Small Amount of Web Reinforcement

41

APPENDIX A
Crack Mapping

BEAM WITH WEB REINFORCEMENT 0.08%

North side
1

2

3

4

5

6

7

8

9

10
D
C
B
A

South side

10
D
C
B
A

9

8

7

6

5

4

3

2

1

BEAM WITH WEB REINFORCEMENT 0.065%

North side
1

2

3

4

5

6

7

8

9

10
D
C
B
A

South side

10
D
C
B
A

9

8

7

6

5

4

3

2

1

BEAM WITH WEB REINFORCEMENT 0.05%

North side

1

2

3

4

5

6

7

8

9

10
D
C
B
A

South side

10
D
C
B
A

9

8

7

6

5

4

3

2

1

BEAM WITH WEB REINFORCEMENT 0.035%

North side
1

2

3

4

5

6

7

8

9

10
D
C
B
A

South side

10
D
C
B
A

9

8

7

6

5

4

3

2

1

APPENDIX B
Crack Deformation at Shear Cracking of Each Beam

Reinforced concrete beam with web reinforcement ratio 0.08%

Element NumberNorth (A9)Crack Angle
10 degree
Applied Load
(tf)
42
45
51
54
55.5
57.24
58.9
60.47
62.04
63.5
65
66.4
67.95

Increment deformation in beam coordinate Increment deformation in cracking coordinate
a1
a2
a3
x
y
xy
0.001
0.002
-0.0170.0018-0.0011-0.0176
0.008
0.003-0.0010.00250.0002-0.0053
0.101
0.152
0.242
0.01580.04750.0998
0.083
0.181
0.23
0.0131
0.05290.0753
0.043
0.19
0.196
0.0051
0.05320.0496
0.038
0.194
0.178
0.00540.0526
0.0316
0.01
0.217
0.13
0.0026
0.0541-0.0199
0.015
0.185
0.178
-0.0016
0.05160.0442
0.172
0.252
0.183
0.04610.0599-0.0409-0.1260.26
0.17-0.0374
0.07090.0308
0.006
0.25
0.217
-0.00430.06830.0419
0.003
0.324
0.102
0.00840.0733-0.1127
0.029
0.578
0.415
0.00190.14990.0109Element Number North (A9)Crack Angle
25 degree
Applied Load
(tf)
42
45
51
54
55.5
57.24
58.9
60.47
62.04
63.5
65
66.4
67.95

Increment deformation in beam coordinate Increment deformation in cracking coordinate
a1
a2
a3
x
y
xy
0.001
-0.008-0.0170.0024-0.0042-0.0052
0.001
0.003-0.0010.00090.0001-0.0027
0.115
0.152
0.242
0.0096
0.05710.0556
0.083
0.181
0.23
0.00640.05960.0255
0.043
0.19
0.196
0.0021
0.0562-0.0052
0.038
0.194
0.178
0.0046
0.0534-0.0199-0.0120.217
0.13
0.00200.0493-0.0700
0.037
0.185
0.178
0.00300.0525-0.0136
0.03
0.252
0.183
0.00940.0611-0.0580
0.016
0.26
0.17
0.00880.0602-0.0729
0.006
0.25
0.217
-0.0046
0.0686-0.0362
0.003
0.324
0.102
0.02690.0549-0.1625
0.029
0.578
0.415
0.01040.1413-0.1386

Reinforced concrete beam with web reinforcement ratio 0.08%

Element Number North (A8)
Crack Angle
20 degree
Applied Load
(tf)
42
45
48
51
54
55.5
57.24
58.9
60.47
62.04
63.5
65
66.4
67.95

Increment deformation in beam coordinate Increment deformation in cracking coordinate
a1
a2
a3
x
y
xy
0.064
0.034
0.104
0.00630.01820.0518
0.105
0.047
0.1
0.02070.01730.0370
0.039
0.115
0.153
-0.0002
0.03870.0338
0.021
0.426
0.486
-0.1032
0.21490.2216
0
0.349
0.351
-0.07460.16180.1460
0.033
0.337
0.127
0.02650.0660-0.1421
0.005
0.228
0.294
-0.02080.07900.0643-0.0010.336
0.244-0.00270.0864-0.0497
0.04
0.239
0.27
-0.00520.07490.0360
0.006
0.366
0.306
-0.00730.1003-0.0238
0.027
0.381
0.249
0.00990.0921-0.0793-0.0150.359
0.365-0.0238
0.10980.0276
0.036
0.414
0.548
-0.03190.14440.1259
0.026
0.698
0.548
-0.0037
0.1847-0.0735

Element Number
Crack Angle

North (B7)50 degree

Applied Load
(tf)
42
45
48
51
54
55.5
57.24
58.9
60.47
62.04
63.5
65
66.4
67.95

Increment deformation in beam coordinate Increment deformation in cracking coordinate
a1
a2
a3
x
y
xy
0.027
0.004
0.024
0.00130.00650.0098
0.006
0.093
0.049
0.01440.0104-0.0428
0.044
0.145
0.191
0.00210.0452-0.0665
0.163
0.534
0.442
0.07220.1021-0.1989
0.07
0.474
0.423
0.0396
0.0964-0.2252
0.053
0.338
0.275
0.03550.0623-0.1541
0.034
0.256
0.222
0.02210.0504-0.1227
0.029
0.344
0.237
0.04100.0522-0.1639
0.041
0.287
0.211
0.03480.0472-0.1293
0.059
0.331
0.285
0.03250.0650-0.1496
0.042
0.4
0.32
0.0386
0.0719-0.1935
0.034
0.366
0.256
0.04340.0566-0.1732
0.063
0.414
0.329
0.04500.0743-0.1885
0.027
0.778
0.519
0.0882
0.1130-0.3900

Reinforced concrete beam with web reinforcement ratio 0.08%

Element Number
Crack Angle

North (C6)30 degree

Applied Load
(tf)
42
45
48
51
54
55.5
57.24
58.9
60.47
62.04
63.5
65
66.4
67.95

Increment deformation in beam coordinate Increment deformation in cracking coordinate
a1
a2
a3
x
y
xy
-0.044-0.065-0.033-0.0170-0.01030.0198
0.046
0.001
0.028
0.00770.00400.0217
0.063
0.14
0.138
0.01270.0381-0.0151
0.137
0.529
0.442
0.03520.1313-0.1152
0.073
0.469
0.402
0.01460.1209-0.1060
0.026
0.303
0.248
0.00570.0765-0.0782
0.034
0.319
0.22
0.01690.0714-0.1017-0.0060.278
0.225-0.0030
0.0710-0.0785
0.038
0.258
0.208
0.01030.0637-0.0653
0.013
0.307
0.234
0.00560.0744-0.0903
0.061
0.423
0.321
0.02080.1002-0.1173
0.016
0.349
0.271
0.00570.0856-0.0999
0.018
0.457
0.34
0.00970.1090-0.1388
0.053
0.807
0.599
0.02380.1912-0.2420

Element Number
Crack Angle

North (C5)
25 degree

Applied Load
(tf)
42
45
48
51
54
55.5
57.24
58.9
60.47
62.04
63.5
65
66.4
67.95

Increment deformation in beam coordinate Increment deformation in cracking coordinate
a1
a2
a3
x
y
xy
0.034
0.063
0.086
0.00260.02160.0130
0.047
0.083
0.085
0.00950.0230-0.0009-0.0050.104
0.126-0.01100.03580.0074-0.0190.458
0.391-0.01630.1260-0.0725
0
0.447
0.44
-0.02150.1333-0.0320-0.0090.316
0.246-0.0055
0.0822-0.0650-0.0060.238
0.212-0.00900.0670-0.0317-0.3520.294
0.227-0.1082
0.0937-0.0829
0.358
0.259
0.204
0.10510.0492-0.0293-0.0150.314
0.262-0.0106
0.0854-0.0537
0.012
0.394
0.304
0.00070.1008-0.0814
0.022
0.34
0.248
0.00690.0836-0.0787-0.0260.432
0.323-0.00900.1105-0.0983
0.014
0.742
0.549
0.00330.1857-0.1689

Reinforced concrete beam with web reinforcement ratio 0.08%

Element Number South (A9)Crack Angle
30 degree
Applied Load
(tf)
42
51
54
55.5
57.24
58.9
60.47
62.04
63.5
65
66.4
67.95

Increment deformation in beam coordinate Increment deformation in cracking coordinate
a1
a2
a3
x
y
xy
0.049
0.01
0.012
0.01360.0011-0.0081
0.086
0.746
0.456
0.05410.15390.26580.074-0.2440.123-0.04640.0039-0.2417
0.008
0.184
0.165
-0.00190.04990.0417-0.0390.218
0.157-0.0083
0.05310.0775
0.043
0.169
0.19
0.00040.05260.0126
0.012
0.191
0.183
-0.0035
0.05420.0368
0.048
0.298
0.191
0.02370.06280.0993
0.019
0.322
0.247
0.00710.0781
0.0930
0.016
0.303
0.241
0.00430.07550.0835
0.026
0.309
0.334
-0.01190.09560.0393
0.029
0.878
0.585
0.03180.19490.3019Element Number South (A8)
Crack Angle
27 degree
Applied Load
(tf)
42
45
48
51
54
55.5
57.24
58.9
60.47
62.04
63.5
65
66.4
67.95

Increment deformation in beam coordinate Increment deformation in cracking coordinate
a1
a2
a3
x
y
xy
0
0
0
0.00000.00000.0000
0.049
0.081
0.082
0.01050.02200.0030
0.06
0.136
0.233
-0.0084
0.0574-0.0486
0.064
0.484
0.466
-0.00120.13820.0570-0.0030.494
0.275
0.01890.10390.1837
0.005
0.307
0.384
-0.0293
0.1073-0.0119
0.01
0.24
0.063
0.02690.03560.1295-0.0020.379
0.256
0.00550.08880.1144
0.001
0.205
0.49
-0.0675
0.1190-0.1450-0.0040.323-0.0370.05560.02420.2478-0.0010.474
0.295
0.01240.10590.1578-0.0130.251
0.253-0.0167
0.07620.0280-0.0020.464
0.352-0.00100.11650.1174-0.0240.797
0.595-0.0059
0.19910.2095

Reinforced concrete beam with web reinforcement ratio 0.08%

Element Number South (A8)
Crack Angle
42 degree
Applied Load
(tf)
42
45
48
51
54
55.5
57.24
58.9
60.47
62.04
63.5
65
66.4
67.95

Increment deformation in beam coordinate Increment deformation in cracking coordinate
a1
a2
a3
x
y
xy
0
0
0
0.00000.00000.0000
0.049
0.081
0.082
0.01160.02090.0141
0.06
0.136
0.233
-0.0101
0.05910.0237
0.064
0.484
0.466
0.01530.12170.1888-0.0030.494
0.275
0.04750.07520.2441
0.005
0.307
0.384
-0.02160.09960.1263
0.01
0.24
0.063
0.04370.01880.1209-0.0020.379
0.256
0.02540.06890.1824
0.001
0.205
0.49
-0.07310.1246
0.0610-0.0040.323-0.0370.0845-0.0047
0.1831-0.0010.474
0.295
0.03840.07990.2301-0.0130.251
0.253-0.00700.06650.1173-0.0020.464
0.352
0.0216
0.09390.2191-0.0240.797
0.595
0.03410.15920.3865

Element Number
Crack Angle

South (B7)
27 degree

Applied Load
(tf)
42
45
48
51
54
55.5
57.24
58.9
60.47
62.04
63.5
65
66.4
67.95

Increment deformation in beam coordinate Increment deformation in cracking coordinate
a1
a2
a3
x
y
xy
0.017
0.002
0.015
0.00230.0025-0.0093
0.027
0.062
0.306
-0.0455
0.0678-0.1317
0.065
0.115
-0.09
0.0563-0.01130.1214
0.13
0.509
0.426
0.03130.1284
0.0975
0.075
0.491
0.553
-0.0143
0.15580.0215
0.042
0.332
0.108
0.04290.05060.1644
0.038
0.229
0.191
0.0081
0.05860.0470
0.033
0.324
0.252
0.00910.08020.0795
0.036
0.25
0.212
0.00650.06500.0501
0.043
0.324
0.261
0.01020.0816
0.0731
0.034
0.376
0.26
0.01590.08660.1110
0.029
0.337
0.282
0.00370.08780.0723
0.046
0.459
0.344
0.01530.11100.1200
0.048
0.843
0.62
0.01960.20310.2320Reinforced concrete beam with web reinforcement ratio 0.08%

Element Number
Crack Angle

South (C6)30 degree

Applied Load
(tf)
42
45
48
51
54
55.5
58.9
60.47
62.04
63.5
65
66.4
67.95

Increment deformation in beam coordinate Increment deformation in cracking coordinate
a1
a2
a3
x
y
xy
0.029
0.001
-0.0055
0.0099-0.0024-0.0019-0.001
0.004
0.009-0.00160.0023-0.0016
0.002
0.11
0.099
-0.00210.03010.0253-0.0180.51
0.417-0.0085
0.13150.1431-0.0230.448
0.39-0.0147
0.12100.1152-0.0590.307
0.229-0.0146
0.07660.1060-0.0340.276
0.208-0.00800.06850.0907-0.0020.267
0.204
0.00080.06540.0807-0.0140.346
0.457-0.04400.12700.0104
-0.03
0.359
0.057
0.0401
0.04220.2222
0.116
0.337
0.282
0.03080.08250.0679-0.0770.423
0.321-0.0200
0.10650.14250.025-0.0080.666-0.1382
0.1424-0.3430

Element Number
Crack Angle

South (C5)35 degree

Applied Load
(tf)
42
45
48
51
54
55.5
57.24
58.9
60.47
62.04
63.5
65
66.4
67.95

Increment deformation in beam coordinate Increment deformation in cracking coordinate
a1
a2
a3
x
y
xy
0.07
-0.0390.088-0.00850.0162-0.0760
0.032
0.064
0.067
0.00620.01780.0085
0.049
0.206
0.155
0.01870.04500.0644
0.113
0.391
0.346
0.02900.09700.0985
0.094
0.399
0.357
0.02260.10060.1055
0.043
0.307
0.213
0.02350.06400.1110
0.076
0.251
0.207
0.02320.0586
0.0673
0.011
0.264
0.198
0.01790.04500.1952
0.013
0.337
0.196
0.02500.06250.1450-0.0280.388
0.227
0.01620.07380.1794
0.068
0.172
0.262
-0.00780.06780.0003
0.006
0.37
0.254
0.01590.07810.1485-0.1180.402
0.329-0.03070.10170.1804
0.081
0.9
0.643
0.05180.19350.3326

Reinforced concrete beam with web reinforcement ratio 0.065%
Element NumberNorth (A8)
Crack Angle
16 degree
Applied Load
(tf)
42
45
48
51
52.846
53.94
55.2
56.64

Increment deformation in beam coordinate Increment deformation in cracking coordinate
a1
a2
a3
x
y
xy
0.027
0.002
0.009
0.00700.00020.0020
0.112
0.184
0.232
0.01820.05580.0522
0.002
1.235
0.877
-0.01030.3196-0.1075-0.0040.647
0.40.00100.1598-0.1059-0.0010.32
0.226-0.0030
0.0827-0.0287
0.033
0.432
0.333
0.00250.1137-0.0205-0.0090.393
0.316-0.01100.1070-0.0014
0.018
0.494
0.384
-0.0034
0.1314-0.0176

Element Number North (A8)
Crack Angle
25 degree
Applied Load
(tf)
42
45
48
51
52.846
53.94
55.2
56.64

Increment deformation in beam coordinate Increment deformation in cracking coordinate
a1
a2
a3
x
y
xy
0.027
0.002
0.009
0.00670.00060.0060
0.112
0.184
0.232
0.0151
0.05890.0264
0.002
1.235
0.877
0.00600.3032-0.3061-0.0040.647
0.4
0.01300.1477-0.1989-0.0010.32
0.226
0.00130.0784-0.0802
0.033
0.432
0.333
0.00680.1094-0.0882-0.0090.393
0.316-0.00800.1040-0.0743
0.018
0.494
0.384
0.00120.1268-0.1000

Element Number North (AB67)
Crack Angle
35 degree
Applied Load
(tf)
45
48
51
52.846
53.94
55.2

Increment deformation in beam coordinate Increment deformation in cracking coordinate
a1
a2
a3
x
y
xy
0
0
0
0.00000.00000.0000-0.0091.63
1.374
0.00020.4051-0.5773-0.0110.77
0.3-0.0055
0.3850-0.3941
0.021
0.414
0.323
0.01050.2070-0.1486
0.008
0.522
0.376
0.00400.2610-0.2035
0.047
0.557
0.419
0.02350.2785-0.1996

Reinforced concrete beam with web reinforcement ratio 0.065%
Element Number
Crack Angle

North (B6)
36 degree

Applied Load
(tf)
42
45
48
51
52.846
53.94
55.2
56.64

Increment deformation in beam coordinate Increment deformation in cracking coordinate
a1
a2
a3
x
y
xy
0.022
-0.0460.024-0.0089
0.00290.0435
0.131
0.36
0.361
0.0251
0.0977-0.0732
0.183
1.538
1.253
0.06950.3608-0.5231
0.075
0.807
0.585
0.04770.1728-0.3036
0.039
0.39
0.307
0.01810.0892-0.1383
0.04
0.524
0.394
0.02520.1158-0.1955
0.021
0.49
0.369
0.24500.1845-0.1879
0.06
0.461
0.4
0.01650.1138-0.1476

Element Number
Crack Angle

North (C5)
31 degree

Applied Load
(tf)
42
45
48
51
52.846
53.94
55.2

Increment deformation in beam coordinate Increment deformation in cracking coordinate
a1
a2
a3
x
y
xy
0.008
-0.0040.027-0.00430.00530.0170
0.117
0.203
0.335
-0.00370.08370.0442
1
1.566
1.187
0.3087
0.3328-0.2949
0.057
0.744
0.545
0.02790.1723-0.2355
0.022
0.386
0.295
0.0096
0.0924-0.1180
0.033
0.511
0.376
0.01700.1190-0.1622
0.023
0.494
0.36
0.0146
0.1147-0.1603

Element Number
Crack Angle

North (C5)37 degree

Applied Load
(tf)
42
45
48
51
52.846
53.94
55.2

Increment deformation in beam coordinate Increment deformation in cracking coordinate
a1
a2
a3
x
y
xy
0.008
-0.0040.027-0.0051
0.00610.0127
0.117
0.203
0.335
-0.0050
0.08500.0069
1
1.566
1.187
0.32430.3172-0.2985
0.057
0.744
0.545
0.04170.1585-0.2904
0.022
0.386
0.295
0.01660.0854-0.1499
0.033
0.511
0.376
0.02650.1095-0.2011
0.023
0.494
0.36
0.02400.1052-0.1984

Reinforced concrete beam with web reinforcement ratio 0.065%

Element Number South (A9)Crack Angle
20 degree
Applied Load
(tf)
45
48
51
52.846
53.94
55.2

Increment deformation in beam coordinate Increment deformation in cracking coordinate
a1
a2
a3
x
y
xy
0
0
0
0.00000.00000.0000-0.0940.724
0.606-0.04630.20380.0400-0.0040.205
0.342-0.03370.0839-0.1178-0.0060.177
0.141-0.00510.04780.0163-0.0140.269
0.25-0.0149
0.0787-0.0029-0.0240.259
0.207-0.01210.07090.0224

Element Number South (A8)
Crack Angle
13 degree
Applied Load
(tf)
45
48
51
52.846
53.94
55.2

Increment deformation in beam coordinate Increment deformation in cracking coordinate
a1
a2
a3
x
y
xy
0
0
0
0.00000.00000.00000.244
1.19031.066
0.03470.3238-0.1061
0.094
0.67370.514
0.01660.17540.0101
0.054
0.316
0.271
0.00740.0851-0.0199
0.06
0.431
0.352
0.00800.1147-0.0144
0.063
0.413
0.324
0.01080.1082-0.0006

Element Number South (AB78)
Crack Angle
10 degree
Applied Load
(tf)
42
45
48
51
52.846
53.94
55.2

Increment deformation in beam coordinate Increment deformation in cracking coordinate
a1
a2
a3
x
y
xy
0.283
-0.459-3.4670.3541-0.39813.0483-0.0780.452
3.925-0.3351
0.4286-3.4219
0.194
1.661
1.291
0.02850.4353-0.0907
0.1
0.758
0.598
0.01550.1990-0.0463
0.028
0.368
0.013
0.02540.07360.2320
0.06
0.523
0.4
0.00920.1365-0.0228
0.045
0.486
0.363
0.00620.1265-0.0162

Reinforced concrete beam with web reinforcement ratio 0.065%

Element Number
Crack Angle

South (B7)
43 degree

Applied Load
(tf)
42
45
48
52.846
53.94
55.2

Increment deformation in beam coordinate Increment deformation in cracking coordinate
a1
a2
a3
x
y
xy
0.009
0.043
-2.6
0.6611-0.64810.2001
0.174
0.4
3.062
-0.62230.7658-0.0808
0.153
1.628
1.28
0.1126
0.33260.7085
0.08
0.395
0.315
0.03730.08140.1517
0.088
0.524
0.4
0.04930.10370.2109
0.087
0.484
0.359
0.0496
0.09320.1929Element Number
Crack Angle

South (B6)30 degree

Applied Load
(tf)
42
45
48
51
52.846
53.94
55.2

Increment deformation in beam coordinate Increment deformation in cracking coordinate
a1
a2
a3
x
y
xy
0.033
0.04
0.067
0.00210.0162-0.0122-0.0190.427
0.422-0.02410.12610.0841
0.015
1.589
1.25
0.00510.39590.4576
0.011
0.755
0.579
0.00680.18470.2242-0.0150.407
0.313-0.00270.10070.1242
0.004
0.51
0.422
-0.00310.13160.1366-0.0050.49
0.115
0.05730.06400.2781Element Number
Crack Angle

South (C5)
13 degree

Applied Load
(tf)
42
45
48
51
52.846
53.94
55.2

Increment deformation in beam coordinate Increment deformation in cracking coordinate
a1
a2
a3
x
y
xy
0.029
0.033
0.042
0.00610.0094-0.0090
0.111
0.376
0.354
0.01900.1028-0.0412
0.112
1.545
1.168
0.00890.40530.0090
0.057
0.736
0.554
0.0056
0.19270.0073
0.029
0.397
0.299
0.00250.10400.0034
0.038
0.508
1.384
-0.1063
0.2428-0.8955
0.029
0.487
-0.64
0.11150.01750.9075

Reinforced concrete beam with web reinforcement ratio 0.065%

Element Number
Crack Angle

South (C5)55 degree

Applied Load
(tf)
42
45
48
51
52.846
53.94
55.2

Increment deformation in beam coordinate Increment deformation in cracking coordinate
a1
a2
a3
x
y
xy
0.029
0.033
0.042
0.00530.01020.0056
0.111
0.376
0.354
0.04620.07550.1623
0.112
1.545
1.168
0.18860.2256
0.7894
0.057
0.736
0.554
0.09120.10710.3729
0.029
0.397
0.299
0.04880.05770.2023
0.038
0.508
1.384
-0.17270.30920.6008
0.029
0.487
-0.64
0.2950-0.1660-0.0919

Reinforced concrete beam with web reinforcement ratio 0.05%

Element NumberNorth (A9)Crack Angle
5 degreeApplied Load(tf)
42
44.66345.66446.49847.49848.36649.3
50.2
51.10152.20253.46955.20456.538

Increment deformation in beam coordinate Increment deformation in cracking coordinate
a1
a2
a3
x
y
xy
0.022
0.179
0.048
-0.0070-0.01930.0597-0.1090.129
0.245-0.03700.04200.2108-0.2350.511-0.764-0.0182
0.0872-0.9531
0.345-0.1631.08
0.04240.00311.0181
0.163
0.088
0.084
0.04240.0203-0.0344
0.144
0.062
0.059
0.03780.0137-0.0362-0.0680.142
0.137-0.0209
0.03940.0802
0.07
0.186
0.124
0.01790.0461-0.0140
0.037
0.063
0.087
0.00770.01730.0342
0.028
0.088
0.083
0.00600.02300.0194
0.135
0.072
0.047
0.03610.0157-0.0502-0.0570.168
0.174-0.01900.0467
0.0972
0.148
0.263
0.224
0.03640.06630.0082Element Number North (A8)
Crack Angle
16 degree
Applied Load(tf)
42
44.66345.66446.49847.49848.36649.3
50.2
51.10152.20253.46955.20456.538

Increment deformation in beam coordinate Increment deformation in cracking coordinate
a1
a2
a3
x
y
xy
0.006
0.219
0.096
0.00770.0485-0.0704
0.857
0.411
0.523
0.2205
0.09650.0240-0.742-0.0021.542-0.4250
0.23901.42710.026
0.653-1.0820.2067-0.0370-1.3716
0.074
0.177
0.161
0.01580.04700.0028-0.0770.136
0.109-0.02570.04050.0110-0.0860.196
0.264-0.0438
0.07130.1025
0.084
0.216
0.098
0.03040.0446-0.0791
-0.05
0.085
0.105
-0.02150.03030.0384
0.015
0.206
0.155
0.00150.0538-0.0129
0.014
0.055
0.085
-0.00240.01970.0320-0.0010.325
0.235-0.00370.0847-0.0245-0.054-0.7020.274-0.1122-0.0768
0.7246

Reinforced concrete beam with web reinforcement ratio 0.05%

Element Number
Crack Angle

North (B7)
33 degree

Applied Load(tf)
42
44.66345.66446.49847.49848.36649.3
50.2
51.10152.20253.46955.20456.538

Increment deformation in beam coordinate Increment deformation in cracking coordinate
a1
a2
a3
x
y
xy
0.096
0.058
0.222
-0.01190.05040.0763
0.147
0.56
0.499
0.03410.1426-0.1295
0.035
0.352
0.333
0.0004
0.0964-0.0881
0.089
0.264
0.212
0.0271
0.0611-0.0655
0.019
0.139
0.093
0.01050.0290-0.0491-0.0180.115
0.005
0.01530.0090-0.0784
-0.03
0.22
0.255
-0.0255
0.0730-0.0491
0.058
0.193
0.157
0.01730.0454-0.0489
0.097
0.144
0.126
0.02650.0338-0.0192-0.0450.148
0.053
0.00270.0230-0.0875
0.075
0.116
0.195
-0.0009
0.04870.0217-0.0160.276-0.0850.0668-0.0018-0.2208
0.63
0.33
0.572
0.11420.12580.1745

Element Number
Crack Angle

North (B7)42 degree

Applied Load(tf)
42
44.66345.66446.49847.49848.36649.3
50.2
51.10152.20253.46955.20456.538

Increment deformation in beam coordinate Increment deformation in cracking coordinate
a1
a2
a3
x
y
xy
0.096
0.058
0.222
-0.01630.05480.0341
0.147
0.56
0.499
0.04680.1299-0.1902
0.035
0.352
0.333
0.00950.0872-0.1430
0.089
0.264
0.212
0.03300.0552-0.0833
0.019
0.139
0.093
0.01470.0248-0.0582-0.0180.115
0.005
0.02120.0030-0.0707
-0.03
0.22
0.255
-0.0193
0.0668-0.1076
0.058
0.193
0.157
0.02180.0410-0.0638
0.097
0.144
0.126
0.02810.0321-0.0228-0.0450.148
0.053
0.01000.0158-0.0958
0.075
0.116
0.195
-0.00140.0491-0.0100-0.0160.276-0.0850.0821-0.0171-0.1677
0.63
0.33
0.572
0.10100.13900.1588

Reinforced concrete beam with web reinforcement ratio 0.05%

Element Number
Crack Angle

North (C6)
20 degree

Applied Load(tf)
42
44.66345.66446.49847.49848.36649.3
50.2
51.10153.46955.20456.538

Increment deformation in beam coordinate Increment deformation in cracking coordinate
a1
a2
a3
x
y
xy
0
0
0
0.00000.00000.0000
0.006
0.581
0.39
0.00280.1439-0.1109
0.063
0.313
0.307
0.00390.09010.0108-0.0050.176
0.279-0.0271
0.06980.0901-0.0130.234
0.02
0.01850.0367-0.1487-0.1730.15
0.166-0.0623
0.05660.0322
0.151
0.201
0.127
0.04710.0409-0.0536-0.0760.359
0.024
0.0126
0.0581-0.2298
0.005-0.0230.01-0.0026-0.00190.0236-0.0620.109-0.0820.00650.0053-0.1358
0.13
0.291
0.266
0.02830.0770-0.0092
0.005
0.252
0.436
-0.0409
0.10520.1562Element Number
Crack Angle

North (C5)
47 degree

Applied Load(tf)
42
44.66345.66446.49847.49848.36649.3
50.2
51.10152.20253.46955.20456.538

Increment deformation in beam coordinate Increment deformation in cracking coordinate
a1
a2
a3
x
y
xy
0.115
0.068
0.154
0.00690.03890.0191
0.065
0.506
0.419
0.04190.1008-0.2293
0.225
0.372
0.253
0.08730.0620-0.0701-0.0610.191
0.178-0.0097
0.0422-0.1336
0.058
0.119
0.091
0.02200.0222-0.0306-0.1680.092
0.171-0.05940.0404-0.1443
0.17
0.127
0.063
0.05810.01620.0274-0.1220.254
0.118
0.00680.0262-0.1912
0.19
0.167
0.285
0.01790.07140.0040-0.1050.116-0.0520.0176-0.0149-0.1062-0.0540.124
0.094-0.0044
0.0219-0.0929
0.037
0.293
0.33
0.00230.0802-0.1392
0.026
0.264
0.127
0.04280.0297-0.1175

Reinforced concrete beam with web reinforcement ratio 0.05%

Element Number South (A9)Crack Angle
17 degree
Applied Load(tf)
42
44.66345.66446.49847.49848.36649.3
50.2
51.10152.20253.46955.20456.538

Increment deformation in beam coordinate Increment deformation in cracking coordinate
a1
a2
a3
x
y
xy
-0.0070.009-0.356
0.0485-0.04800.3004-0.0010.115
0.174-0.01410.0426-0.0646
0.038
0.14
0.175
-0.0003
0.0448-0.0428-0.0190.075
0.447-0.06130.0753-0.32110.001
0.068-0.3040.0490-0.0318
0.2994
0.004
0.065
0.05
0.00010.01710.0042
0.01
0.14
0.137
-0.00340.0409-0.0151
0.063
0.106
0.126
0.01090.0314-0.0224-0.3450.097
0.073-0.10430.0423-0.0397
0.291
0.085
0.072
0.08460.00940.0386
0.006
0.065
0.076
-0.00290.0207-0.0171
0.013
0.145
0.124
-0.00020.0397-0.0004
-0.2
0.171
0.177
-0.0688
0.0616-0.0550

Element Number South (A8)
Crack Angle
17 degree
Applied Load(tf)
42
44.66345.66446.49847.49848.36649.3
50.2
51.10152.20253.46955.20456.538

Increment deformation in beam coordinate Increment deformation in cracking coordinate
a1
a2
a3
x
y
xy
0.082
0.089
0.092
0.01970.0230-0.0034
0.131
0.418
0.424
0.01800.1193-0.0437
0.051
0.566
0.241
0.03320.12110.2000
0.074
-0.08
0.185
-0.01110.0096-0.1989-0.0230.119
0.105-0.01070.0347-0.0076
0.01
0.102
0.114
-0.0036
0.0316-0.0224
0.034
0.23
0.196
0.00370.06230.0017
0.03
0.157
0.136
0.00430.04250.0003
0.022
0.143
0.136
0.00060.0406-0.0105
0.016
0.126
0.104
0.00170.03380.0034
0.021
0.137
0.109
0.00350.03600.0076
0.034
0.24
0.199
0.00420.06430.0062
0.041
0.291
0.246
0.00440.07860.0036

Reinforced concrete beam with web reinforcement ratio 0.05%

Element Number
Crack Angle

South (B7)45 degree

Applied Load(tf)
42
44.66345.66446.49847.49848.36649.3
50.2
51.10152.20253.46955.20456.538

Increment deformation in beam coordinate Increment deformation in cracking coordinate
a1
a2
a3
x
y
xy
0
0
0
0.00000.00000.0000
0.15
0.591
0.535
0.05150.13370.2205
0.077
0.312
0.261
0.03200.0652
0.1175
0.36
0.239
0.189
0.10250.0473-0.0605-0.3070.149
0.121-0.06970.03020.2280
0.021
0.14
0.108
0.01330.02700.0595
0.03
0.216
0.202
0.01100.0505
0.0930
0.052
0.191
0.12
0.03080.03000.0695
0.02
0.122
0.121
0.00530.03020.0510
0.015
0.17
0.116
0.01730.02900.0775
0.07
0.108
0.118
0.01500.02950.0190-0.0220.266
0.204
0.01000.05100.1440
0.048
0.327
0.266
0.02730.06650.1395

Element Number
Crack Angle

South (C6)
25 degree

Applied Load(tf)
42
44.66345.66446.49847.49848.36649.3
50.2
51.10152.20253.46955.20456.538

Increment deformation in beam coordinate Increment deformation in cracking coordinate
a1
a2
a3
x
y
xy
0
0
0
0.00000.00000.0000-0.0110.637
0.512-0.01190.16840.1203
0.012
0.345
0.266
0.00110.08810.0713-0.0140.221
0.165-0.00480.05650.0505
0.018
0.136
0.113
0.00290.03560.0221-0.0250.116
0.101-0.0106
0.03330.0183
0
0.194
0.171
-0.0055
0.05400.0267-0.0020.074
0.117-0.01260.0306-0.0230
0
0.198
0.106
0.00750.04200.0713-0.0010.123
0.106-0.0033
0.03380.0186
0
0.146
0.111
-0.00080.03730.0315-0.0030.243
0.195-0.00410.0641
0.04600
0.332
0.26-0.0032
0.0862
0.0667

Reinforced concrete beam with web reinforcement ratio 0.05%

Element Number
Crack Angle

South (C5)
40 degree

Applied Load(tf)
42
44.66345.66446.49847.49848.36649.3
50.2
51.10152.20253.46955.20456.538

Increment deformation in beam coordinate Increment deformation in cracking coordinate
a1
a2
a3
x
y
xy
0.044
0.079
0.088
0.0081
0.02270.0126
0.137
0.519
0.468
0.03920.12480.1638
0.047
0.297
0.247
0.0191
0.06690.1101
0.04
0.194
0.152
0.01730.04120.0698
0.042
0.137
0.103
0.01700.02780.0444
0.013
0.164
0.099
0.01630.02800.0725
0.031
0.144
0.156
0.0026
0.04120.0437
0.02
0.148
0.111
0.01160.03040.0583
0.016
0.133
0.103
0.00910.02820.0527
0.017
0.124
0.102
0.00750.02770.0472
0.017
0.129
0.106
0.00770.02880.0494
0.028
0.241
0.184
0.01680.0504
0.0963
0.031
0.319
0.252
0.01850.06900.1284

Element Number
Crack Angle

South (C5)
13 degree

Applied Load(tf)
42
44.66345.66446.49847.49848.36649.3
50.2
51.10152.20253.46955.20456.538

Increment deformation in beam coordinate Increment deformation in cracking coordinate
a1
a2
a3
x
y
xy
0.044
0.079
0.088
0.00850.0222-0.0161
0.137
0.519
0.468
0.02370.1403-0.0421
0.047
0.297
0.247
0.00670.0793-0.0126
0.04
0.194
0.152
0.00810.05040.0023
0.042
0.137
0.103
0.01020.03450.0087
0.013
0.164
0.099
0.00400.04020.0237
0.031
0.144
0.156
0.00170.0421-0.0368
0.02
0.148
0.111
0.00370.03830.0038
0.016
0.133
0.103
0.00240.03490.0000
0.017
0.124
0.102
0.00220.0331-0.0049
0.017
0.129
0.106
0.00210.0344-0.0051
0.028
0.241
0.184
0.00430.06300.0022
0.031
0.319
0.252
0.00300.0845-0.0061Reinforced concrete beam with web reinforcement ratio 0.035%

Element NumberNorth (A8)
Crack Angle
30 degree
Applied Load
(tf)
42
45
47
47.9
48.86
49.76
50.7
51.6
52.4
53.3
54.27
55.2

Increment deformation in beam coordinate Increment deformation in cracking coordinate
a1
a2
a3
x
y
xy
-0.89300.016-0.2676
0.0443-0.1554-0.012-0.011-0.0310.0013-0.0070-0.0102
0.669
3.049
2.578
0.16030.7692-0.6711
0.049
0.346
0.243
0.02100.0778-0.1059
0.041
0.198
0.162
0.01090.0489-0.0467
0.017
0.204
0.177
0.00150.0537-0.0477
0.042
0.21
0.14
0.01800.0450-0.0657
0.011
0.101
0.206
-0.02410.05210.0360
0.019
0.234
0.18
0.0066
0.0566-0.0663
0.079
0.255
0.182
0.02750.0560-0.0687
0.002
0.327
0.224
0.00790.0743-0.1110
0.028
0.28
0.257
0.00040.0766-0.0576

Element Number
Crack Angle

North (B7)23 degree

Applied Load
(tf)
42
45
47
47.9
48.86
49.76
50.7
51.6
52.4
53.3
54.27
55.2

Increment deformation in beam coordinate Increment deformation in cracking coordinate
a1
a2
a3
x
y
xy
0.025
0.014
0.021
0.00560.00420.0050-0.0030.019
0.019-0.00190.0059-0.0003
0.456
3.268
2.241
0.15320.7778-0.7481
0.03
0.318
0.15
0.02280.0642-0.1203
0.033
0.237
0.153
0.01280.0547-0.0609
0.021
0.209
0.168
0.00290.0546-0.0308
0.026
0.215
0.167
0.00540.0549-0.0357
0.017
0.237
0.178
0.00350.0600-0.0437
0.039
0.278
0.209
0.00980.0695-0.0509-0.0060.257
0.201-0.00500.0678-0.0421
0.147
0.284
0.255
0.03490.0729-0.0218-0.0780.261
0.217-0.02910.0749-0.0347

Reinforced concrete beam with web reinforcement ratio 0.035%

Element Number
Crack Angle

North (B7)57 degree

Applied Load
(tf)
42
45
47
47.9
48.86
49.76
50.7
51.6
52.4
53.3
54.27
55.2

Increment deformation in beam coordinate Increment deformation in cracking coordinate
a1
a2
a3
x
y
xy
0.025
0.014
0.021
0.00400.00580.0044-0.0030.019
0.019
0.00060.0034-0.0145
0.456
3.268
2.241
0.52190.4091-1.4386
0.03
0.318
0.15
0.06360.0234-0.1218
0.033
0.237
0.153
0.04000.0275-0.1005
0.021
0.209
0.168
0.02620.0313-0.1074
0.026
0.215
0.167
0.02910.0311-0.1052
0.017
0.237
0.178
0.03130.0322-0.1212
0.039
0.278
0.209
0.04020.0390-0.1297-0.0060.257
0.201
0.02750.0352-0.1508
0.147
0.284
0.255
0.05180.0559-0.0786-0.0780.261
0.217
0.01140.0343-0.2059

Element Number
Crack Angle

North (C6)30 degree

Applied Load
(tf)
42
45
47
47.9
48.86
49.76
50.7
51.6
52.4
53.3
54.27
55.2

Increment deformation in beam coordinate Increment deformation in cracking coordinate
a1
a2
a3
x
y
xy
-0.0040.013
0.007-0.0005
0.0027-0.0061
0.023
0.005
0.02
0.00330.00370.0108
0.357
3.206
2.504
0.11090.7799-0.8724
0.032
0.309
0.243
0.00960.0756-0.0837
0.026
0.249
0.164
0.01470.0540-0.0833
0.026
0.199
0.159
0.00720.0490-0.0517
0.02
0.232
0.165
0.00980.0532-0.0723
0.005
0.249
0.174
0.00630.0572-0.0822
0.008
0.224
0.196
-0.0018
0.0598-0.0535
0.049
0.267
0.193
0.01830.0607-0.0769-0.0090.295
0.213
0.0016
0.0699-0.0966
0.024
0.283
0.252
0.00090.0759-0.0629

Reinforced concrete beam with web reinforcement ratio 0.035%
Element Number
Crack Angle

North (C6)45 degree

Applied Load
(tf)
42
45
47
47.9
48.86
49.76
50.7
51.6
52.4
53.3
54.27
55.2

Increment deformation in beam coordinate Increment deformation in cracking coordinate
a1
a2
a3
x
y
xy
-0.0040.013
0.007-0.0003
0.00300.0015
0.023
0.005
0.02
0.00200.00500.0090
0.357
3.206
2.504
0.26470.6260-1.4245
0.032
0.309
0.243
0.02450.0608-0.1385
0.026
0.249
0.164
0.02770.0410-0.1115
0.026
0.199
0.159
0.01650.0398-0.0865
0.02
0.232
0.165
0.02170.0413-0.1060
0.005
0.249
0.174
0.02000.0435-0.1220
0.008
0.224
0.196
0.00900.0490-0.1080
0.049
0.267
0.193
0.03070.0483-0.1090-0.0090.295
0.213
0.0182
0.0533-0.1520
0.024
0.283
0.252
0.01370.0630-0.1295

Element Number North (CD56)
Crack Angle
47 degree
Applied Load
(tf)
45
47
47.9
48.86
49.76
50.7
51.6
52.4
53.3
54.27
55.2

Increment deformation in beam coordinate Increment deformation in cracking coordinate
a1
a2
a3
x
y
xy
0
0
0
0.00000.00000.0000-0.0593.267
2.425-0.04940.8514-0.4400-0.0010.292
0.248-0.00820.0809-0.0156
0.001
0.208
0.144
0.00000.0523-0.0363
0.002
0.116
0.149
-0.0106
0.04010.0323-0.0040.309
0.163
0.00650.0698-0.0926-0.0030.115
0.166-0.0150
0.04300.0463
0.003
0.244
0.2
-0.00450.0662-0.0189-0.0270.246
0.19-0.01170.0665-0.0261-0.0030.293
0.221-0.00430.0768-0.0369
0.078
0.361
0.155
0.03810.0716-0.1404

Element Number North (CD45)
Crack Angle
20 degree
Applied Load
(tf)
45
47
47.9
48.86
49.76
50.7
51.6
52.4
53.3
54.27

Increment deformation in beam coordinate Increment deformation in cracking coordinate
a1
a2
a3
x
y
xy
0.053
0.02320.027
0.01420.00490.00110.34
2.77382.243
0.04590.7325-0.2566
0.018
0.298
0.231
0.00100.0780-0.0341
0.016
0.188
0.156
0.00040.0506-0.0139
0.017
0.213
0.16
0.00280.0547-0.0285
0.069
0.202
0.148
0.0191
0.0486-0.0332-0.0170.186
0.153-0.00930.0516-0.0128
0.012
0.266
0.189
0.00240.0671-0.0433
0.018
0.247
0.18
0.0036
0.0627-0.0372
0.033
0.212
0.218
-0.00190.06310.0156

Reinforced concrete beam with web reinforcement ratio 0.035%

Element Number South (A9)Crack Angle
20 degree
Applied Load
(tf)
46.9
47.9
48.86
49.76
50.7
51.6
52.4
53.3
54.27
55.2

Increment deformation in beam coordinate Increment deformation in cracking coordinate
a1
a2
a3
x
y
xy
0.511
1.735
1.626
0.08270.47880.0081
0.056
0.197
0.165
0.01190.05130.0158
0.015
0.325
0.119
0.02100.06400.1387
0.025
0.099
0.108
0.00100.0300-0.0115
0.024
0.071
0.11
-0.0027
0.0264-0.0328
0.025
0.062
0.134
-0.0072
0.0290-0.0574
0.039
0.348
0.135
0.02820.06860.1441
0.023
0.162
0.131
0.00360.04260.0152
0.032
0.008
0.172
-0.01710.0271-0.1242
0.044
0.188
0.208
0.00040.0576-0.0242Element Number South (A8)
Crack Angle
25 degree
Applied Load
(tf)
42
45
47
47.9
48.86
49.76
50.7
51.6
52.4
53.3
54.27
55.2

Increment deformation in beam coordinate Increment deformation in cracking coordinate
a1
a2
a3
x
y
xy
0.014
0.005
0.011
0.00280.0019-0.0044
0.01
0.029
0.039
-0.0004
0.0101-0.0053-0.0052.952
2.528-0.07120.80790.4548
0.006
0.298
0.252
-0.00460.08060.0476
-0.03
0.205
0.109
-0.00110.04490.0762
0.001
0.2
0.163
-0.0028
0.05310.0360
0.025
0.181
0.161
0.00210.04940.0225-0.0260.22
0.175-0.0105
0.05900.0441
0.001
0.218
0.194
-0.0062
0.06100.0288-0.0120.244
0.181-0.00400.06200.0563
0.003
0.282
0.236
-0.00470.07590.0468-0.0010.147
0.219-0.02160.0581-0.0372

Reinforced concrete beam with web reinforcement ratio 0.035%

Element Number
Crack Angle

South (B7)45 degree

Applied Load
(tf)
42
45
47
47.9
48.86
49.76
50.7
51.6
52.4
53.3
54.27
55.2

Increment deformation in beam coordinate Increment deformation in cracking coordinate
a1
a2
a3
x
y
xy
-0.0140-1.2920.3195-0.3230
0.0070
0.042
0.019
1.303
-0.3105
0.3258-0.0115
0.521
3.137
2.593
0.26630.6482
1.3080
0.012
0.311
0.246
0.01930.06150.1495
0.023
0.212
0.164
0.01780.04100.0945
0.053
0.2
0.154
0.02480.03850.0735
0.018
0.203
0.173
0.01200.0432
0.0925
0.028
0.233
0.183
0.01950.04570.1025
0.03
0.251
0.198
0.02080.04950.1105
0.033
0.241
0.178
0.02400.04450.1040
0.044
0.302
0.235
0.02780.05870.1290
0.027
0.193
0.224
-0.00100.05600.0830Element Number South (BC67)
Crack Angle
27 degree
Applied Load
(tf)
45
47
47.9
48.86
49.76
50.7
51.6
52.4
53.3
54.27
55.2

Increment deformation in beam coordinate Increment deformation in cracking coordinate
a1
a2
a3
x
y
xy
0
0
0
0.00000.00000.0000
0.621
3.209
2.624
0.14520.81230.6301
0.055
0.262
0.203
0.01540.06380.0576
0.034
0.204
0.179
0.0051
0.05440.0335
0.044
0.208
0.154
0.01380.04920.0499
0.05
0.208
0.169
0.0126
0.05190.0404
0.045
0.235
0.187
0.01150.05850.0492
0.052
0.256
0.196
0.01500.06200.0578
0.036
0.241
0.191
0.00890.06030.0521
0.068
0.269
0.219
0.0171
0.0671
0.0516
0.059
0.31
5.902
-1.1287
1.2210-3.2591

Reinforced concrete beam with web reinforcement ratio 0.035%

Element Number
Crack Angle

South (C5)
18 degree

Applied Load
(tf)
42
45
47
47.9
48.86
49.76
50.7
51.6
52.4
53.3
54.27
55.2

Increment deformation in beam coordinate Increment deformation in cracking coordinate
a1
a2
a3
x
y
xy
-0.0030.04-0.0020.00330.00600.0292
0.009
-0.02
0.021
-0.0023-0.0004-0.0300
0.434
3.023
2.443
0.06530.79890.1828
0.032
0.31
0.231
0.00580.07970.0332
0.018
0.199
0.16
0.00130.05300.0115
0.043
0.194
0.15
0.00970.04950.0189
0.014
0.204
0.165
-0.00020.05470.0105
0.024
0.233
0.175
0.00420.06010.0238
0.029
0.243
0.18
0.00590.06210.0273
0.026
0.131
0.195
-0.00810.0474-0.0634
0.035
0.211
0.176
0.0052
0.05630.00880.026
0.294-0.0020.03670.04330.2098Element Number
Crack Angle

South (C5)30 degree

Applied Load
(tf)
42
45
47
47.9
48.86
49.76
50.7
51.6
52.4
53.3
54.27
55.2

Increment deformation in beam coordinate Increment deformation in cracking coordinate
a1
a2
a3
x
y
xy
-0.0030.04-0.0020.00640.00290.0289
0.009
-0.02
0.021
0.0090-0.0200
0.0210
0.434
3.023
2.443
0.11560.7486
0.7638
0.032
0.31
0.231
0.01240.07310.0904
0.018
0.199
0.16
0.00470.0496
0.0526
0.043
0.194
0.15
0.01340.04590.0496
0.014
0.204
0.165
0.00330.05120.0543
0.024
0.233
0.175
0.00900.05530.0672
0.029
0.243
0.18
0.0111
0.05690.0707
0.026
0.131
0.195
-0.01220.0514-0.0128
0.035
0.211
0.176
0.00830.05320.04970.026
0.294-0.0020.05830.0217
0.1970Reinforced concrete beam with web reinforcement ratio 0.035%

Element Number
Crack Angle

South (C4)30 degree

Applied Load
(tf)
42
45
47
47.9
48.86
49.76
50.7
51.6
52.4
53.3
54.27
55.2

Increment deformation in beam coordinate Increment deformation in cracking coordinate
a1
a2
a3
x
y
xy
0.023
0.022
0.003
0.00990.00130.0093-0.0180.016
0.038-0.01080.0103-0.0048-0.0092.764
2.262-0.02040.70920.7585-0.0090.284
0.209
0.0006
0.0682
0.0911-0.0180.2
0.156-0.00490.05040.0619
0.007
0.189
0.147
0.00250.04650.0543
0.019
0.195
0.157
0.00490.0486
0.0512
0.025
0.22
0.167
0.00880.05240.0622-0.0430.235
0.182-0.01200.06000.0774-0.0050.224
0.175-0.00110.05590.0664
0.009
0.287
0.228
0.00230.07170.0804
0.001
0.37
0.165
0.02780.06500.1700