EVALUATION OF SHEAR STRENGTH OF
PRESTRESSED CONCRETE BEAMS WITH HIGH
STRENGTH MATERIALS
Ferri Eka Putra∗
MEE07159

Supervisor:

Susumu KONO∗∗

ABSTRACT
Shear behavior of prestressed concrete beams has been studied for many years and some equations
were already proposed in several codes. However, these equations do not correctly evaluate which
resulting in conservative design or dangerous design. One hundred sixty-eight (168) data from 14
papers published in Japan during 1981 to 2004 are used in analysis. Equations provided by Japanese
PC standard 1998 and ACI 318-05 are studied using these data. Based on analysis, equation 71.2a ofJapan PC standard 1998 was the best design equation to predict shear strength of prestressed concretebeams among six equations. However, the equation has following shortcomings. It is quite
conservative for I-section and un-bonded rectangular section. On the other hand, it overestimate shear
strength of pre-cast prestressed concrete and rectangular beam using high strength steel as shear
reinforcement. To improve its performance, the arch contribution was isolated by choosing beams with
un-bonded tendon or specimens without shear reinforcement. Then, the modification was made to; theeffect of prestressing force on the compressing strength of concrete strut, confining effect provided byshear reinforcement on the compressive strength of concrete strut and the effective width of the web
for I-sections. On the other hand, truss contribution was not modified except for I-section. Proposed
modification has produced satisfying results. Average experiment-to-prediction ratio is getting closer
to 1.0 and scatter of data is getting small.

Keywords: Prestressed concrete, Japanese PC standard 1998.
INTRODUCTION
Use of prestressed concrete (PC) has became popular for building structure recently because it
provides advantages such as high performance, high durability and other benefits from economic pointof view. In developed country such as America and Japan, this type of structure already applied formany types of buildings such as residents, offices, universities, hospitals and sports facilities.
On the other hand, research in PC field has been conducted by several institutions andresearchers around the world. Moreover, some equations were already proposed to predict behavior of
PC beams. However, these equations do not correctly evaluate especially for shear behavior.
Comparing with flexural behavior which is already able to be predicted easily, the exact shear
behavior of member is still unpredictable. Due to that reason, most of codes in the world prefer to use
higher safety factor due to accommodate the uncertainty behavior of shear. Therefore, evaluation ofequation provided by code should be done to achieve more efficient design.
The purpose of this thesis is to gain more understanding about shear behavior of PC beams
and to propose some consideration for modifying existing standard to get more efficient andreasonable result in predicting shear strength of PC beams∗

Research and Development Center for Human Settlements, Department of Public Works, Indonesia
Assoc. Professor, Graduate School of Engineering, Kyoto University, Japan.

∗∗

1

STATE OF THE ART
Shear Mechanism
According to the discussion of America Concrete Institute (ACI) committee 445 in 1998, shear
resistance capacity for concrete members without shear reinforcements is contributed by fivecomponents, namely un-cracked concrete and flexural compression zone (Vc), aggregate interlock (Va),dowel action of longitudinal reinforcement (Vd), residual tensile stress across crack and Arch action.
The amount of contribution will depend on geometry of members, type of loading, steel ratio, type of
web reinforcement and prestressed reinforcement (CTA, 1976). On the other hand, shear
reinforcement plays important role to ensure that shear failure will not occur. The contribution of shear
reinforcement is to improve contribution of dowel action, suppressing flexural tensile stresses, limiting
the opening of diagonal cracks, providing confinement and preventing the breakdown of bond whensplitting cracks develop in anchorage zones (Park and Paulay, 1997).
Equation 71.2a of Japanese PC Standard 1998
Equation 71.2a (presented by eq.1) was developed based on superposition of arch and truss
mechanism. Left part of equation presents truss mechanism and right part presents arch mechanism.
Qu = b0 j 0 p ww f y +b0 Dν f 'c − 2 p ww f y tan θ
2

(

)

(1)

where, b0 is width of beam, j0 is distance between compressive and tensile strength, D is depth of beam.
Besides, pw is reinforcement ratio of stirrup, f’c is compression strength of concrete, ν factor iseffectiveness factor. This factor is influenced by the shear span to depth ratio, axial force which
represented by effective prestressed stress and compressive strength of concrete. It amount is limited
to the range of 0.65 to 1. This limitation introduced to represent maximum, 100 % of compressive
strength of concrete can contributes in arch mechanism. Besides, shear reinforcement effect isconsidered in terms of shear reinforcement ratio.

DATABASE
There are 14 prestressed concrete research had been done during 1981 to 2004 in Japan and at least
168 specimens were tested. 50 specimens are prestressed beam without shear reinforcement and 118
specimens with shear reinforcement. Compressive strength of concrete which was used for the
specimens is varied from 27 MPa to 110 MPa. The ratio of longitudinal reinforcement varies from
0.35 to 2 %. The ratio of prestressed reinforcement is varying from 0.2 up to 3.6 % and ratio of shear
reinforcement is from 0.2 to 1.4 %. The shear spans to depth ratio of specimen varies from 1 to 3.2.
Besides, specimens can be divided into two group; rectangular shape and “I” shape. At least 107
specimens are rectangular shape section and 60 specimens are “I” shape section. Rectangular section
can be divided into 2 types; bonded prestressed reinforcement (90 data) and un-bonded prestressedreinforcement (17 data). Meanwhile, “I” shape section only use bonded prestressed reinforcement.
Data in this study were compiled from journal as shown in table 1.
Table 1. Source of database
AuthorFunakoshi M, Okamoto T
Funakoshi M, Okamoto T, Tanaka E
Fukui, Ookuma A, Hamahara M, Suetsugu H,
Fukui T, Nagasawa T, Hamahara M, Suetsugu H
Funakoshi M, Tanaka H, Tani M
Matsizaki Y, Hirayama A, Kobayashi A, Sakai M

Publisher
1
1
2
2
1
2

2

Volume and Page
Vol.3, 365-368
Vol.4, 297-300
C-2, Structures IV, 877-880
Structures II, 1023-1028
Vol.6, 469-472
Vol.59, 1683-1684

Year
1981
1982
1996
1994
1984
1984

Table 1. Source of database (continued)
AuthorPublisher
Volume and Page
Matsizaki Y, Suzuki K, Matsutani T, Shirahama S,
2
C-2, Structures IV, 887-890
Takayama Y
Muguruma Hiroshi, Watanabe Fumio, Fujii Masanori
2
Vol.58, 2549-2550
Ogawa T, Saito A, Iida S, Fukui T, Suetsugu H,
2
C-2, Structures IV, 1077-1082
Sakiyama K, Hamahara M
Ohtaka K, Yuasa N, Hamahara M
2
C-2, Structures IV, 991-994
Ookuma A, Fukui T, Hamahara M, Suetsugu H
2
C-2, Structures IV, 847-852
Wakamatsu S, Takizawa K, Takagi H, Shiraishi I
2C-2, Structures IV, 1039-1044
Yuasa N, Ogawa T, Kamakura M, Fukui T, Uchida R,
C-2, Structures IV, 965-970
Hamahara M
Yuasa N, Ohtaka K, Fukui T, Hamahara M
2
C-2, Structures IV, 955-960
1
Proceedings of the Japan Concrete Institute.
2
Summaries of technical papers of Annual Meeting Architectural Institute of Japan.

Year
1996
1983
1999
2002
1997
1998
2000
2001

RESULTS AND DISCUSSION
Modification of the Arch Contribution in Equation 71.2a of PC standard 1998
In general, equation 71.2a of PC standard 1998 shows best performance in predicting behavior ofshear strength among six studied equations. However, the equation has following shortcomings andsome modification should be done for applying this equation to rectangular un-bonded and “I” shapesection beams to get more efficient design.
Modification of the arch contribution for rectangular un-bonded beams

e/cTruss mechanism can not exist for un-bonded specimen because it comes from internal force which
resulted from bonded condition. Therefore, absence of bond will distinguish truss mechanism. Figure
1 show specimens without shear reinforcement or with pw = 0% has ratio of experiment-to- prediction
in range of 1 to 1.75. Average
2.5experiment-to-prediction ratio is 1.5
and it standard deviation is 0.5.
2.0
Therefore, some modification should
1.5
be done to get more accurate
prediction result. Circled data in
1.0
figure 1 will be neglected for next
analysis since it has different
0.5
specification than other data.
0.0
The effect of axial force of
0.00
0.20
0.40
0.60
0.80
1.00
prestressed reinforcement could be
pw %one factor which produces higher
contribution of compressive strength
Figure 1. Performance of equation 7 with varying shearof concrete in arch mechanism.
reinforcement ratio for un-bonded rectangular section
Therefore, term of ν factor in eq.
specimens
71.2a to represent the contribution of
axial force may not be correctly evaluated. For instant, neglect the upper limitation of ν factor as
shown in eq. 2 can be used to represent higher contribution of axial force. Meanwhile, bottom limit ofthis equation is kept to guarantee at least 65 % of its compressive strength will contribute in resisting
shear force. Proposed equation to obtain modified ν factor is presented by eq. 2.
ν mod = α L r 1 +σ 'g 

0.65 ≤ νmodf c 

3(2)

Contribution of shear reinforcement in resisting shear for un-bonded specimen could be
neglected due to the absence of bond which play important role for truss mechanism. However, basedon experiment result, the contribution of shear reinforcement still exists. It is expected that shear
reinforcement confines the core of concrete and compression strength of the concrete increases
Amount of shear reinforcement contribution in un-bonded specimens can be solved by using eq. 3.
However, advanced research should be done to evaluate this equation.
c1 = 1.7 pw + 1(3)

where, c1 is coefficient of confining contribution and pw is shear reinforcement ratio in percent.
Finally, we propose equation 4 as modification of equation 71.2a to predict more accurate ofshear strength rectangular un-bonded beams.
Qu =

b0 D
c1ν mod f 'c tan θ2(4)

By using equation 4, prediction result becomes closer to experiment result. Average ratio ofexperiment-to-prediction decrease from 1.5 to 1.1 and it standard deviation decrease from 0.50 to 0.13.
It can be said that our proposal is working well to predict shear strength of un-bonded PC beams.
Figure 2 shows performance of proposed equation for un-bonded rectangular section.

Modification of the arch contribution for“I” shape section beams

3.0

Average e/c = 1.1
Std.deviation = 0.13

2.5

e/c

2.0
1.5
1.0
0.5
0.0
0.00

0.20

0.40

0.60

0.80

pw %

Figure 2. Performance of eq. 4 in predicting shear
strength of un-bonded rectangular section specimens
3.5
3.0

e/c

2.5
2.0
1.5
1.0
0.5
0.0
0.00

0.50

pw

1.00

1.50

Figure 3. Performance of eq.71.2.a with varying
shear reinforcement ratio for I section specimens

Based on figure 3, we can observe that theaccuracy of this equation is getting worse for
pw = 0 %. Ratios between experiment andprediction are located in range 2 to 3. It
means the results are too conservative. For
specimens without shear reinforcement,
shear strength of PC member is only
contributed by arch action.
Therefore, we can assume that
equation for arch action part is not correctly
evaluated. Based on statistic approach, we
get an average of experiment-to-prediction
ratio is 2.1 and standard deviation is 0.34.
We were trying to apply the same
modification for ν factor as well as we didfor un-bonded specimens. Equation 2 is used
to replace original term of ν factor in
original equation. On the other hand, we aremodifying term of b0 since using this term incalculation produced conservative results.

We found that ratio of gross section area to the depth of specimens is resulting best result toreplace term of b0. This ratio can be called as effective width (beff). Finally, we proposed equation 5 topredict shear strength of I-section PC beams.

4Qu = beff j 0 p ww f y +beff Dν mod f 'c − 2 p ww f y tan θ
2(Average e/c = 1.2
Std.dev.
= 0.18

e/c

2.5
2.0
1.5
1.0
0.5
0.0
0.00

0.50

pw

1.00

(5)

Equation 5 produces better result. Averageexperiment to prediction ratio was reduced from 2.1
to 1.2 and it standard deviation was reduced from
0.34 to 0.18. Moreover, data of experiment toprediction ratio which less than 0.95 is less than 5%.
Therefore, this proposal is efficient and safe to be
adopted. Figure 4 shows performance of equation 5.
We can observe that the results are more consistent
with specimens with and without shear reinforcement
which represented by small gradient of its results.
This tendency shows arch and truss mechanisms are
represented well by using this consideration.

3.5
3.0

)

1.50

Figure 4. Performance of proposed
modification (eq.5) for “I” section specimens

Rectangular bonded beams
Equation 71.2a works well in predicting shear strength of bonded rectangular specimens. Average
ratio of experiment-to-prediction result is 1.0 and its standard deviation is 0.22. However, 16 of 90
specimens can not be predicted well by this equation. The equation tends to overestimates shear
strength of specimens which using high strength steel (960 MPa). On the other hand, from 8 data
which can be collected, all of them have the ratio of experiment-to-prediction results in range of 0.5 to
0.6. Another situation which could not be predicted well by equation 71.2a is using of pre-cast system.
Equation 71.2a tends to overestimate shear strength of pre-cast specimens using strand 70 asprestressed tendon. Three of 4 specimens have shows this tendency. The ratio of experiment toprediction result for these specimens is around 0.75 to 0.85. Besides, for pre-cast system using highstrength bar as prestressed reinforcement, equation 71.2a produces conservative results.
Based on previous approach in increasing performance of equation 71.2a, similar modification
was applied to bonded rectangular section specimens. Term of modified “ν” factor which obtained
from equation 2 was used. Therefore, modified equation 71.2a can be written as eq. 6.
bo D
ν mod f 'c − 2 p ww f y tan θ
2(Figure 5 show performance ofequation 6 for monolithic rectangular bonded
specimen only.
Equation 6 produce slightly change inpredicting shear strength of rectangular bonded
specimen. By using equation 6, averageexperiment-to-prediction ratio was reduced
from 1.08 to 1.00 and it standard deviation is
0.14. Therefore, equation 6 is proposed only
for monolithic rectangular bonded PC beams.
Figure 5 shows performance equation 6 withvarying shear reinforcement ratio formonolithic bonded rectangular beams.

)

(6)

2.0

Average e/c = 1.00
Std.deviation = 0.14e/c

Qu = bo j 0 p ww f y +

1.0

0.0
0.00

0.50

1.00

1.50

pw

Figure 5. Performance of equation 6 formonolithic bonded rectangular section specimens

5CONCLUSIONS
Shear behavior of prestressed concrete beams has been studied for many years and some equations
were already proposed by several codes. However, these equations were not correctly evaluated. One
hundred sixty-eight (168) data from 14 papers published in Japan during 1981 to 2004 were used inanalysis. The equations provided by Japanese PC standard 1998 and ACI 318-05 were studied using
these data. Based on analysis, equation 71.2a of Japan PC standard 1998 was the best design equation
to predict shear strength of prestressed concrete beams among six equations. Average ofexperiment-to-prediction ratio for all types of specimens is 1.8. Besides, its standard deviation is 0.6.
However, the equation has following shortcomings. It is quite conservative for I-section andun-bonded rectangular section. On the other hand, it overestimates shear strength of pre-cast PC andbeam using high strength shear reinforcement.
In order to improve the performance of equation 71.2a, the arch contribution was isolated bychoosing beams with un-bonded tendon or specimens without shear reinforcement. The modification
was made to; the effect of prestressing force on the compressing strength of concrete strut, confining
effect provided by shear reinforcement on the compressive strength of concrete strut and the effective
width of the web for I-sections. Meanwhile, truss contribution was not modified except for I-section.
Equations for I-sections and rectangular section were separated and then modifications for each typeof beam were proposed. For rectangular un-bonded specimens, we proposed neglecting of upper limitof ν factor and considering confining effect by shear reinforcement. Average ratio of experiment-to
prediction for proposed modification has decreased from 1.5 to 1.1 and it standard deviation has
decreased from 0.50 to 0.13. For I-shape section, using of effective width and neglecting upper limitof ν factor were proposed. Effective width can be obtained from ratio of gross area of section by
depth of beams. Its average experiment-to-prediction ratio was reduced from 2.1 to 1.2 and it standard
deviation was reduced from 0.34 to 0.18. For rectangular-bonded specimens, equation 6 is proposed.
However, the equation is valid only for monolithic rectangular bonded beams. Averageexperiment-to-prediction ratio was reduced slightly from 1.08 to 1.00 and it standard deviation isremain 0.14

RECOMMENDATION
Some modifications have already been proposed to increase performance of equation 71.2a of JapanPC Standard 1998. However, some conditions still can not be satisfied by proposed modifications.
Shear strength of specimens using of high strength shear reinforcement and shear strength of pre-cast
PC beams needs to be evaluated. Moreover, contribution of axial force provided by prestressing force
is not clearly determined. Furthermore, truss contribution was not evaluates except for I-section.
Therefore, further study to observe these conditions should be done.

AKNOWLEDGEMENT
I would like to express my heartiest gratitude to my advisor, Dr. Shin Koyama for giving very usefuladvices during my individual study. I also would like to thanks to my colleague in Kono’s Laboratoryof Kyoto University, Ms. Y Ichioka and Dr. M Sakashita who help me learn Japanese code.

REFERENCES
American Concrete Institute, 2005, ACI.
American Concrete Institute, 1998, ACI
Architectural Institute of Japan, 1994, AIJ
Architectural Institute of Japan, 1998 AIJ
Concrete Technology Associates, 1976, Technical Bulletin 76-B11/12
Park, R. and Paulay, T, 1973, John Wiley & Sons

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