Shear Capacity of Post-Tensioning Pre-Stressed Concrete ... · Shear Capacity of Post-Tensioning Pre-Stressed Concrete Beams with High Strength Stirrups . Hye-Sun Lim 1, Byung-Koo

Shear Capacity of Post-Tensioning Pre-Stressed

Concrete Beams with High Strength Stirrups

Hye-Sun Lim1, Byung-Koo Jun

2, Dong-Ik Shin

1, and Jung-Yoon Lee

1

1Department of Civil, Architectural and Environmental System Engineering, SungKyunKwan University, Suwon-si,

Republic of Korea 2Department of Global Construction Engineering, SungKyunKwan University, Suwon-si, Republic of Korea

Email: [email protected], {jbghello, s91120}@naver.com, [email protected]

Abstract—The yield strength of stirrups is limited to

420MPa and 600MPa in the ACI 318-14 standard and EC2-

02 respectively. In this study, four beams were tested to

investigate the influence of high strength stirrups on the

shear behavior of PSC beams. For extension of this study,

simulations to obtain the shear behavior of prestressed

concrete beams with various yield strength of stirrups was

conducted using a finite element analytical program

(RCAHEST). The experimental and analytical results

indicated that the limitation on the yield strength of shear

reinforcement for prestressed concrete beams in the ACI

318-14 design code was too conservative. The simulation

result also indicated that it could be possible to increase the

yield strength of shear reinforcement in the ACI 318-14

design code up to 610MPa. The shear strength of

prestressed concrete beams with high strength stirrups did

not proportionally increase with the increase of yield

strength of stirrups.

Index Terms—failure modes, high strength steel bars,

prestressed concrete beams, shear strength

I. INTRODUCTION

Recently, special structures such as high-rise buildings,

long-span bridges and nuclear power plants are rapidly

being constructed to meet the demands of the times.

However, raw materials including steel keep insufficient

and its prices are drastically increasing; hence, high

performance materials which are compatible to these

structures should be needed. To apply high strength

materials to Reinforced Concrete (RC) and prestressed

concrete (PSC) members, various material properties

must be examined.

In ACI 318-14 [1], the yield strength of flexure and

axial reinforcement is limited to 550MPa and the yield

strength of web reinforcement is limited to 420MPa.

EC2-02 [2], meanwhile, allows the yield strength of web

reinforcement to 600MPa.

In this study, four PSC beams with high strength

stirrups were tested. The test results were compared with

the shear behavior of PSC beams analyzed by two

analytical methods. In addition, some simulations are

conducted to figure out the behavior of beams with high

yield strength of web reinforcement.

Manuscript received December 4, 2015; revised May 4, 2016.

II. TEST PROGRAM AND MESUREMENTS

A. Test Program

To evaluate shear behaviors according to the yield

strength of web reinforcement for prestressed concrete,

the four simply supported PSC beams were made. These

specimens had rectangular section shape and were

distinguished by the yield strength of web reinforcement.

The ratio of nominal shear strength with respect to

nominal flexure strength was less than 0.7 to induce shear

failure of all beams prior to flexural failure.

The cross sectional dimensions of specimens were 370

× 500mm and the shear span-depth ratio (a/d) of all the

beams was designed, 2.3 (Fig. 1). D29 deformed steel

bars with 501.9MPa yield strength were placed as the

longitudinal bars at compressive and tensile side each

four. D10 deformed steel bars for the web reinforcement

were used to be perpendicular to longitudinal axis.

According to the yield strength of web reinforcement,

specimens varied RB-0, RB-280, RB-450 and RB-500. In

Table I and Fig. 2, the overall dimensions of these

specimens are shown. Strain gauges were attached to the

longitudinal bars, web reinforcement bars, and strands to

examine shear failure mode.

The prestressing strands are seven-wire strands with a

nominal diameter of 12.7 mm (Ap=98.71mm2). All the

specimens have five tendons at compressive and tensile

side both to apply prestressing force by post-tensioning

systems. The yield strength and ultimate strength of the

strands were 1580.4 and 1853.9MPa, respectively. The

applied prestressing force was 1014kN. On the basis of

ACI 318-14 [1] standard, the total prestress loss was

consider. Initial and effective prestressing forces of these

specimens are shown in Table II.

B. Loading System and Measurements

The locations of the Linear Variable Differential

Transducers (LVDTs) are shown in Fig. 2. Six LVDTs

were attached to each face of the beam near the shear

critical region to measure the displacement at longitudinal

and transverse of each region. Two LVDTs were attached

to each bottom surface of loading point to measure the

deflection of the beams as well.

258© 2016 Int. J. Struct. Civ. Eng. Res.

International Journal of Structural and Civil Engineering Research Vol. 5, No. 4, November 2016

doi: 10.18178/ijscer.5.4.258-264

The PSC specimens were simply supported and

subjected to two-point concentrated loads. In the test, a

strain-controlled with 0.02 mm displacement per 1 sec

test procedure was adopted. When testing the specimens,

magnifying glass used every 10 tons to measure the state

of diagonal cracks. The test was continued when the

shear strength of beams reached their 90% of maximum

shear strength.

Figure 1. Overall dimensions of PSC beams

Figure 2. Test setup and instrument of test beams

TABLE I. SPECIFICATION OF PSC BEAM SPECIMENS

Beams a/d fck

(MPa)

Longitudinal tensile bar Shear steel bars

No.rebar

& diameter

fyl

(MPa)

𝜌l

(%) Φ

S

(mm)

fyt

(MPa)

𝜌t

(%)

RB-0 2.3 56.9 4-D29 501.9 1.67 D10

RB-280 2.3 56.9 4-D29 501.9 1.67 D10 100 281.8 0.39

RB-450 2.3 56.9 4-D29 501.9 1.67 D10 100 448.8 0.39

RB-500 2.3 56.9 4-D29 501.9 1.67 D10 100 499.5 0.39

TABLE II. SPECIFICATION OF PRESTRESSING TENDONS

Beams

Prestressing tendons Pi

(kN)

Pe

(kN) Pe/bdfck No. strand

& diameter

fpy

(MPa)

𝜌pt

(%)

RB-0 5-Φ12.7 1580.4 0.32 828.0 745.0 0.073

RB-280 5-Φ12.7 1580.4 0.32 917.0 826.0 0.081

RB-450 5-Φ12.7 1580.4 0.32 857.0 771.0 0.076

RB-500 5-Φ12.7 1580.4 0.32 812.0 730.0 0.072

TABLE III. TEST RESULTS

Beams

Test results Vcal

(kN) Vmax/Vcal

Failure

modes

Vmax

(kN)

Δmax

(mm)

RB-0 582.5 11.15 385.8 1.51 SF RB-280 726.5 13.29 552.3 1.32 SYCF

RB-450 793.5 20.58 653.4 1.21 SYCF

RB-500 813.6 20.67 683.2 1.19 SYCF

where, SF : Shear failure, SYCF : Shear failure after the yielding of shear reinforcement



Figure 3. Load versus deflection curves

III. TEST RESULTS

Specimens of RB-0 and RB-280 failed in shear

without the flexural yielding of the longitudinal

reinforcements. In case of the specimen RB-450 and RB-

500 failed in shear nearly simultaneously when

longitudinal tensile bars reached their yield strain. In the

all PSC beams, flexural cracks occurred firstly at the

middle of span in the maximum moment region. As the

load increased, flexural-shear cracks and diagonal cracks

appeared gradually. The number of diagonal cracks

increased as load level increasing.

Fig. 3 represents the load-deflection curves of

specimens. As the yield strength of shear reinforcement

increased, the maximum shear strength and deflection

also increased. In case of RB-0 with no shear

reinforcement and RB-280 with relatively low yield

strength, shear strength has a tendency of dramatic

reduction after reached their maximum load. In case of

the specimens with high strength shear reinforcement,

shear strength reduced moderately after reached their

maximum load. It can be considered that the specimens

with high strength shear reinforcement have low ratio of

nominal shear strength and nominal flexure strength than

the beams having lower yield strength of shear

reinforcement. All the test results such as maximum load,

maximum deflection, and failure mode are shown in

Table III.

IV. PREDICTION OF THE SHEAR BEHAVIOR OF TESTED

PSC BEAMS

In order to predict the structural behavior of tested

four PSC beams, two analytical methods, Rotating-Angle

Softened Truss Model (RA-STM) [3], [4] and

RCAHEST [5], were adopted in this paper.

A. Rotating-Angle Softened Truss Model (RA-STM)

RASTM (Fig. 4) is a method to predict the shear

behavior of RC or PSC beams, based on the mechanism

of materials (equilibrium of forces, compatibility

equations, and stress vs. strain relations of concrete and

steel bars). The stress-train curve of concrete must reflect

two characteristics. First, is the nonlinear relationship

between stress and strain and the second, and perhaps

more important, is the softening of concrete in

compression, caused by cracking owing to tension in the

perpendicular direction. Consequently, a softening

coefficient will be incorporated in the equation for the

compressive stress-strain relationship of concrete.

Figure 4. Flow chart of RA-STM

In view of the crucial importance of the softening

effect on the biaxial constitutive laws of reinforced

concrete, this model has been named the ‘softened truss

model’. The word ‘softened’ implies two characteristics:

first, the analysis must be nonlinear and, second, the

softening of concrete must be taken into account.

Equilibrium Equations

2sinl l l lp lp d rf f (1)

2cost t t tp tp d rf f (2)

( )sin coslt d r r (3)

where, lp , tp = prestressing steel ratios in the l and t

directions, respectively,

lpf , tpf = stresses in prestressing steel in the l and t

directions, respectively.

Compatibility Equations

2 2cos sinl r r d r (4)

2 2cos sint r r d r (5)

( )sin cos2

ltr d r r

(6)



Constitutive Law of Concrete in Compression

- Ascending branch

2

' 2 d dd c

o o

f

/ 1d o (7a)

- Descending branch

2

( / ) 1' 1

(2 / ) 1

d od cf

/ 1d o (7b)

0.9

1 600 r

(8)

Constitutive Law of Mild Steel

l s lf E l ly (9a)

l lyf f l ly (9b)

t s tf E t ty (10a)

t tyf f t ty (10b)

where sE = 200,000 MPa (29,000,000 psi).

Constitutive Law of Pre-stressing Steel

0.7p puf f ( )p ps dec sf E (11a)

0.7p puf f 1

' ( )

' ( )1

ps dec s

p

m m

ps dec s

pu

Ef

E

f

(11b)

where pf = stress in prestressing steel - pf becomes

lpf or tpf when applied to the longitudinal and

transverse steel, respectively;

s = strain in the mild steel - s becomes l or t ,

when applied to the longitudinal and transverse steel,

respectively;

dec = strain in prestressing steel at decompression of

concrete;

psE = elastic modulus of prestressed steel, taken as

200,000MPa (29,000ksi);

'psE = tangential modulus of Ramberg-Osgood curve

at zero load, taken as 214,000MPa (31,060ksi);

puf = ultimate strength of prestressing steel;

m =shape parameter (taken as 4).

dec pi i

where

pi = initial strain in prestressed steel after loss;

i = initial strain in mild steel after loss.

dec is approximately equal to 0.005 for grades

1723MPa (250ksi) and 1862MPa (270ksi) prestressing

strands.

Shear behavior of tested beams predicted by RA-STM

The analysis results from Rotating-Angle Softened-

Truss Model (RA-STM), which uses the constitutive

equations based on the actual, observed stress-strain

relationships of concrete and steel, are compared to test

results. In Fig. 5 and Table IV, test results and analytical

results are shown.

Figure 5. Test results and analytical results from RA-STM

From analytical values, shear stress and shear strain

can be obtained. The values of shear stress and shear

strain for the test specimens can be obtained from Lee’s

analytical method [6] to predict shear deformation. Lee’s



analytical model can predict the full load- shear

deformation response of a beam until it reaches its

maximum shear strength by adopting an incremental

analytical method.

TABLE IV. T VALUES AND ANALYTICAL RESULT

VALUES

Beams Analytical Value

VRA-STM (kN)

VRA-STM/Vtest

RB-280 994.6 1.37

RB-450 1037.9 1.31

RB-500 1054.3 1.30

Prestressing (compression to axis direction) is changed

to normal stress along longitudinal axis.

RA-STM cannot be used for beams with no stirrups;

hence, the comparison with the specimen which does not

have stirrup was ruled out.

The applicability of RA-STM for concrete members is

investigated by comparison with analytical results and

test results.

The values of VRA-STM/Vtest are 1.37, 1.31 and 1.30

according to RB-280, RB-450 and RB-500 each. The

average value of these is 1.326. Shear stress from

analysis was higher than the maximum shear stress of

test results and analytical method predicts somewhat

excessive.

RA-STM assumes the model is under only pure shear

state, thus, the moment influence is not considered. The

beam in practice is subjected to flexure and shear

coincidently. Finally, there are differences between

analytical results and test results.

B. Rcahest

RCAHEST is a finite element analysis program

developed for the purpose of research and education by

Taylor, Berkeley University. It can be defined one-

dimension, two-dimensions, and three dimensions

component network. Not only it has various linear and

nonlinear analysis algorithms, but also shows the results

graphically. In addition to linear or nonlinear solid

components, two-dimension or three-dimension frame

components, and panel or shell components, constitutive

equation for linear, viscoelasticity and plasticity, etc, is

included. Besides, user can develop components and add

those, as well as use the combination. (Taylor, 2000 [7])

The accuracy of nonlinear infinite element analysis for

RC structure depends on how exact nonlinear analysis

model (the components of beam, column, and shell for

RC and PSC) can describe the mechanical behavior. For

more rational and realistic prediction of behavior

properties of structures, more deliberate and efficient

model for nonlinear analysis is needed. Hence, the team,

Structural Analysis Laboratory, SungKyunKwan

University, which made RCAHEST, develop and verify

nonlinear analysis model and this model is applied for

RCAHEST, which includes RC plane stress component,

joint component, elasticity component, beam and column

component, shell component, footing component,

structural component considering geometrically

nonlinear and expansion joint, etc,.

Shear Behavior of Tested Beams Predicted by

RCAHEST

The analysis results from finite element analysis

program RCHEST, are compared to test results. Fig. 6

and Table V show the test results and analytical results.

Figure 6. Test results and analytical results from RCAHEST



EST ESULTS R

Load-displacement curves from analytical results can

be obtained. The values of load-displacement from test

specimens are compared to analytical result values.

As test specimens are prestressed concrete beams,

prestressed concrete elements were selected for

RCAHEST analysis accordingly. The element of

application to analytical prestressed concrete method in

RCAHEST is “Reinforcing or Prestressing Bar Element”.

It is used with “beam element, 2 dimension stress

element and shell element”, etc.

The applicability of RCAHEST for concrete members

is investigated by comparison with analytical results and

test results.

The values of VRCAHEST/Vtest are 0.93, 1.10, 1.07 and

1.03 according to RB-0, RB-280, RB-450 and RB-500

each. The average value of these is 1.033. In the only

case of specimen RB-0, the value of VRCAHEST/Vtest is

under 1.0. Analysis results are much similar with test

results and the analytical maximum load is very close to

the test result values. Analytical method predicts safety

side except for RB-0 which has no stirrup.

It is concluded that comparing with RA-STM,

RCAHEST analysis is considered not in pure shear state,

but under flexure and shear, both. Also, RCAHEST

divides beams to infinite elements, resulting in better

result, hence makes good agreement to test specimens.

TABLE V. TEST RESULTS VALUES AND ANALYTICAL RESULT

VALUES

Beams Analytical Value

VRCAHEST (kN)

VRCAHEST/Vtest

RB-0 1087.6 0.93

RB-280 1593.5 1.10

RB-450 1703.0 1.07

RB-500 1680.2 1.03

C. Simulations for Beams with Higher than 500MPa of

Stirrups

1) Shear behavior of simulation beams predicted by

RCHAEST

Through the previous analysis results, it is shown

RCAHEST can predict better than RA-STM. RCAHEST,

which is finite element analysis program, is determined

as more rational method to analyze than RA-STM where

only pure shear is considered. To estimate the behavior

of post-tensioned prestressed concrete beams with

stirrups strength higher than 500MPa, RCAHEST is

adopted as a simulation tool. In these simulations, the

target specimens have same properties with previous test

specimens in practice, except the yield strength of

stirrups, which varies 550MPa, 600MPa and 610MPa.

Fig. 7 and Table VI represent simulation results. The

predicted maximum shear strength of beams with stirrups

strength of 550MPa was 1702.4kN and it had shear-

tension failure where beam failed after its stirrups

yielded. The predicted maximum shear strength of beams

with stirrups of 600MPa was 1705.0kN and there were a

little difference with stirrup strength of 550MPa. It also

had shear-tension failure. In case of the beam with

stirrups of 610MPa, shear-tension failure is occurred as

well, but beams with greater than 610MPa yield strength

of stirrups failed at the loading points elements with

crushing. Hence, up to 610MPa yield strength of stirrups,

simulations are available.

(a) RB-550

(b) RB-600

(c) RB-610

Figure 7. Simulation results for higher strength stirrups

2) Analysis of load development trend with respect to

yield strength

In Fig. 8 and Table VI the relation load development

trend and yield strength is shown. To figure out progress

of maximum load transition with respect to yield strength

of stirrups, additional simulations of beams with stirrups

of 100MPa, 200MPa, 350MP and 400MPa are conducted.

The rate of increase of maximum load is decreasing

following increasing the yield strength of stirrups, but it

keeps increasing from beams with no stirrups to with

stirrups of 450MPa. Beams with stirrup strength

exceeding 450MPa, there was no clear inclination.



TABLE VI. SIMULATION RESULT VALUES FROM RCAHEST

Beams Simulation Value: VRCAHEST (kN)

RB-100 1288.2

RB-200 1496.0

RB-350 1651.5

RB-400 1680.9

RB-550 1702.4

RB-600 1705.0

RB-610 1702.0

Figure 8. Load development trend with respect to yield strength of stirrups

V. CONCLUSIONS

In this paper, four PSC beams with high strength

stirrups were tested. In addition, a simulation by using a

finite element method was conducted to predict the

structural behavior of PSC beams with high strength

stirrups. The results obtained from the experimental and

analytical study are following below.

Test results indicated that the PSC beams with

high strength stirrups greater than 500MPa

showed shear tension failure. The limitation on

the yield strength of shear reinforcement for PSC

beams in the ACI 318-14 design code is too

conservative.

Simulation results conducted by a finite element

analytical method, RCAHEST, indicated that the

PSC beams with stirrups lower than 610MPa

showed shear tension failure.

The experimental and analytical results indicated

that it could be possible to increase the yield

strength of shear reinforcement in the ACI 318-14

design code up to 610MPa.

The shear strength of PSC beams with high

strength stirrups did not proportionally increase

with the increase of yield strength of stirrups.

ACKNOWLEDGMENT

The support from the Korea Hydro & Nuclear Power

Co. Ltd. (2014151010169B) and the basic research

program of National Research Foundation of Korea

(NRF) (2013R1A1A2006697) is gratefully

acknowledged.

REFERENCES

[1] ACI Committee 318, Building Code Requirements for Structural Concrete (ACI 318-14) and Commentary, Farmington Hills, MI:

American Concrete Institute, 2014, p. 520.

[2] Eurocode 2: Design of concrete structures – Part 1-1: General rules and rules for buildings, EN 1992-1-1, 2004, pp. 227.

[3] T. T. C. Hsu, Unified Theory of Reinforced Concrete, Boca Raton, FL: CRC Press, 1993, pp. 313.

[4] ASCE-ACI Committee 445 on Shear and Torsion, “Recent

Approaches to Shear Design of Structural Concrete,” Journal of Structural Engineering, vol. 124, no. 12, pp. 1375-1417,

December 1998. [5] H. Mok, Shin, Tae hoon, Kim, Jae Geun, Park, Dae Jeong, Seong,

(2008). Nonlinear Finite Element Analysis of Reinforced

Concrete Bridge, 2008, p. 190. [6] Jung-Yoon Lee, “Theoretical prediction of shear strength and

ductility of reinforced concrete beams,” Ph.D. dissertation,

Department of Architectural Engineering, Kyoto University,

Kyoto, Japan, 1998.

[7] O. C. Zienkiewicz and R. L. Talyor, The Finite Element Method, vol. 2 – Solid and Fluid Mechanics, Dynamics and Non-Linearity,

McGraw Hill, Co., 4th ed.

Hye-Sun Lim was born in Uijeongbu, Korea,

on July 12th, 1991. She received his BS from SungKyunKwan University at Suwon,

Republic of Korea in 2015. His research interests include the shear behavior of

prestressed concrete structure.

Now, she is a master’s degree course in the Dept. of Civil, Architectural and

Environmental System Engineering at SungKyunKwan University, Republic of

Korea.

Byung-Koo Jun was born in Pocheon,

Republic of Korea, on May 2th, 1989. He received his Bs from Kyung Hee University

at Yongin, Republic of Korea in 2014. His

research interests include the shear behavior of prestressed concrete structure.

He worked as a military at Imsil , Korea from 2009 to 2011. Now, he is a master’s degree

course in the Department of Global

Construction Engineering at SungKyunKwan University, Republic of Korea.

Dong-Ik Shin was born in Taebaek, Korea,

on January 20th, 1991. He received his BS

from Kyung Hee University at Yongin, Republic of Korea in 2015. His research

interests include Diagonally-Reinforced

Concrete Coupling Beams.

He worked as a military at Taebaek , Korea

from 2011 to 2013. Now, he is a master’s degree course in the Dept. of Civil,

Architectural and Environmental System Engineering at SungKyunKwan University, Republic of Korea.

Jung-Yoon Lee (Corresponding Author)

was born in Buan, Republic of Korea, onSeptember 13th, 1966. He received his PhD

in structural engineering from the Kyoto

University, Kyoto, Japan in 1998. His research interests include the shear behavior

and seismic design of reinforced and prestressed concrete buildings.

He is a Professor in the School of Civil and

Architectural Engineering at SungKyunKwan University, Republic of Korea. He is involved in the committees, Shear

and Torsion and Seismic Design, of the Korean Concrete Institute Committee.



Documents

Shear Capacity of Post-Tensioning Pre-Stressed Concrete ... · Shear Capacity of Post-Tensioning Pre-Stressed Concrete Beams with High Strength Stirrups . Hye-Sun Lim 1, Byung-Koo