Shear Capacity of Post-Tensioning Pre-Stressed Concrete ... · Shear Capacity of Post-Tensioning...
Transcript of Shear Capacity of Post-Tensioning Pre-Stressed Concrete ... · Shear Capacity of Post-Tensioning...
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
259© 2016 Int. J. Struct. Civ. Eng. Res.
International Journal of Structural and Civil Engineering Research Vol. 5, No. 4, November 2016
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)
260© 2016 Int. J. Struct. Civ. Eng. Res.
International Journal of Structural and Civil Engineering Research Vol. 5, No. 4, November 2016
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
261© 2016 Int. J. Struct. Civ. Eng. Res.
International Journal of Structural and Civil Engineering Research Vol. 5, No. 4, November 2016
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
262© 2016 Int. J. Struct. Civ. Eng. Res.
International Journal of Structural and Civil Engineering Research Vol. 5, No. 4, November 2016
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.
263© 2016 Int. J. Struct. Civ. Eng. Res.
International Journal of Structural and Civil Engineering Research Vol. 5, No. 4, November 2016
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.
264© 2016 Int. J. Struct. Civ. Eng. Res.
International Journal of Structural and Civil Engineering Research Vol. 5, No. 4, November 2016