a numerical study on confinement effect by square hoops

7/28/2019 A Numerical Study on Confinement Effect by Square Hoops

http://slidepdf.com/reader/full/a-numerical-study-on-confinement-effect-by-square-hoops 1/10

Mem. Fac. Eng., Osaka City Univ., Vol. 48, pp. 27-36 (2007)

A Numerical Study on Confinement Effect by Square Hoops

in Reinforced Concrete Bridge PiersOsaka City University

Yasuhisa KITADA*, Hajjme OHUCHr**, Hisao TSUNOKAKE *** an d Tomoaki SATO ****

(Received September 28,2007)

Synopsis

This paper presents a numerical study on the confinement effect by square hoops in reinforced concrete

(RC) bridge piers with a square cross section. A Parametric study was carried out on the mechanical behavior of

the structure subjected to monotonic and cyclic horizontal actions with any load directions by means of three

dimensional non-linear finite element analysis. As the result, noticeable results were obtained as follows: First,

under the monotonic loading, confinement effect by square hoops was more effective the larger the angle of the

applied load axis apart from the prescribed design principal axis. Second, under the cyclic loading, the

maximum strength was around 20 % higher than the corresponding design strength when the angle of the load

axis was more than 15 degrees apart from the principal axis. Last, under the triangle loading, , the level of the

peak strength for Y-component is smaller than the uni-axial loading result, and also, higher than the

Y-component of the result with the cyclic loading for f) = 45° . And, the strain of lateral reinforcement greatly

increases after the 4.0 % drift angle.

KEYWORDS: Reinforced Concrete Bridge Piers, Confinement Effect, Numerical Study, Bi-axial Bending

1. IntroductionA lot of researches on the dynamic loading capacity of RC bridge piers under uni-axial horizontal load

have been already conducted. Furthermore, the confinement effect by lateral reinforcement such as hoop tie has

been considered recently in Japanese earthquake resistant design codes 1).2). While, those under bi-axial load are

required to established a rational seismic design method, because an actual ground motion in earthquake is

essentially three-dimensional. A few researches of the RC columns under bi-axial horizontal load, however,

have been conducted , i.e. by Chen et al. 3).4) and Yoshimw-a et al. 5). Moreover, studies on the confinement effect

by lateral reinforcement such as hoop ties under bi-axialloading can be hardly found.

In this paper, therefore, the RC column for bridge pier model under bi-axial horizontal load is examined

numerically by using the three dimensional non-linear finite element analysis. The RC bridge pier model to be

* Sakai City, Japan**

Professor, Div. of Structural & Concrete Engineering***

Research Associate, Div. of Structural & Concrete Engineering**** JIP Techno Science Corp., Japan

27



analyzed herein is designed according to an existing design code'l

. Th e Loading axis has an inclined angle of 15,

30, or 45 degrees from a main standard bending axis. In addition, various loading cases, such as monotonic and

cyclic ones are also considered. The resul t is discussed focusing both on the load - displacement relat ionships and

the strain distribution in the plastic hinge zone ofthe pier.

2. MODEL

Figure 1 shows the front an d cross-section views of the present bridge pier model . This column has a square

cross section of 2.5 m in each s ide length, which is des igned not to fail in shear bu t ductile flexural mode

according with the code I) . Design conditions are as fol lows: the normal stress equals to 1.0 MP a at the bottom of

RC column, and natural period is 1.0 second, and it s tands on the ground Type II which denotes diluvial and

alluvial ground. The longitudinal reinforcement is, thus, decided as follows: the tensile reinforcement ratio is

equal to 1.728 %, and volume ra tio of dual square hoop ties is 0.469 % as shown in Fig. 2.

ooIf )

ooIf )

16*120 150140

Fig. 2 Cross Section

•• • • • • • • • • •- .

. . . . .

- :It • • • • ..... ••••• 1:

==·1-

~-I--l-

It • -I- I--

~ : -I--I-

=:·1--I-

I--

~ : -I-- I

~ : - I- IIe A A A A A A A A A ::::A 1 f-AAAAA

A A A A A I--

I

Unit: mm

150140

SUPERSTRUCTURE0

i f0

, If )

0 ' ..--<

0 _ ~ _ I IIf )

N ::s:::II

0 ...c:000 W =15000.--<

II

::r:: 2500~ > I

CROSS

I ~ECTION

1MPa

Fig. 1 Bridge Pie r Model

3. NUMERICAL ANALYSES

3.1 Idealization of Bridge Pier Model

Th e three dimensional finite element analysis with material non-linearity wa s employed in the present study.

Only the column member of the bridge pier model as s hown in Fig. 1 was analyzed. Furthermore, material

non-linearity was accounted exclusively on its bottom region, which is subjected to a predominant bending action

to control the pier model failure. The height of the region was 1.2 t imes of the side length of the column cross

section as shown in grayed area of Fig. 3. While, the longitudinal reinforcement arrangement of the model in Fig.

2 was put together to representative locations at the solid circles as shown in Fig. 4 with an equivalent total cross

sectional area of the model. Moreover, solid, truss and beam elements were applied for concrete, longitudinal

reinforcing bars and the hoop ties , respectively. The beam element , in particular, was necessary to reveal howthe hoop ties work to generate the confinement effect. As the result, the column with reinforcement of the model

was subdivided into 1923 elements with 1573 nodes.

- 28 -



Pressure Load H"""""Unit: mm

Elastic

oV"lN. -*"l

oooo. -II

::r::

oV"lr -

*oV"l'<j"

*"l

Fig. 4 Cross Section View of Idealization

I I I150

1406*320

I II150

140

Non

Z L L i n e a r

X

Fig. 3 Front View of Column Idealization

3.2 Material Properties

Figure 5 shows the stress - strain relationship for the reinforcing bar that is bi-linear model and the strain

hardening is also considered as 1.0 % of its elastic modules. While, as shown in Fig.5 (b), the Modified Ahmad

Model is used for the stress-strain relationship for the concrete material. Yield conditions of the materials are von

Mises and Ottesen's criterions 6), respectively. The concrete material factors: a to d in Tables 1 were suggested by

Hatanaka et at. 7). Moreover, the tensile softening is also introduced 8). In addition, the material constants for

concrete and reinforcement were shown in Table 2.

Table I Material Factor for Ottesen's ModelFracture Model

OttesenProposer a b c

Hatanaka et al.[7J 1.2560 4.0300 14.6300d

0.9870

Table 2 Material Constants for Concrete or Reinforcement

Concrete

Reinforcement

Compressive Strength ( fc)Tensi Ie Strength ( Ii)

Elastic Modulus ( E c )

Poisson's ratio ( vc )

Yield Strength (h )Elastic Modulus ( E s )

Poisson's ratio ( Vs )

21.0 MP a

1.75 MP a

23.5 GP a

0.167300.0 MP a

200.0 GPa0.3

" - - - - - ' - - - - - - - - - - - - - 88 y

(a) Reinforcement

Fig. 5 Stress-Strain Relationships

(b) Concrete

- 29 -



3.3 Various Loading Cases

In this s tudy, the fol lowing three loading cases were examined as fo llows: Fi rst , the monotonic loadings for

four directions were given as shown in Fig. 6(a). Th e loading directions are 15,30,45 degrees f rom the principal

axis f) = 0° as a basic case. Second, the cyc li c loading cases for t he s am e directions as t he m on ot on ic o ne s were

app lied as shown in Fig. 6(b). Last , the triangle loading cases were adopted as th e state of bi-axial bending is

assumed as shown in Fig. 6(c).

In all cases , the displacement control loading was imposed. Th e applied displacements were se t as magnifications

of the dri ft ang le. A unit drift angle was defined herein as horizontal displacement div ided by the height f rom the

bottom of the pier to the superstructure's centroid of its cr oss se cti on, an d the ma gni fic ati on s eq ual to 0.5 %,

1.0 %, 2.0 %, 3.0 %, 4.0 %, 5.0 %.

y y y

x

(a)Monotonic Loading (b) Cyclic Loading

Fig. 6 Loading Cases

(c) Triangle Loading

3.3.1 Monotonic Loading Cases

Figure 7 shows the load (P ) - displacement ({)) relationships for all loading cases, in which squares, circles

and triangles indicate that the ini tial yie lding, the maximum strength an d the pos t peak strength at 10% reduction

f rom the maximum value, respectively. From this figure, the maximum s trength increased as the angle increased,

and also that wa s around 30% h ig her than the corresponding design strength. No s ignificant load reduction,

furthermore, could be observed up to 5.0 % drift angle .

.Nex t, the normal compressive stress distributions of concrete on the c ross sec tion at heigh t of O.36D (D: section

dep th ) are as shown in Fig. 8 when the pos t peak strength decreased to 90 % of the maximum strength. Th e shape

of the distribution changed to be tr iangle as the ang le for loading increased, though it was rectangle in th e case of

f) = 0°. Th e high level s tresses could be observed somewhat inside area f rom the comer in Fig. 8 (b) - (d), which

was suggested the occurrence of compression failure in concre te . Because the st resses on an d n ear the comer

undergone the largest compressive strain were smaller than those of t he a bov e area, that st resses should be at

compressive sof tening zone after the peak according to th e s tress - s train curve of Fig.5 (b).

Last, the strain distributions of hoop ties are shown in Fig. 9, as well as Fig.8. I t can be confirmed tha t the st rain

distribution of the hoop agreed fairly with that of concrete in Fig. 8.

- 3 0 -



6000 1 - - - - - - - - - - - - - - - - - - - - - - ,

5000

~ 4000g~ 3000-0

'"

:32000

1000

- - - - - - ~ - - - ~- ~ -_ ! . . . ~ -- - ~ - - - - - - - - - ~ - - - - - - - - ~ - - ~ - ~ - ~ - ~ ~ - ~ ~ - -

Yielding Ultimate______ InitiaLYjelcling ,----,0,--=---:0::::: - - - - - - - - ,

- 0 = 150

- 0 =3 0 0

- 0 =45 0

- - Japanese Design Code- - Ultimate Strength

• : Initial Yielding

• : Maximum Strength

... : 10% reduction strength

after the marimum

500(5%)

CrackingO - - - - - - - ' - - - - - - ' - - - - - - - - ' - - - - - - ' - - - - - - - J

o 100(1%) 200(2%) 300(3%) 400(4%)Displacement (6 ) (mm) (Drift Angle %)

Fig. 7 Relationships of Load (P) - displacement (6)

MPa

-21XE-OI

(a) B = 00

(b) B = ISo

-4.OCE-OI

- ..31\E-OI

xH - + - - j - - j - - -

YL1 - + + - - + - t - - t - - - t - + - - F ' T 9

(c) B= 300

(d) B=4So

Fig. 8 Normal Stresses of the Cross Section in Height of 0.36D

- 31 -



I

J ~ - ~(a) 8=30°

1 -~

(a) 8 = 15°

IL - - .

0.002

0.00175

0.0015

OJ)(}\

o (ltll)-:'5

O,(I()05

O.OOO:!5

n . ( l l , ) O ~ 5

-O,(X)()5

·0.00075

-0.001

-0.00125

-0.0015

·0.00175YLx

Fig. 9 Strain of the Lateral Reinforcement in Height of 0.36D

3.3.2 Cyclic Loading Cases

Firs t, the behavior of the model under the cyclic loading to the each direction as same as the monotonic

loading was also analyzed. The load (P ) - displacement (5 ) relat ionships for each loading cases a re shown in

Fig. 10 with the monotonic loading resul ts . The displacement at the maximum strength increased compared with a

monotonic load resul ts for all loading cases . Furthermore, the maximum strengths were around 20% higher than

the corresponding design strength.

Second, the distribution of normal stress a t 3.0 % drift angle is shown in Fig. I I for two loading cases (( ) =

0°,45°). The area those stress should be the state of a compressive softening expands than the case of monotonic

loading as s ho wn in Fig. 8 (d) in case of the loading for () = 45°(Fig. 11 (b)). But , it c loud not be li tt le difference

seen in loading for the () = OO(Fig. 8 (a), and Fig. 11 (a)).

Third, the strain distributions of the hoop ties were also hown in Fig. 12. It differed obviously from those in

Fig. 9 under the monotonic loadings, which showed a strain concentration with smaller intensi ty near the corner of

the hoops. A s to Fig.12 under the cycli c loadings, both no conce nt rat ion a nd larger int ensity beyond yie lding

could be observed.

Last, Fig. 13 shows the tens il e strain a t the corner of the hoop ties a t each dri ft angles. Excep t for () = 0°, it

can be confirmed generally that the strain of the hoops increased as the drift angle increased, although the

numerical solutions were somewhat scattered because of the dual hoop ties arrangement of the pier.

- 32 -



500(5%)00(3%)10 0(- 1%) 100(1%)

D . p ~ c e l r e n '(0) (!Tnn)

-6000 L l----'=======:::'J-500( -5%) -300( -3%)

6000

4000

-0~o

....l -2000

2000~

5 0 ~ 5 % )01X3%)100(-1%) 100(1%)

D " p ~ c e l r e n t(0) (rnrn)

-6000 L L---'=======::J-500 ( -5%) -300( -3%)

4000

-4000

6000 , - - - - - - - - - - - - - , - - - - - - - - - - ~

-0~o

....l -2000

2000~

(a) () = 00 (b) () = 15°

500(5%)00(3%)1 00(-1%) 100(1%)

D . p ~ c e n " n l0 (mm)

-6000 L ~ _ _ = = = = = = = ~-500( -5%) -300( -3%)

4000

-4000

6000 r - - - - - - - - - - , - - - - - - - - - - -

-0~o

....l -2000

2' 20006

500(5%)00(3%)301X-3%) -100(-1%) 100(1%)

D . p ~ c e n " n t(0) (rnrn)

-6000

-500(-5%)

-40001 " " - - - " -

4000

6000 r - - - - - - - - - - - , - - - - - - - - - -

-0~o

....l -2000

2000~

(c) () = 30°

Fig. 10 Load (P) - Displacement (8 ) Relationships

(d) () =45°

(Drift angle %)

Fig. 11 Normal Stresses ofthe Cross Section in Height o f 0.36D

(a) () = 0° (b) () = 45°

MPa

- 33 -



0.002

- ~- -

<==->

•-

-- - -

(b) e =450

0.00175

lUX)\S

n.fXJl25

0.001

O.OIXJ75

n.OO05

IUX)()25

o.

·0.00025

-0.0005

-0.00075

-O.l)()l

-0.00125

-0.0015

-0.00175

Fig. 12 Strain of the Lateral Reinforcement in Height of O.36D -0.002

o V- - - - - ¥ - - - - - ¥ - - - - - - - ' - - - - - - - - - '

40000 , - - - - - - - - - - - - - - - - . . . - . . . ,

• 0 = 0°• 0 = 15°• 0 = 30°• 0 = 45°

30000

10000

~

:l.

" 20000§r/l

••

••••

• 0 = 0°• 0 = 15°• 0 = 30°• 0 = 45°

4 ס ס o o

30000

2-

" 2 ס ס o o'§r/l

1 ס ס o o

04 2 4

Drift angle (0/0) Drift angle (0/0)

(a) Outer Hoop (b) Inner Hoop

Fig. 13 Variability of the Strain of Lateral Reinforcement

3.3.3 Triangle Loading Cases

Based upon the previous described resul ts for the monotonic loading cases drawn that the strength decrease

of the pier under the loading angle of 45 degrees could be observed the earliest loading stage, we examined here

how the strengths and stress distributions of the pier depend on the loading direction angle.

Figure 14, therefore, shows the load (P ) - displacement (6 ') relationships under the triangular loadings as shown

in Fig. 6(c). In this f igure, the obtained gross horizontal force intensi ty and the displacement were decomposed

into two components along the two crossing direct ions of X and Y as shown in Figs. 4 and 6. The maximum

strength of X-component under the triangle loading indicated the red curve in Fig. 14 (a) was equal to tha t under

cyclic loading for e = OOof the black curve additionally drawn in Fig. 14(a). I t could be said, thus, that there was

a little influence of a combined loading of Y-direction on the maximum strength of X-direction.On the contrary,

the maximum strength of Y-direction under the triangle loading as shown in Fig. 14 (b) was smaller than that

under cyclic loading for e = 0 0•

Furthermore, Fig. 15 shows the tensi le strain at the comer of the hoop ties a t each drif t angles under the tr iangle

- 34 -



loading and also under the cyclic loading for e = 45°. As for the outside hoop ties , no dependency of the strain

on the loading cases could be observed up to 3.0 % drift angle. B ey ond 4 % dri ft ang le , however, a dis tinct

dependency could be recognized. Otherwise, no remarkable dependency for llmer hoop ties was not confirmed.

(-4%) (-3%) ( -2%) H% )6000

(1%) (2%) (3%) (4%) (-4%) (-3%) (-2%) (-1%) (1%) (2%) (3%) (4%)6000 I - - - - - _ _ ---, ~ . . . . : . . _ _ . : . . : : _ = . : . _ _ _ _ _ . : . _ . : . . ;

Displacement for X-Direclion (oX) (rom)100 200 300 400

-6000 L - - - - I . _ - - ' = = = : = : : : ' ~ = ~ = : : ' J

-400 -300 -200 -100

D5placcmem for V-Direction (8 Y) (nun)

~ 4000

co~o.

'"~ -2000

5 2000

."

..3 -1000

100 200 300 400

-2000

-4000

-6000 L - l ~ ~ = = = = ~ ~-400 -300 -200 -) 00

~ 4000

~- 2000

(a) X-Component (b)

Fig. 14 Relationships of Load (P) - Displacement (0)Y-Component

(Drift Angle %)

7 ס ס o o, - - - - - - - - - - - - - - - - - ~ 70000 , - - - - - - - - - - - - - - - - - -

6 ס ס o o

5 ס ס o o

60000

5 ס ס o o

_j +Triangle

I +Cycl icO=45°

1-- - - - - - - - - - - - - - -

"- 4 ס ס o o

"'£ 3 ס ס o oell

..=; 4 ס ס o o

"~ 3 ס ס o oci5

2 ס ס o o2 ס ס o o

•< ; ; ; L - - - - - ¥ - _ _ __,o"_____ _ _'______ס ס o o

J

Drift angle (% )

4

1 ס ס o o

•v - - - - _ ¥ _ _ _ - o t ~ - - ~ - -

Drift angle (% )

(a) Outer Hoop (b) Iwer Hoop

Fig. 15 Variability of the Strain of Lateral Reinforcement

4. Concluding Remark

In this paper, numerical s tudies for the RC columns with a square cross sec tion are car ried out sub jec ted to

the three loading cases. As the result, the noticeable results were obtained as follows:

First, under the monotonic loading, the maximum s trength increased as the angle increased, and also that was

around 30 % higher than the corresponding design strength. No significant load reduction, furthermore, could be

observed up to 5.0 % drift angle. And, from the normal compressive stress dis tr ibut ions resul ts , it was confirmed

that the shape of the dis tr ibut ion changed to be triangle as the angle for loading increased, though it was rectangle

in the case of e = 0°.

Second, under the cyclic loading, the d is pla ceme nt at m axim um s tren gth increased comp ared with the

monotonic load results for all load ing cases. Moreover, the maximum strengths were also around 20% higher

- 35 -



than the corresponding design strength. I t could be also found that the tensi le strains of the hoop ties increased

more extensively than those under the monotonic loadings. And, it can be confirmed general ly that the tensi le

strain of the hoops at the comer increased as the drift angle increased, although the numerical solutions were

somewhat scattered because of the dual hoop ties arrangement of the pier.

Last, under the triangle loading as the bi-axial bending, it could be observed that there was a little influence

of a combined loading of Y-direction on the maximum strength of X-direction. On the contrary, the maximum

strength ofY-direction under the triangle loading was smaller than that under cyclic loading for () = 0°. And, from

the tensile strain at the corner of the hoop ties at each drift angles, as for the outside hoop ties, no dependency of

the strain on the loading cases could be observed up to 3.0 % drift angle. Beyond 4 % drift angle, however, a

distinct dependency could be recognized.

Acknowledgement

The authors wish to thank Mr. Shimabata of Osaka City University for his help with the analyses.

5. References

1) Japan Road Association, Specifications for Highway Bridges: Part V- Seismic Design, Maruzen, Tokyo, 2002

2) Japan Society of Civil Engineers, Earthquake Resistant Design Codes in Japan, Maruzen, Tokyo, 2000

3) Chen, W. F & Shoraka, M. T.: Tangent stiffness method for biaxial bending of reinforced concrete columns,

IABSE Publications, 35(1), pp. 23-44, 1975

4) Chen, W. F & Atsuta, T.: Theory of Beam-Columns, Vol. 2: Space behavior and design, McGraw-Hill, New

York,pp.253-260, 1977

5) Yoshimura M. et al.: Analysis of reinforced concrete structure subjected to two-dimensional forces, Journal of

structural and construction engineering, Architectural Institute of Japan, No. 298, pp. 31-41, 1980

6) Chen, W. F : Plasticity in Reinforced concrete, Trans. M. Shikibe, M. Kawakado, H. Adachi, Maruzen, Tokyo,

pp. 207-252,1985.

7) K. Naganuma: Stress-strain relationship for concrete under triaxial compression, Journal of structural and

construction engineering, Architectural Institute of Japan, No.4 74, pp. 163-170, 1995

8) 1. Izumo, H. Shima, H. Okamura: Analytical Model for RC Panel Elements Subjected to In-Plane Forces,

Concrete Library International, JSCE, No. 12, pp. 155-181, 1989

- 36 -

a numerical study on confinement effect by square hoops

Documents