Kobayashi, Harutoshi

Sato, Tomoaki

Shimada, Isao

Osaka City University Repository

Mem. Fac. Eng., Osaka City Univ., Vol. 43, pp.59-67 (2002)
A Simple Numerical Method for Biaxial Bending Moment-Curvature Relations
of Reinforced Concrete Column Sections
Tomoaki SATO,* Isao SHIMADA,** and Harutoshi KOBAYASHI***
(Received September 30, 2002)
Synopsis
Proposed in this paper is a simple numerical method for computing the bending moment -
curvature relations for reinforced concrete rectangular cross section of bridge piers under
biaxial bending action, which is essentially required at their seismic design considering a
horizontal force due to earthquake in arbitrary direction. The cross section is divided by a
number of rectangular finite areas of concrete and reinforcing bar areas. And then
numerical integration technique is developed to evaluate the stress-strain relation for each
rectangular finite area divided. Numerical results are presented to show validity of the
present method with satisfactory accuracy compared with those by the method
recommended in the Japanese seismic design code [1].
Key Words: Reinforced concrete, Elasto-plasticity, Biaxial bending, M-ep relations,
Numerical Method
1. Introduction
The current Japanese seismic design code for bridge structures [1] treats that the seismic force for bridge
piers should be considered under horizontal actions individually parallel and perpendicular to the direction of
bridge axis. It is, however, natural and rational to evaluate interactively those actions. The bending moment -
curvature relation, i.e. M-f/J relation for cross sections of bridge pier necessary in the seismic design is also
shown to be evaluated numerically using the method of multi-layered model [1] that essentially requires a
sufficient number of divided layers resulting in complicated computational effort.
In this paper, therefore, we propose a simple numerical method for computing M-f/J relations for
reinforced concrete rectangular cross section of bridge piers under a biaxial bending action. The cross section
is divided by a number of rectangular finite areas of concrete and reinforcing bar areas. And then numerical
integration technique is developed to evaluate the stress-strain relation for each rectangular finite area divided.
Their validity is discussed based upon the numerical results obtained comparing with those by the method
recommended in the code.
2. Outline of Existing Estimation Method
In the seismic design code [1], fifty-layered model is recommended to evaluate M-f/J relations for
reinforced concrete rectangular cross section of bridge piers under a horizontal action, namely uni-axial
bending. In case of the action in arbitrary direction, for example biaxial bending, the code has not given an
obvious evaluation method for M-f/J relations.
Chen and co-workers [2], [3] have developed the portioning model for cross sections in biaxial bending as an
extension of the layered model. They have obtained various kinds of the M-ep curves relating axial compression
and biaxial bending moments for a standard concrete cross section specified by ACI code, however, their models
are sophisticated for practical design work, so that more simplified methods are earnestly awaited.
* Docter Course Student, Division of Civil Engineering, Department of Urban Engineering
** Head of Research Section, Toyo-Giken Consultants Co. Ltd., Osaka
*** Professor, Division of Civil Engineering, Department of Urban Engineering
59
3.Simplified Estimation Method
3.1 Required accuracy
In case of uni-axial bending action, the M-ep relations obtained by the proposed method should not be
exceed 10% deviation from those by the evaluation method recommended in the code [1] with respect to the
following essential view points: initial yielding point of reinforcement, peak strength point and maximum
deformation capacity. Furthermore, in case of biaxial bending action, the same accuracy as the former should
also be guaranteed.
3.2 Numerical idealization
Divided layered number of fifty in the vertical direction as described in the code as shown in Fig. l(a) is
decreased to three as shown in Fig. 2, in which same subdivision in the horizontal direction is also shown.
The area of cover concrete located outsides of reinforcements is ignored because the area becomes
ineffective owing to cracking or peeling off even before the initial yielding point of reinforcement. The yield
of reinforcement is the first-coming essential phase on the M-¢ relations.
h
y
----4--------~=~
-..---- ===--e--=
------
~--:=== ------e-:
::::te:=:--- -========-=
• .--- _.--=--~- X
.....---._-J..-__............~
I
b
h
y
~~ ~lTi~' : f: ::~
~~=+i .. '.l_ tt-ttm:;
~.'T. " ".- •
.. I .. , ~ :~
.f ".. ':ttm
•••••. I".'
.~~4-+4-~-+4.+: ., +Hi+-.
(a) Layered model for
uni-axial bending
(b) Finite element model for
biaxial bending
Fig. 1. Layered Model for Cross Section
h
••• • ••
• •
• •
• I I •
_·__ L __ l-_~_
• I I •
• I I • X
• I
• I I •~.- - +- - - +- - ~-
• •
• I I •
, • I I •
I • I I • I
••••••••••• 1
,
b
Fig. 2. Model for Proposed Method
60
3.3 Numerical integration technique
The stress-strain relations of cross section are evaluated numerically by dividing the cross section into
the several rectangular finite elements shown in Fig. 2. Number of finite elements Nc considered is 9 ( 3 X 3 )
for the section. In figure, the mark 0 present the calculation points the (xo, Yo) for numerical integration
that are located at the gravity center of each finite element, and those except central finite element are
defined by
(x,y) =a~Ao (xo, Yo) (1)
. A
where a is coefficient to be taken as 0.8, which is valid in case of the ratio of cross sectional height to
breadth being 1.0 to 2.0; A is the effective area surrounded by reinforcement arrangement for each finite
element; Aois the whole area including cover concrete area for each finite element; and (xo,Yo) are the
location at the gravity center of each finite element.
Setting curvature: 4' - 4' + ~4'
Asumption of neutral axis
Setting of coordinates of sampling points
for nu~erical integration
en
~ .~
0
.Q ~
.....
... ...
c:: =Q) Q)
8 c:Q)
0 .s
.5 en
Q)
.~
.....
Q)
= ~...
Cd
~
=U
Caluculation of stress at sampling points by using the
stress-strain relations, the stresses were given by
Eq.(3)(4) 0Cij =Oc(Eij ) , aSk -as(Ek)
Integration to caluclate for normal force
-N - ~ ~aCij M ij + ~ a'kMk6t~ 61
No Judgement
Integraed Normal Force:
External Normal Force
yes
END
Integration to caluclate for bending moment
M. - ~ ~ acijYijM;j + ~ a'k YkMk61# ~
My - ~~ a CijXijM;j + ~ a'k XkM.t61~ 61
Fig. 3. Flow Chart for Numerical Calculation
61
3.4 Numerical procedure
The elasto-plastic behavior of RC cross section members is formulated on the basis of the following
assumptions:
1. The strain distribution across the section is assumed to be linear, varying in proportion to the distance
from the neutral axis, Le. it is assumed that the plane section before bending remains the plane after the
application of loads.
2. The strain and stress in are computed at the divided sectional areas. And in divided each areas, stresses
are treated as constant.
3. Cover concrete area is considered as separated after the yield of reinforcement. And it is to be ignored
for flexural rigidity, but to be considered to calculate the coordinate point (x, y) for obtaining
stress-strain relationship.
The flow chart for the present numerical procedure is shown in Fig. 3. In this flow, the strain of each
integration point is calculated by using Bernoulli's assumption. In case of the elastic state (Fig. 4a), the
coordinate is measured from the centroid of cross section. If considering the partially yielded cross section
for the elasto-plastic state (Fig. 4b), the coordinate is measured from the neutral point.
b
•
.. ~ ...
P (1J,~):
:/(t) •
_.J~ ._..,
/1
/ I
oal --..;I""'8- ·""'O"--.x
· / : .I
, .
r-.-- I O'(xn , Yp ) --.-~ 11
0: 0 : 0 •
I I •
I I
• I I •;···r···r···
h
.:
·t···l
P (x,y).
:t(t) •
--J~---
/1 •
/ 1 •
bI~
./
./
////./ Y
L----,,-+---T----
,···t·
• I
· :
• 0:
• I:------r--
• I
· :I. 0:
• 0I
• I I •:- ·----r------r----.--
:.0: 0 :0. 1 //
• I I • II I ./
• I I • /
·..,...,... ///
--' , _----//
h
(a) Elastic state (b) Plastic state
Fig. 4. Calculation of Strain at Integration Points
The strain of each integration point for the case of elastic and plastic states are expressed as follows:
(a) Elastic state
£ij = £0 + lPyXij -l/JxYij for concrete (2a)
Ek =Eo +lPyXk -l/JxYk for reinforcement (2b)
where £0 is the axial strain; l/Jx and l/Jyare the curvatures with respect to x- and y-axes respectively;
(Xij, Yij) is the coordinate of the integration point of each finite element for concrete; and (Xk, Yk) is the
coordinate of each reinforcement.
(b) Plastic state
Cij=£O+l/Jy(Xij+Xn)-l/Jx(Yij+Yn) for concrete (3a)
Ck =co +l/Jy(xk +xn)-l/Jx(Yk + Yn) for reinforcement (3b)
where (xn, Yn) is the coordinate of centroid of cross section for neutral point; and the other characters
are the same to the case of elastic state one.
After calculation of strains, the stresses of concrete and reinforcement are calculated according to the
Japanese seismic design code [1]. The stress-strain relationship of concrete shown in Fig. 5(a) is given by the
62
following expression.
(4a)
where
(4b)
and where a e is the concrete stress; a is the confined concrete stress with lateral reinforcement· c is theIT , e
concrete strain; cce is the strain at peak stress; Celt is the ultimate strain; Ec is the modulus of elasticity of
concrete, and Edes is the falling inclination after the peak stress.
0.80cc -----
E c
£cc Ecu
(a) Concrete (b) Reinforcement
Fig. 5. Stress-Strain Relations
The stress-strain relationship of reinforcement shown in Fig. 5 (b) is given by
as = Eses when e slesyl
as = a sy when E > Esy
as = -asy when E < -Esy
(5a)
(5b)
(5c)
where a sy is the yield strength of reinforcement; as is the stress of reinforcement; Es is the modulus of elasticity
of reinforcement; Csy is the yield strain of reinforcement, and E.'l is the strain of reinforcement.
4. Numerical Examples
Several numerical examples are shown to verify the present method. In the numerical computing, the width of
the cross section is fixed as constant value, Le. 2.0 m, and the height has been changed from 2.0 m to 4.0 m. The
z My f/JjY
Y
h
Fig. 6. Definition of Positive Directions for Bending Moments and Curvatures
63
0.040.030.02
cP (11m)
0.01
! ..........., Existing
Proposed
! I
I
I
I.....
.... I.. -r I ,- -i Ii;
I :o
o
10
40
50
'¥ 30
Z
6
i 20
(a) f)'=O°
50 ......---~----.....,.---------.....
••••••••••• , Existing
Proposed
40 ~-----+-----T---
EZ 30
~
'::E..... 20 1-------+-------+---------;---------1
'::E.~
0.010.00750.005
cP (11m)
0.0025
10 I------~-.......~hr-~----_+_----___t
50 r-----------~------------.
•••••••••••, Existing
Proposed
40 1-------------------'------...,-------1
------------~-----------------
E 30 t------~--------~----.------­
Z
~
'::E..... 20
10 A I
-~
T I -I I
I0
0 0.01 0.02 0.03 0.04
cP (11m)
(c) e= 90°
Fig. 7. Moment-Curvature Relations (h/b = 1.0 )
64
I
I
I
I
!
I
I
I
I
0.03 0.040.02
cP (11m)
0.01
~ 20 r I~ ; I
10 ~ I !
, I !
I I
I I
o ......----....I----.......:.I------...----~
o
50 r------r-
I
, -----,....-,----_-------.
••••••••••• , Existing
i Proposed
40 r------t-,-----+1------,..------1
! I
e 30 ..-----+-1-------+--: --+ --1~ .-~ ~ __.-----l---.
.: I
50 r-----~----_---------.....
••••••••••• , Existing
Proposed
40 I------~------+------__r_----_I
0.010.00750.005
cP (11m)
0.0025
ez 30
6
~~ 20 I-----=---------------........--------~----~----------I
~~
50 -----..------......-----------..
••••••••••• , Existing
Proposed
40 I--------+-------------~-
e 30
z
6
~~ 20
~ ......
10 --------------------- ------------
0.040.030.02
cP (11m)
0.01
o~--------- ........---------...
o
Fig. 8. Moment-Curvature Relations (h/b = 1.5 )
65
0.040.030.02
¢> (11m)
0.01
I
, , '" " .....~~. ........• ••••••••••• 1 Existing
I Proposed
r
! I
I io
o
10
40
50
e 30
Z
6
~~ 20
50
40
e
:Z 30
6
~;>., 20 Mx
~~
10
••••••••••• Existing
Proposed
66
details of dimension of the cross sections are presented in Table 1. The cross section is first loaded axially by
compression force P that is equal to the normal stress 0.98 N/mm2, and keeping the compression force P, and then
the curvature l/Je is increased proportionally in magnitude from zero. Definition of positive directions for
bending moments Mxand My and curvatures l/Jx (= l/Je-o 0), l/Je-45 0,and l/Jy (= l/Je=90 0) are shown in Fig. 6.
Table 1. Analysis Model
Height h Width b h/b Reinforcement(m) (m) Ratio (%)
2.0 2.0 1.0 0.56
3.0 2.0 1.5 0.59
4.0 2.0 2.0 0.62
The numerical results for moment-curvature relations subjected to the above mentioned conditions are
presented in Figs. 7-9 compared with the evaluation method in the code [1]. In these figures, marked points
denote the following events; • yielding point of reinforcement, A peak strength points, and - maximum
deformation capacity respectively.
5. Concluding Remarks
The result of compared simplified estimation method with regulated method, the maximum error is
occurred with 10% at ultimate deformation capacity in case h/b=2.0. These results show that the required
accuracy is mostly satisfied irrespective of each bending direction.
References
[1] Japan Road Association: Specifications for Highway Bridges; Part V-Seismic Design, April 2002
[2] W. F. Chen and M. T. Shoraka: Tangent stiffness method for biaxial bending of reinforced concrete,
IABSE Publications, International Association for Bridge and Structural Engineering, Vol. 35-1, pp. 23-44,
1975
[3] S. I. AI-Noury and W. F. Chen: Behavior and design of reinforced and composite concrete sections,
Journal of the Structural Division, Proceedings of the American Society of Civil Engineers, Vol. 108, No.
ST6,pp. 1266-1284, 1982
67


A Simple Numerical Method for Biaxial Bending Moment-Curvature Relations of Reinforced Concrete Column Sections

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A Simple Numerical Method for Biaxial Bending Moment-Curvature Relations of Reinforced Concrete Column Sections

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