Kitoh, Hiroaki

Kuramoto, Wataru

Ohuchi, Hajime

Osaka City University Repository

Numerical Study on Compressive Concrete as a Strain Softening Material
Wataru KURAMOTO*, Hiroaki KITOH** and Hajime OHUCHI***
(Received September 30,2009)
Synopsis
The prediction of post peak behavior of compressive concrete is significant especially for ductility based on
seismic design of reinforced concrete members. The present paper describes numerical modeling of post peak
behavior of compressive concrete as a strain softening material with constitutive law based on incremental elasto
plastic theory formulated in a strain space. A series of numerical parametric studies have been conducted for the
existing experimental results of compressive concrete including useful post peak behavior test.
KEY WORDS: Post-Peak Behavior, Compressive Concrete, Strain-Softening, Strain Space Fonnulation
strain
Fig. I Stress-strain relation of concrete
under uniaxial compression
Vl eak stress
Vl - -/-"' •••••v " .....••
t: '" ···· contined concrete
Vl '" ...
/
'" " '" '"u~confined concrete
'" [;
Hatanaka".
1. Introduction
It is significant to predict plastic deformation capacity of reinforced concrete members, i.e. how compressive
concrete can deform and also how it can sustain further compressive stress. Post peak behavior of compressive
concrete is deeply related to existence of confinement stress due to lateral reinforcement.
As illustrated in Fig.I, typical stress strain relationship under uniaxial compression, unconfined concrete
provides steep descending branch. However, the slope of that branch can be improved milder shown as the dotted
line, once effective lateral confinement is provided so that volumetric expansion is restricted by lateral
reinforcement such as hoops or ties. Appropriate numerical modeling for such phenomena can be effective to
predict rigorously the post peak behavior of reinforced
concrete members.
The present study investigates nonlinear constitutive
model based on incremental plastic theory with a loading
function defined in strain space, proposed by Mizuno and
The purpose of the study is to verify effectiveness of
the model through numerical finite element simulations
for several existing experimental results with strain
softening behavior.
*
**
***
Student, Master Course of Dept. of Civi I Engineering
Associate Professor, Dept. of Civil Engineering
Professor, Dept. of Civil Engineering
- 37 -
2. Existing Experiments on Strain Softening of Compressive Concrete
There are many experimental studies on the strain softening behavior of compressive concrete with various
lateral pressures, those inform us their stress strain relations and also failure envelops in various combination of
stress states. Form the viewpoint of existence of lateral pressure, the two of the results were adopted as
examples for the numerical simulations. One is the uniaxial compressive test without lateral pressure by
Barnard'), and the other is the tri-axial compressive test by Hatanaka", et al.
First, Barnard's specimen is shown as Figure 2, and its dimensions are listed in Table 1. Figure 3 is an
example of the obtained stress-strain relations.
Table I Dimensions of specimens
height II.5in. (29.2Icm)
diameter(edge) 4.5in. (11.43cm)
ditto.(center) 2.5in. ( 6.35cm)
gauge length 4.0in. (l0.16cm)
30 ,---------,-------,-----,---------,------,
100008000600040002000
OL--------'-------L-- i.-.__---' --.!
o
Strain(llJ
Fig. 2 Specimens') Fig. 3 Stress-strain relations')
Next, Hatanaka's specimen and test setup is shown as Fig. 4, in which biaxial lateral pressure was given
passively dependent on the lateral dilatation of the concrete specimen. Testing parameters and an example of the
results are shown in Table 2 and Fig. 5, respectively.
Table 2 Outline of experimeneJ
water-cement ratio height/width Lateral confining stress a L
W/C H/D (MPa)
0.45
0.55
0.70
1.0
1.3
2.0
o
0.1
0.3
0.6*1
1.2*2
*1 :anly far H/D=l
*2:anly far W/C=0.55 ofH/D=1
average strain rate of2 Xl 0·3/min
- 38-
t1
..... ;J;:I
0 r;}
:;-."
~
\,,>J
til
8:
0 ~
M
Ul
fn
20n12 16
S'l'.l:;;AIN (it 10-3.l
Fig. 4 Specinlen and loading setup3) Fig. 5 Stress-strain relations3)
(1)
(2)
(3)
3. Elasto-Plastic Model for Strain-Softening
The determinations of loading conditions defined in stress space are given in Eqs.(1)-(3). Figures 6 and 7
show the conceptual two-dimensional views of the loading conditions of stress and strain spaces, respectively.
When strain hardening and strain softening condition in the stress space are to be defmed, it should be difficult
to detennine the condition of loading, neutral loading or unloading, because the detennination equations, such
as Eqs.(2) and (3), naturally involve multiple conditions.
df > 0 : Loading (hardening)
df < 0 : Loading (softening), Unloading
df = 0 : Loading (perfect plasticity), Neutral loading
Loading function :/
--,
"
\
I
J
Loading function: F
--,
.....-..... "'...... ,~
~~ ~
..~..
~..
dF <{}':
Fig. 6 Loading condition of stress space Fig. 7 Loading condition of strain space
The corresponding determination equations in strain space are also given in Eqs.(4)-(6). It can be found out
that a consistent detennination of the loading conditions is described without the above difficulty. Moreover, a
comprehensive comparison of elasto plastic incremental theory between in stress and strain spaces is illustrated
in Table 3.
- 39 -
dF > 0 : Loading (hardening, softening, perfect plasticity)
dF < 0 : Unloading
dF = 0 : Neutral loading
Table 3 Comparison between stress and strain space formulation
(4)
(5)
(6)
Loading
function
Flow rule
Stress-strain
relationsh ip
Stress space
Drucker's postu late
Compatibility condition
Strain space
I1'yushin's postulate
Compatibility condition
dF= of dc =0
oCi) IJ
Figure 8 shows the loading function in three dimensional principal strain space. Equations (7) and (8)" are
loading function:F and plastic potential function:G, respectively.
Fig.8 Loading function in a principal strain space
(-- )3 [ [p J'" ][(AT + a)3 I (-- 'rn- --JF= All +a - 27+/" AII(~a 127 -"3 AI, +a)BJ2 +CJ3 =0
[ [ j"I][(--)3 ~-- 3 P AI + a 1 -- ---G=(AI I +a) - 27 +77 2 a I -~(AII +a}BJ2 +CJ3 =0All + a 27 J
- 40-
(7)
(8)
where 11' J 2 and J 3 = the first invariant of strain tensor and the second and third invariants of the
deviatoric strain tensors, respectively; Pa = the atmospheric pressure; In = the shape parameter; a = a
magnitude of movement to the tensile direction along the hydrostatic pressure axis; and fp = the loading
parameter. Moreover, A =3K, B=4)12 and C=8)13, where K and)1 are bulk and shear modulus,
respectively.
Substituting by those into incremental stress-strain relation, the elasto plastic constitutive matrix in finite
element formulation can be formed.
4. Numerical simulations
4.1 Preliminary
Compressive strength of concrete and also lateral confinement stress intensity are fundamentals for the
behavior of the compressive concrete, so that their influences were examined preliminary.
First, the obtained results with the various compressive strengths are shown in Fig. 9. It could be found that
the slope of the falling branch becomes steep as the given compressive strength is high, while the branches
came together beyond the compressive strain of 1OOOO~l at last.
ext, Fig. 10 shows the results with the various confinements, In which the vertical axis indicates a
nonnalized compressive stress by the cOITesponding compressive strength. From the results, the confinement
intensity could considerably improve of the post-peak behavior, while it could not affect the peak stress.
40
~ 30
'"5
'"~ 20
Vl
10
t;.'=33.8MPa
=27.3MPi!
"-'21.6MPa
1.2
.r;
en
~ 1.0
Vl
.~ 0.8 -
'"~
Q.
E 0,6
o
V
0.2
...
....
•••')..,,~•••• ~ •••••• ¥ .
.... .,
........ cr,.I.!" =.0.9%
............ //-_..~
... "0.0%
' ..... ..P
o 20000 o sooo 10000
Strain (;i)
Fig. 9 Stress-strain relations with
various compressive strengths
Fig. 10 Stress-strain relations with
various confinement stresses
4.2 Material parameters
Material parameters of the material nonlinear model" as described in the previous chapter 3 are listed in
Table 4. It is noted that,[p relevant to '7 1 is the most essential to control the strain-softening behavior in the
model. For example, the three of the parameters to be examined were 77 1 , JirJppeak and Y2 , those are related
closely to a uniaxial compression state. Figures II. 12 and 13 are the results varying each of the corresponding
three parameters, where each vertical axis denotes the normalized stress as same as that in Fig. 10. From these
figures, it could be recognized in particular that the peak stress increased as 77 1 increases. as well as the post-
peak branch arises as rrJl'l'eak and Y2 increases.
- 41 -
Table 4 Material parameters of load function
'7 1 :The peak value of parameter fa 165
m :The shape parameter 0
:A magnitude of movement to the tensile
0.31/ ca direction along the hydrostatic pressure
p :The value obtained by using least 0.099
square method
I :The value obtained by using least 0.867
square method
o :The plastic work corresponding to the 0.345W
ppeak non lateral pressure
:The value obtained by using least
0.1021Yl
square method
:The value obtained by using least
2.41Y?
- square method
.r: 1.0
0,
c
'"~ 0.8
<1!
,;:
'"~ 0.6
n
E
o
v
". 0.4
'"'"e!
v; 0.2
11 J
---- 175
--165
--- 160
~--~ 155
--145
80002000 4000 60()O
Strain (p)
Fig.11 Stress-strain relations with various 77
1
o
.r:: 1.0g,
~~ 0.8 ""
~
'Vi
'" 06 _.0.'
E
o
'::: OA -
'"
'"~
Vi 0.2 .
o 2000
.r:: LO
0>
c
~
v; 0.8
~
'in
~ 0.6
a.
E
o
'::: OA
~
~
V; 0.2
o
Fig. I2 Stress-strain relations with various Wppeak Fig.13 Stress-strain relations with various 1 2
- 42 -
Fig.14 Stress-strain relations according
to the parameter values in Table 4
2000 4000 6000
Strain (p)
8000
•••• ., Exp.12)if"cc=27.2MPa)
-NlIm.l
..... Exp.22J (f'c=22.9MPa)
-- Num.2
•••••. Exp. 121(1"(=27 ,2MPa)
,-- Num.l
...... Exp.2<)((<:=22,9MPa)
- Num.2
••• -- Exp,3IlV'c=24.6MPa)
- Num.3
I
o
25
20 '......
5
20
25 -
10
4.3 Application to existing results
The numerical simulations based on the
material nonlinear modeling were conducted on two
of the existing experimental evidences as introduced
in the chapter2. One was under uniaxial
compression only, and the other was under the
compression with lateral confinement pressure,
so-called triaxial compression.
First, the results for the former under uniaxial
compression are shown in Fig.14, in which the
values of the material parameters in Table 4 were
used. Because the parameters were adjusted for
another concrete material from Ref.2), there were
naturally some differences between the
experimental and numerical curves. Thus, another
simulation have been carried out, modifYing the set
of the parameters as 771 = 155, FVppeak=O.02 and
Y2 = 2.41. As the modified results, better
agreements could be obtained as shown in Fig.15,
although slight differences on the peak stresses were
hold.
0.0 0
3)
0', (MPa) Exp. Numer.
8000
o
Fig.15 Stress-strain relations using
modified parameter values
o
10
30
o
pressure.
Consequently, the numerical modeling discussed
here could be effective to evaluate the post-peak
behavior of compressive concrete with lateral
Next, the results for the latter under triaxial
compression are shown in Fig.16. Comparing the
experimental and numerical results, a satisfactory
agreement of the peak stress point could be obtained,
although a slight leftward sift of the point appeared
for the case with the highest lateral pressure.
Furthermore, as to the post-peak branch, the numerical
results could trace well along the corresponding
experimental ones, except for the case of no lateral
confinement stress. o 5000 10000 1500005train (p)
20000
Fig.16 stress-strain relations with various
lateral confinement stresses
- 43-
5. Concluding Remarks
from the parametric studies on compressive concrete as a strain softening material, the followings are
concluded.
(I) Essential material parameters for the load function were obtained.
(2) Satisfactory solutions have been obtained for the existing experimental results.
(3) Confinement effect can be also evaluated including post-peak region.
6. References
I) Mizino and Hatanaka: Compressive softening model for concrete, Journal of Engineering Mechanics, ASCE,
Vol. I 18, No.8, pp.1546-1563, (1992)
2) P. R. Barnard: Researches into the complete stress-strain curve for concrete, Magazine of Concrete Research,
Vo1.l6, No.49, pp.203-21 0, (1964)
3) Hatanaka, S., Kosaka, Y and Tanigawa, Y; Plastic deformational behavior of axially loaded concrete under
low lateral pressure -An evaluation method for compressive toughness of laterally confined concretes (part I),
Transactions ofAU, No.377, pp.27-40, (1987)
- 44-


Numerical Study on Compressive Concrete as a Strain Softening Material

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Numerical Study on Compressive Concrete as a Strain Softening Material

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