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STATIC ANALYSIS OF R
EINFORCED CONCRETE W
ALLS WITH
RESPECT OF NON
-
LINEAR MATERIAL BEHA
VIOUR


Brožovský J. jr. (Brno, Czech Republic)



Introduction


To obtain precise results of static analysis of reinforced concrete buildings, real material properties
(co
ncrete, steel) must be respected. Behaviour of real reinforced concrete is non
-
linear and depends
on many factors, but the most important is load level. Exact description of reinforced concrete
behaviour is impossible, so simplified material models must be

used.


Material model (ba
sed on works of Bažant and Červenka) has been prepared. Problem is solved as
two
-
dimensional (plane stress), using finite element method. Material characteristics are obtained
from equvivalent uniaxial load diagram. For better results, uniaxial diagram par
ameters are
computed from Kupfer's biaxial criteria for concrete. Step
-
based solution (arc
-
length method) is
used. A computer program is developed for practical usage of described method.



1. Finite elements


Four finite element types are used: three
-
node

plane element [6] and family of isoparametric plane
elements (four, eight and nine
-
node elements) [7]. All elements have two parameters (x and y
translations) in each node. Three
-
node element is very simple, isoparametric elements can give
better results.

Reinforcement can be modelled with link elements.



2. Solution of linear systems


The preconditioned Bi
-
Conjugate Gradient Stabilized Method (BiCGS) [8] is used for solution of
linear systems. Typical system of linear equations in Finite Element Method

(FEM) looks like:








F
r
K

.
,


where
[K]

is stiffness matrix,
(r)
is dispacement vector and
(F)
is load vector. Normally
[K]

is
symmetric and positive definite and many effective methods (Gauss Elimination Method, Conjugate
Gradient Method
etc) can be used for solution. But in some cases (if using Arc
-
Lenght Method for
example)
[K]

can be nonsymmetric. BiCGS was developed for systems with nonsymmetric
[K]

matrix, so can be used.



2. Solution of non
-
linear equations


Constitutive equations
are nonlinear, so computation of system of nonlinear equations is needed.
Solution of large nonlinear systems if difficult. But solution of nonlinear system can be tranformed
to iterative solution. In this case we need to solve only linear systems. Widely
used is Newton
-
Rapshon method. In this method, solution is controlled by load multiplier λ, given as input. But
better results can be obtained if λ is computed during iteration process, as is if the Arc
-
Lenght
Method (ALM) [8]. If using ALM we have to solv
e linear system like this in each iteration:





























a
g
r
F
F
r
F
K
T
T
)
(
)
(
.
)
)(
(
2
)
(
2
]
[
2
|





It means, we need to solve linear system with nonsymetric matrix. There are some variants of ALM
(Spheric ALM, Cyllindric ALM, Linearized ALM). In those variants only solution of linear
systems
with symmetric metrices are needed. But those methods are more complicated and their
convergence may be poor. We prefer usage of unmodified ALM and solution of nonsymmetric
linear system with BiCGS method.



3. Material model


Developed material m
odel of reinforced concrete is based of works of Bažant [4] and Červenka [5].
Model is developed for plane stress only. Smeared crack concept is used. "Cracks" are modelled by
modification of constitutive equations (e.g. only material stifness matrix is c
hanged).


Reinforcement can be computed as discrete (using line elements) or smeared (if added to material
stifness matrix of concrete). In both alternatives, reinforcement behaviour is ideally elasto
-
plastic.


Concrete behaviour is initially linear isotro
pic. Material stifness matrix is:















)
1
(
5
,
0
0
0
0
1
0
1
1
]
[
2




E
D


"Cracked concrete" can be viewed as ortothropic material with this material stifness matrix:















d
C
E
E
E
E
d
D
cr
0
0
0
0
]
[
1
1
1
1




To obtain material model characterictics, computed stresses and strains are
transformed to
equivalent stress
-
strain relation (similar to Červenka in [5]). In tension, concrete softening is
predicted. But concrete behaviour in compression is predicted as elasto
-
plastic (for simplicity).
Parameters of equivalent relation are compu
ted from Kupfer biaxial criteria for concrete.



4. Size effect influence


If described material model is used, results are dependent on size of finite elements. Minimal unit of

structure is finite element, so size of "cracked" region depends of size of

"cracked" element. This is
no problem for elasto
-
plastic model, but this is a big problem if model with softening is used (as is
in tension). Results from two analyses of the some structure, but with different element size can be
absolute different. To av
oid this problem fracture mechanics can be used. Fracture energy in
softening must be constant for any piece of material:


.
.
const
L
A
G
f



diagram.jpg













Fig.1 Equivalent stress
-
strain realtion for concrete



Where
G
f

is fracture energy, A

is area under softening line if equvalent stress
-
strain relation and
L

is
width of finite element in direction perpendicular to "cracks" (width of "crack band"). Because
element size can be changed, equivalent relation must be corrected to get constant
G
f
. This method
is sometimes named as "crack band model" and was initially developed by Bažant.


References


[1] Brožovský, J.: Modelování fyzikálně nelineárního chování železobetonových konstrukcí,
pojednání o tématu disertační práce, FAST VUT, Brno, 2001.

[2] Brožovský, J.: Fyzikálně nelineární modelování stěnových železobetonových konstrukcí, In
sborník semináře Problémy lomové mechaniky, FAST VUT, Brno, 2001.

[3] Brožovský, J.: Některé aspekty návrhu programu pro analýzu stavebních konstrukcí metodou
konečných prvků, In sborník semináře Problémy modelování, FAST VŠB
-
TUO, Ostrava, 2002,
ISBN

80
-
214
-
2017
-
0.

[4] Bažant, Z. P., Planas, J.: Fracture and Size Effect in Concrete and Other Quassibrittle Materials,
CRC Press, Boca Raton 1998

[5] Červenka, V.:
Constitutive Model for Cracked Reinforced Concrete, ACI Journal, Titl.82
-
82,
1985

[6] Kolář V., Kratochvíl, Leitner, Ženíšek: Výpočet plošných a prostorových konstrukcí metodou
konečných prvků, Praha, 1979.

[7] Servít, R., Drahoňovský, Z., Šejnoha, J.,
Kufner, V.: Teorie pružnosti a plasticity II., SNTL,
Praha, 1984

[8] Wempner, G. A.: Discrete aproximationrelated to nonlinear theories of solids, Intl. Journal of
Solids ans Structs., 7, pp. 1581
-
1599, 1971

[9] http://www.netlib.org/templates



Author

:

Brožovský Jiří,
VŠB
-
Technical University Ostrava, Faculty of civil engineering, Departement of structures, Czech
Republic, assistant, VŠB
-
Technická univerzita Ostrava, Fakulta stavební, Katedra konstrukcí, Ludvíka Podéště 1875,
708 33 Ostrava
-
Poruba, +
420

69

7321321, +420

69

7321358, jiri.brozovsky@vsb.cz.