Modeling and influence of shear retention parameter on the response of reinforced concrete structural elements
To obtain the complete solutions describing the balance of a reinforced concrete structure, it is necessary to introduce a behavioral law characterizing the physical properties of material. The goal of this work is to study the response of reinforced concrete elements by taking into account th...
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
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| Cite this: | Modeling and influence of shear retention parameter on the response of reinforced concrete structural elements / L. Dahmani, A. Khennane, S. Kaci // Проблемы прочности. — 2009. — № 4. — С. 108-116. — Бібліогр.: 9 назв. — англ. |
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| citation_txt | Modeling and influence of shear retention parameter on the response of reinforced concrete structural elements / L. Dahmani, A. Khennane, S. Kaci // Проблемы прочности. — 2009. — № 4. — С. 108-116. — Бібліогр.: 9 назв. — англ. |
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| description | To obtain the complete solutions describing the
balance of a reinforced concrete structure, it is
necessary to introduce a behavioral law characterizing
the physical properties of material. The
goal of this work is to study the response of reinforced
concrete elements by taking into account
the variation of the shear retention parameter (aggregate
interlock) and the mesh density. The concrete
was assumed as elastic-plastic material and
follow Drucker-Prager failure criterion with associated
flow rule, the steel reinforcements were
assumed to be elastic-perfectly plastic. The numerical
results obtained are compared with other
results available in the literature.
Для отримання повних розв’язків із метою опису рівноваги залізобетонної
конструкції необхідно використати рівняння, що характеризують фізичні
властивості матеріалів. Вивчалася поведінка залізобетонних елементів з урахуванням
зміни ретенційного параметра зсуву (множинного блокування) і
щільності сітки. Постулювалось, що бетон являє собою пружно-пластичний
матеріал, для якого правдиві критерій руйнування Друкера-Прагера та закон
асоційованої течії, в той час як стальні елементи арматури припускалися
пружно-ідеально-пластичними. Отримані результати числових розрахунків
порівнювалися з наведеними у літературних джерелах.
Для получения полных решений с целью описания равновесия железобетонной конструкции
необходимо использовать уравнения, характеризующие физические свойства материала.
Изучено поведение железобетонных элементов с учетом изменения ретенционного параметра
сдвига (множественной блокировки) и плотности сетки. Постулировалось, что бетон
является упругопластическим материалом, для которого справедливы критерий разрушения
Друкера-Прагера и закон ассоциированного течения, тогда как стальные элементы арматуры
предполагались упруго-идеально-пластичными. Полученные результаты численных расчетов
сравнивались с приведенными в литературных источниках.
|
| first_indexed | 2025-12-07T16:19:53Z |
| format | Article |
| fulltext |
UDC 539.4
Modeling and Influence of Shear Retention Parameter on the Response
of Reinforced Concrete Structural Elements
L. D ahm ani,a A . K hen n an e,b and S. K acia
a Mouloud Mammeri University, Tizi-Ouzou, Algeria
b University of Southern Queensland, Toowoomba, Australia
УДК 539.4
Моделирование и оценка влияния ретенционного параметра
сдвига на поведение конструкционных элементов из железобетона
Л . Д ахм ан и а, А . Х ен н ан е6, С. К аци а
а Университет им. Мулуда Маммери, Тизи-Узу, Алжир
6 Университет Южного Квинсленда, Тувумба, Австралия
Для получения полных решений с целью описания равновесия железобетонной конструкции
необходимо использовать уравнения, характеризующие физические свойства материала.
Изучено поведение железобетонных элементов с учетом изменения ретенционного пара
метра сдвига (множественной блокировки) и плотности сетки. Постулировалось, что бетон
является упругопластическим материалом, для которого справедливы критерий разрушения
Друкера-Прагера и закон ассоциированного течения, тогда как стальные элементы арма
туры предполагались упруго-идеально-пластичными. Полученные результаты численных рас
четов сравнивались с приведенными в литературных источниках.
К л ю ч е в ы е с л о в а : моделирование, армированный бетон, растрескивание, ретен-
ционный параметр сдвига.
Introduction . The intact concrete is supposed to be isotropic and linear
elastic, w hile the Rankine criterion is used to detect the crack initiation. In an
initially intact integration point, the principal stresses and their directions are
evaluated. I f the maximum principal stress exceeds the tensile strength, a crack
appears in the plane perpendicular to the direction of this stress and the concrete
becom es anisotropic [1]. The shearing retention parameter characterizes shear
behavior o f an elem ent o f volum e o f cracked reinforced concrete. The reduction
factor i o f the initial rigidity modulus G is used, in order to take into account a
certain redistribution o f the shear stress in the cracking plane (agregate interlock)
[2]. The best fit values to be adopted depend on the type o f the problem, but the
best fit results are obtained i f the parameter i values are within the range from 0.3
to 0.5 [2]. In the reinforcement zone, the dow el action can be superimposed with
the effect o f aggregate interlock, thus conferring a certain rigidity o f additional
shearing on the crack. The shear retention parameter also depends on the crack
opening, and the assessm ent o f its evolution during the loading process becom es
com plicated i f the latter is alternate. In the version o f the m odel used in this work,
© L. DAHMANI, A. KHENNANE, S. KACI, 2009
108 ISSN 0556-171X. Проблемы прочности, 2009, № 4
Modeling and Influence o f Shear Retention Parameter
various values o f i were adopted in order to suitably simulate the reinforced
concrete structures [3].
C om pression M odel Behavior. A linear elastic m odel is used for the
reversible part o f the strain and an approach based on plasticity w ith isotropic work
hardening is em ployed for the irreversible part o f the strain. The total strain rate is
thus broken up into an elastic strain rate d e e and a plastic strain rate de C
associated to the com pression [4, 5],
de = de e + de p ( 1)
or
= D ede e = D e ( d e - de p ), (2)
where D elastic matrix defined by H ooke’s law.
The m odel requires the definition o f a load surface w hich characterizes the
plastic criterion, plastic flow rule, work-hardening rule, and collapse condition.
The load function for the concrete under a biaxial stress state is generally supposed
to depend on two invariants o f the stress tensor. A load function o f the Drucker
Prager type, w hich depends on the first invariant o f the stress tensor 1 1 and the
second invariant o f the deviatoric stress tensor J 2 , was thus adopted [4]:
C J J 2 + D I ,
F = ^ J J K V - ' = a- (3)
The experiment has shown that dependence o f the load function in 1 1 and J 2
gives quite satisfactory results and, moreover, such form sim plifies calculation.
The constants C and D are given as follows:
V3(2n — 1)C = ----- )_, (4 )
U
u — 1
D = V - e>
f b
f * = f ’ (6)
where f c is com pressive strength and f bc is biaxial com pressive strength.
For u w e adopt the value o f 1.16, according to the experimental results o f
Kupfer et al. [6 , 7].
Traction M odel Behavior. The two-dim ensional behavior o f the concrete is
based on the Rankine criterion for traction [8 ]. The evolution o f cracking state is
taken into account by setting to zero the elastic modulus according to the cracked
direction and by the redistribution o f the corresponding stresses. The use o f a
ISSN 0556-171X. npoôëeMbi npounocmu, 2009, N9 4 109
L. Dahmani, A. Khennane, and S. Kaci
shearing parameter function o f the crack opening in the elasticity matrix o f the
cracked elem ent makes it possible to simulate the aggregates interlock. Bonds
between the concrete and the steel reinforcement are considered to be perfect. The
cracked concrete is treated like an orthotropic material, w hose orthotropic axes are
parallel and normal to the crack direction [9]. The Poisson ratio effect is negligible
because o f the lack o f interaction between the two orthogonal directions after
cracking, and the elastic modulus o f the concrete normal to the crack direction is
reduced to zero.
The total stresses after cracking are given in respect to the axes o f local
coordinates (n , t) by
a n Eb 0 0 £ n £ n
a t = 0 0 0 ■ « t = [D c ]■ £ t
X nt 0 0 bG_ y nt y nt.
(7)
where E b is elastic modulus o f the concrete, ft is shear retention parameter o f the
concrete (0 < f t < 1), G is shear modulus o f the concrete, and [D c ] is elastic matrix
o f the cracked elem ent in the local coordinates (n , t).
The shear modulus is reduced by shear retention parameter w hich lies
between 0 and 1. In various applications, the value o f ft is taken equal to 0, when
the crack is open, and equal to 1, when the crack is close. This im plies that there is
no agregate interlock when the crack is open and a perfect healing w hen the crack
is closed. In order to transform the concrete stresses from local coordinate to global
coordinate system (Fig. 1), the follow ing procedure is used
[D ] = [P ]T [ D c ] [P ],
where
[P ] =
c s cs
2 2s c —cs
- 2 c s 2 cs c 2 — s 2
(8)
(9)
where
c = cos ̂ , s = sin
[P ] is transformation matrix, [D ] is elastic matrix after cracking in the global
coordinate ( X , Y), and ^ is angle betw een the crack direction and O X axis.
The residual stress vector after cracking is given by the follow ing relation:
{a 0 } = [I ]-
2 2 c s
c 3 s
2 2 c s
cs~
2 c 3 s
2 cs 3
2c 2 s 2
a
' xy)
(10)
where {a 0 } is stress vector adjusted after cracking and [I ] is identity matrix o f
order 3 x 3.
110 ISSN 0556-171X. npo6n.eMH npounocmu, 2009, N 4
Modeling and Influence o f Shear Retention Parameter
Fig. 1. Representation of smeared crack.
The incremental relation (stress vs strain) in the cracked concrete is given as
follows:
[ d a } = [D c ]{d£}. (11)
The total released stresses after cracking w ill be distributed in the adjacent
elements (Fig. 2). The total variation o f the stresses w ill be te following:
{Aa } = {da} - { a o } = [D c ] { d £ } ~ { a o } (12)
where [D c ] is elastic matrix o f the cracked concrete and { o 0 } indicates released
stresses after cracking o f the concrete.
Fig. 2. Stress-strain model of the cracked concrete.
N on linear C alcu lation Procedure. The calculation stages are the following:
(i) introduction o f the necessary data for a m esh generation;
(ii) introduction o f the boundary conditions;
(iii) m esh generation by taking into account the longitudinal and vertical
reinforcements;
(iv) applying a load increment A f) ;
(v) start o f the iterative (Newton-R aphson) procedure;
(vi) evaluation o f the residual stress vector {o 0 } and the residual force vector
{ f o};
ISSN 0556-171X. npoôëeMbi npounocmu, 2009, N 4 111
L. Dahmani, A. Khennane, and S. Kaci
(vii) calculation o f the norm o f the residual force vector {f 0 }:
1) i f the norm o f { f 0 } is less than the tolerance level, convergence is
checked. I f the final loading is not reached, apply a new load increment, and repeat
the procedure from the step (4);
2) i f the norm o f { f 0 } is greater than the tolerance leel, convergence is not
checked and i f the iteration lim it is not reached, repeat the procedure from the step
(5);
3) i f the m aximum iteration is reached, w hich corresponds to the horizontal
stage o f the stress-strain curve, and then the load thus found corresponds to the
ultimate load;
(viii) posting o f the results.
V alidation and N um erica l A pplication E xam ples. The m odel developed for
the plane stress calculation o f the reinforced concrete structural elements by the
finite elem ent m ethod is applied to the study o f reinforced concrete panel [9] and a
beam [8], where dimensions, reinforcement, and the loading are given in Tables 1
and 2 and illustrated in Figs. 3 and 7, respectively. For reasons o f symmetry, only
h alf o f the panel and the beam are m odeled. Figures 4, 5, 8, and 9 present the
response o f the elements (panel and beam) in term o f the diagram relating the
midspan vertical displacements with the applied forces and compare the numerical
results with the experimental results. The general pace o f the numerical results
agrees rather w ell with the experiment. The refinement o f the mesh, as awaited,
leads to a more flexible response which approaches the experiment w ell. In order
to appreciate the influence o f the shear parameter y3, a simulation with various
values was carried out. The load-displacem ent curves obtained are represented (see
Figs. 4, 5, 8, and 9) at the same time as the experimental curve. It is noted that the
ultimate load is better approached by the values o f between 0.3 and 0.5.
T a b l e 1 T a b l e 2
Reinforced Concrete Panel [9] Reinforced Concrete Beam [8]
Mechanical property Concrete Steel
E (MPa) 30,000 207,000
V 0.2 0.3
Fc (MPa) 56 -
Ft (MPa) 6 -
Fy (MPa) - 320
Mechanical property Concrete Steel
E (MPa) 20,400 192,000
V 0.2 0.3
Fc (MPa) 26.7 -
Ft (MPa) 3.4 -
Fy (MPa) - 360
In order to study the influence o f the m esh refinement on the finite element
solution, test simulations were carried out with two different meshes: The simulation
results are also presented. Figures 6 and 10 show the midspan displacements o f the
elements for a load o f 90 kN for values o f shear parameter ranging from 0 to
1.0. It is noted that for the value o f ranging between 0.3 and 0.5, displacements
are almost constant and agree w ell w ith the experimental value o f 1.6235 mm for
the panel and 3.7125 mm for the beam. So one can conclude that the best choice o f
the shear parameter considerably affects the response o f the reinforced concrete
structural elements.
112 ISSN 0556-171X. npo6n.eMH npounocmu, 2009, N 4
Modeling and Influence o f Shear Retention Parameter
Fig. 3. Geometry, loading, and panel reinforcements. (Dimensions in cm.)
Displacement {mm)
Fig. 4
Displacement (mm)
Fig. 5
Fig. 4. Load vs displacement at panel midspan for the different values of 5 (panel of 20 elements).
Fig. 5. Load vs displacement at panel midspan for the different values of 5 (panel of 77 elements).
Influence o f M esh D ensity. In finite elem ent m odeling, a finer m esh typically
results in a more accurate solution. However, as a m esh is made finer, the
computation time increases. In order to get a m esh that provides a satisfactory
balance between accuracy and computing resources, one w ay is to perform a m esh
convergence study as follow s:
1. Create a m esh using the fewest, reasonable number o f elements and analyze
the model.
2. Recreate the m esh with a denser elem ent distribution, re-analyze it and
compare the results to those o f the previous mesh.
3. Keep increasing the m esh density and reanalyzing the m odel until the
results converge satisfactorily.
ISSN 0556-171X. npodxeMbi npounocmu, 2009, N 4 113
L. Dahmani, A. Khennane, and S. Kaci
Shear parameter p
Fig. 6. Displacement at panel midspan for the different values of 5 .
Fig. 7. Geometry, loading, and beam reinforcements. (Dimensions in cm.)
Displacement (mm)
Fig. 8
Displacement (
Fig. 9
Fig. 8. Load vs displacement at the beam midspan for the different values of 5 (panel of 28 elements).
Fig. 9. Load vs displacement at the beam midspan for the different values of 5 (panel of 60 elements).
114 ISSN 0556-171X. npo6n.eMH npounocmu, 2009, N 4
Modeling and Influence o f Shear Retention Parameter
Shear parameter |
Fig. 10. Displacement at beam midspan for the different values of 5 .
This type o f m esh convergence study provides an accurate solution with a
m esh that is sufficiently dense and not overly demanding o f computing resources.
Figures 11 and 12 describe the midspan displacement versus the number o f
elements for the reinforced concrete panel and beam, respectively. The results
converge towards the objective value for a grid o f 50 elements.
Fig. 11 Fig. 12
Fig. 11. Midspan displacement vs number of elements reinforced concrete panel.
Fig. 12. Midspan displacement vs number of elements reinforced concrete beam.
This can be explained by the fact that the isoparametric quadrilateral element
used for m odeling, reconstitute correctly the deformation required by the beam
theory in general.
C onclusions. We present a numerical calculation m odel for evaluation o f the
response o f the reinforced concrete elements under the action o f static loads in the
elastoplastic field. The results o f digital simulation agree w ell w ith the
experimental results. It is also noted that the best choice o f a shear retention
ISSN 0556-171X. npo6n.eMH npounocmu, 2009, N 4 115
L. Dahmani, A. Khennane, and S. Kaci
parameter play a significant role in the total response of the structure. The value of
fl ranging between 0.3 and 0.5 (Fig. 6 and 10) give a better result. The shear
parameter used in this model is constant and ranges between 0 and 1.0.
For the future research an improvement can be obtained by introducing a
shear retention parameter fl, which varies in function of the opening and the
closing of the crack, into the model.
Р е з ю м е
Для отримання повних розв’язків із метою опису рівноваги залізобетонної
конструкції необхідно використати рівняння, що характеризують фізичні
властивості матеріалів. Вивчалася поведінка залізобетонних елементів з ура
хуванням зміни ретенційного параметра зсуву (множинного блокування) і
щільності сітки. Постулювалось, що бетон являє собою пружно-пластичний
матеріал, для якого правдиві критерій руйнування Друкера-Прагера та закон
асоційованої течії, в той час як стальні елементи арматури припускалися
пружно-ідеально-пластичними. Отримані результати числових розрахунків
порівнювалися з наведеними у літературних джерелах.
1. Z. P. Bazant and S. S. Kim, “Plastic fracturing theory of concrete,” in: J. Eng.
M ech. D iv . A S C E , 105, No. EM3, June, Proc. Paper 14653, pp. 407-428.
2. W. F. Chen and D. J. Han, P la s tic ity f o r S tru c tu ra l E n g in eers, Springer
Verlag, New York (1988).
3. K. Mourad, A n a ly se p a r É lém en ts F in is d e s P an n eau x en B éton A rm é, Thèse
de Magister, Département de Génie Civil, Université Mouloud Mammeri de
Tizi-Ouzou (2000).
4. D. C. Drucker and W. Prager, “Soil mechanics and plastic analysis gold limit
design,” Q uart. A ppl. M a th , 10, 157-165 (1952).
5. Y. R. Rashid, “Analysis of prestressed concrete nuclear reactor structures,” in:
Unpublished notes presented at Conference on Prestressed Concrete Nuclear
Reactor Structures, University of California, Berkley (1968).
6. H. B. Kupfer and K. H. Gerstle, “Behaviour of concrete under biaxial
stresses,” N e w sp a p e r Eng. M ech . D iv . (ASCE), 99, No. 4, 853-866 (1973).
7. H. Kupfer, H. K Hilsdorf, and H. Rusch, “Behaviour of concrete under biaxial
stresses,” Int. N ew sp a p er , Aug., 656-666 (1969).
8. D. Ngo, and A. C. Scordelis, “Finite element analysis of reinforced concrete
beams,” N e w sp a p e r A m er. C o n cre te Inst., 54, No. 3, 152-163 (1967).
9. V. Cervenka and R. Pulk, “Computer models of concrete structures,” Struct.
Int. Eng. (IABSI), 2, 103-107 (1992).
Received 12. 09. 2007
116 ISSN 0556-171X. Проблеми прочности, 2009, № 4
|
| id | nasplib_isofts_kiev_ua-123456789-48413 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0556-171X |
| language | English |
| last_indexed | 2025-12-07T16:19:53Z |
| publishDate | 2009 |
| publisher | Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| record_format | dspace |
| spelling | Dahmani, L. Khennane, A. Kaci, S. 2013-08-19T12:49:45Z 2013-08-19T12:49:45Z 2009 Modeling and influence of shear retention parameter on the response of reinforced concrete structural elements / L. Dahmani, A. Khennane, S. Kaci // Проблемы прочности. — 2009. — № 4. — С. 108-116. — Бібліогр.: 9 назв. — англ. 0556-171X https://nasplib.isofts.kiev.ua/handle/123456789/48413 539.4 To obtain the complete solutions describing the balance of a reinforced concrete structure, it is necessary to introduce a behavioral law characterizing the physical properties of material. The goal of this work is to study the response of reinforced concrete elements by taking into account the variation of the shear retention parameter (aggregate interlock) and the mesh density. The concrete was assumed as elastic-plastic material and follow Drucker-Prager failure criterion with associated flow rule, the steel reinforcements were assumed to be elastic-perfectly plastic. The numerical results obtained are compared with other results available in the literature. Для отримання повних розв’язків із метою опису рівноваги залізобетонної конструкції необхідно використати рівняння, що характеризують фізичні властивості матеріалів. Вивчалася поведінка залізобетонних елементів з урахуванням зміни ретенційного параметра зсуву (множинного блокування) і щільності сітки. Постулювалось, що бетон являє собою пружно-пластичний матеріал, для якого правдиві критерій руйнування Друкера-Прагера та закон асоційованої течії, в той час як стальні елементи арматури припускалися пружно-ідеально-пластичними. Отримані результати числових розрахунків порівнювалися з наведеними у літературних джерелах. Для получения полных решений с целью описания равновесия железобетонной конструкции необходимо использовать уравнения, характеризующие физические свойства материала. Изучено поведение железобетонных элементов с учетом изменения ретенционного параметра сдвига (множественной блокировки) и плотности сетки. Постулировалось, что бетон является упругопластическим материалом, для которого справедливы критерий разрушения Друкера-Прагера и закон ассоциированного течения, тогда как стальные элементы арматуры предполагались упруго-идеально-пластичными. Полученные результаты численных расчетов сравнивались с приведенными в литературных источниках. en Інститут проблем міцності ім. Г.С. Писаренко НАН України Проблемы прочности Научно-технический раздел Modeling and influence of shear retention parameter on the response of reinforced concrete structural elements Моделирование и оценка влияния ретенционного параметра сдвига на поведение конструкционных элементов из железобетона Article published earlier |
| spellingShingle | Modeling and influence of shear retention parameter on the response of reinforced concrete structural elements Dahmani, L. Khennane, A. Kaci, S. Научно-технический раздел |
| title | Modeling and influence of shear retention parameter on the response of reinforced concrete structural elements |
| title_alt | Моделирование и оценка влияния ретенционного параметра сдвига на поведение конструкционных элементов из железобетона |
| title_full | Modeling and influence of shear retention parameter on the response of reinforced concrete structural elements |
| title_fullStr | Modeling and influence of shear retention parameter on the response of reinforced concrete structural elements |
| title_full_unstemmed | Modeling and influence of shear retention parameter on the response of reinforced concrete structural elements |
| title_short | Modeling and influence of shear retention parameter on the response of reinforced concrete structural elements |
| title_sort | modeling and influence of shear retention parameter on the response of reinforced concrete structural elements |
| topic | Научно-технический раздел |
| topic_facet | Научно-технический раздел |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/48413 |
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