Plastic-Damage Macro-Model For Non-Linear Masonry Structures Subjected To Cyclic or Dynamic Loads
Plastic-Damage Macro-Model For Non-Linear Masonry Structures Subjected To Cyclic or Dynamic Loads
Plastic-Damage Macro-Model For Non-Linear Masonry Structures Subjected To Cyclic or Dynamic Loads
Summary
In the paper the plastic–damage material model for concrete (based on continuum
damage mechanics and the theory of elasto-plasticity), proposed by J. Lubliner et. al. (1988)
and modified by L. Fenves et. al. (1998), is adapted to masonry structures. A manner of
determining model parameters determination for homogeneous masonry structures is
explained. Some results of the numerical analysis of a prismatic specimen subjected to cyclic
loading are presented, which show a good compatibility with laboratory tests (Singha et. al.).
Additionally, a seismic simulation of the response (time history analysis) of a masonry
structure to tremors is submitted.
Keywords: dynamics of structures, masonry constitutive macro-model, non-local damage
mechanics, theory of plasticity, time history analysis.
1. Introduction
The task of modelling masonry structures belongs to relatively complex numerical
problems. The mechanical behaviour of masonry depends on the composite nature of the
material, constituted by the fine dimensions, of natural or clay blocks connected by cement or
cement-lime mortar joints. Both these mentioned materials differ from each other, among
others: in the compression modulus, tensile (compressive, shear) strength. The emergency
state of a masonry wall is manifested by crack expansions mainly near the joints.
The complex mechanical behaviour of masonry structures requires so-called two-
material micro-models with discontinuous discrete elements (modelling the blocks) and
interface elements (modelling the mortar joints). However this type of these models (see eg.
[5]) have a restricted range of application, limited to constructions which are not too large and
whose geometry is rather simple. In numerical analyses, discontinuities of displacement fields
are taken into account in FEM by interface elements. Also some new methods, based on the
discrete element method (DEM) [6], is becoming more and more popular.
In the case of modelling geometrically complicated structures significant computational
complications leads to the formulation of a simplified description of the mechanical properties
of masonry structures, using a macro–model. The macro-modelling approach makes no
distinctions between blocks and joints, e.g. masonry material is treated as a fictitious
homogeneous and continuous equivalent material, the mechanical parameters of which can be
determined on the basis of laboratory tests.
3
3
σ1
1
1 2
(2)
2
σt0 σt0 biaxial
tension
3 (4)
1 3
2
1
2
(3) σc0
(σb0 ,σb0)
biaxial uniaxial
compression compression
Figure 1 presents the shape of the yield surface of the BM in the plane stress space that
consists of four functions (1):
− the 1st quarter of the co-ordinate system: one quarter of the circle with a radius equal to
uniaxial tensile strength σt 0 :
− the 2nd quarter:
1
f (σ , κ ) ≡ (q − 3α p + β σ 1 ) − σ c (κ ) = 0 (1’)
1−α
− the 3rd quarter:
1
f (σ , κ ) ≡ ( q − 3α p ) − σ c (κ ) = 0 (1’’)
1−α
− and the 4th one:
1
f (σ , κ ) ≡ (q − 3α p + β σ 2 ) − σ c (κ ) = 0
1−α
where α i β are dimensionless parameters of the model – eqs. (2) - which are expressed by the
following yield stresses: for uniaxial compression σ c 0 = σ c (0) , for biaxial compression σb 0 and
for the tension (the same one for uni- and biaxial acting) σ t 0 = σ t (0) . The denotations p and q
are stress tensor invariants. The overline (e.q. q ) denotes the effective value which depends
on the actual level of material degradation.
σ b0 − σ c0 σ c (κ )
α= , β (κ ) = (1 − α ) − (1 + α ) (2)
2σ b 0 − σ c 0 σ t (κ )
a) b)
σt
1D
σc stress 1D
σ cu ultim ate
stress
hardening
σ t0 Failure
stress
stress softening
tension stiffening
σ c0 plasticity
d c′ , d c′′ → d c (ε%cp ) curve
lim it
E0
E0 (1- d c′ ) E 0
(1- d t′) E0 d t′, d t′′→ d t (ε%tp )
(1 - d c′′) E 0
(1- d t′′) E0
ε%tp ε%td ε%te ε%t
ε%cp ε%cd ε%ce
ε%c ε%tin
ε%cin
σcu
0.25 0.25
σ2 σ 2 α = 22.5 ο
σ1
0.50 σ1 0.50
σ1 / fmx
σ1 σ1
σ2 σ2
0.75 0.75
σ1 / fmx
σbu
1.0 1.0
1.25 1.25
A A
σ2 / fmx
c)
1.0 0.75 0.50 0.25 Biaxial compressive strength of masonry
0
Ganz, Thurlimann (1982), Page (1981),
Lurati, Graf (1989)
0.25
α = 45ο Soild clay brick masonry
σ σ ( fmx= 8.0 MPa )
2
0.50
σ1 / fmx
proposal
0.75
for σcu = 0.50 fmx
1.0 fmx
1.25
Fig. 3. Proposition of the biaxial strength envelope for masonry (dashed line)
The envelope of the elastic states for biaxial compression of masonry (yield surface)
was evaluated on the basis of experimental results obtained by Naraine & Singha [12, 13].
There were assumed the following values of uniaxial yield stresses has been assumed: for
compression σco = 0.67 σcu = 0.34 fmx , and for tension σtu =σt0 = 0.03 fmx
The hardening rule and degradation functions are determined for uniaxial material
loading. Their specification can be found on the basis of the results of the cyclic compression
of the masonry wall obtained by Naraine & Singla w [11] – Fig. 4a (full line). Real hysteresis
loops of loading should be changed by straight lines (dashed straight lines on Fig. 4a). It was
assumed that the compressive yield stress is equal to σco = 2.25 MPa for an elastic strain equal
ε ce = 0.1 percent.
The hardening function for compression was expressed – according to an
implementation of the BM in the finite element software ABAQUS – by non-elastic strains
ε cin = ε c − ε ce (see fig. 2). The function of material degradation dc was determined on the basis
of changes of Young’s modulus (different slope of the dashed line in Fig. 4a) during
successive cycles of loading. The discrete values and their approximation are presented in Fig.
4b).
a)
-6.0 b)
dc
0.5
-5.0 0.40
0.4 0.38
0.43
0.35
0.3 0.28
-4.0 0.22 dc
0.2
stress [MPa]
-1.0
0.0
εce = 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0
Strain εc [0/00]
These parameters and functions of the BM are used during the numerical reconstruction of
Naraine’s [12] experiment. Results of this process are presented in Fig. 5. and seem to be
good – the exactness of the reconstruction is very high.
σc 0=2.25
2.0
laboratory test
0.0 ε [0/00]
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0
Fig. 5. Comparison between the laboratory [12] and the numerical results
4. Simulation of seismic tremor in building
The main aim of this simulation is: 1) to test the explicabilities of BM in non-elastic
seismic or paraseismic analyses, 2) comparison between building responses in two cases –
elastic and non-elastic models of wall material. A 3D finite element shell model of two-storey
building was analyzed with brick walls and floors and headers made of concrete (Fig. 6).
Front view g= 20 cm
a) 10
b)
mur
5.6 3 0 cm
20 x 30 cm
40 x 50 cm
6.0
m
4.0
ax
Material code
front wall
concrete
masonry
The assumed values of yield points (σc0 , σt0), ultimate stresses (σcu , σtu) and initial modulus
of elasticity E are as follows
− for concrete: σc0 = 5.33MPa, σcu = 16.0 MPa, σt0 = σtu = 1.60 MPa, E = 27.0 GPa
− for masonry: σc0 = 1.99 MPa, σcu = 3.0 MPa, σt0 = σtu =0.10 MPa, E = 2.21 GPa.
The deadweight of the building is taken into account at the first step of the history
analysis then the influence of the known seismic tremors is added as a kinematical forcing –
second step (step time from 0 to 3.73 sec) – imposed horizontally - on the bottom edge of the
model (see Fig. 6a where ax is a component of acceleration). The course of the real
acceleration of the ground (recorded in the Polish copper mining district) which was used in
simulation is presented in Fig. 7a). To receive a distinct plastic response of the building, this
acceleration is multiplied by a coefficient equal to 2.5. The time history method is used to find
numerical results - 752 integration steps are needed and a total number of iterations amounts
to 776.
The diagram of displacements of one point (No 10 on fig. 6a) from the upper edge of the
building model, concerning plastic-damage and elastic models, are presented in Fig. 7b).
Differences between these two curves result from taking into consideration the dissipation of
the energy caused by material plasticity and its degradation. Additionally, in Fig. 8 the final
pictures of degradation are described by two variables: dt – for tension degradation and d - for
global degradation.
5. Conclusions
The presented BM of masonry can be used for non-linear dynamic or cyclic analyses.
There is a relatively good agreement between numerical simulations and laboratory
experiments (see [1, 3, 9]). One of the main disadvantage of the BM is the assumption
concerning the isotropy of masonry. Nowadays attempts are made ones work to modify the
plane isotropic BM model for orthotropic material.
Results which are received with the application of masonry BM in comparison with
elastic solutions the following features:
− the field of displacement is characterised by a lower instantaneous value of the
displacement amplitude;
− there are no fictitious stress concentration which exceed the ultimate yield stresses for
compression and tension (see Fig. 9);
− local damages of material are clearly described by degradation variables (Fig. 8).
displacement [mm]
3.0 Elastic model
1.0 2.0
acceleration [m/s2]
1.0
0.5
0.0
0.0 -1.0
-2.0
-0.5 -3.0 step time [s]
step time [s] -4.0
-1.0
0 0.5 1 1.5 2 2.5 3 3.5 4 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
0.45
0.11
Increase
0.00 0.00
a) layer n- b) layer n-
(non-realistic values)
Fig. 9. The horizontal tensile stresses result from elastic and plastic-damage models
(for masonry part of model only)
6. Acknowledgements
The financial assistance of the Ministry of Scientific Research and Information
Technology within the grant number 7 T07E 021 28 is gratefully acknowledged herewith.
The numerical calculation were carried out in the Academic Computer Centre
CYFRONET-AGH within the grant number KBN/SGI2800/PŚląska/023/2003.
7. References
[1] CIŃCIO A., Numeryczna analiza dynamicznej odporności niskiej zabudowy na wstrząsy
parasejsmiczne z zastosowaniem przestrzennych modeli wybranych obiektów. PhD
Thesis, Silesian Univ. of Technology., 2004 (in Polish).
[2] CIŃCIO A., WAWRZYNEK A.: Obliczenia dynamiczne budowli z zastosowaniem
plastyczno-degradacyjnego modelu materiału, Proc. 3rd International Conference on
New Trends in Static and Dynamic of Buildings, Bratislava 21-22.10.2004, pp. 293-296
(in Polish).
[3] CIŃCIO A., WAWRZYNEK A.: Zastosowanie plastyczno-degradacyjnego modelu
materiału w obliczeniach dynamicznych budowli, Materiały 50-tej Konferencji
Naukowej KILiW PAN i KN PZITB, Krynica 12-17 09.2004, t.II, pp. 29-36 (in Polish)
[4] FENVES L., LEE J., A plastic-damage concrete model for earthquake analysis of dams.
Earthquake Eng. and Structural Dynamics, vol. 27, 1998, pp. 937-956.
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via interface models, Computer Method in Applied Mechanics and Engineering
(Elsevier), vol. 190, 2001 pp.6493-6511.
[6] JING L.: Formulation of discontinuous deformation analysis (DDA) an implicit discrete
element model for brick systems, Eng. Geology (Elsevier), vol. 49, 1998, pp. 371-381.
[7] JIRÁSEK M.: Nonlocal models for damage and fracture: Comparison of approaches,
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[8] LEE J., FENVES G.L., Plastic-damage model for cyclic loading of concrete structures,
Journal of Eng. Mechanics, vol 124, No.8, 1998, pp. 892-900.
[9] LUBLINER J., OLIVER J., OLLER S., OÑATE E., A plastic-damage model for
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[10] LUBLINER J., OLIVER J., OLLER S., OÑATE E., Finite element nonlinear analysis of
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