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4. Uluslararası Deprem Mühendisliği ve Sismoloji Konferansı
11-13 Ekim 2017 – ANADOLU ÜNİVERSİTESİ – ESKİŞEHİR
MASİF BETON YAPILARDA HASAR İLERLEMESİ SİMÜLASYONU,
ÖRTÜŞMELİ KAFES MODELİ VE SONLU ELEMANLAR YAKLAŞIMLARI
KARŞILAŞTIRMASI
B. F. Soysal1, Y. Arıcı2, B. Binici3 and K. Tuncay3
1Araş. Gör., İnşaat Müh. Bölümü, Orta Doğu Teknik Üniversitesi, Ankara 2 Doç. Dr., İnşaat Müh. Bölümü, Orta Doğu Teknik Üniversitesi, Ankara
3 Prof. Dr., İnşaat Müh. Bölümü, Orta Doğu Teknik Üniversitesi, Ankara
Email: [email protected]
ÖZET:
Hesaplama kuvvetinin ciddi derecede artması beton ağırlık barajları gibi masif beton yapılar için performansa
dayalı değerlendirme olanağını ortaya çıkarmıştır. Bu değerlendirmeler genelde geleneksel sonlu eleman
yöntemleri ile yapılmaktadır. Bu gibi çalışmalarda öncelik gövdedeki çatlak ilerlemesi ve açılmasının tahmini olup
sonlu eleman yöntemlerinin devamlılık bazlı matematiksel formülasyonları hesaplamalar açısından çok masraflı
ağ yapısı yenileme yöntemleri kullanılmaması durumunda tartışılabilecek sonuçlara yol açmaktadır. Elde edilen
hasar bölgelerinin gerçek ayrık çatlaklara göre çok yaygın olması ve hasarın ileri aşamalarının simüle edilememesi
sonlu eleman yöntemleri ile çalışan araçların ciddi sınırlamaları olarak ortaya çıkmaktadır. Buna karşın, masif
beton yapıların oluşan ayrık çatlakların modellemesinde blokların yapıdan ayrılabilmesi ve salınımını da
gösterebilen yeni bir yaklaşım olarak örtüşmeli kafes modeli büyük potansiyel içermektedir. Bu makalede sonlu
eleman araçlarının masif beton yapıların davranışlarını yansıtma yeterliliği örtüşmeli kafes modeli yaklaşımı ile
karşılaştırılmaktadır. Bu amaçla laboratuvar koşullarında değişik tip yüklemelerle gerçekleştirilen üç değişik test
iki yöntemle de simüle edilmiştir. Yöntemlerin hasar belirlemede ve çatlak ilerlemesini tahmindeki sınırları ve
yeterlilikleri belirlenmiştir.
ANAHTAR KELİMELER: Örtüşmeli kafes modeli, çatlak ilerlemesi, beton ağırlık barajı, laboratuvar testleri,
denektaşı modelleri
SIMULATION OF DAMAGE PROPAGATION ON MASSIVE CONCRETE
STRUCTURES, THE OVERLAPPING LATTICE MODEL VS. THE FINITE
ELEMENT APPROACH
ABSTRACT:
The increase in the computational power enables the performance based assessment of very large structures like
concrete gravity dams. These assessments are usually carried out with the traditional finite element tools. The
continuum based formulation of these analyses in the absence of costly re-meshing operations cast doubt on the
performance assessment of such massive plain concrete structures given the primary output assessed for these
analyses should be the crack propagation and opening on the dam body. Smearing of the cracking in contrast to
the discrete cracks on these systems, and the failure to simulate advanced stages of damage is a significant
limitation for the finite element methods. On the other hand, a new approach in the form of overlapping lattice
model shows a great potential for modeling of discrete cracking on plain concrete structures including the
separation and rocking of individual components on the body. In this paper, the capabilities of the finite element
tools for simulation of large damage on plain concrete structures are compared to the overlapping lattice model
(OLM) approach. A set of three laboratory tests conducted with different types of loading were simulated using
4. Uluslararası Deprem Mühendisliği ve Sismoloji Konferansı
11-13 Ekim 2017 – ANADOLU ÜNİVERSİTESİ – ESKİŞEHİR
both tools. The limitations and capabilities of both approaches in predicting damage in the nature of crack
propagation was established.
KEYWORDS: Overlapping lattice model, crack propagation, concrete gravity dam, laboratory testing,
benchmark models
1. INTRODUCTION
In order to evaluate the older infrastructure for higher design standards and for the design of new dams, the use of
nonlinear tools for the prediction of the performance of concrete structures is quite popular (Guanglun et al., 2000;
Pan et al., 2011; Zhang et al., 2013). The common tool utilized for these analyses is the finite element method.
Smeared crack model, which was first proposed by Rashid (1968), is an important constitutive model for engineers
conducting nonlinear analysis for concrete structures. Main advantages are the ease of formulation &
implementation, and the robust convergence behavior (Rots, 1988). The cracked concrete is analyzed as a
continuum in smeared crack models (Rashid, 1968) despite the heterogeneous nature of the concrete. Various
versions of this model were implemented by researchers specifying the post-peak response, reloading-unloading
behavior and the effect of confinement (Rots, 1988; Hordijk, 1991; Selby and Vecchio, 2013). Smeared crack
models were also used to predict the behavior of concrete dams in a number of studies (Guanglun et al., 2000;
Espandar et al., 2003; Hariri-Ardebili et al., 2016). A review of these studies reveals that the model calibration
was usually conducted to match the observed damage in the Koyna Dam or to match the behavior of a specimen
tested in the laboratory. However, the finite element method has significant limitations regarding the use for
simulation of the behavior of such massive unreinforced plain concrete structures given the discrete nature of
cracking violating the continuum assumptions. Moreover, the performance levels for these systems at the extreme
scenarios require separation, rocking or sliding phenomena which is challenging to simulate in the continuum
finite element setting.
The overlapping lattice model (OLM); on the other hand, is a promising alternative for studying the nonlinear
behavior of the concrete gravity dams under dynamic loading. The overlapping lattice approach employs pin
connected bar elements extending over a predefined horizon to discretize the continuum similar to the concept
used in peridynamics and hence can model the separation, opening of the cracks and crack propagation much
better compared to the conventional finite element approach. OLM considers cracking by employing brittle or
elastic-softening force displacement models (Madenci and Oterkus, 2014; Aydın et al., 2016). Hence OLM has
the potential to model crack initiation, opening and propagation naturally.
In this study, the capabilities of the finite element tools for simulation of large damage on plain concrete structures
are compared to the OLM approach. A set of three laboratory tests conducted with different types of loading were
simulated using both tools. The first benchmark study is modeling the experiment on a scaled dam monolith tested
under lateral pushover type loading (Carpinteri et al., 1992). Then, the lateral pushover loading test performed
after a pseudo dynamic testing on a scaled monolith was simulated (Aldemir et al., 2013). The final benchmark
study used is a well-documented shake table testing of a scaled gravity dam model constructed by Tinawi et al.
(2000). The limitations and capabilities of both approaches in predicting damage in the form of crack propagation
was established.
2. MATERIAL MODELS
2.1. Finite Element Model
The material model employed herein follows the rules described by Selby and Vecchio (2013). The behavior of
concrete is described by elastic isotropic stress-strain relationship prior to cracking. Upon cracking, the material
is treated as orthotropic (Rots, 1988). The tensile, ft, and compressive strengths, fc, and the shape of the post-peak
response are the key characteristics of the stress-strain models. Mesh dependence of such models was addressed
4. Uluslararası Deprem Mühendisliği ve Sismoloji Konferansı
11-13 Ekim 2017 – ANADOLU ÜNİVERSİTESİ – ESKİŞEHİR
by using the fracture energy, Gf, determining the stress-strain response with respect to the characteristic size of the
finite element (h) (Figure 1a). The general purpose finite element program DIANA was used as the simulation tool
in this study (DIANA, 2014).
(a) (b)
Figure 1. Concrete tension softening models (a) FEM (b) OLM
2.2. Overlapping Lattice Model In the overlapping lattice model, in order to take the nonlocal effects into account, each node interacts with points
within a predetermined distance called horizon (δ) (Figure 2a and Figure 2b). For initially uniformly distributed
particles separated by a grid spacing of d in x, y and z directions, δ is commonly taken slightly more than three
times d. Therefore, for problems in a two-dimensional setting, a particle located away from boundaries initially
interacts with 28 neighboring points as presented in Figure 2b.
Figure 2. (a) and (b) OLM
Concrete exhibits tension softening beyond the critical strain; therefore, the elements can transfer further tension
by softening as shown in Figure 1b. Nonlinear tension softening was assumed to be in the form of a stepwise linear
softening function as shown in Figure 2Figure 1b. Force-deformation curves for direct tension tests for a specific
gauge length were employed to calibrate the input force-deformation response of the truss elements (Aydın, 2017).
For elements with different sizes, the length scale was then used to adjust the input force-deformation function
similar to the approach used in the mesh regularization in finite element simulations (Bazant and Oh, 1983) as
shown in Figure 1b. A classical structural analysis approach with explicit time integration was used in the
simulations (Chung and Lee, 1994).
3. COMPARISON OF FINITE ELEMENT METHOD AND OVERLAPPING LATTICE MODEL
3.1. Pushover Loading of a Notched Model Concrete Dam
A 1:40 scale concrete gravity dam was tested by (Carpinteri et al., 1992) under hydrostatic, monotonically
increasing loading conditions. A 30 cm deep notch was cut on the upstream face of the dam at a height of 0.6 m
from the base in order to determine the location of crack initiation on the specimen. The tensile strength, modulus
of elasticity and the fracture energy of the specimen were reported as 3.6 MPa, 35.7 GPa and 184 N/m,
4. Uluslararası Deprem Mühendisliği ve Sismoloji Konferansı
11-13 Ekim 2017 – ANADOLU ÜNİVERSİTESİ – ESKİŞEHİR
respectively. The load-crack mouth opening displacement (CMOD) curves from the test as well as the propagation
of the crack were used as benchmark values to compare FEM and OLM.
In FEM, displacement controlled analyses were performed using the arc-length control on the crack mouth opening
displacement in order to evaluate the behavior of the specimen. The modulus of elasticity and the fracture energy
was increased by 1.58 and 1.36, respectively in the analytical simulations to overcome initial low stiffness and
capacity of the model. Exponential form of tension softening was used in the simulations. The finite element mesh
constituted in plane stress setting consisted of 1503 elements. For the OLM analyses, the modulus of elasticity
was increased by 1.58 and the tensile strength by 1.38. This model consisted of 17343 elements with horizon
δ=3.01d. The proportional-integral-derivative (PID) control method was employed in the OLM simulation. The
cracking pattern for the two methods are compared in Figure 3a and b together with the experimental result (shown
with the red line). Furthermore, the load-CMOD graph is presented in Figure 3c.
(a) (b) (c)
Figure 3. Comparison of the results (a) FEM cracking pattern (b) OLM cracking pattern (c) load-CMOD graphs
The FEM and OLM simulated the length of the cracking less than the experiment (Figure 3a and b); however, the
orientation of cracking was close to the laboratory observations. The load vs. CMOD behavior of the two models
were also similar. Both models successfully simulated the experimental behavior. However, the load capacity
estimation of OLM was better than the FEM model (Figure 3c).
3.2. Pseudo-Dynamic Earthquake Simulation of a Scaled Gravity Dam
A 1/75 scaled model of the 120 m high Melen Dam was tested by (Aldemir et al., 2015) using three different
scaled ground motions in a pseudo-dynamic setup. After the completion of the earthquake loading, the specimen
was pushed to failure by increasing the lateral load on the system in a static fashion. The special setup of the test
enabled the use of only the bottom half of the dam section, the inertial and hydrodynamic load effects were
simulated using a special loading apparatus. The tensile strength and modulus of elasticity used were 1.0 MPa and
15 GPa, respectively. The flexibility at the base of the specimen was taken into account by using spring elements
at the base for both FEM and OLM models. In the FEM analyses, the performance of the specimen was simulated
by applying a lateral force at the top controlled using the arc-length method. The linear tension softening function
was used in FEM simulations. The model in OLM was constituted with horizon δ=1.5d. For both FEM and OLM,
the simulation was conducted on a damaged system in which the cracking occurring during the transient motions
were reflected by an equivalent reduction in the strength of those members. The simulated load-displacement
behavior together with the cracking pattern for the specimen are compared with the experimental results in Figure
4.
4. Uluslararası Deprem Mühendisliği ve Sismoloji Konferansı
11-13 Ekim 2017 – ANADOLU ÜNİVERSİTESİ – ESKİŞEHİR
(a) (b)
(c) (d)
Figure 4. Prediction of pushover testing (a) cracking on the specimen (b) FEM simulated cracking pattern (c)
OLM simulated cracking pattern (d) load-displacement behavior
It can be observed from Figure 4b and c that the crack propagation of FEM model was better than the OLM model
as the crack at the body of the dam joined to the downstream side, unlike the OLM model. Given the capacity was
estimated higher by the OLM model (Figure 4d), the crack propagation was less than that of the experiment. The
FEM model exhibited a stiffer behavior than the experiment (Figure 4d). The base shear capacity was estimated
quite close to the laboratory result; however, the displacement capacity was lower than the experiment. The OLM
model, on the other hand, had stiffness that matched the laboratory observation. Although the base shear was
overestimated, the displacement capacity matched with the experiment (Figure 4d).
3.3. Shake Table Testing of a Scaled Gravity Dam
The experimental results of Tinawi et al. (2000) were reproduced in this study in order to determine the effect of
the modeling assumptions for the analytical prediction of the behavior of concrete gravity dams. In this experiment,
a plain concrete gravity dam specimen having a height of 3.4m was tested using a pulse excitation at three different
intensity levels. The pulse was scaled with respect to its first peak acceleration (FPA) to three levels: 0.87, 0.94
and 0.98g. The tests with the FPA of 0.94g and 0.98 are referred as the first and second cracking tests, respectively
(Tinawi et al., 2000), as the first test with the FPA of 0.87 did not lead to any cracking on the specimen. The
specimen had notches both in the upstream and downstream sides. The elasticity modulus, tensile strength and
fracture energy was reported as 18.5 GPa, 3.73 MPa and 105 N/m, respectively. It should be noted that the dynamic
amplification factor for tensile strength and fracture energy was 1.75 (Tinawi et al., 2000). The natural frequency
of the specimen was obtained as 16.4 Hz in the laboratory testing.
The FEM simulations were conducted with element size of 10 cm. The elasticity modulus and tensile strength was
assumed as 18.5 GPa and 3.78 MPa, respectively and linear tension softening was used. The fracture energy was
55 N/m, i.e. no dynamic amplification factor was employed. The natural frequency of the FEM model matched
with the experimental result. Similar to the analytical model named as M2 by Tinawi et al. (2000), the FEM and
OLM models had springs at the base. The material properties used in the OLM model was 18.5 GPa, 3.0 MPa and
54 N/m for the elasticity modulus, tensile strength and fracture energy, respectively. The model was constituted
with horizon δ=1.5d. The first frequency was slightly higher than the experiment: a natural frequency of 18.3 Hz
was obtained. For both FEM and OLM models, the notch was simulated by removing the elements at the notch
locations. For the first and second cracking tests, the cracking schemes for FEM and OLM methods are given in
4. Uluslararası Deprem Mühendisliği ve Sismoloji Konferansı
11-13 Ekim 2017 – ANADOLU ÜNİVERSİTESİ – ESKİŞEHİR
Figure 5b and c. Furthermore, the CMOD time history for the second cracking test was compared with the
experimental results in the same figure (Figure 5d).
First Cracking Test
Second Cracking Test
(a) (b) (c) (d)
Figure 5. Analysis results (a) Experimental result (Tinawi et al., 2000) (b) FEM cracking scheme (c) OLM
cracking scheme (d) CMOD time history
As shown in Figure 5a, in the first cracking experiment, a partial crack with a length of approximately 350 mm
with an initial angle of about 300 downward was observed at the downstream notch. In the second cracking
experiment the crack at the downstream notch joined to the upstream notch (Tinawi et al., 2000). In the FEM
model, the inclination of cracking in the first test could not be simulated correctly; moreover, the obtained crack
length was higher than that of the experiment (Figure 5b). The OLM model, on the other hand, was successful in
simulating the direction and length of the cracking in the first test (Figure 5c). In the second test, in both models
the downstream crack propagated towards and joined to the upstream notch. The CMOD time histories for the
downstream notch opening were compared in Figure 5d for both models. In the FEM model, the cracking initiated
earlier than the experiment; the peak value was obtained as 3 mm, a little lower than the peak CMOD observed in
the test (3.5 mm). The CMOD behavior was simulated better with the OLM model. Both the timing of the cracking
and the maximum value of CMOD matched the experimental results.
4. SUMMARY AND CONCLUSIONS
In this study, the capabilities of the finite element tools for the simulation of extensive damage on plain concrete
structures are compared with the capabilities of the Overlapping Lattice Models (OLM). Three laboratory tests
conducted with different types of loading were simulated using both tools and the following conclusions were
drawn:
The first benchmark study was modeling the experiment on a scaled dam monolith tested under lateral
pushover type loading (Carpinteri et al., 1992). The FEM and OLM simulations yielded similar results in
propagation of cracking and the load-crack mode opening behavior. The load capacity estimation of OLM,
however, was slightly better than the FEM model.
The lateral pushover loading test performed after a pseudo dynamic testing on a scaled monolith was
simulated (Aldemir et al., 2013). The base shear capacity was estimated quite close to the laboratory result
with the FEM model; however, the displacement capacity was lower than the experiment. The OLM
model, on the other hand, had stiffness that matched the laboratory observation. Although the base shear
was overestimated, the displacement capacity matched the experiment. Since the base shear was estimated
higher, the crack propagation was less in the OLM method. On the other hand, for this experiment, the
FEM model could simulate the crack propagation correctly.
The final benchmark study used was the shake table testing of a scaled gravity dam model constructed by
Tinawi et al. (2000). In the first cracking test, the inclination and propagation of the cracking could not be
exactly matched with the FEM model. Conversely, the OLM model was successful in simulation of this
4. Uluslararası Deprem Mühendisliği ve Sismoloji Konferansı
11-13 Ekim 2017 – ANADOLU ÜNİVERSİTESİ – ESKİŞEHİR
test. In the second cracking test, in both FEM and OLM models, the crack propagation matched with the
laboratory observation. The CMOD behavior was simulated nearly perfectly by the OLM model and
satisfactorily with the FEM counterpart. The timing of cracking and the maximum value of CMOD
matched the experimental results very well using OLM.
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11-13 Ekim 2017 – ANADOLU ÜNİVERSİTESİ – ESKİŞEHİR
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