이윤
(Yun Lee)
1†
Copyright © The Korea Institute for Structural Maintenance and Inspection
키워드
지진하중, 철근콘크리트, 전단벽, 유한요소해석, 콘크리트 균열
Key words
Seismic loading, Reinforced concrete, Shear wall, Finite element analysis, Concrete crack
1. Introduction
Because seismic design has become increasingly more important recently for nuclear
power plants, the American Society of Civil Engineers (ASCE) compiled the ASCE 43-05
report (ASCE/SEI 43, 2005) asserting that there is a need to assess nuclear power plant structures taking into
consideration the impact of latent cracks, and it was also reported by the American
Concrete Institute (ACI) that cracking reduces the stiffness of a structural member,
and cracking should be considered in stiffness assumptions so that drift caused by
wind or earthquakes is not grossly underestimated (ACI 349, 2001).
Additionally, the U.S. Nuclear Regulatory Commission (US NRC) also mandated that project
owners should design a new nuclear power plant taking into consideration the impact
of latent cracks. Accordingly, based on the above assertions and requirements for
evaluating nuclear power plant safety, the impact of latent cracks has been recognized
to be an important factor as seen in the AP1000 and US-APWR examinations for obtaining
the U.S. design certificate (DCD, 2011; Arel et al., 2017; NUREG/CR 6926, 2007; Sogbossi et al., 2017; US-APWR, 2011; Wang et al., 2018). On the other hand, when experimentally verifying a rectangular box-shaped shear
wall system tested under the JNES/NUPEC program in Japan, it was reported that the
frequency and shear stiffness of a shear wall decreased with an increase in the seismic
loading history (Choubey et al., 2014; Habasaki et al., 1999; Hiroshi et al., 2001; Kusama et al., 2003; Mistri et al., 2016; Shojaei et al., 2017; Syed et al., 2015; Torita et al., 2004; Zhou et al., 2018).
Thus, the aim of this study was to analytically identify the phenomenon of stiffness
reduction caused by the presence of latent cracks brought about by seismic loads in
the shear walls of reinforced concrete structures. The reinforced concrete shear wall
was selected in this study because it is a major structural member used extensively
in all Nuclear Power Plants to provide seismic load resisting capability.
In this study, seismic time history analysis was carried out on RC shear walls to
investigate the impacts of cracks on structures with nonlinear finite element analysis.
To quantitatively reflect the influence of cracks on a seismic design, the influence
of the cracks according to the level of the seismic load was investigated in terms
of the natural frequency and shear stiffness by classifying the possible seismic loads
by grade.
By varying the input motions, concrete strength and shear wall thickness, the seismic
responses of the shear wall were considered with nonlinear time history analysis,
and the progressive cracking behavior and corresponding stiffness degradation were
investigated.
The paper is organized into four sections. Section 1 is the introduction, and Section
2 provides a brief description of the numerical framework used in this study. The
cracking and failure description, the numerical results of an individual time history
analysis and their comparisons are discussed in Section 3. Finally, conclusions are
provided in Section 4.
2. Numerical Framework for the Nonlinear Analysis of a Shear Wall
2.1 Preparation of Numerical Analysis
2.1.1 Finite element analysis program
To identify the shear stiffness degradation behavior from concrete cracks, the numerical
analysis should first adopt real earthquake data and carry out a nonlinear time history
analysis. It should also be able to effectively simulate cracks generated in a concrete
structure by seismic loads. To this end, this study selected ANSYS V.13.5 which is
the common software used to do a nonlinear time history analysis. Because cracks in
concrete should be simulated throughout the whole analysis process, the appropriate
material model should be used for an applied finite element program when considering
concrete cracks. ANSYS V.13.5 provides SOLID65 as a material model for concrete which
can take into consideration different damages and cracks in concrete. Accordingly,
this study selected SOLID65 from the ANSYS V.13.5 software as the material model for
concrete which can simulate gradual damages and cracks in concrete.
2.1.2 Mesh modeling of a shear wall
In this analysis, an RC shear wall structure, for which a vibration test was successfully
done shown in Fig. 1, was selected as the analysis target. Based on the specification of the selected
shear wall and the details of the reinforcing bars, mesh modeling was done on a target,
and the reinforcing bars in shear wall and the flange were the modeled structure shown
in Fig. 2. Additionally, the bottom of the shear wall was fixed with an assigned boundary condition.
To reflect the effect of the vertical load on the shear wall, a stress of 0.442 MPa
was vertically loaded onto the top surface of the shear wall, and a mass of 50 tonf
was given to the mass center of the vertical load using the point mass function of
ANSYS V.13.5 to reflect the effect of mass on the top of the shear wall. The concrete
shear wall was modeled with SOLID65, an eight-node solid element, provided by ANSYS,
and the reinforcing bars were modeled with the LINK180 element, a two node truss element.
Fig. 1 Shear wall configuration and dimensions of the Rebar Configuration (Unit: mm)
Fig. 2 Mesh model and rebar configuration
2.1.3 Concrete material model
In cases in which a cyclic load such as a seismic load should be simulated with the
nonlinear behavior of concrete, analysis programs are generally designed to enable
a concrete material model to represent the gradual degradation of stiffness and strength
using a damage model for which the tangent modulus of elasticity decreases according
to the stress history. Generally, existing cracks in a structure influence the stiffness
and strength of a structure and change the various aspects of failure, which should
be appropriately described in the numerical model. This study used SOLID65 from ANSYS
V.13.5 as a numerical model to simulate damage effects caused by concrete cracks.
SOLID65 can effectively simulate a multi-axial cyclic history behavior, and adopt
a smeared crack approach to simulate concrete cracks to be considered in this study.
The smeared crack model can be easily engrafted on a finite element method and has
the advantage of overcoming the greatest weaknesses of a discrete crack model in which
crack initiation and propagation have to be determined beforehand. A smeared crack
model can represent the effects of cracks by changing the material constitutive relations
through a uniform distribution of cracks inside a finite element. With the implementation
of SOLID65, the crack occurrence in a concrete shear wall can be visually identified
in the post-process provided by ANSYS V.13.5, and the cracking patterns according
to an increasing seismic load can be investigated in this study.
2.1.4 Applied load
As a basic input motion, the time-acceleration history, in which the maximum acceleration
is scaled to 0.3 g, the safe shutdown earthquake (SSE) value, is implemented shown
in Fig. 3. The time interval and its time duration are 0.01second and 24 seconds, respectively.
The minimum time increment in the time history analysis was adopted based on 1/10
of the natural period known as the stability limit of dynamic analysis. The natural
frequency of a given shear wall structure was assessed approximately at 23 Hz through
modal analysis, and the minimum time increment was considered as 0.004 seconds smaller
than 0.0043 seconds which is estimated as the stability limit of the dynamic analysis.
Fig. 3 Basic input motion scaled to a PGA of 0.3 g
2.2 Classification of the Performed Numerical Analysis
The purpose of this study was to identify the influence of concrete cracks inside
a shear wall structure with a relatively low tensile strength under a small to large
seismic load on the stiffness reduction of a structure. To this end, the numerical
cases done by this study are summarized as follows:
2.2.1 Modal analysis
Modal analysis was done on a RC shear wall for a concrete strength of 21MPa modeled
as mentioned in section 2.1 using ANSYS V.13.5. Because a concrete shear wall shows
nonlinear behavior due to the nonlinearity of the concrete itself, natural frequency
according to modal analysis is not the same as the natural frequency of a concrete
shear wall; however, based on this fact, the modal analysis of a shear wall in an
elastic state was done to determine the time interval and appropriate mesh size in
the time history analysis by assessing the natural frequency and mode of a shear wall
not taking into account the nonlinear effect.
2.2.2 Static analysis
The main scope of this study was to identify the seismic behavior of RC shear walls
through a dynamic analysis. However, an analysis of a static loading case in lateral
displacement on a target member was conducted to predict the cracking load and ultimate
load of a target shear wall and to compare them with dynamic loading cases. The static
analysis was done and designed to be compared with a hysteresis curve under a dynamic
analysis.
2.2.3 Increasing seismic load cases for RUN-1 ~ RUN-5
Based on basic input motions, a nonlinear time history analysis for an earthquake
duration up to 24 seconds was conducted by scaling peak ground acceleration (PGA)
for a concrete strength of 21 MPa from RUN-1 to RUN-5 for 5 phase loads, 0.1, 0.2,
0.3, 0.5 and 0.7 g, shown in Fig. 4. Through this analysis, the overall pattern and extent of concrete cracks are identified,
and the dynamic features of a shear wall under a seismic load is identified through
an FFT analysis with a maximum response.
Fig. 4 Increasing seismic load cases
3. Analysis Results and Discussion
3.1 Modal Analysis
As described in section 2.2.1, modal analysis was conducted to determine the size
of the elements for mesh modeling and to analyze the frequency in an elastic state.
Fig. 5 shows the mesh configuration for a concrete strength of 21 MPa. According to the
analysis results, when the size for an element of concrete was 40 mm, the natural
frequency of direction Z, which is the direction of an input earthquake, was 22.55
Hz, and when the element
size was 80 mm, which is about three times the maximum size of the aggregate, the
natural frequency was 22.57 Hz. It was observed that there was little difference in
the natural frequency according to the size and number of element. Considering analysis
time saving and computer memory capability, the mesh size was determined as 80 mm,
and the relevant numerical analysis was conducted.
Fig. 5 Comparison of mesh configurations via modal analysis
3.2 Static analysis
Fig. 6 shows a load-displacement curve derived from a static analysis for a concrete strength
of 21MPa. The first cracking load after loading was 0.294 MN; the local maximum load
was 0.339 MN, and the ultimate failure load was 0.428 MN. By investigating the load
that generates cracks and the ultimate failure load through static analysis, the magnitude
level of the seismic load for the increasing load cases in section 2.2.3 was determined.
Fig. 7 shows a cracking pattern after placing a static load. It was found that typically,
diagonal cracks occur in a concrete shear wall which confirm that the failure of a
concrete shear wall, which is the target of this study, is controlled by the shear
failure mode at the ultimate state.
Fig. 6 Load-displacement curve from a static analysis on the concrete strength of 21 MPa
Fig. 7 Cracking pattern from a static analysis on the concrete strength of 21 MPa
3.3. Increasing seismic load cases for RUN-1 ~ RUN-5
As mentioned in section 2.2.3, a nonlinear time history analysis was conducted based
on gradually increasing the magnitude of the seismic load from RUN-1 through RUN-5
for a concrete strength of 21 MPa (modulus of elasticity of 20 GPa; tensile strength
of 1.5 MPa) Figs. 8 to 12 show the acceleration response at the top surface of a shear
wall by each phase in the frequency domain using Fast Fourier Transform (FFT) and
the load-displacement curve after an individual time history analysis.
According to the load-displacement curve in Fig. 8, energy loss hardly occurred due to the applied load, and stiffness reduction also
was hardly caused by the cyclic loading history. Moreover, concrete cracking did not
occur according to the numerical analysis result. In the case of RUN-2 with a PGA
of 0.2 g in Fig. 9, the maximum load was about 0.15 MN, which is equivalent to about 50% of the first
cracking load of 0.294 MN for the static loading case. This means that concrete cracking
and the ensuing stiffness reduction hardly occurred.
Fig. 8 Results of seismic analysis for phase load RUN-1 on a concrete strength of 21 MPa
Fig. 9 Results of seismic analysis for phase load RUN-2 on a concrete strength of 21 MPa
In the case of RUN-3 with a PGA of 0.3 g, as shown in Fig. 10, the energy loss and stiffness reduction according to the cyclic loading history
in the load-displacement curve were caused by concrete cracking at the bottom of the
shear wall and flange. The maximum load was about 0.2 MN, which is equivalent to about
60% of the first cracking load of 0.294 MN for the static loading case.
Fig. 10 Results of seismic analysis for phase load RUN-3 on a concrete strength of 21 MPa
Fig. 11 Results of seismic analysis for phase load RUN-4 on a concrete strength of 21 MPa
Fig. 12 Results of seismic analysis for phase load RUN-5 on a concrete strength of 21 MPa
In the case of RUN-4 with a PGA of 0.5 g, as shown in Fig. 13, cracks generated at the bottom of the shear wall and flange expanded upwards. Consequently,
the energy loss and stiffness reduction according to the cyclic loading history in
the load-displacement curve were clearly induced by the concrete cracking. Additionally,
in the case of RUN-4, the maximum load was about 0.35 MN, which was higher than the
first cracking load of 0.294 MN and approached the local maximum load of 0.339 MN
in the static loading case.
Fig. 13 Cracking pattern for phase load RUN-4 for a concrete strength of 21 MPa
In the case of RUN-5 with a PGA of 0.7 g, convergence was not created any more at
3.8 seconds after the start of the time history analysis. As shown in Fig. 14, concrete cracks were distributed in a longitudinal direction below the middle of
the shear wall and flange, and accordingly, energy loss and stiffness reduction clearly
occurred according to the cyclic loading history. In the case of RUN-5, the maximum
load at failure was 0.42 MN, and the failure occurred at a load very similar to 0.428
MN which is the failure load for the static loading case. Therefore, it is estimated
that the ultimate earthquake acceleration, where a failure in the target shear wall
occurs, is about 0.5 to 0.7 g.
Table 1 shows the frequencies at the maximum acceleration responses derived from the FFT
for the time history of the acceleration response at the top of the shear wall by
each phase of the seismic load.
Fig. 14 Cracking pattern for phase load RUN-5 for a concrete strength of 21 MPa
In the case of the input motions with a PGA of 0.1 and 0.2 g in which energy dissipation
was hardly caused by concrete cracking, the frequency was about 21.4 Hz, and in the
case of an input motion with a PGA of 0.3 g for which energy loss and stiffness reduction
were produced by cracks at the bottom of the shear wall, the frequency at the maximum
acceleration response decreased by about 0.2 Hz.
In a seismic motion with a PGA of 0.5 g for which energy loss and stiffness reduction
clearly occurred due to concrete cracking at the bottom of the shear wall, the frequency
at the maximum acceleration response was about 14.42 Hz with a remarkable decrease
of about 36% compared with the modal analysis result, and it was found that the shear
stiffness was reduced by about 60% compared with the no-cracked sound structure. In
the case of the input motion with a PGA of 0.7 g at which the analysis abruptly stopped
because of the occurrence of failure during the analysis, the frequency at the maximum
acceleration response was 13.68 Hz, which implies about a 63% reduction in the shear
stiffness compared with no- damaged shear wall.
Table 1 Frequency results
|
Increasing load cases
|
Frequency at the peak response in
the FFT of the acceleration response
|
|
Modal Analysis
Run-1 (0.1 g)
Run-2 (0.2 g)
Run-3 (0.3 g)
Run-4 (0.5 g)
Run-5 (0.7 g) (Failure)
|
22.57 Hz
21.38 Hz
21.37 Hz
21.17 Hz
14.42 Hz
13.68 Hz
|
4. Conclusions
This study was conducted to evaluate the influence of stiffness reduction in a concrete
shear wall caused by cracks. The study results are summarized as follows:
(1) From the results of the nonlinear analysis of a gradual increase in the seismic
load, it was found that as the magnitude of a seismic load became large, shear cracks
occurred and progressed in the concrete shear wall and that energy loss and shear
cracks clearly occurred according to the cyclic history.
(2) For the increasing seismic load cases, as the magnitude of the seismic load approached
the ultimate seismic capacity of the shear wall, the frequency decreased by about
36% compared to the intact state, and shear stiffness decreased by about 60% compared
to the crack-free sound state.
(3) It was found that in cases where concrete cracks occurred in the shear wall due
to a seismic load, an considerable stiffness reduction occurred compared with the
virgin load case, and it is suggested that the effect of concrete cracks in shear
walls on the shear stiffness should be appropriately considered in the seismic design
of shear walls.
Acknowledgements
This work was supported by Korea Institute of Nuclear Safety(KINS).
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