ABSTRACT
This paper presents a
numerical procedure which is called the Vector form intrinsic finite element method.
It is used to analyze the nonlinear behavior of structural collapse under
seismic excitation. Nonlinear dynamic analysis was performed due
to the large amount of literature
supporting it as the superior method of analysis
as compared to nonlinear static
analysis . The accuracy of
the multistory frame
model was also
confirmed through comparison of
its response under blast loading
with that of an existing study in the literature, and the results were in good
agreement . The whole progressive collapse processes of the framed structures in various
conditions are also simulated. The numerical procedures in Vector form
intrinsic finite element method provided for new concept to deal a large
deformation and rotation from continuous state to discontinuous state. The
geometry nonlinear and nonlinear materials have been considered. The damage
indexes of the element are adopted to decide failure criteria of element
joints. In order to simulate the progressive collapse behavior of the structures,
the compression between of the numerical simulation Vector form intrinsic
finite element method and shake table experiments demonstrate the accuracy of
the Vector form intrinsic finite element
method.
The information
presents applications of proposed collapse methodology to the development of
collapse fragility curves and the
evaluation of the mean annual frequency of collapse. The previous of the problems;
we can analyze and short out of the problems related to the collapse. Because,
mainly part of the collapse.
KEYWORDS
Vector form intrinsic
finite element method, Progressive collapse, Damage index, Collapse
analysis, Discontinuous displacement, Collision/Impact, Geometric nonlinearity.
1. INTRODUCTION
To prevent the immeasurable losses of
human lives and social properties due to earthquakes and terrorist attacks,
resistance evaluation and retrofitting of civil infrastructures have become an
important issue of many countries in the world. Great attention has been
focused on a type of failure known as “progressive collapse” since the Ronan
Point apartment collapse in London in 1986 (Griffiths et al., 1986). In recent
years, however, terrorist attacks have also become evident as seen in the
Alfred P. Murrah Building in Oklahoma City in 1995 and the World Trade Center
in New York, 2001 which did not withstand the fires from terrorist attacks that
have eventually induced progressive collapse of the building. The 27 May 2006
earthquake has hit the provinces of Yogyakarta and Central Java in Indonesia
which lead to RC and brick building’s collapse. Besides experimental and
theoretical studies, numerical simulation is another way to assist engineers to
understand the nonlinear dynamic failure behavior of structure under an
earthquake excitation. Recently, the V-fife method has been proposed by Ting,
et al. (2004a, 2004b) and Wang and Ting (2004). This method applies a unique
approach to compute the effects of rigid motion, allowing the simulation of
extremely large deformation of elastic motion structures. The V-fife method
will be explained in greater detail later in this paper. It redefines a set of
deformation coordinates on the element in each time step. Therefore, it removes
each element’s rigid body rotations and displacements. The V-fife method adopts
a convicted material frame and the explicit time integration method for solving
the equations of motion. This method has been successfully applied to the
nonlinear motion analysis of the 2D elastic frame (Wu et al., 2006), the
dynamic stability analysis of the space truss structure (Wang et al., 2006),
and the elastic-plastic analysis of a space truss structure (Wang et al. 2005).
The key objective of this study is to construct the V-fife method for the
analysis of RC frame subjected to extremely large deformation having inelastic
material properties. In this paper, analysis of collapse structures included
multiple frame elements motion for large deformation and
rotation from continuous states to discontinuous state. The comparison between
numerical simulation of V-5 method and shake table experiments demonstrates the
accuracy of V-5 method. Based on the survey around the heavily damaged area,
three key problems in collapse analysis of structures were identified as
description of discontinuous displacements, collision/impact between structural
elements and adjacent buildings, and geometric nonlinearity of structures. Some research work trying to solve the
solutions is also introduced.
2. VECTOR FORM
INTRINSIC FINITE ELEMENT METHOD
In this study, the V-5
method is extended in order to analyze elastic-plastic systems containing
multiple deformable bodies with the following characteristics:
(1) Interact
with each other.
(2) Discontinuous.
(3) undergo
large deformations and arbitrary rigid body motions.
Since the conventional finite
element method (FEM) is an energy-based method, it does not require the balance
of forces within each element. Since these unbalanced residual forces will do
some work under virtual rigid body motion it will cause inaccuracy and un-convergence
of the computed results. In order to solve these problems, the V-5 method has
been proposed.
There are vector form intrinsic finite element
method includes four main procedures:
(1) Construct the equation of motion using Newton’s
Law at the mass points.
(2) Update
the material frame.
(3) Compute
the fictitious reversed rotations.
(4) Determine
the deformation coordinates.
These
aforementioned computation procedures have some points in common with the
concept of the modern FEM. However, the key concept of the V-5 is that V-5
maintains the intrinsic nature of the original FEM. The V-5 method makes use of
the strong form of equilibrium at each element. All the forces are balanced
within each element. These forces are obtained from the principle of virtual
work. The associated nodal displacements satisfy the compatibility condition.
3. FINITE ELEMENT ANALYSIS MODELING OF CONCRETE FRAMES
Finite element analysis software program was used to model a three-story reinforced Concrete frame building. To simulate blast loading condition, a time history air pressure wave was applied to the frame. Nonlinear dynamic time history response of the frame model was validated with that of an existing study in the literature (Jayashree et al. 2013) and they were in good agreement.
3.1 GEOMETRY AND MATERIAL PROPERTIES OF THE FRAME STRUCTURES
The
simulated frame consisted of two bays in A one bay
in B, and three story elevations in C
Directions of the global coordinate system.
All bays were equally spaced at 8 m in
x and y directions. The floor to floor height of each story was
equal to 3 m . Base rectangular model with x y and z coordinates b) plan view of
rectangle c) plan view of the square model d) plan view of the L-shaped.
The beams were assumed to be 45 x 25 cm reinforced
concrete sections. All story columns were 300*300 mm RC sections with eight longitudinal bars (29
mm). The hoop reinforcement bars Were assumed to be 12 mm with 15 cm longitudinal
spacing and concrete cover of 40mm. The floors diaphragm were
treated as thin shell sections with
two layers of the reinforcement.
The thickness of the slab was 100mm with 1 mm of concrete cover. The
concrete compressive strength and modulus of elasticity were 25 MPa and 23,158
MPa, respectively. The bars were made of grade
60, with the
yield strength of
414 MPa and modulus
of elasticity of
2x10^5 MPa, respectively.
3.2. LOADING AND BOUNDARY CONDITIONS
Consideration of the structures
joints, moment connections while all columns of the ground
were modeled as fixed supports.
Additionally, the end length offset option in SAP2000 was used to connect the beams and columns properly.
All floor systems were assumed to work as
diaphragms.
3.3. ANALYSIS PROCEDURE
The blast load was applied to the validated reinforced concrete frame by
performing nonlinear time history analysis.
P-M2-M3 and M3 plastic hinges
were applied to both ends of columns and both ends of the beams, respectively. P-M2-M3 plastic hinges have a moment rotation
curve which is used to describe a combination of axial and bending behavior of
column elements (SAP2000 2014). ‘Automatic plastic hinges’ option in SAP2000
software program was used for this purpose. All plastic hinge definitions are
based on tables 6-7 and 6-8 of FEMA 356 (FEMA 2000. A 5 percent constant
damping ratio was assumed. The analysis was performed for 2 seconds with 250.
3.4. NON-LINEAR SOLUTION AND FAILURE CRITERION
Iteration method was
used to predict the required load value to form the first plastic hinges as they
exceeded the collapse prevention condition.
Based on the GSA, for the nonlinear dynamic analysis, the moment and
rotation values should not exceed the collapse prevention condition. The
magnitude of blast load was changed by increasing or decreasing the load scale-factor. Based
on the choice of referring co-ordinates, methods regarding geometric
formulation of deformable bodies may be categorized there are two types:
Lagrangian Formulation and Eulerian Formulation. In 1979 Argyris and Doltsinis
employed the current unstressed configuration to deal with large deformation
and inelastic problems, where are iteration needed since the current configuration
remains unknown. In structural analysis the geometric nonlinearity related with
large deformation are usually avoided by assuming concentrated plasticity of
materials. Thus a general only large displacement/large rotation with small
deformation are considered in structural analysis.
3.4.1 LARGE
DESPLACEMENT ROTATION
The consideration of the geometric non-linearity frame structure P and Delta effects of the structures. The introducing a geometric stiffness matrix, the coupling between bending moment and axial force can be taken into account in terms of the effective stiffness. These methods, however, can be not suitable for collapse analysis of framed structures. For example, the variation of axial forces cannot be considered appropriately in dynamic analysis.
3.4.2 FAILURES DUE TO THE BUCKLING
The well-known three-hinged buckling problem can also be solved using the current unstressed configuration. Referring to the current unstressed configuration might be a good choice considering buckling failure in a consistent manner with other geometric nonlinearity.
4. DISCONTINUOUS DISPLACEMENT
An intact structure
under extreme conditions usually will subject to various failures such as
cracking, yielding and even fracture. As far as the distribution of
displacement and deformation is concerned, it may be termed as having
discontinuous displacement. A typical example is the failure of a
beam-column member in the formation of a plastic hinge – the slope along the
member has some discontinuities, which has been widely studied over the past
decades.
4.1 DISCONTINUOUS DISPLACEMENT IN BEAM-COLUMN MEMBERS
These displacement discontinuities are most
common in damage investigation as shown in Fig.1 and Fig.2, where some failures
with discontinuities in slope (for example, Fig.1) can be successfully
described by plastic hinge model. There are plenty of literatures on using
plastic hinges in collapse analysis and seismic assessment such as
Fig-1 fig-2
4.2 DISCONTINUOUS DISPLACEMENT OF SLAB/WALL ELEMENTS
It is well known that most of the casualties
are caused by collapses in strong earthquakes. In fact it might more
appropriate to say that most of the casualties resulted from collapse of floor
slabs and walls. Such, show as figures.
Unfortunately such kind of behavior of slab/wall under extreme conditions cannot be simulated appropriately using available models. The application of DEM and its extended form is also prohibited by similar problems in dealing with beam-column members as described in Section 2. In current design codes, the roles that slab/wall plays in the whole structural system carrying external loads is usually simplified in modeling.
In the
multi-storey frame structure buildings are damages from earthquake generally
initiates at position of structural weaknesses. The openings present in slabs
are often providing discontinuities in distribution of loads. But when these
openings are provided at suitable position the of structure to damage can be
avoided. Structural engineers have developed confidence in the design of
buildings. In the present work openings are enclosed by shear walls at placed
different positions.
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