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Collapse Analysis of RC Structures using
Improved Applied Element Method
Said ELKHOLY1, Mohamed GOMMA2and Adel AKL3
1
Assistant Professor, Fayoum University, Egypt
[email protected]
2
Ph.D. Candidate, Cairo University, Egypt
3
Professor, Cairo University, Egypt
ABSTRACT
In this paper, the Improved Applied Element Method (IAEM), which was
originally developed as an effective analysis technique of large-scale steel
structures up to complete failure under different hazard loads, has been
progressively developed to carry out modeling of the behavior of plane frame
reinforced concrete (RC) and composite structures. By adopting the fiber-beamelement model and the multi-layer-element model, which are developed for RC
and composite framed structures, the extreme nonlinear behavior of RC and
composite structural elements can be accurately simulated. The technique is an
effective tool for the numerical evaluation of the behavior of structures under
extreme loading conditions and their collapse mechanisms. Moreover, the
proposed model's high simulation capability for the cycle behavior under coupled
axial force-bending moment-shear force, the breakdown of structural elements at
ultimate states, and the contact between structural elements during the collapse
are demonstrated. Finally, the method is used to study the vulnerability of school
buildings in Egypt to progressive collapse by conducting a detailed dynamic
collapse analysis for an old school building. The analysis is carried out to
investigate the cause of the collapse and the method that can be adopted to
enhance the structure so that the collapse can be prevented.
Keywords: Progressive collapse, Improved Applied Element Method, Multi
Layered Element, School buildings, Retrofitting technique.
1. INTRODUCTION
Reinforced concrete is one of the most widely used modern building materials.
With the rapid growth of urban population, RC frame construction has been
widely used for residential construction in both the developing and industrialized
countries. One of the major causes of seismic vulnerability associated with RC
buildings is that, in the developing countries, a large number of the existing RC
frame buildings have been designed without engineering input and have been built
by inadequately skilled construction workers. This paper considers issues related
to earthquake induced progressive collapse of RC buildings in extreme events.
This paper concentrates on developing an efficient numerical technique based on
the IAEM. The IAEM is a newly developed method for structural analysis of
large-scale steel framed structures (Elkholy and Meguro, 2003) It can trace the
0ctober 2010, Kobe, Japan
behavior of steel structures up to the complete failure stage with high accuracy in
reasonable CPU (Elkholy and Meguro, 2004). In IAEM, each structural member
is divided into a proper number of rigid elements connected by pairs of normal
and shear springs distributed on the boundary line between elements. Unlike the
conventional AIM (Tagel-Din and Meguro, 2000), the element in IAEM has been
modified in order to use different thickness per each connecting spring to follow
the change of thickness of the cross-sectional width in the case of modeling nonrectangular cross-sections.
This research is concerned with the development of an efficient and accurate
numerical technique based on the IAEM to analyze the collapse of large-scale RC
and composite structures under hazardous loads, such as earthquakes and
explosions.
2. MODELING OF RC AND COMPOSITE STRUCTURES USING IAEM
In IAEM, each structural member is divided into several segments. Each segment
represents certain cross section properties. A new element type (multi-layered
element) is introduced to IAEM to simulate non-homogenous cross sections
(Figure 1). This new element is composed of several layers. Each of these layers
has its own material properties. All identical layers in the nearby elements are
assumed to be connected to each other by sets of normal and shear springs
distributed on the boundary line. These sets of springs represent the microscopic
material properties of the layer. Moreover, each spring has its own thickness
according to the average thickness in the area represented by this spring and its
own shear and normal stiffness.
Figure 1: Modeling RC framed structures with the new element type
The value of normal stiffness ( K ni ) and shear stiffness ( K si ) for each pair of
springs in the Lth layer can be determined as:
New Technologies for Urban Safety of Mega Cities in Asia
Collapse Analysis of RC Structures using Improved Applied Element Method
E .d i .(T i )
(K )L = L L n L
a
i
n
,
( )
G L .d Li . Tsi
(K )L =
a
i
s
L
(1)
where: the subscripts n and s refer to normal and shear springs, respectively; d Li is
the distance served by the ith spring in the Lth layer; a is the length of the
representative area; EL and GL are the Young’s and shear’s modules of the Lth
layer, respectively; Tni and Tsi are the thicknesses represented by the pair of
springs “i” for the normal and shear cases, respectively.
The corresponding condensed stiffness matrix of each element is obtained by
summing the effects of each spring pair at each layer that the element is composed
of. This matrix is then transformed from the local (element) co-ordinate system to
the global (structure) co-ordinate system. Equation (2) shows the components of
the upper left quarter of the stiffness matrix. All used notations in this equation
are shown in Figure 2.
(2)
Figure 2: Contact points and DOF
2.1 Material Modeling
The Maekawa compression model for concrete is adopted in the IAEM (Figure
3a). The tangent modulus is calculated according to the strain at the spring
location. In tension, the stress-strain curve for concrete is approximately linear
elastic up to the maximum tensile strength. After this point, the concrete cracks
and its strength decreases gradually down to zero. The steel I-beam and steel bars
are modeled as a classical elastic plastic material with strain hardening using
bilinear stress–strain relationship in both compression and tension (Figure 3b).
Finally, the stress strain relation of all types of FRP material is assumed linear up
to failure (Figure 3c).
Collapse Analysis of RC Structures using Improved Applied Element Method
0ctober 2010, Kobe, Japan
(a) Concrete
(b) Steel
(c) FRP
Figure 3: Material Models used in IAEM
3. MODIFICATION IN DYNAMIC PROPERTIES
The general differential equation of motion, governing the response of the
structure in a small displacement range can be expressed as:
[M ]{ΔU&&}+ [C ]{ΔU& }+ [K ]{ΔU } = Δf (t ) − [M ]{ΔU&&G }
(3)
where: [M] is mass matrix; [C] is the damping matrix; [K] is the nonlinear
stiffness matrix; Δf(t) is the incremental applied load vector; {ΔU&&} , {ΔU& } , {ΔU }
and {ΔU&& } are the incremental acceleration, velocity, acceleration, and gravity
acceleration vectors, respectively.
G
Equation (3) is solved numerically using an implicit time-stepping method based
on the Newmark-Beta technique (Chopra, 2000). In this technique, the
displacement, velocity and acceleration vectors at time t are used to obtain a
displacement vector after a time interval (Δt) based on the following equation.
⎡ 1
⎤
⎡ 1
⎤
⎛ γ
⎞
⎡ 1
γ
γ ⎤
M+
C + K ⎥ ΔU = Δf + ⎢
M + C ⎥U& + ⎢
M + ⎜⎜
− 1⎟⎟ΔtC ⎥U&&
⎢
2
βΔt
β ⎦
⎝ 2β
⎠
⎣ β Δt
⎣ 2β
⎦
⎣ β (Δt )
⎦
(4)
where the parameters β and γ define the variation of acceleration over a time-step
and determine the stability and accuracy.
Adopting the multi-layered element in the IAEM allows the user to define
material specific weights for each layer, with which the distributed self-mass of
the structure can then be obtained using the Materials and Sections facilities. The
individual structural mass matrix of each element is calculated assuming that
masses lumped at the element centroid. The corresponding lumped mass in each
DOF direction is calculated by summing the effect of the small segmental mass
represented by each spring considering the change of the springs’ thickness.
Equation (5) represents the value of lumped mass in each DOF direction. In
addition, non-structural mass is added to the model to define any mass not
associated to the self-weight of the structure (e.g. slab, finishing, walls, etc).
(5)
where: a and b are the element dimensions; ρ j is the density of material of the jth
layer considered; nlyr is the number of layers; nsp is the number of connecting
springs; mix and miy are the ith spring mass per unit area of the element in x and y
New Technologies for Urban Safety of Mega Cities in Asia
Collapse Analysis of RC Structures using Improved Applied Element Method
directions respectively; and t xj and t jy are the ith spring thickness at a certain level
in the jth layer in x and y directions respectively.
4 ASSESSMENT OF OLD EXISTING SCHOOL BUILDINGS
The vulnerability of schools and other educational facilities has been observed in
every recent destructive earthquake all over the world. The work in this section
aims to identify the causes of the collapse of one of the typical types of old
existing school buildings in Egypt to evaluate its seismic risk, to evaluate the
efficiency of different retrofitting techniques, and to propose a national and public
policies to reduce the risk of this type of schools. In this section, the multi-layered
IAEM is applied to investigate the validity of the proposed method in simulating
progressive collapse of RC school buildings under hazardous load conditions. A
detailed analysis of the collapsing process of the selected structure with and
without retrofitting under severe ground motion conditions is presented.
4.1 School buildings in Egypt
School buildings in Egypt are typically one to five-storey long narrow buildings
with an open corridor on the school courtyard side and classrooms on the
longitudinal side. The courtyard side of a classroom typically has large windows;
and the back wall is solid up to about 70 cm from the upper floors, where long and
narrow longitudinal windows are located to allow for ventilation and extra
lighting. School class blocks are often aligned in an L-shape or U-shape around
the school courtyard. The typical structure consists of a moment resistant concrete
frame in the longitudinal direction and a concrete frame with unreinforced
masonry infill walls in the transversal direction.
4.2
Structural modeling and verification
Figure 4 shows the concrete dimensions and reinforcement details of one of the
typical school buildings that were designed and constructed according to the
Egyptian specifications issued before the October 1992 earthquake. The structure
consists of a series of single bay, 3-story RC frames with a cantilever. Using the
multi-layered element features, the school building was modelled using only 87
elements for the superstructure. Moreover, three retrofitting techniques for this
type are also investigated using the same number of elements (Figure 5).
columns
Figure 4: Concrete dimensions and reinforcement details
Collapse Analysis of RC Structures using Improved Applied Element Method
0ctober 2010, Kobe, Japan
RC Jacketing 1st floor
CFRP warping
RC Jacketing all floors
Figure 5: The three proposed retrofitting techniques
Mode 1
Mode 2
Mode 3
ω = 6.422 (rad/sec)
ω = 22.207 (rad/sec)
ω = 43.608 (rad/sec)
ω = 6.25 rad/sec
2.693%
ω = 21.48 rad/sec
3.370%
ω = 40.66 rad/sec
7.264%
FEM
Multi-layered
IAEM
To evaluate the accuracy of the modelling, the frame is simulated both in IAEM
and SAP2000 (CSI, 2000). A modal analysis was performed to calculate the
natural frequencies of the structure. The first three modes are basically
coincidentally close and a maximum difference of 7.26 % has been observed.
These three mode shapes are shown in Figure 6.
Difference
Figure 6: Comparison between the first three modes obtained by IAEM and FEM
4.3 Seismic response
After verifying the method, the inelastic dynamic collapse analysis has been
performed by introducing a displacement time history of three ground motion
records at the supports, as shown in Table 1. The acceleration time history and 5%
damping elastic response spectra curves of the selected ground motions with the
buildings fundamental periods illustrated on the curve by vertical lines are shown
in Figure 7 which shows a distinctive difference among each ground motion. Each
record has been scaled up and down to different levels in order to determine the
PGA value at which a certain building type can resist without collapse.
Table 1: Ground motions characteristics
Earthquake
Kobe
Northridge
Park field
Date
16/1/1995
17/1/1994
28/6/1966
Component
KAK090
NWH090
C02065
PGA (g)
0.345
0.583
0.476
PGV (cm/s)
27.6
75.5
75.1
PGD (cm)
9.6
17.57
22.49
A/V
1.247
0.772
0.634
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Collapse Analysis of RC Structures using Improved Applied Element Method
Figure 7: Selected ground motions and their response spectrum
4.4
Analysis results
The main objectives of the assessment are to investigate the collapse mechanism
of each model and to obtain the minimum value of the PGA of each ground
motion record used in this analysis that can produce the collapse. These minimum
values are used to evaluate the vulnerability of each model and to assess the
efficiency of each retrofitting strategy. Figure 8 summarizes the results of the
analysis for the un-retrofitted and the retrofitted models. The figure demonstrated
that the un-retrofitted school building has a very poor performance in the short
direction (investigated direction). Therefore, even a small magnitude earthquake
posed a threat. It was also founded that retrofitting the columns by CFRP warping
increased the seismic capacity by around 40%, however, using RC jacketing
increased the seismic capacity of the building by at least 80% to 150% based on
the height of the jacketing.
Figure 8: Minimum values of the PGA that produced complete collapse
Figure 9 shows the seismic collapse mechanism of the un-retrofitted school. The
figure shows that the collapse initiated at the mid of the ground floor column due
to combined actions of axial and flexural moments. The insufficient ductility of
the members due to insufficient confinement, and low level of strength due to
Collapse Analysis of RC Structures using Improved Applied Element Method
0ctober 2010, Kobe, Japan
Northridge earthquake (PGA = 0.233 g)
insufficient cross-sectional dimensions, longitudinal reinforcement ratio and low
strength of longitudinal bars were the main reasons of the failure.
(a) 10.05 (sec)
(b) 10.20 (sec)
(c) 10.25 (sec)
(d) 10.30 (sec)
(e) 10.35 (sec)
(f) 10.50 (sec)
(g) 10.55 (sec)
(h) 10.60 (sec)
Figure 9: Collapse mechanism of the building without repair
The rehabilitation using FRP
Figure 10 illustrates the enhancement in the columns capacity due to using CFRP
warping which increased the resistance capacity of the building. The FRP
composite materials do not significantly affect the initial stiffness of the concrete
members but improves the strength. Therefore the rehabilitation using FRP
warping for the columns does not significantly change the dynamic response of
the frame. FRP-strengthening significantly changes the damage location and the
collapse mechanism of the frame.
The rehabilitation using RC jacketing
The effect of using RC jacketing for columns in the first floor only is shown in
Figure 11. However, the seismic capacity of the building increased by more than
80%, Figure 11 shows one of the undesirable collapse mechanisms. Due to the
sudden change of the cross-section of columns, the building had a non-ductile
soft-story collapse mechanism at the second floor level. The max story
deformation occurs at the second story and plastic hinges developed at the ends of
columns.
Finally the analysis shows that using the RC jacketing for all columns in all floors
improves the seismic performance of the structure (Figure 12). According to the
figure, the inelastic deformation demand and energy dissipation spread throughout
the structure. Beams commence yield before columns and a beam side-sway
mechanism occurred. Plastic hinges form at critical sections of all beams in the
frame and the bottom of the first story columns.
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Northridge earthquake (PGA = 0.292 g)
Collapse Analysis of RC Structures using Improved Applied Element Method
(a) 9.50 (sec)
(b) 9.60 (sec)
(c) 9.85 (sec)
(d) 9.95 (sec)
(e) 10.50 (sec)
(f) 10.70 (sec)
(g) 11.00 (sec)
(h) 11.50 (sec)
Northridge earthquake (PGA = 0.758g)
Figure 10: Collapse mechanism of FRP Retrofitted building
(a) 23.50 (sec)
(b) 24.00 (sec)
(c) 24.50 (sec)
(d) 24.75 (sec)
(e) 25.00 (sec)
(f) 25.20 (sec)
(g) 25.40 (sec)
(h) 25.60 (sec)
Figure 11: Collapse mechanism of the building with RC jacketing columns in the
ground floor only
5. CONCLUSIONS
The numerical method, IAEM, presented in this paper shows a good capability to
study the total behavior of RC and composite structural buildings from the early
stage of loading until the total collapse occurs. The analysis demonstrated the
Collapse Analysis of RC Structures using Improved Applied Element Method
Northridge earthquake (140% PGA = 0. 0.816 g)
0ctober 2010, Kobe, Japan
(a) 10.00 (sec)
(b) 15.00 (sec)
(c) 20.00 (sec)
(d) 23.00 (sec)
(a) 23.50 (sec)
(b) 23.75 (sec)
(c) 23.85 (sec)
(d) 24.35 (sec)
Figure 12: Collapse mechanism of building with RC jacketing in all floors
reliability of the new IAEM in modeling large scale RC framed structures using
the minimum number of elements. The validity of the developed code has been
demonstrated by a numerical example. Moreover, less computational effort and a
wider applicability for structural analysis have been observed than those of
conventional discrete element methods. The proposed method can provide a better
understanding of the failure mechanism of buildings due to severe ground
motions. Simple two-dimensional analysis tools such as that adopted in this paper
can be used to judge in a qualitative and quantitative manner the damage tolerance
of buildings. Evaluation of the seismic performance of a sample of old existing
school buildings in Egypt is presented. Approaches to reduce the vulnerability of
the building are also highlighted.
REFERENCES
Elkholy, S., H. Tagel-Din, and K. Meguro. Structural Failure Simulation due to
Fire by Applied Element Method. in The 5th Japan Conference on Structural
Safety and Reliability. 2003. Tokyo, Japan.
Elkholy, S. and K. Meguro. Numerical Simulation of High-rise Steel Buildings
using Improved Applied Element Method. in 13th World Conference on
Earthquake Engineering. 2004. Paper No. 930, Vancouver.
Tagel-Din, H. and K. Meguro, Applied Element Method for Simulation of
Nonlinear Materials: Theory and Application for RC Structures. Structural
Engineering and Earthquake Engineering, 2000. 17(2): p. 137-148.
Sap2000 v.12, Structural Analysis Program, Computers and Structures, Inc., CA.,
USA, 2000
Chopra, A.K., 2000. Theory and Applications to Earthquake Engineering. 2nd
edition, ed. P.H. Education.: Prentice Hal
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