A Finite Element Approach for
Structural Collapse Analysis of Buildings
byDaigoro Isobe
University of Tsukuba,Japanisobe@kz.tsukuba.ac.jp

tructural designers usually do not
expect a loss of structural members
in buildings, but these unexpected events
do occur from time to time. However,structural collapse phenomena ofbuildings are very difficult to reproduce
in experiments. They are also difficult tobe simulated numerically, since they arepacked with many mechanical interactions
and strong nonlinear behaviors.

SFigure 1:
Linear Timoshenko beam element and its
physical equivalence to RBSM

The possibilities of simulations consideringa loss of structural members are immeasurable, because the risk of lives often
depend on the behaviors of structures
after they had lost their strengths. Fromthese strong demands, new techniques
are introduced into a finite element code
utilizing linear Timoshenko beam elements
to cope with such discontinuous problems.
The main technique adopted in the codeis called the adaptively shifted integration(ASI) – Gauss technique [1]; it candrastically reduce the computational
cost without losing high levels ofaccuracy, and, at the same time, canhandle fractured sections in structuralcomponents of buildings.

ASI-Gauss TechniqueIn the ASI-Gauss technique, high accuracy
with small number of element subdivision
is achieved by shifting numerical integration points in beam elements adaptively,
when a fully plastic section is determinedin the element, in order to express a plastic hinge exactly at that section in the element. This shifting is operated according
to a relation between the locations of anumerical integration point and a plastichinge [2], which is obtained by considering
the equivalence conditions between thestrain energy approximations of a linear
Timoshenko beam element and a rigidbody-spring-model (RBSM) (Figure 1).

Figure 2:
Locations of thenumerical integration andstress evaluation points
in the elastic range(a) Conventional FEM
iacm expressions 42/18 6(b) ASI-Gauss techniqueFigure 3:
Bird view and upper view of the WTC 2 during aircraft collision(a) 0.12 s after impact

When the plastic hinge is determined to
be unloaded, the corresponding numericalintegration point is shifted back to its initiallocation. By these processes, the plastic
behavior of the element is simulated
appropriately, and the converged solution
can be obtained with a small numberof elements per member. It should benoted here, that the initial location of
the numerical integration point in theconventional finite element method (FEM)
utilizing linear Timoshenko beam element
is at the midpoint, and is always at
the midpoint regardless of the presentcharacteristic phase of the member(Figure 2(a)). The location is considered
to be optimal for one-point integration
when the entire region of the element
behaves elastically. However, the bending
deformations in the elastic range areless accurate when the number ofelements per member is small, since
the displacement functions of the finite
element are defined by linear functions.
Therefore, a simple means is implemented
to improve the accuracy for the elastically(b) 0.28 s after impact(c) 0.56 s after impact

deformed member in the ASI-Gauss
technique; two consecutive elements
are considered as a subset as shown in
Figure 2(b), and the numerical integration
points of an elastically deformed member
are placed such that the stress evaluation
points coincide with the Gaussian integration points of the member. This meansthat the stresses and strains are evaluatedat the Gaussian integration points ofthe elastically deformed members.
These locations are optimal for two-point
integration in the Gaussian quadrature,and the accuracy of the bending
deformation defined by a cubic function
is mathematically guaranteed. In this way,
the ASI-Gauss technique takes advantageof two-point integration while using
one-point integration per element in theactual calculations. Please refer to book[3] and papers [4-11] for further details
on member fracture and elemental contactalgorithms, and some verification andvalidation results.

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Figure 4:
Progressive collapse
sequence of a high-rise
tower with an outrigger
truss system
Applications
Several numerical simulations of high-risebuildings were conducted using thenumerical code utilizing the ASI-Gauss
technique to identify the specific structuralcause of the high-speed total collapse of
the World Trade Center (WTC) towers,
which occurred during the 9.11 terrorist
attacks. Figure 3 shows the numericalFigure 5:
Blast demolition sequence of ahigh-rise tower

result on full-scale aircraft impact
simulation of the WTC tower 2. The
colors in the figure indicate yield function
values with red color indicating that theelement is yielded. The simulation wasconducted to examine the damage andthe dynamic unloading phenomena, a socalled “spring-back phenomena”, that
occurred in the core columns during theimpact. The spring-back phenomena
might have caused the destruction of
the splices between column sections,and consequently, triggered the following
total collapse of the tower with a highpossibility.
A fire-induced progressive collapse
analysis of high-rise buildings withoutrigger truss systems was carried out toqualitatively demonstrate the effects of fire(a) Blast plan(b) Demolition sequence

iacm expressions 42/18 8

Figure 6:
Collapse behaviors of theNuevo Leon buildings
under long-period
ground motion

Figure 7:
Seismic collapse
analysis of theCTV building(a) Ruins of the CTV building

patterns and structural parameters on
the progressive collapse behaviors ofbuildings (Figure 4). The effects of fire
patterns and structural parameters on the
redundant strengths were surveyed byobserving the collapse initiation time: theduration from the beginning of the fire until
collapse initiation. It was confirmed thatcollapse initiation times were significantlyaffected by the member joint strengths
if the axial force ratio was small on thecondition that the fire pattern was nearly(b) Collapse sequence of the CTV building

symmetrical, and the load paths to andfrom the outrigger truss system weresufficiently protected.
Figure 5 shows an example of blast
demolition analysis conducted using thecode. The building was planned to be
demolished in its own wake, by carefully
selecting the blast time interval and thecolumns of specific floors to be removed.
Half of the columns on either side of thefloors were removed sequentially from the9 iacm expressions 42/18

Figure 8:
Behavior of a building without walls under the water line
subjected to tsunami and debris collision(a) Before impact(b) After impact(c) Disintegration of debris

upper floor with a time interval of 0.5 s;
27F (t=0.0 s), 19F (t=0.5 s), 11F (t=1.0 s), 4F(t=1.5 s). As can be seen from the figure,it is successfully demolished without giving
any damages to the concourse area which
stands next to the building. However,there are some remains piled up higher
on its wake, which may become a serious
problem when clearing up the area.
A seismic pounding analysis wasperformed on a full model of the Nuevo
Leon buildings, which consisted of threesimilar buildings built consecutively withnarrow expansion joints between thebuildings, and collapsed completely with
one building remaining in the 1985
Mexican earthquake. It was aimed tounderstand the impact and collapsebehavior of structures, built near eachother, under long-period ground motion.
It can be told from the numerical resultsthat the difference of natural periods
between the adjacent buildings caused byprevious earthquakes may have triggered
the collisions and the collapse (Figure 6).
An investigation was conducted on thefactors that caused the collapse of theCanterbury Television (CTV) building
during the Lyttelton aftershock on
February 22nd, 2011, in New Zealand,
which collapsed with only the north wallcomplex left standing. The collapse
behavior with a clear twist mode vibration
iacm expressions 42/18 10

around the north wall complex wasobserved by carrying out a seismiccollapse analysis (Figure 7). The periodof the twist mode vibration at the southeast corner of the building coincidentally
matched the predominant period of theseismic wave in EW direction, which might
had led to the deterioration of the columns
at the location.
Figure 8 shows a behavior of a steel
frame building in tsunami, which wasobtained by applying seismic groundmotion, fluid forces and debris collision,
continuously in a single simulation.
Since there are no walls under water,
the building withstands the fluid force oftsunami and the following impact force
applied by the debris. These results canprovide information on appropriate designfor tsunami refuge buildings and can also
lift up a discussion to avoid the approachof large debris to such buildings.

Conclusions
The ASI-Gauss code can practicallysimulate the collapse behaviors ofbuildings with very low computationalcost, and the most attractive point of thiscode is that it can be used in any
personal computers with smaller memory
resources. Although member fracture,contact and release are not considered,the developed numerical codes can bedownloaded from the author’s website[12, 13].
Some of the applications described here
belong to real events occurred in thepast. The observed data, if present, arevaluable for the validation of the numericalcode. One must always realize thatvalidations are mostly important forthese kinds of phenomena with strong
non-linearity, because they are packed
with unexpected mechanical interactions.
However, it is evident that the numericalcode based upon a finite elementapproach can provide useful informationsuch as sectional forces acting in eachmember, deformation occurring in thestructures, and so on, of which othermethods based upon non-continuum
mechanics cannot. The applicable fieldof the numerical code is now expanding
from collapse analysis of structures tomotion analysis of indoor non-structuralcomponents, and it is expected to expand
further more. lReferences“ ...it is[1]evident thatthe numericalcode based
upon a finite
element
approach canprovide useful
information ...
deformation
occurring inthe structures... of which
other methods
based uponnon-continuum
mechanics
cannot.”

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]
[13]

K.M. Lynn and D. Isobe. Finite Element Code for Impact Collapse Problemsof Framed Structures, International Journal for Numerical Methods inEngineering, Vol. 69, No. 12, pp. 2538-2563, 2007.
Y. Toi. Shifted Integration Technique in One-Dimensional Plastic Collapse
Analysis Using Linear and Cubic Finite Elements, International Journal for
Numerical Methods in Engineering, Vol. 31, pp. 1537-1552, 1991.
D. Isobe. Progressive Collapse Analysis of Structures: Numerical Codesand Applications, Elsevier, eBook ISBN: 9780128130421, Paperback
ISBN: 9780128129753, 2017.
Y. Toi and D. Isobe. Adaptively Shifted Integration Technique for Finite
Element Collapse Analysis of Framed Structures, International Journal for
Numerical Methods in Engineering, Vol. 36, pp. 2323-2339, 1993.
Y. Toi and D. Isobe. Finite Element Analysis of Quasi-Static and Dynamic
Collapse Behaviors of Framed Structures by the Adaptively Shifted
Integration Technique, Computers and Structures,Vol. 58, No. 5, pp. 947-955, 1996.
D. Isobe and Y. Toi. Analysis of Structurally Discontinuous Reinforced
Concrete Building Frames Using the ASI Technique, Computers and Structures,Vol. 76, No. 4, pp. 471-481, 2000.
D. Isobe and M. Tsuda. Seismic Collapse Analysis of Reinforced Concrete
Framed Structures Using the Finite Element Method, Earthquake Engineeringand Structural Dynamics, Vol. 32, No. 13, pp. 2027-2046, 2003.
D. Isobe, L. T. T. Thanh and Z. Sasaki. Numerical Simulations on theCollapse Behaviors of High-Rise Towers, International Journal of Protective
Structures, Vol. 3, No. 1, pp. 1-19, 2012.
D. Isobe, T. Ohta, T. Inoue and F. Matsueda. Seismic Pounding and Collapse
Behavior of Neighboring Buildings with Different Natural Periods,
Natural Science, Vol. 4, No. 8A, pp. 686-693, 2012.
D. Isobe, W.S. Han and T. Miyamura. Verification and Validation of a Seismic
Response Analysis Code for Framed Structures using the ASI-GaussTechnique, Earthquake Engineering and Structural Dynamics,
Vol. 42, No. 12, pp. 1767-1784, 2013.
D. Isobe. An Analysis Code and a Planning Tool Based on a Key Element
Index for Controlled Explosive Demolition, International Journal of High-Rise
Buildings, Vol. 3, No. 4, pp. 243-254, 2014.
Website of ASIFEM. Finite element code for framed structures using
ASI-Gauss technique, http://www.kz.tsukuba.ac.jp/~isobe/asifem_e.html.
Website of ASIRES. Seismic response analysis code for framed
structures using ASI-Gauss technique,http://www.kz.tsukuba.ac.jp/~isobe/asires_e.html.

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