SEISMIC POUNDING ANALYSIS OF NEIGHBORING BUILDINGS UNDERLONG-PERIOD GROUND MOTION
Daigoro ISOBE, Department of Engineering Mechanics and Energy, University of Tsukuba
1-1-1 Tennodai, Tsukuba-shi,Ibaraki 305-8573, JAPAN
Email: isobe@kz.tsukuba.ac.jp
Tokiharu OHTA, Collaborative Research Center, Ashikaga Institute of Technology, JAPANTomohiro INOUE, Science Programs Division, Japan Broadcasting Corporation, JAPANFujio MATSUEDA, Kyoryo Consultants Ltd. , JAPAN
Keywords: seismic pounding, long-period ground motion, collapse analysis, FEM, ASI-Gauss technique
SUMMARY
Seismic pounding phenomena, namely the collision of neighboring buildings under long-period ground motion,
are becoming a major issue in Japan. We focused on a specific apartment house called Nuevo Leon buildings inTlatelolco district of Mexico City, with three same type buildings consecutively built with very narrow
expansion joints in between. Two out of the three collapsed totally in 1985 Mexican Earthquake. Using a finite
element code based upon the adaptively shifted integration (ASI)-Gauss technique, a seismic pounding analysis
is carried out on a simulated model of Nuevo Leon buildings to realize the impact and collapse behavior of thebuildings built in the vicinity of each other. The numerical code used in the analysis provides higher
computational efficiency than the conventional code in this kind of problems, and enables us to cope withdynamic behavior with strong nonlinearities including phenomena such as member fracture and elemental
contact. Contact release and re-contact algorithms are also developed and implemented in the code to realize
complex behaviors of structural members during seismic pounding and collapse sequence. According to thenumerical results, collision of buildings may well be generated by the difference of natural periods between theneighboring buildings. This was actually detected in the same type of buildings due to the damages caused by thepast earthquakes. By setting the natural period of the north building to be 25 % longer than the others, the groundmotion with relatively long period of 2 s first caused the collision between the north and the middle buildings.
This eventually led the middle to collapse, followed by the destruction of the north.

1. INTRODUCTION
Seismic pounding phenomena, namely the collisionof neighboring buildings under long-period groundmotion, are becoming a major issue in Japan. In 1985
Mexican Earthquake, many apartment houses
collapsed due to long-period ground motion. It was
crucial especially in Mexico City, which was about
400 km away from the epicenter (see Fig. 1). We
focused on a specific apartment house called Nuevo
Leon buildings in Tlatelolco district of Mexico City,
with three same type 14-story buildings consecutively
built with very narrow gaps of 10 cm in between (seeFig. 2). Two out of the three collapsed totally in thegreat earthquake (see Fig. 3).
We carried out a seismic pounding analysis on asimulated model of Nuevo Leon buildings to realize
the impact and collapse behavior of the buildings
built in the vicinity of each other. We used a finite
element code based upon the adaptively shifted
integration (ASI)-Gauss technique [Lynn and Isobe,
2007] which provides higher computational

zMexico City
400km
Epicenter
Ms8.1

Figure 1 Epicenter of the 1985 Mexican earthquake

Figure 2 Nuevo Leon buildings before the earthquake
efficiency than the conventional code in this kind ofproblems, and enables us to cope with dynamic

integration point and the stress evaluation point
where a plastic hinge is actually formed is expressedas [Toi, 1991]
r = -sFigure 3 Collapse of the Nuevo Leon buildings(south building at the far side, pictureby Marco Antonio Cruz)
behavior with strong nonlinearities including
phenomena such as member fracture and elemental
contact. Contact release and re-contact algorithms arealso developed and implemented in the code torealize complex behaviors of structural members
during seismic pounding and collapse sequence. In
the analysis, we set the natural period of one buildingout of the three to be 25% longer than the others, as
the difference of natural periods was actuallyobserved in the same type buildings caused by thedamage that remained after past earthquakes.
2. NUMERICAL METHODS
General concept of the ASI-Gauss technique incomparison with the earlier version of the technique,the ASI technique [Toi and Isobe, 1993] is explainedin this section. Moreover, algorithms considering
member fracture and elemental contact, andincremental equation of motion for excitation at fixed
points, are described.
2.1 ASI-Gauss TechniqueFigure 4 shows a linear Timoshenko beam element
and its physical equivalence to the rigid bodies-springmodel (RBSM). As shown in the figure, therelationship between the locations of the numerical
-1

s1

0
s

θ1
u1

1
θ2
u2

Linear Timoshenko beam element

θ1

r1
r

θ2

u1

u2
Rigid bodies-spring model (RBSM)
Numerical integration point
Rotational and shear spring connecting rigid bars
(Plastic hinge including the effect of shear force)

Figure 4 Linear Timoshenko beam element andits physical equivalent(1)

In the above equation, s is the location of thenumerical integration point, and r the location wherestresses and strains are actually evaluated. We refer tor as the stress evaluation point later in this paper. s
and r are nondimensional quantities, which takevalues between -1 and 1.
In both the ASI and ASI-Gauss techniques, thenumerical integration point is shifted adaptively whena fully plastic section is formed within an element toform a plastic hinge at exactly that section. When theplastic hinge is determined to be unloaded, thecorresponding numerical integration point is shiftedback to its normal position. Here the normal positionmeans the location where the numerical integration
point is placed when the element acts elastically. By
doing so, the plastic behavior of the element issimulated appropriately, and the converged solution isachieved with only a small number of elements per
member. However, in the ASI technique, thenumerical integration point is placed at the midpointof the linear Timoshenko beam element, which isconsidered to be optimal for one-point integration,
when the entire region of the element behaveselastically. When the number of elements per member
is very small, solutions in the elastic range are notaccurate enough because one-point integration is used
to evaluate the low-order displacement function of
the beam element.
The main difference between the ASI and
ASI-Gauss techniques lies in the normal position of
the numerical integration point. In the ASI-Gauss
technique, two consecutive elements forming amember are considered as a subset, and the numericalintegration points of an elastically deformed member
are placed such that the stress evaluation points
coincide with the Gaussian integration points of themember. This means that stresses and strains areevaluated at the Gaussian integration points ofelastically deformed members. Gaussian integration
points are optimal for two-point integration, and theaccuracy of bending deformation is mathematicallyguaranteed [Press et al, 1992]. In this way, the
ASI-Gauss technique takes advantage of two-point
integration while using one-point integration in actual
calculations.
Figure 5 shows the locations of the numericalintegrations points of elastically deformed elementsin the ASI and ASI-Gauss techniques. The elemental
stiffness matrix, generalized strain and sectional force
increment vectors in the elastic range for the ASI
technique are given by Eqs. (2) and those for the
ASI-Gauss technique by Eqs. (3).

MemberMember

-1

-1/2

0

1/2

1

-1

1

0

-1 -1

0

1

1

Element 1

r g 0 sg

Element 2

Numerical integration point

0

1/√3

1

-1 -1

sg 0 rg1-1/√3

Element 1

Element 2

sg = 1-(2/√3), rg = -1+(2/√3)

Stress evaluation point(b) ASI-Gauss technique(a) ASI techniqueFigure 5 Locations of numerical integration and stress evaluation points in elastic range[K]=L[B(0)]T[D(0)][B(0)](2a){∆ε(0)}=[B(0)]{∆u}(2b){∆σ(0)}=[D(0)]{∆ε(0)}(2c)[K]=L[B(sg)]T[D(rg)][B(sg)](3a){∆ε(rg)}=[B(sg)]{∆u}(3b){∆σ(rg)}=[D(rg)]{∆ε(rg)}(3c)

sg = 1 −2
3,

rg = − 1 +

2

f=

s=s
s=s
s=－1

Plastic hinge

s=0

Node
(a) Elastic stage

ì2

+

í

My
My0

ì2

+

í

N
N0

ì2

Ä 1 = fy Ä 1 = 0(4)

2.2 Member Fracture and Contact Algorithm
A plastic hinge is likely to occur before it develops toa member fracture, and the plastic hinge is expressedby shifting the numerical integration point to the
opposite end of the fully-plastic section. Accordingly,
the numerical integration point of the adjacentelement forming the same member is shifted back to
its midpoint where it is appropriate for one-point
integration. Figure 6 shows the locations of numericalintegration points for each stage in the ASI-Gauss
technique.
Member fracture is determined, in this study, bymeans of curvatures, shear strains and axial tensile
strain occurred in the elements as shown in thefollowing equation.

Here, {∆ε}, {∆σ} and {∆u} are the generalized strain
increment vector, generalized stress (sectional force)increment vector and nodal displacement increment
vector, respectively. [B] is the generalizedstrain-nodal displacement matrix, [D] is thestress-strain matrix, and L is the length of theelement.
The plastic potential used in this study isexpressed by

s=s

Mx
Mx0

Here, fy is the yield function, and Mx , My and
N are the bending moments around the x- and y- axes
and axial force, respectively. Those with the subscript
0 are values that result in a fully plastic section in anelement when they act on a cross section
independently. The effect of torsion and shear force
is neglected in the yield function.

(3d)

3

í

s=0

Fractures=0

Numerical integration point(b) Plastic stage(c) Member fracture

Figure 6 Locations of numerical integration points in each stage

å
å
å
å
å
å
å îx ååååååå Ä 1 ï 0 or å îy å Ä 1 ï 0 or å çxz å Ä 1 ï 0
åîx0 å
åîy0 å
åçxz0 åååå çyz åå Ä 1 ï 0 or èz Ä 1 ï 0
or åå
çyz0 å
èz0[M ]f u°g t+1 + [K ]fÅug = fE g t+1 Ä fF g t(5)where κx, κy, γxz, γyz, εz, κx0, κy0, γxz0, γyz0 and εz0 are
the curvatures around the x- and y-axes, the shear
strains for the x- and y-axes, the axial tensile strain
and the critical values for these strains, respectively.
We fixed the critical values using the informationfrom actual experimental data [Hirashima et al, 2007].
Contact determination is done by examining thegeometrical locations of the elements and once twoelements are determined to be in contact, they arebound with a total of four gap elements between thenodes [Lynn and Isobe, 2007]. The sectional forces
are delivered through these gap elements to theconnecting elements. To express contact release, thegap elements are automatically eliminated at the time
when the mean value of the deformation of gap
elements is reduced to a specified ratio.
2.3 Incremental Equation of Motion
The dynamic equilibrium equation at time step t=t
can be formulated as(6)[M ]f°
ugt = fEgt Ä fF gt

ugt , fEgt and fF g t are the mass
where [M], f°matrix, acceleration vector, nodal external force
vector and internal force vector at time step t=t,respectively.
The following equation is substituted into Eq. (6)at t=t+1 in the implicit code:(7)

fF gt+1 = fF gt + [K ]fÅug

Then the following incremental stiffness equation isevaluated:(8)

where [K] is a stiffness matrix at time step t=t. By
neglecting residual forces, an implicit code is
obtained by evaluating the following incremental
equation of motion:[M ]fÅ u°g + [K]fÅug = 0(9)Consequently, the incremental equation of motion fora structure under excitation at fixed points, which isused in this paper, yields to:[M1 ]fÅ u°g + [M2 ]fÅ u°b g + [K1 ]fÅug + [K2 ]fÅub g = 0(10)

The subscript '1' indicates the coupled terms between
free nodal points, '2' indicates the coupled terms
between free nodal and fixed nodal points, and 'b'
indicates the components at fixed nodal points.
ug and fÅug are the nodal acceleration
Vectors fÅ°
increment and the nodal displacement increment,
respectively.
Under the assumption that the displacements atfree nodal points are estimated by adding quasi-static
displacement increments fÅus g and dynamic
displacement increments fÅud g , the displacements
at free nodal points are given asfÅug = fÅus g + fÅud g(11)fÅus g is evaluated, by neglecting inertia force, as

follows:fÅus g = Ä[K1 ]Ä 1 [K2 ]fÅub g(12)

Substituting Eq. (11) and Eq. (12) into Eq. (10), thefollowing equation is obtained:

South building
Middle building
North building
Figure 7 Numerical models of three connected buildings

Table 1 Sectional properties of the structural members
Columns

Columns(6~10 Fl.)

(11~12 Fl.)

□330×330×10□280×280×9□230×230×7

Beams

Floor slabs

H292×730.0×16.2×11.6□230×230×7

Columns (1~5 Fl.)
Section(mm)
Section(mm)

Table 2 Natural period of each building model
NS
1.5 s
1.2 s
1.2 s

North
MiddleSouth

EW
1.72 s1.65 s1.65 sTable 3 Natural period of Chihuanua, the same type building
NS

EW

Building No.

1

2

3

1

2

3

Natural period (s)

1.39

1.11

1.13

1.94

1.63

1.77

1.25

1.0

1.02

1.19

1.0

1.09

Ratio of period to
No.2 building[M1 ]fÅ u°d g + [K1 ]fÅud g = ([M1 ][K1 ]Ä 1 [K2 ] Ä [M2 ])fÅ u
°b g

(13)

In this scheme, equivalent forces are calculated bysubstituting nodal acceleration increments at fixed
points into the right side of the above equation.
3. SEISMIC POUNDING ANALYSIS OF
NEIGHBORING BUILDINGS
A seismic pounding analysis is carried out, using thenumerical code shown above, on neighboringbuildings to investigate the effects of collisions
between them during a long-period ground motion.
We modeled a specific apartment house, as shown in
Fig. 7, with three same type 14-story buildings
consecutively built with very narrow gaps of 10 cm
in between. The model is 42.02 m high and 12.4 mwide, with a total length of 160 m. By referring to thedesign guideline of Mexico in 1985, the base shear
coefficient is set to 0.06 and the axial force ratio on
the 1st floor to 0.5. The dead load for each floor is set
to 4.0 kN/m2 and the damping ratio to 5 %. The

Nuevo Leon buildings are originally built withreinforced concrete (RC) members; however, themodel constructed in this study is intentionally madewith steel members to easily verify the influence ofstructural parameters such as member fracture strains.
Critical curvatures for fracture are set to 3.333×10-4,
critical shear strains to 3.380×10-4 and critical axial
tensile strain to 0.17, respectively. The critical shear
strains are lowered than those of steel members toconsider the characteristics of RC beams. The
sectional properties of the structural members areshown in Table 1. The time increment is set to 1 ms,
and the calculation is done for a total of 90,000 steps.
The analysis takes approximately 4 days with apersonal computer (CPU: 2.93 GHz Xeon).
In this case, we set the natural period of the northbuilding model to be 25% longer than the others, as
shown in Table 2, by lowering the structural strengthsof the columns. The difference of natural periods was
actually observed in the same type buildings built
near the site (see Table 3), caused by the damage thatremained after past earthquakes [Ciudad de Mexico,

0

EW

100
0

0

50

100

Time [s]

150

UD

20
0

-20

-100

-100

A ccelera tion [gal]100

A cce le ration [gal]A cce leration [ga l]

40

NS

0

50

100

150

-40

Time [s]Figure 8 Ground acceleration (SCT wave)

0

50

100

Time [s]150SouthMiddleNorth59.9 sbetween neighboring buildings. The north building
with a longer natural period first collides with the
middle, consequently changing the natural period of
the middle, which ends up generating other collisions
between the south and the middle. The middle
building first initiates collapse due to continuous
collisions given from both sides, around 73 s from thestart of seismic activity. Although the north building
collapses a few seconds after the middle, the southbuilding withstands the collisions and resists tocollapse.
4. CONCLUSIONSThe numerical result shown in this paper clarifies thepossibility that the long-period ground motion mightcause extra damage on high-rise architectures due tointer-building collisions if the distances between
them are not sufficiently secured. Extra cautions
might be needed if the natural periods of theneighboring buildings are different, which can beeasily fulfilled, for example, by the difference of theirheights.

73.4 s78.8 s84.8 sFigure 9 Collapse behaviors of neighboring buildings
under long-period ground motion
1986]. EW, NS and UD components of SCT seismic
wave shown in Fig. 8 are subjected to the fixed points
on the ground floor. The intensity period of theseismic wave was about 2 s due to the reclaimed soft
soil of Mexico valley. According to the numericalresults such as in Fig. 9, the collision of buildings
may be generated by the difference of natural periods

5. REFERENCES
Ciudad de Mexico. (1986), Programa deReconstruccion Nonoalco/Tlatelolco, Tercera
Reunion de la Comision Tecnica Asesora (theThird Meeting of the Technical Commission
Advises).
Hirashima, T., Hamada, N., Ozaki, F., Ave, T. andUesugi, H. (2007), "Experimental Study on Shear
Deformation Behavior of High Strength Bolts at
Elevated Temperature", Journal of Structural andConstruction Engineering, Vol.621, pp.175-180, inJapanese.
Lynn, K.M. and Isobe, D. (2007), "Finite Element
Code for Impact Collapse Problems of Framed
Structures", International Journal for Numerical
Methods in Engineering, Vol.69, No.12, pp.
2538-2563.
Press, W.H., Teukolsky, S.A., Vetterling, W.T. and
Flannery, B.P. (1992), Numerical Recipes in
FORTRAN: The Art of Scientific Computing, New
York: Cambridge University Press.
Toi, Y. (1991), "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.
Toi, Y and Isobe, D. (1993), "Adaptively Shifted
Integration Technique for Finite Element Collapse
Analysis of Framed Structures", International
Journal for Numerical Methods in Engineering,
Vol.36, pp.2323-2339.