COMPUTATIONAL MECHANICS
New Trends and Applications
S.Idelsohn, E. Oñate and E. Dvorkin (Eds.)©CIMNE, Barcelona, Spain 1998

ASI FINITE ELEMENT ANALYSIS OF DYNAMIC COLLAPSEBEHAVIORS OF FRAMED STRUCTURES CONSIDERINGMEMBER FRACTURE
Daigoro Isobe* and Yutaka Toi*** Institute of Engineering Mechanics
University of Tsukuba
1-1-1 Tennodai Tsukuba-shi, Ibaraki 305-0006, Japane-mail: isobe@kz.tsukuba.ac.jp, web page: http://www.kz.tsukuba.ac.jp/~isobe/** Institute of Industrial Science
University of Tokyo
7-22-1 Roppongi Minato-ku, Tokyo 106-0032, Japane-mail: toi@iis.u-tokyo.ac.jp

Key words: Adaptively Shifted Integration Technique, Explosive Demolition Analysis,
Seismic Damage Analysis, Member Fracture, Reinforced Concrete Building, Finite Element
MethodAbstract. The Adaptively Shifted Integration (ASI) technique, which produces the highest
computational efficiency in the finite element analyses of framed structures including staticand dynamic collapse problems, is applied to the explosive demolition analysis and theseismic damage analysis of a reinforced concrete building. By expressing an explosion or amember fracture by a plastic hinge located at the exact position with a simultaneous releaseof resultant forces in the element, discontinuous problems such as these dynamic collapse
problems can be easily analyzed even by the conventional finite element code with thedisplacemental form. By using the algorithms described in this paper, sufficiently reliable
solutions for the practical use have been obtained in the explosive demolition and seismic
damage analyses of a five stories-five span reinforced concrete building. The present
technique can be easily implemented with a minimum effort into the existing finite element
codes utilizing the linear Timoshenko beam element.

1Daigoro Isobe, Yutala Toi

1. INTRODUCTION
The technology used in the demolition of old aged buildings has always been a major interestin the civil engineering field. Conventional demolition techniques using hydraulic concrete
crusher, concrete cutter or non-explosive demolition agent required a longer period of workand a higher cost. With a strong demand, a demolition technique by controlled explosion
using high explosives has developed in recent years. In Japan, the explosive demolition
technique has recently been used only in a few demolition cases such as buildings of theinternational exhibition in Tsukuba or an unused hotel in Kyoto. The explosive demolition has
a high risk of damaging other neighboring buildings especially in urban area though it
increases work efficiency. Therefore, the pre-assumption by computational analysis has
become an essential matter in the process.
Great Hanshin-Awaji Earthquake, which occurred in January 1995, had given major damage
over the area. Thus the structural design guidelines for buildings especially against verticalseismic wave has been thoroughly reconsidered. Therefore, the arrival of a convenienttechnique to analyze collapse modes of structural objects under three-directional excitation, is
now desired.
However, there are few numerical examples applying computational schemes such as theDistinct Element Method (DEM)1 or the Discontinuous Deformation Analysis (DDA)2 toseveral demolition analyses or seismic damage analyses3,4,5,6,7,8. On the other hand, generally
used finite element code needs some extra complicated processes to simulate these kinds ofdynamic collapse problems which contain strong nonlinearity and discontinuity, such as
fracture occurred in flexural damage or shear damage in reinforced concrete members.
In this paper, the Adaptively Shifted Integration (ASI) technique with the linear Timoshenko
beam element, which can be easily implemented with a minimum effort into the existing finite
element codes, is applied to the explosive demolition analysis and the seismic damage
analysis of a reinforced concrete building. As shown in the previous papers10,11,12,13, thehighest computational efficiency in the finite element analyses of framed structures including
the geometrically nonlinear, elasto-plastic analysis as well as the geometrically linear, plastic
collapse analysis and the dynamic collapse analyses have been produced by using the ASI
technique.
In this technique, the numerical integration points in an elastically deformed element areplaced at the optimal points for linear analysis (midpoint in the linear Timoshenko beam andGaussian integration points in the cubic beam element), while the integration points are
shifted immediately after the occurrence of a fully-plastic section in the element, using thepreviously established relations between the locations of numerical integration points andthose of plastic hinges9 , to form a plastic hinge exactly at the position of that section. Thus thetechnique produces higher computational accuracy with fewer elements than the conventional
finite element code. An explosion or a member fracture can be expressed by a plastic hinge
located at the exact position with a simultaneous release of resultant forces in the element.
The expression of a member fracture at one side of member section is possible by utilizing atleast two Timoshenko beam elements as a member. With the proposed technique,discontinuous problems such as those mentioned above can be easily analyzed even by theconventional finite element code with the displacemental form.
2. ASI TECHNIQUE FOR DYNAMIC COLLAPSE ANALYSESIn this chapter, the algorithm of the ASI technique, which is applied to the dynamic collapse
analyses, is explained. By considering the equivalence conditions between the strain energy

2Daigoro Isobe, Yutala Toi

approximations of the linear Timoshenko beam element and the physical model, the rigidbodies spring model (RBSM), the relation between the location of a numerical integration
point (s1 ) and that of occurrence of a plastic hinge (r1 ) in the linear Timoshenko beam element( − 1 ≤ r1 , s1 ≤ 1 ) is obtained. Referring to Fig.1, it is expressed by the following equation:
s1 = −r1orr1 = −s1(1)where s1 and r1 are the position of the numerical integration point and the position of theplastic hinge or the member fracture to be formed, respectively.
When the entire region in an element behaves elastically, the midpoint of the element (s1 =0)
is the most adequate integration point from the point of view of accuracy and symmetry. Thus
the internal force vector at incremental step n based on the Updated Lagrangian Formulation(U.L.F.) is expressed as[ ][ ][]{}{nn F} = ∫ nl 0 T . u T . n B L (0 ) . nn R (0 ) dl
T

T

n

T

(2 )

R1
where the value in a parenthesis in [nn Bl ] indicates the location of the integration point, and thatin {nn R} indicates the point at which stresses are evaluated, respectively. According to theelementary beam theory, the relations between bending moments R1 , R 2 and shear forces R 5 ,R 6 are expressed as
d R2
R5 = −
(3a )
dz
d R1
R5 = −(3b)dz

3

Daigoro Isobe, Yutala Toi

Thus the distributions of the bending moment increments ∆ n R1 (s ) and ∆ n R 2 (s ) along theelement length can be approximated by the following equations using the bending moment
increments ∆ n R1 (s ) , ∆ n R 2 (s ) and the shear force increments ∆ n R5 (s ) , ∆ n R 6 (s ) at the midpoint in
the element:n∆ n R6 (0) ls∆ n R1 (s ) = ∆ n R1 (0 ) −(4 a )
2∆ R (0)n ls∆ n R2 ( s ) = ∆ n R2 (0) − n 5(4b )2
where n l is the element length at t = t n . Equations (4) show that the bending moments are
subject to a linear change in an element and are likely to take the maximum value on either
ends ( s = Ü 1) . As other resultant forces have constant values in the element, a fully-plastic
state can be determined with a yield function by comparing the calculated distributions fromequations (4).
In the dynamic collapse analyses using the ASI technique, member fracture is expressed byshifting the numerical integration point according to equation (1) immediately after theoccurrence of a fractured section on either end of the element, and by reducing the resultant
forces of the element simultaneously. For instance, if the fully-plastic section or the fractured
section has first occurred at the left edge in the element ( r = Ä 1) , the numerical integration
point is shifted immediately to the right edge in the element ( s = 1) according to equation (1).
The released force vector, which operates to the element at the following incremental step isthen expressed by the following equation:[ ][ ][]{}{nn F} = ∫ nl 0 T . u T . n B L (1) . nn R(− 1) dl
T

T

T

n

(5)

Similarly, if the fully-plastic section or the fractured section has first occurred at the rightedge in the element ( r = 1) , the numerical integration point is shifted to the left edge in theelement ( s = Ä 1) .
In case of the implicit scheme, the following incremental stiffness matrices and initial stressmatrices are used in the algorithm:[ K ] = ∫ [ T ] ⋅ [ T ] [ B (0)] [D (0)][ B (0)] [ T ] ⋅ [ T ] dl
[ K ]= ∫ [ T ] ⋅ [ T ] [ G(0)] [ S (0)][ G(0)] [ T ] ⋅ [ T ] dl
n
n

L

n
n

NL

T

u

T

u

t

T n
n

0

nl

0

nl

e

L

t n
n

T n
n

n
n

t

L

t

n
n

T

0

T

0

T

u

T

u

(6a )
(6b )

if the entire region in an element behaves elastically, and[ K ] = ∫ [ T ] ⋅ [ T ] [ B (1)] [D (− 1)][ B (1)] [ T ] ⋅ [ T ] dl
[ K ]= ∫ [ T ] ⋅ [ T ] [ G(1)] [ S (− 1)][ G (1)] [ T ] ⋅ [ T ] dl
n
n

L

n
n

NL

T

u

u

nl

T

t

T n
n

0

nl

0

L

T n
n

t n
n

p

n
n

n
n

t

L

t

0

T

0

T

T

u

u

T

(7 a )(7b )if the fully-plastic section or the fractured section has first occurred at the left edge in theelement (r ) = −1 .
The location of the numerical integration point in the linear Timoshenko beam element alongwith the rotational and shear springs in the RBSM at the explosion or member fracture phaseis shown in Fig.2.

4

Daigoro Isobe, Yutala Toi

At the member fracture phase in an element, the numerical integration point is shifted to the
opposite end of the fractured section from the center of the element. Therefore, an imaginary
plastic hinge occurs at the fractured section according to the algorithm explained in thischapter. Generally, to release the resultant forces instantly is considered not the best way as
the longitudinal wave might occur in the process. In such cases, a gradual release technique ora technique considering artificial viscosity is used to stabilize the numerical calculation.
However, in this paper, an instant release technique is used to make the algorithm simple.
Rebounds and insertions to the ground, or contacts between elements are neglected in thisalgorithm.
It is also to be noted that as elements and node points are still treated as a continuous modelin the calculation process, new imaginary node points for the fractured sections are needed tobe established at the post processing stage. The elements with the imaginary node points arethen visualized as rigid bars thereafter.
3. DYNAMIC COLLAPSE ANALYSES OF REINFORCED CONCRETE BUILDINGIn this chapter, the explicit nonlinear code based on the U.L.F. with the ASI technique isimplemented into the existing finite element code, and the code has been applied to the
explosive demolition analysis of a five stories-five span reinforced concrete building. Also theseismic damage analysis has been conducted to the same model by the implicit nonlinear code
based on the U.L.F. with the ASI technique.
3.1 Strength characteristics of reinforced concrete member
In this section, strength characteristics of the reinforced concrete member used in theanalyses, is shown.
Generally, many parameters such as shape, steel bar arrangement, material characteristics aswell as stress condition or stress history, contribute big influence to the strengthcharacteristics of a reinforced concrete member. Consequently, the limits of application range
occur with nature for proposal formulas and experimental data. In this paper, the formulas
which are partially excluded for these restrictive matters are used14 .
A tri-linear type model, which has a crack point and an yield point, is employed. The
formulas used as crack and yield strengths are shown below:
Flexural crack strength;
M c = 1.8 Fc ⋅ Z e +

NZ e
Ac

(8)

Flexural yield strength;N M y = 0.8at ⋅ s f y ⋅ D + 0.5 ND1 − bDFc (9 )Shear crack strength;
N 
0.085

Qc = 1 +bj
 K c (500 + Fc )M Qd + 1.7
 150bD 

5(10)

Daigoro Isobe, Yutala Toi

Shear ultimate strength; 0.115k u ⋅ k p (180 + Fc )N 
Qy = 
+ 2.7 p w⋅ s f wy + 0.1
bj
M Qd + 0.115
bD 


(11)

Decrease ratio of stiffness at yield point; d 
M
α y =  0.043 + 1.65n ⋅ pt + 0.043+ 0.33η0  Qd D 

2(12)

where b: Column width, D: Column height, h0 :Inside measurement height of column, d: Validheight =0.9D, Ac : Section area of reinforced concrete member =bD, Fc : Compressive strengthof concrete, sf y: Yield stress of tension reinforcement, sf wy: Tension yield stress of shear
reinforcement, Ze : Section modulus considering reinforcement =1.1Z=1.1 bD2 /6, n: Ratio of
Young's modulus between concrete and reinforcement, N: Normal force, M/Qd: Shear span
ratio =h0 /2d, ë0 : Normal force ratio =N/ AcFc, j: Distance between tension and compression
resultants =7d/8, k c: Coefficient of member height =0.7, k u : Coefficient of member dimension=0.7, kp : Correction factor of tension reinforcement ratio =0.82pt 0.23, pt: Tension
reinforcement ratio =at /bD, pw : Shear reinforcement ratio = at bχ , at : Section area of tension
reinforcement, aw : Section area of one pair of shear reinforcement and ü : Distance between
each shear reinforcements, respectively.
3.2 Explosive demolition analysis of a reinforced concrete building
In this section, the explicit nonlinear code based on the U.L.F. with the ASI technique isemployed to the explosive demolition analysis of a five stories-five span reinforced concretebuilding as shown in Fig.3. The members are divided into two Timoshenko beam elements.
Dead loads are given statically to the node points at initial steps of the analysis, followed by

6

Daigoro Isobe, Yutala Toi

the dynamic sequenced explosions of the member sections in order and time as shown in the
figure.
A summary of the analyzed model is as follows; Young's modulus ofconcrete Ec = 2 .1 × 105 [kg cm2 ] , Poisson's ratio of concrete óc = 0: 1 7 , density of reinforced concrete
member ρ = 2 .4 × 10−3 [kg cm3 ] , compressive strength of concrete Fc = 240[kg cm2 ] , yield stress oftensionreinforcement s f y = 4.0 ×103 [kg cm2 ] ,tensionyield
stressofshear
reinforcement s f wy = 3.0 × 103 [kg cm2 ] . Fracture of the members is assumed to occur at the criticalcurvature point ( kcr = 1 .0 × 10−3 [1 mm] ). The column section of 1st floor is 80[cm]× 80[cm] , and that ofover 2nd floor is 60[cm]× 60[cm], both squarely shaped. A 40[cm]× 40[cm] reinforced concrete
member is used for every lateral beam. Main reinforcement ratio of the columns is 4% for
below 3rd floor, 0.8% (minimum allowable ratio) for upper floors, and 5% for lateral beams,
respectively. The diameter of shear reinforcement in every member is set to 1:3[cm ] , and the
distance between each shear reinforcements ü is given as 30[cm ] .
The theoretical maximum time increment Å t c r for this model is ∆tcr = 6.15 × 10−4 [sec] , and the limit
value of time increment taken in this analysis is ∆t = 5.00 × 10−4 [sec] . The total number of time steps
used in the analysis is 10000 steps and the computing time on the EWS (Sun SPARC station
5) is about 110 minutes.
The sequence of explosions is positively controlled in the analysis to detect the faster and thesafer demolition mode. One of the requirements for a successful demolition is that the floorslabs should become fragmented into small pieces. Therefore, the explosions are sequenced tooccur from the central core outwards as shown in Fig.3, thus causing a destructive shearing
distortion of the stories. The deformed configurations for the reinforced concrete building isshown in Fig.4. As explosion occurs one after another at several beam sections, whole
structure collapses inwards as expected. Appearances which differ from actual phenomenonsuch as beams going through a floor, is due to the disregard of contacts between elements.
However, it is estimated that, the expression of the explosive demolition process by using thecomputational algorithm shown in this paper, is possible. The computational efficiency andthe practicability by using the explicit nonlinear code, are also confirmed.
3.3 Seismic damage analysis of a reinforced concrete building
In this section, the implicit nonlinear code based on the U.L.F. with the ASI technique isemployed to the seismic damage analysis of the same model used in the previous section.
Similarly to the explosive demolition analysis, dead loads are given statically to the nodepoints at initial step of the analysis. The process is followed by dynamic analysis under
excitation at fixed points by three-directional seismic wave (Great Hanshin-Awaji Earthquake,
1995, JMA Kobe), as shown in Fig.5. Newmark's β method ( δ = 1 / 2 ), well known for its
practicability and usefulness, is used for the direct numerical integration scheme in thisanalysis. However, results tend to emit in range of 0 < β ≤ 1 / 4 for sake of its strong
nonlinearity of the problem. Therefore å = 0:4 , which is in range of 1 / 4 < β < 1 / 2 where solutions
are likely to be stable though precision falls a little15 , is used in the analysis. Time increment
used in this analysis is ∆t = 7.50 × 10−3 [sec] , and input time of the seismic wave is 30[sec]( 4000st eps) .
The computing time on the EWS (Sun SPARC station 5) is about 190 minutes.
Figure 6 shows the collapse process of a reinforced concrete building under the Great
Hanshin-Awaji seismic wave. Member fracture occurs first at a column member on second
floor, which has lost their strength by repetition of yielding and unloading, followed by acollapse chain reaction of columns and beams supporting third floor and above. Appearanceof the mode collapsing from upper floor is inevitable because the main reinforcement ratio of
the columns upper floor is reduced to the minimum allowable ratio, which is actually acommonly used value in Japan. The mode which differ from actual phenomenon such as

7

Daigoro Isobe, Yutala Toi

beams going through a floor also appears in this analysis, which is due to the disregard ofcontacts between elements.
As a result of the analysis, it is estimated that the expression of the seismic damage processof reinforced concrete building has become possible by using the algorithm shown in thispaper. On the other hand, the algorithm is confirmed to be practical and economic by using
the Conjugate Gradient method as solver, though the implicit nonlinear code is used.
4. CONCLUDING REMARKSIn the present paper, the nonlinear finite element code using the ASI technique, which can
express a plastic hinge or a fractured section exactly at the end of an element, is applied to the
explosive demolition analysis and the seismic damage analysis of a reinforced concretebuilding.
By using the explicit nonlinear code based on the U.L.F. with the ASI technique, sufficiently
reliable solution for the practical use has been obtained in the explosive demolition analysis ofa five stories-five span reinforced concrete building. The practical information such as falling
patterns or the influence of reinforcements can be obtained from the analyses by studying thesequence of explosions.
On the other hand, the implicit nonlinear code based on the U.L.F. with the ASI technique isapplied to the seismic damage analysis of a five stories-five span reinforced concrete building
under excitation by the accelerogram of Great Hanshin-Awaji Earthquake. The practicalexpression of the damage process has been obtained in the analysis. It is confirmed that thealgorithm can be applied to the structural design process of buildings.
It is confirmed that the strong nonlinear problems, in which complicated processes areneeded in the conventional finite element code to analyze, can be easily analyzed by
implementing the ASI technique to the finite element codes utilizing the linear Timoshenko
beam element.
This research is partially in support of the University of Tsukuba Research Projects..
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