Seismic Damage Finite Element Analysis of Framed Structures
Considering Member Fracture
Daigoro ISOBE 1) and Yutaka TOI 2)
1) Assistant Professor, Institute of Engineering Mechanics, University of Tsukuba, Dr.Eng.
2) Professor, Institute of Industrial Science, University of Tokyo, Dr.Eng.

SUMMARY
The Adaptively Shifted Integration (ASI) technique, which produces the highest computational efficiency in the finite element
analyses of framed structures including static and dynamic collapse problems, is applied to the seismic damage analysis of areinforced concrete building. By expressing member fracture by a plastic hinge located at the exact position with a simultaneous
release of resultant forces in the element, discontinuous problem such as this kind can be easily analyzed even by the conventional
finite element code with the displacemental form. By using the algorithms described in this paper, sufficiently reliable solutions forthe practical use have been obtained in the seismic damage analysis 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. INTRODUCTION
The Great Hanshin-Awaji Earthquake, which occurred inJanuary 1995, had given major damage over the area. Thus thestructural design guidelines for buildings especially against
vertical seismic wave has been thoroughly reconsidered.
Therefore, the arrival of a convenient technique 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 the Distinct Element Method(DEM)[1][3] or the Discontinuous Deformation Analysis(DDA)[2][4] to seismic damage analyses. On the other hand,
generally used finite element code needs some extra
complicated processes to simulate this kind of dynamic collapse
problem, which contains strong nonlinearity and discontinuity
such as member 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 canbe easily implemented with a minimum effort into the existing
finite element codes, is applied to the seismic damage analysis
of a reinforced concrete building. As shown in the previous
papers[6][7], the highest computational efficiency in the finite
element analyses of framed structures including thegeometrically nonlinear, elasto-plastic analysis as well as thegeometrically linear, plastic collapse analysis and the dynamic
collapse analyses have been produced by using the ASItechnique.
In this technique, the numerical integration point in anelastically deformed element is placed at the optimal point for
linear analysis(midpoint in the linear Timoshenko beam), while
the integration point is shifted immediately after the occurrenceof a fully-plastic section in the element, using the previously
established relation between the location of numericalintegration point and that of plastic hinge, to form a plastichinge exactly at the position of that section. Thus the techniqueproduces higher computational accuracy with fewer elements
than the conventional finite element code. Member fracture canbe expressed by a plastic hinge located at the exact position
with a simultaneous release of resultant forces in the element.
The expression of a fracture at one side of member section ispossible by utilizing at least two Timoshenko beam elements asa member. With the proposed technique, discontinuous
problems can be easily analyzed even by the conventional finite
element code with the displacemental form.
2. ASI TECHNIQUE FOR DYNAMIC COLLAPSEANALYSIS
In this chapter, the algorithm of the ASI technique, which isapplied to the dynamic collapse analysis, is explained. By

considering the equivalence conditions between the strainenergy approximations of the linear Timoshenko beam element
and the physical model, the rigid-bodies spring model (RBSM),
the relation between the location of a numerical integration
point (s1) and that of occurrence of a plastic hinge (r1) in thelinear Timoshenko beam element (      ) isobtained. Referring to Fig.1, it is expressed by the followingequation:      where s1 and r1 are the position of the numerical integration
point and the position of the plastic hinge or the fracture to beformed, respectively.
When the entire region in an element behaves elastically, themidpoint 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 theUpdated Lagrangian Formulation (U.L.F.) is expressed as           
where the value in a parenthesis in    indicates thelocation of the integration point, and that in   indicatesthe point at which stresses are evaluated, respectively.
According to the elementary beam theory, the relations between
bending moments, and shear forces  , areexpressed as        

Thus the distributions of the bending moment increments  and  along the element length can beapproximated by the following equations using the bending
moment increments  ,   and the shear force         , at the midpoint in theincrements   element:             
where  is the element length at    . Equations (2.4)
show that the bending moments are subject to a linear change inan element and are likely to take the maximum value on eitherends    . As other resultant forces have constant values
in the element, a fully-plastic state can be determined with ayield function by comparing the calculated distributions fromequations (2.4).

In the dynamic collapse analyses using the ASI technique,member fracture is expressed by shifting the numericalintegration point according to equation (2.1) immediately after
the occurrence of a fractured section on either end of theelement, and by reducing the resultant forces of the element
simultaneously. For instance, if the fully-plastic section or thefractured section has first occurred at the left edge in theelement    , the numerical integration point is shifted
immediately to the right edge in the element    according
to equation (2.1). The released force vector, which operates tothe element at the following incremental step is then expressedby the following equation:             



 

Similarly, if the fully-plastic section or the fractured section has
first occurred at the right edge in the element    , thenumerical integration point is shifted to the left edge in theelement    .

In case of the implicit scheme, the following incremental
stiffness matrices and initial stress matrices are used in thealgorithm:                            

if the entire region in an element behaves elastically, and                             



if the fully-plastic section or the fractured section has firstoccurred at the left edge in the element    .
At the 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 this chapter. Generally, to release the resultant
forces instantly is considered not the best way as thelongitudinal wave might occur in the process. In such cases, agradual release technique or a technique considering artificial
viscosity is used to stabilize the numerical calculation.
However, in this paper, an instant release technique is used tomake the algorithm simple. Rebounds and insertions to theground, or contacts between elements are neglected in thisalgorithm.
It is also to be noted that as elements and node points are stilltreated as a continuous model in the calculation process, newimaginary node points for the fractured sections are needed tobe established at the post processing stage. The elements withthe imaginary node points are then visualized as rigid bars
thereafter.
3. SEISMIC DAMAGE ANALYSIS OF A REINFORCED
CONCRETE BUILDINGIn this chapter, the implicit nonlinear code based on theU.L.F. with the ASI technique is implemented into the existing
finite element code, and the code has been applied to theseismic damage analysis of a five stories-five span reinforced

concrete building as shown in Fig.2.
Generally, many parameters such as shape, steel bar
arrangement, material characteristics as well 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, areused [8].
Dead loads are given statically to the node points 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, JMAKobe), as shown in Fig.3. A summary of the analyzed model isasfollows;
Young’smodulusofPoisson’s ratio ofconcrete  ,concrete , density of reinforced concrete member , compressive strength of concrete , yield stress of tension reinforcement  , tension yield stress of shear reinforcement   . Fracture of themembers is assumed to occur at the critical curvature point). The column section of 1st floor( is  , and that of over 2nd floor is                         
bothsquarely
shaped.
A  ,   reinforced concrete member is used forevery 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, and the distance between each shear reinforcements
is given as .
Newmark’s
method (), well known for its
practicability and usefulness, is used for the direct numericalintegration scheme in this analysis. However, results tend toemit in range of   for sake of its strong     

nonlinearity of the problem. Therefore, which is inrange of     where solutions are likely to bestable though precision falls a little [9], is used in the analysis.
Timeincrement
usedinthisanalysisis   , and input time of the seismic waveis  . The computing time on the EWS(Sun SPARC station 5) is about 190 minutes.
Figure 4 shows the collapse process of a reinforced concretebuilding 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 a collapse chain reaction of columns and beams       

supporting third floor and above. Appearance of 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 a commonly usedvalue in Japan. The structural members falling from upper floor
receive greater reaction force in the process, and thus often end
up flying about in different directions, and finally reaching theground. The mode which differ from actual phenomenon suchas beams going through a floor appears in this analysis, which isdue to the disregard of contacts between elements.
As a result of the analysis, it is estimated that the expressionof the seismic damage process of reinforced concrete building
has become possible by using the algorithm shown in this paper.
On the other hand, the algorithm is confirmed to be practicaland economic by using the Conjugate Gradient method assolver, 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 afractured section exactly at the end of an element, is applied tothe seismic damage analysis of a reinforced concrete building.
The case of five stories-five span reinforced concrete building
under excitation by the accelerogram of Great Hanshin-Awaji
Earthquake is studied. The practical expression of the damage
process has been obtained in the analysis. It is confirmed thatthe algorithm can be applied to the structural design process ofbuildings.
The strong nonlinear problems, in which complicatedprocesses are needed in the conventional finite element code toanalyze, 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 ofTsukuba Research Projects.

REFERENCES[1] P.A.Cundall, A Computer Model for Simulating
Progressive, Large-scale Movement in Blocky Rock System,

Proceedings of the International Symposium on RockMechanics, II-8,129-136,(1971).
[2] G.H.Shi and R.E.Goodman, Discontinuous Deformation
Analysis, Proceedings of 25th U.S. Symposium on RockMechanics, 269-277,(1984).
[3] K.Meguro and M.Hakuno, Simulation of Collapse Processof Structures due to Earthquake, Proceedings of Symposium on
Computational Methods in Structural Engineering and Related
Fields, Vol.15,325-330,(1991).[in Japanese][4] M.Y.Ma, P.Barbeau and D.Penumadu, Evaluation of active
thrust on retaining walls using DDA, Journal of Computing inCivil Engineering, Vol.1,820-827,(1995).
[5] Y.Toi, Shifted Integration Technique in one-dimensional
plastic collapse analysis using linear and cubic finite elements,
International
Journalfor
Numerical
Methods
inEngineering,Vol.31,1537-1552,(1991).
[6] Y.Toi and D.Isobe, Adaptively Shifted Integration
Technique for Finite Element Collapse Analysis of Framed
Structures, International Journal for Numerical Methods inEngineering,Vol.36,2323-2339,(1993).
[7]Y.Toi and D.Isobe, Finite Element Analysis of Quasi-staticand Dynamic Collapse Behaviors of Framed Structures by the
Adaptively Shifted Integration Technique, Computers andStructures,Vol.58,947-955,(1996).
[8] Architectural Institute of Japan, Guidelines on ultimate
strength design of reinforced concrete,70-71.[in Japanese][9] N.Shimizu, Response analysis using personal
computers,Kyouritu-Shuppan,267-268.[in Japanese]