Development of Seismic Damage Finite Element Code forReinforced Concrete Framed Structures
D. Isobe1 and M. Tsuda2

Summary
A new finite element code using the Adaptively Shifted Integration (ASI) technique with a linear
Timoshenko beam element, which can express member fracture by a plastic hinge located at an exact
position with a simultaneous release of the resultant forces in the element, is applied to the seismic
damage analysis of reinforced concrete (RC) framed structures. Contact between members is also
considered in order to obtain results that agree more closely with actual behaviors such as intermediatelayer failure. By using the proposed code, sufficiently reliable solutions have been obtained and theresults reveal that this technique can be used in the numerical estimation of structural reliabilities.

Introduction
In the conventional design of a building, only static analysis in the horizontal and uniaxial
directions is commonly carried out, in order to minimize calculation costs. This approach may ensure
the structural strength of the building, if there is sufficient strength to support the load in verticaldirection. However, the mass system model replaces the building layer in dynamic analysis, and thecomplicated dynamic behavior of the structure at member level is not sufficiently examined. Therefore,
the development of a more precise and more efficient dynamic analysis code is strongly desired.
Recently, significant advances in the field of computers have been removing the calculation cost
restrictions, and various dynamic analysis codes are being developed. In this study, the Adaptively
Shifted Integration (ASI) technique [1] is implemented into the finite element code in order to develop
a more precise and less calculation- time-consuming seismic response analytical tool.
The purpose of this study is to verify the validity of the ASI technique in seismic response analysis
and to construct a highly efficient structural design tool for RC structures. Analyses considering verticalseismic excitation or the phase difference of seismic-wave propagation, and those involving structural
discontinuities such as member fracture are carried out as numerical examples. A contact algorithm isadded to the code to reproduce phenomena such as intermediate-layer failure.

Adaptively Shifted Integration techniqueIn the ASI technique, the numerical integration points in an elastically deformed beam element areplaced at the optimal points for linear analysis (midpoint in the linear Timoshenko beam), and are
immediately shifted after the occurrence of a fully plastic section in the element, using previously
established relation between the location of a numerical integration point and that of a plastic hinge [1],to form a plastic hinge exactly at the position of the fully plastic section. In this manner this techniqueproduces a higher computational accuracy with fewer elements than the conventional finite element
code. Member fracture can also be considered by shifting the numerical integration point with asimultaneous release of resultant forces in the element. In addition, the implementation of the techniquein the finite element code is reasonably easy. Numerical tests concerning the geometrically nonlinear,

1
2

Institute of Engineering Mechanics and Systems, University of Tsukuba, Tsukuba-shi, Ibaraki 305-8573, Japan
Hitachi Software Engineering Co., Ltd., Kanagawa, Japan200

2elem./memb.
4elem./memb.
8elem./memb.
16elem./memb.
32elem./memb.

100

0P,δ

300

load P [kN]load P [kN]

300012200
Kanakubo et al.
2elem./memb.
4elem./memb.
8elem./memb.
16elem./memb.
32elem./memb.

100

03displacement δ [cm]0123

displacement δ [cm](a)Conventional code(b) ASI techniqueFig.1 Simply supported column subjected to shear force

elasto-plastic analysis as well as the geometrically linear, plastic collapse analysis and the dynamic
collapse analysis, are shown in papers [1~3].

Static and quasi-static analyses of RC structures
First, the validity of the ASI technique in the static and quasi-static analyses is verified. In thestatic and quasi-static analyses for a frame member, it is demonstrated that the converged solution canbe obtained by only two-element subdivision per member in the ASI technique, while more elements
are required in order to obtain the converged solution in the conventional finite element code (Fig.1).
Furthermore, it is verified by comparison with the experimental value [4] that the degrading tri-linear
model, which is applied for RC modeling, is appropriate (Fig.2).
As a result of static analysis on a twelve-story, three-span existing structure, it is revealed that themodeling and analysis by the ASI technique are appropriate based on the consideration of output suchas the plastic chart (Fig.3) and the drift angle distribution (Fig.4). The computing time using SUN ultra
5 (CPU: 270MHz, memory: 128MB) is approximately 10 minutes.

Seismic response analysis of RC structures
Second, a seismic response analysis is carried out to verify the validity of the proposed code.
Seismic excitation is applied to the fixed nodal points. Although comparison with the experimental
200

200load P [kN]load P [kN]P,δ10001000-100-100-200-200-2.0 0
2.0 4.0
displacement δ [cm](a) Experiment [4]-2.0 02.0 4.0
displacement δ [cm](b) ASI technique (2elem./memb.)

Fig.2 RC column under repeated quasi-static load

design limit state
ultimate limit state

1211
109

story

87
6
5
4

1/72

1/110

3
2
1

: flexural crack
0: plastic hinge

Fig.3 Plastic chart

0.005

0.01

0.015

drift angle [rad]Fig.4 Drift angle distribution

value reveals that attention is needed for handling the stiffness, inertial mass and damping caused by
the existence of floors and walls, etc., a practically useful solution can be obtained by using theproposed code.
It has been mentioned that very large vertical ground excitation was one of the factors in which thedamage expanded in the Great Hanshin-Awaji Earthquake. By carrying out an analysis under a threedirectional seismic wave, an increase in the number of plastic hinges as well as the differences ofdeformed configurations is observed. As shown in Fig.5, this phenomenon depends upon the ratio ofthe vertical amplitude against the horizontal amplitude. Therefore, the results obtained by dynamic

UD / NS =(333gal/226gal)= 1.47(1-a) NS+EW (25kine) (1-b) NS+EW+UD (25kine)

UD / NS =(333gal/451gal)= 0.74(2-a) NS+EW (50kine) (2-b) NS+EW+UD (50kine)
Fig.5 Plastic chart obtained from three-directional excitation analysis (JMA Kobe)analysis only under horizontal loads may be overestimated to the safe side, and further investigation isrequired.

Seismic response analysis considering the phase difference of seismic-wave propagation
As an application of the proposed code, the effect of the phase difference of seismic-wave
propagation on a structure is estimated. A horizontal seismic wave is assumed to spread from ahypocenter at a constant rate in a concentric circular plane in this study. An input seismic wave is madeto produce a time difference by calculating the time difference in the arrival of the seismic wave bydividing the distance between each fixed nodal point by the horizontal seismic-wave speed. Interactions
between the ground and the structure, the attenuation of the seismic wave and so forth, are notconsidered. Figure 6 shows the plastic charts of an eight-story, three-span building subjected to seismic
excitation with no phase difference, in the cases of wave propagation speeds of 10km/sec and of5km/sec. The burden of the beam increased with the change in the differential arrival time of the input
seismic wave, and the influence against the generated number of plastic hinges is confirmed. Therefore,
the results obtained in the analyses not considering phase difference must be handled carefully in thedesign stage, since there is a possibility of overestimating to the safe side.

EWVsNSPlan: plastic hinge(a) Normal(b) Vs=10 km/s(c) Vs=5 km/s

Fig.6 Seismic response analysis considering the phase difference of wave propagation

Seismic damage analysis considering member fracture and contact
Phenomena with strong nonlinearity and discontinuities such as member fracture are easily
analyzable using the proposed code. However, behavior such as the penetration of a member through afloor could be observed in the analyses [3], since contact between members was not considered.
Actually, some structures observed in the Great Hanshin-Awaji Earthquake were collapsed in theintermediate layer and as a result, the upper layer piled up on the layer. Although there are somenumerical examples using the Distinct Element Method (DEM) that consider contact between members,
there are no application examples using the FEM that can continuously analyze the behavior from theelastic stage to the collapse stage. Thus, a contact algorithm is added to the proposed code and applied
to the seismic damage analysis to reproduce the intermediate-layer failure phenomena.
Member fracture is determined by examining the ductility factor and the shear strain in allelements, and is expressed by shifting the numerical integration point of the fractured element with asimultaneous release of the resultant forces in the element, as shown in Fig.7. The distance between thefractured element and the other element is calculated, and the contact is determined from the distance
between each node and by the condition under which all four nodal points exist on an identical plane.

Once two elements are judged to be in contact, a total of four gap elements are fixed between the nodalpoints, as shown in Fig.8.

(a) Linear Timoshenko
beam element(b) RBSM

Fig.7 ASI technique dealing with member fracture

Figure 9(a) shows a case in which only member fracture is considered, and Fig.9(b) shows theresult when the contact algorithm is added to the code. The total number of elements is 464, and that of
nodal points is 340. The computing time using SUN ultra 5 (CPU: 270MHz, memory: 128MB) is
approximately 50 minutes. By considering the contact between members, the reproduction of seismic
damage observed in actual earthquakes such as intermediate-layer failure became possible.
Gap element
Element in contact

Fractured element

Fig.8 Binding condition of gap element

Concluding remarks
In this paper, a nonlinear finite element code using the ASI technique is used to verify the validity
in static and dynamic response analyses, and it is applied to seismic damage analyses including
structural discontinuities. The fracture of a section is modeled by shifting the numerical integration

2.6 s

3.4 s

4.2 s

5.0 s

(a) Without using contact algorithm

2.6 s

3.4 s

4.2 s

5.0 s

(b) Using contact algorithm
Fig.9 Seismic damage analysis of RC framed structure

point with a simultaneous release of the resultant forces. The proposed code is improved by considering
the contact between members in order to obtain results that agree more closely with actual behavior.
The results reveal that this technique can be used in the numerical estimation of structural reliabilities.

Reference
1 Toi, Y. and Isobe, D. (1993): “Adaptively Shifted Integration Technique for Finite Element
Collapse Analysis of Framed Structures”, International Journal for Numerical Methods inEngineering, Vol. 36, pp. 2323-2339.
2 Toi, Y. and Isobe, D. (1996): “Finite Element Analysis of Quasi-static and Dynamic Collapse
Behaviors of Framed Structures by the Adaptively Shifted Integration Technique”, Computers andStructures, Vol. 58, No.5, pp. 947-955.
3 Isobe, D. and Toi, Y. (2000): “Analysis of Structurally Discontinuous Reinforced Concrete
Building Frames Using the ASI Technique”, Computers and Structures, Vol. 76, No.4, pp. 471-481.
4 Kanakubo, T. and Sonobe, Y. (1992): “Study on Seismic Performance of Reinforced Concrete
Members Using High-Strength Light-Weight Concrete”, Journal of Structural and Construction
Engineering, No.441, pp. 63-72, in Japanese.