## IDARC: Computer Program for Inelastic Damage Analysis of Reinforced Concrete Structures |

## Abstract

Objective and Approach

Program for Structural Modelling and Evaluation of Buildings

Identification of Component Model Parameters

Program Validation

Personnel and Institutions

Collaboration

Technical References

Publications

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- Reinhorn, A., Kunnath, S.
*IDARC: Computer Program for Inelastic Damage Analysis of Reinforced* *Concrete Structures.*Research Accomplishments, 1986-1994: The National Center for Earthquake Engineering Research, pages 29-40. (Buffalo : National Center for Earthquake Engineering Research, September 1994)

The need to evaluate structures which experience nonelastic behavior as designed by current seismic codes motivated the development of a specialized computer program which models such behavior in an efficient way. The primary modeling technique employed in the computer program IDARC (Inelastic Damage Analysis of Reinforced Concrete Structures) is the representation of the overall behavior of components in terms of macromodels. Each component of the structural system is discretized into a series of macroelements: beam-column elements whose inelastic behavior is characterized by a primarily flexural response, shear-wall elements that may deform inelastically in flexure or shear, floor-slabs that are capable of deforming inelastically in the plane of the diaphragm, and inelastic rotational springs that may be attached at any node. Special-purpose elements, such as viscoelastic braces and other friction devices for hysteretic energy dissipation were also developed. The reliability of the macroelements is enhanced through the introduction of distributed flexibility models which account for the effects of spread plasticity. Nonlinear material behavior is specified by means of a generic hysteretic force-deformation model that incorporates stiffness degradation, strength deterioration and pinching or bond-slip effects. Solution modules for nonlinear static, monotonic, quasistatic cyclic, and transient seismic loads were implemented. The final response quantities are expressed in terms of damage indices that provide engineers with a qualitative interpretation of the analysis.

All modules are incorporated into a state-of-the-art computer program that executes in
personal computer, workstation, mainframe and supercomputer environments. The program has
been used in a variety of applications ranging from the design of laboratory experiments
and shaking table tests to the seismic evaluation of buildings and bridges. Most
importantly, IDARC continues to grow as a result of the wide interest generated by its
concepts and procedures. Extended versions of IDARC for (1) fully three-dimensional
analysis of structures subjected to multi-directional seismic input and (2) analysis of
bridge systems are currently in the final stages of development.

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The need for efficient and reliable analytical tools to support experimental research on the seismic behavior of reinforced concrete structures led to the development of IDARC. Additionally, NCEER research on nonductile concrete connections and subassemblages required the development of element models with complex hysteretic loops for use in evaluation studies of buildings in the eastern United States.

The development of member and material models followed a "performance-based" approach wherein the models were conceived from and verified against observed experimental behavior. All models were incorporated into a computer program to facilitate analysis of components, subassemblages and entire structures under quasistatic cyclic and seismic loads. A review of available literature on experimental and analytical advances in reinforced concrete indicated that, for the case of a material like reinforced concrete, complex modeling schemes [finite elements or fiber models] do not necessarily result in any significant improvement in the accuracy of predicted behavior. Consequently, it was decided that it would be more prudent to resort to simple yet reliable modeling schemes [macromodeling] that permitted efficient analysis of entire structural systems in the nonlinear range.

This research task is part of NCEER's **Building Project**. Task numbers are:
88-1005B, 89-1201B, 90-l2OlB, 91-3115A, 92-3105, 92-3105A, 93-3101 and 93-3103.

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The first choice facing the IDARC development was a modeling issue. In a general sense, it may be stated that there are two primary approaches to structural modeling of reinforced concrete (RC) structures: the first is based on traditional finite element discretizations (micromodels); and the second is based on macromodel representations. Micromodeling schemes are normally unsuitable for inelastic analysis of entire structural systems primarily because of the immense demand on computational needs. Macromodeling schemes, on the other hand, offer an attractive alternative both in the economy of computation and in the flexibility of modeling. It is possible to account for almost any type of behavior pattern in an equivalent sense. This ability of a macromodel to capture the overall behavior pattern using simplified extensions from the "micro" level makes it ideal for RC modeling. Therefore, the modeling and solutions were based on macromodel approaches with suitable validation via experiments.

It is common knowledge that inelastic deformations in reinforced concrete members do not concentrate in critical sections, but spread across a finite region known as the plastic hinge length. Based on studies of different spread plastic element models, new models were proposed (Kunnath et al., 1990). This model is both an extension and a simplification of the Takizawa (1976) model. Figure 1 presents a qualitative illustration of the distributed element model. The assumption of linear flexibility combined with the possibility of varying the contraflexure point makes it both simple and versatile. The basic incremental moment-rotation relationship is established from the integration of the curvature (M/EI) diagram. These integrals, which eventually result in the element stiffness matrix, can be obtained in closed form and directly used in a computer program. Shear deformations can be directly incorporated into the above formulation. The contraflexure point may be fixed or allowed to vary. In either case, the resulting stiffness matrix is symmetric. The values for stiffness (EI) are obtained directly from the moment-curvature hysteresis model which are governed by rules for material nonlinearities as described in the next section. Improved versions of such distributed flexibility models have also been investigated. A nonlinear distribution was developed for tapered elements (Kunnath et al., 1992) and a multilinear distribution with nonsymmetric properties was developed for the 3D model (Lobo et al., 1992, 1994).

In the macromodel approach, the inelastic behavior is described using force-deformation rules which attempt to capture overall member behavior. In theory, it is possible to construct force-deformation curves using constitutive models. However, constitutive laws hold true only for a microscopic point in the material. For an inhomogeneous material such as reinforced concrete, it will take a very fine discretization of the cross-section to represent the material behavior in terms of local concrete-steel interaction. Such an approach is both tedious and computationally intensive and was not considered as an effective approach in the IDARC formulations.

Force-deformation relationships, on the other hand, are different in character from the underlying constitutive equations since they reflect member behavior as a whole and also include geometrical effects (Ozdemir, 1976). The basis of development of force-deformation models is experimental testing. From observed member behavior under cyclic loading, it is possible to set up a mathematical model of characteristic behavior as described below:

The basic model uses three primary parameters and some secondary related parameters to establish the rules under which inelastic loading reversals take place (figure 2). A variety of hysteretic properties can be achieved through the combination of a nonsymmetric trilinear curve and those control parameters which characterize stiffness degradation, strength deterioration and pinching, respectively.

** Stiffness Degradation: **Experimental evidence indicates
that stiffness degradation is best expressed as a function of attained ductility. The
degradation is not obvious at small ductilities. To model the reduced stiffness, all
unloading branches were directed towards a common target point as shown in figure 2.

** Strength Deterioration:** The modeling of strength decay is accomplished
using two control parameters: ductility and dissipated hysteretic energy. Both parameters
control the amount of strength loss per cycle till the previous maximum deformation is
exceeded. Figure 2 also shows the modeling of
strength decay.

** Pinching Behavior: **Pinched loops are typical in cyclic RC member behavior
due to the presence of high shear forces, or from the opening and closing of cracks, or
the result of rebar slippage at beam column interfaces. This behavior is modeled using a
third primary control parameter which reduces the target force as the load path crosses
the zero force axis as shown in figure 2.

The force-deformation model of member behavior requires the identification of the trilinear monotonic envelope at member-joint interfaces, and the specification of the hysteretic parameters which describe the loading and unloading rules. The specification of hysteretic rules is rather empirical and should be based on available experimental data. Certain default data sets were established for typical connections, which represent average parameters. In a realistic design situation, a parametric study may be necessary to determine the limits of the response. The trilinear envelope, on the other hand, is easily established from a fiber model analysis of the cross-section using section properties and reinforcement data. Complete details of the fiber model analysis procedure is outlined in a recent paper (Kunnath and Reinhorn, 1994).

The component models described above are used to assemble a three-dimensional structural system composed of columns, shear-walls, beams, and inter-connecting slabs and transverse elements. An important aspect of the proposed structural model is its capability to integrate ductile moment-resisting frames with shear-wall models with in-plane and out-of-plane behavior.

The 2D version of IDARC assumes that the floor slabs are sufficiently rigid in their own planes. This limitation is superseded in the 3D version. The rigid fioor approximation permits the individual lateral frame degrees-of-freedom to be reduced to a single master degree-of-freedom. Also, identical frames can simply be lumped together. A special version of the 2D program, IDARC2 (Reinhorn et al., 1988) provided for flexibility of the floor diaphragms by allowing each frame to displace in its own plane. In plane rotations of the floor slab system was also facilitated. Studies on the effects of diaphragm flexibility has been published in a separate paper (Kunnath et al., 1991).

The development of a fully three-dimensional model was facilitated by the formulation of a material model for inelastic biaxial bending interaction (Kunnath and Reinhorn, 1990). The implementation of 3-D geometry with biaxial bending interaction was developed for IDARC3D (lobo et al, 1994). The models for various elements consider inelastic interaction only in cases of comparable characteristics, otherwise the elements are assumed to be governed by uniaxial behavior or elastic interaction.

This module may be considered to be a postprocessor to the main inelastic response analysis. The objective is to provide engineers a qualitative interpretation of the results of the dynamic response analysis. The current release versions of IDARC are equipped with the Park damage model (Park and Ang, 1985). Efforts are currently underway to include two additional damage parameters, one of which has been incorporated through a collaborative effort with Princeton University, and another one, generic, based on experimental data and mechanical principles.

The damage index quantity represented by the Park model can be used directly to determine damage at a joint or component. To extend the concept for the entire building, two additional indices were developed: a story level damage index; and an overall structural damage index. This was accomplished through the use of a weighing factor based on dissipated hysteretic energy.

The additional overturning moments generated by relative inter-story drift are generally referred to as P-delta effects. It arises essentially due to gravity loads and is usually taken into consideration by evaluating axial forces in the vertical elements and computing a geometric stiffness matrix which is added to the element stiffness matrix. In the IDARC formulations, P-delta effects are represented by equivalent lateral forces, equal in magnitude to the overturning moment caused by eccentric gravity forces due to interstory drift.

The use of a distributed flexibility model and the hysteretic force-deformation model present formidable challenges in element state determination. The flexibility matrix for a member needs to be updated for one or both of the following reasons: (a) a transition in stiffness as prescribed by the hysteresis model; and (b) a shift in the contraflexure point. All such changes lead to unbalanced forces between two solution steps. Item (a) can be dealt with using an event-to-event strategy which can be extremely time-intensive. The difficulties associated with a varying contraflexure point is not associated with any predefined event change. Hence, an iterative approach to ensure stability of the final solution is necessary. To expedite the solution process, a predetermined fixed contraflexure point based on the initial loading step (not necessarily at the center of the member) is used.

The solution of the dynamic equation of equilibrium is accomplished by a direct step-by-step integration procedure using Newmark's beta method for constant average acceleration. The step-by-step procedure assumes that the properties of the structure do not change during the time step of analysis. However, since the stiffness of some element is likely to change state during some calculation step, the new configuration may not satisfy equilibrium. A compensation procedure is adopted to minimize this error by applying a single-step unbalanced force correction

In the 3D analysis, the volume of operations required at every step increases substantially. A two-stage analysis is performed to reduce the problem size. First, a push-over (shake down) analysis is performed to identify the degrees of freedom affected and unaffected by inelastic behavior. In a second stage, a condensation of the degrees of freedom unaffected by inelastic behavior is performed and the step-by-step computations are made on the reduced problem.

The sequence and interaction of the various modules of the IDARC program is shown in the flow chart in figure 3.

A compact scheme is used to store the banded symmetric stiffness matrix wherein the main diagonal is offset to the first column and only the remaining half band width is saved. Stiffness matrices are stored at the element level. Element sub-matrices are stored in a manner to enable direct computation of inelastic end moments at the face of the element across the rigid panel zone. The updating of stiffness matrices is carried out only in the event of a stiffness change.

The current version of the program comes with the following analysis options:

nonlinear static analysis for computation of initial stress states under dead and live loads;

push-over failure mode analysis under monotonic lateral loading;

quasi-static cyclic analysis under load or displacement control;

incremental dynamic response analysis under horizontal and vertical seismic excitations; and

a comprehensive damage analysis.

The push-over analysis is a simple and efficient technique to predict seismic response behavior prior to a full dynamic analysis. The usefulness of this method for practicing engineers led to the enhancement of the push-over analysis options in the latest IDARC release. The shake down (push-over) is performed via self adapting loads distributed proportionally to the inelastic resistance of structure.

The program was written in standard FORTRAN-77 and operates in a PC environment under
MS-DOS or MS-Windows; on UNIX or VMS workstations; and on mainframe or supercomputers.

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The behavior of lightly reinforced concrete frame structures has been the subject of numerous investigations at the State University of New York at Buffalo and at Cornell University (see Evaluation and Retrofit of Lightly Reinforced Concrete Buildings in this volume). A 1:3 scaled model was constructed, tested, retrofitted, and retested using simulated earthquake motion generated by the shaking table at the University at Buffalo. IDARC played a significant role in the planning and design of the shaking table tests. Complete details of the IDARC model and analysis of the model building is reported in Kunnath et al. (1992), Bracci et al, (1993a,b). The model was tested by a sequence of ground (shaking table) motions reflecting a low level earthquake (PGA=0.05g), a moderate earthquake (PGA=0.20g) and a severe earthquake (PGA=0.30g). The ground motion was obtained by scaling the acceleration time history of Taft (1952) N21E component. The main purpose of this study was to investigate the effectiveness of using identified component properties from separate sub-assemblage tests in predicting the dynamic response of the total structure. Input data for the analysis was derived entirely from the results of separate interior and exterior beam-column subassemblage tests which provided information on yield strengths and hysteretic behavior. No attempt was made to fit the observed shaking table response. The comparison of response displacements for the top story for the moderate ground motion is shown in figure 4.

A separate study was conducted to simulate the response of a ten story model structure tested at the University of Illinois (Cecen, 1978). Again, details of the modeling may be found in Kunnath et al. (1992). A sample simulation comparing the observed response at the top story with analytical prediction under severe inelastic deformations is shown in figure 5.

The collapse of the Cypress Viaduct during Loma Prieta earthquake in 1989
provided an excellent opportunity to verify IDARC in seismic damage evaluation of an
existing structure. Eleven types of bents were used in the construction of the viaduct.
Fifty-three of these bents were designated as Type B 1, which consists of two portal
frames, one mounted on top of the other. The upper frame is connected to the lower by
shear keys (hinges). B1 bents suffered the most damage and seemed to have failed in the
same consistent manner throughout the freeway. A typical B1 bent was modeled using a
combination of tapered column, shear-panel and beam elements. The pedestal region was
modeled as a squat shear wall so that its impending shear failure could be monitored. The
influence of gravity loads on the structure was simulated by imposing a ramp load in the
form of a vertical excitation with magnitude of 1 g. The actual ground motions were
introduced after the resulting free vibrations had damped out. The IDARC model of the bent
is shown in figure 6. The IDARC analysis revealed
that the first element to fail was the left-side pedestal after approximately 12.5 seconds
into the earthquake. A plot of the damage history of this pedestal is shown in figure 7, in which the horizontal input motion and
the pedestal shear history are also shown for reference. Complete details of the analysis
of the Cypress Viaduct using IDARC is reported in a separate publication (Gross and
Kunnath, 1992).

The case studies presented here are meant to show a representative sample of IDARC
capabilities. The task of modeling different structures vary from case to case, depending
upon the degree of complexity in structural configuration and member connections. While
IDARC must still be regarded as a special-purpose program, it can be used with generality
in the analysis of structures ranging from buildings to bridges and partial subassemblages
used in laboratory testing. The input parameters to the program are obtained directly from
engineering drawings or from separate computations of member properties. The only
exceptions are the input of hysteretic parameters which may be obtained directly from
component tests when available, either by actual testing or from the literature of past
testing of similar configurations and details. The choice of hysteretic parameters is
critical only in the prediction of local failures at a beam-column interface. For systems
with a large number of elements, the overall response is less sensitive to local behavior.
Consequently, the prediction of global damage states is more reliable for single
components, such as single bridge piers, and structures where the damage is more evenly
distributed.

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IDARC developments were motivated by the needs of experimental research and simulation studies of nonductile frame structures performed collabortively by the members of NCEER. Consequently, most of the element and material models, including input and output features, were developed as a result of close interaction with other researchers. These include:

Damage indexing was incorporated as a result of the widespread interest in damage modeling. Collaboration with Princeton University resulted in the implementation of Dr.A. Cakmak's softening model. Collaboration with Cornell University resulted in an expanded definition of damage, including more response parameters. Collaborating with the University of Naples led to important improvement in energy based components.

Joint work with Rice University resulted in a special purpose hysteretic model for analysis of non-ductile flat plate buildings.

Options for quasistatic cyclic testing were developed following the needs for experimental research at Cornell University, Rice University and the University at Buffalo.

Interaction with industry resulted in revised versions of the program with enhanced features for pushover analysis, and availability of a personal computer version.

IDARC was considerably expanded and enhanced following a collaborative effort with the National Institute of Standards and Technology in which IDARC was used in the failure analysis of the Cypress viaduct.

Dames and Moore adopted a modified version of IDARC2 for developing damage limiting seismic design guidelines for gypsum roof structures.

Interaction with University of Naples resulted in first verification of 3D models, enhancements of pushover approach and effective solutions.

Cooperations with National University of Mexico resulted in more efficient numerical solution techniques.

In the development of this analysis package several students completed their Ph.D. work (Kunnath, S., Lobo, R.) and several used its capabilities to define and identify engineering issues in their MS work (Seidel, M., Bracci, J., Hoffman, G., Jenne, C., and Vladescu, A.). A larger number of students were instructed on use of program for further transfer of technology.

This program was developed with user friendly access and distributed to the research
and engineering communities through National Information System for Earthquake Engineering
(NISEE) from University of California at Berkeley. Currently a Users Group was established
and the latest version of the program (Version 3.1) is distributed with support and
interaction with users.

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Florida

Richard White

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Allahabadi, R. and Powell, G.H.,"**;DRAIN-2DX User Guide.**" Technical
Report UCB/EERC 88/06, University of California, Berkeley, 1988.

Cecen, H., "**Response of Ten Story Reinforced Concrete Model Frames to Simulated
Earthquakes **", Ph.D. Dissertation, Department of Civil Engineering, University of
Illinois, Urbana, 1978.

Kanaan, A. and Powell, G.H., "**General Purpose Computer Program for Inelastic
Dynamic Response of Plane Structures,**" Technical Report UCB/EERC 73/06,
University of California, Berkeley. 1973.

Kunnath, S.K. and Reinhorn, A.M., "** Efficient Modeling Scheme for Transient
Analysis of Inelastic RC Structures.**" Microcomputers in Civil Engineering (in
press), 1994.

Kunnath, S.K. and Reinhorn, A.M., "**Model for Inelastic Biaxial Bending
Interaction of Reinforced Concrete Beam-Columns,**" ACI Structural Journal, Vol.87,
No. 3, 1990.

Kunnath, S.K., Reinhorn, A.M. and Lobo, R.E, "**IDARC Version 3.0: A Program for
Inelastic Damage Analysis of RC Structures**" ,Technical Report NCEER-92-0022,
National Center for Earthquake Engineering, Buffalo, New York, August 31, 1992.

Kunnath, S.K., Reinhorn, A.M. and Park, Y.J.,"**Analytical Modeling of Inelastic
Seismic Response of R/C Structures**", Journal of Structural Engineering, ASCE,
Vol. 116, No. 4, 1990.

Lobo R.E, Kunnath, S.K. and Reinhorn, A.M. "**3D Inelastic Dynamic Analysis of RC
Buildings.**"Proceedings, Eighth ASCE Conference on Computing in Civil Engineering,
Dallas, 1992

Ozdemir, H ,"**Nonlinear Transient Dynamic Analysis of Yielding Structures**",
Ph.D. Dissertation, University of California, Berkeley, 1976.

Park, Y.J. and Ang, A. H-S. ,"**Mechanistic Seismic Damage Model for Reinforced
Concrete**", Journal of Structural Engineering, ASCE, Vol. 111, No. 4, 1985.

Park, Y.J., Reinhorn, A.M. and Kunnath, S.K.,"**IDARC: Inelas tic Damage Analysis
of Reinforced Concrete Frame Shear-Wall Structures.**" Technical Report NCEER
87-0008, National Center for Earthquake Engineering Research, Buffalo, New York, July 20,
1987.

Takizawa, H.,"**Notes on Some Basic Problems in Inelastic Analysis of Planar R/C
Structures (Parts I and II)**", Transactions of the Architectural Institute of
Japan, No. 240, 1976.

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Park, Y.J., Reinhorn, A.M. and Kunnath, S.K.,"**IDARC: Inelastic Damage Analysis
of Reinforced Concrete Frame Shear Wall Structures, **"Technical Report NCEER
87-0008, July 20, 1987.

Reinhorn, A.M., Kunnath, S.K. and Panahshahi, N., "**Modeling of RC Building
Structures with Flexible Floor Diaphragms (IDARC2),**" Technical Report NCEER
88-0035, September 7, 1988.

Kunnath, S.K. and Reinhorn, A.M., "**Inelastic Three-Dimensional Reinforced
Concrete Building Structures: Part I - Modeling (IDARC-3D),**" Technical Report
NCEER 89-0011, January 29, 1989.

Bracci, J.M., Reinhorn, A.M., Mander, J.B. and Kunnath, S.K., "** Deterministic
Model for Seismic Damage Evaluation of Reinforced Concrete Structures,**" Technical
Report NCEER 89-0033, September 27, 1989.

Kunnath, S.K., Reinhorn, A.M. and Lobo, R.E, "**IDARC Version 3.0: Inelastic
Damage Analysis of Reinforced Concrete Structures,**" Technical Report NCEER
92-0022, August 31, 1992.

Hoffmann, G.H., Kunnath, S.K., Mander, J.B and Reinhorn, A.M.,"**Gravity Load
Designed RC Buildings: Seismic Evaluation of Existing Construction and Detailing
Strategies for Improved Seismic Resistance,**" Technical Report NCEER 92-0016, July
15, 1992.

Bracci, J.M., Reinhorn, A.M. and Mander, J.B.,"**Seismic Resistance of Reinforced
Concrete Frame Structures Designed Only for Gravity Loads: Part I - Design and Properties
of One-Third Scale Model Structure,**" Technical Report NCEER 92-0027, December 1,
1992.

Bracci, J.M., Reinhorn, A.M. and Mander, J.B., "**Seismic Resistance of
Reinforced Concrete Frame Structures Designed Only for Gravity Loads: Part III -
Experimental Performance and Analytical Study of a Structural Model,**" Technical
Report NCEER 92-0029, December 1, 1992.

Bracci, J.M., Reinhorn, A.M. and Mander, J.B.,"**Evaluation of Seismic Retrofit
of Reinforced Concrete Frame Structures: Part II - Experimental Performance and Analytical
Study of a Retrofitted Structural Model,**" Technical Report NCEER 92-0031,
December 8, 1992.

Lobo, R.E, Bracci, J.M., Shen, K., Soong, T.T. and Reinhorn, A.M., "**Inelastic
Response of R/C Structures with Viscoelastic Braces,**" Technical Report NCEER
93-0006, April 5, 1993.

Bracci, J.M., Reinhorn, A.M. and Mander, J.B., "**Seismic Resistance of
Reinforced Concrete Frame Structures Designed Only for Gravity Loads: Performance of
Structural System,**" ACI Structural Journal (in press), 1994.

Bracci, J.M., Reinhorn, A.M. and Mander, J.B.,"**Seismic Retrofit of Reinforced
Concrete Frame Structures Designed Only for Gravity Loads: Performance of Structural
System,**" ACI Structural Journal (in press), 1994.

Kunnath, S.K., Hoffmann, G.H., Reinhorn, A.M. and Mander, J.B.,"**Gravity Load
Designed RC Buildings: Seismic Evaluation of Existing Construction,**" ACI
Structural Journal (in press), 1994.

Kunnath, S.K., Reinhorn, A.M., Hoffmann, G.H. and Mander, J.B.,"**Gravity Load
Designed RC Buildings: Evaluation of Detailing Enhancements,**" ACI Structural
Journal (in press), 1994.

Lobo, R.E, Bracci, J.M., Shen, K., Soong, T.T. and Reinhorn, A.M., "**Inelastic
Response of R/C Structures with Viscoelastic Braces,**" Earthquake Spectra, Vol. 9,
No. 3, 1993, pp.419-446.

Valles, R., Kunnath, S.K. and Reinhorn, A.M.,"**Simplified Drift Evaluation of
Wall-Frame Structures,**" Microcomputers in Civil Engineering, Vol. 8, 1993, pp.
233-246.

Reinhorn, A.M., Kunnath, S.K. and Mander, J.B.,"**Seismic Design of Structures
for Damage Control,**" Nonlinear Seismic Analysis and Design of RC Structures
(Fajfar and Krawinkler, editors), Elsevier Science Publishers, U.K., 1992, pp. 63-76.

Kunnath, S.K., Reinhorn, A.M. and Abel, J.E, "**A Computational Tool for
Evaluation of Seismic Performance of RC Buildings,**" Computers and Structures,
Pergamon Press, Vol. 41, No. 1, 1992, pp. 151-173.

Kunnath, S.K., Panahshahi, N. and Reinhorn A.M., "**Seismic Response of RC
Buildings with Inelastic Floor Diaphragms,**" Journal of Structural Engineering,
ASCE, Vol. 117, No. 4, 1991, pp. 1218-1237.

Kunnath, S.K., and Reinhorn, A.M.,"**Model for Inelastic Biaxial Bending
Interaction of RC Beam Columns,**"ACI Structural Journal, Vol. 87, No. 3, 1990, pp.
284-191.

Kunnath, S.K., Reinhorn, A.M., Abel, J.E and Park, YJ.,"**A Macromodel Approach
to Practical Analysis of RC Buildings under Seismic Excitations, **" in Computer
Aided Analysis and Design of Concrete Structures, (N. Bicanic and H. Mang, editors),
Pineridge Press, Vol. 2, 1990, pp. 1155-1167.

Kunnath, S.K., Reinhorn, A.M. and Park, YJ.,"**Analytical Modeling of Inelastic
Seismic Response of RC Structures,**" Journal of Structural Engineering, ASCE, Vol.
116, No. 4, 1990, pp. 996-1017.

Reinhorn, A.M., Mander, J.M., Bracci, J.M. and Kunnath, S.K., "**Simulation of
Seismic Damage of RC Buildings in Eastern US,**" Structural Safety and Reliability
(Ang, Shinozuka and Schueller, editors), ASCE Publishing, New York, 1990, pp. 407-414.

Kunnath, S.K., Reinhorn, A.M. and Park, YJ.,"**Seismic Evaluation of RC
Frame-Wall Buildings,**" Concrete International Design and Construction, ACI
Publication, August Issue, 1989, pp 57-61.

Reinhorn, A.M., Kunnath, S.K., Bracci, J.M. and Mander, J.B., "**Normalized
Damage Index for Evaluation of RC Buildings,**" Seismic Engineering: Research and
Practice (Kircher and Chopra, editors),ASCE Publishing, New York, 1989, pp.507-516.

Kunnath, S.K. and Jenne, C., "**Seismic Damage Assessment of Inelastic RC
Structures.**" Proceedings, Fifth US National Conference on Earthquake Engineering,
Chicago, 1994.

Lobo, R.E, Reinhorn, A.M. and Kunnath, S.K., "**Inelastic Response of 3D RC
Structures to Multidimensional Seismic Input,**" Proceedings, Fifth U.S. National
Conference on Earthquake Engineering, Chicago, 1994.

Kunnath, S.K., "**Distributed Flexibility Models for Nonlinear Analysis of RC
Structures,**" Proceedings, Conference on Analysis and Computation, ASCE Structures
Congress, Atlanta, 1994.

Kunnath, S.K., Durrani, A. and Luo, Y.,"**Hysteresis Model for Seismic Analysis
of Nonductile Flat-Plate Buildings,**" Proceedings, 1993 National Earthquake
Conference, Memphis, 1993.

Lobo, R.E, Kunnath, S.K. and Reinhorn, A.M., "**3D Inelastic Dynamic Analysis of
RC Buildings.**" Proceedings, Eighth ASCE Conference on Computing in Civil
Engineering, Dallas, 1992.

Valles, R., Kunnath, S.K. and Reinhorn, A.M., "**Rapid Evaluation of Multistory
Building Drift,**" Proceedings, Ninth National Conference on Microcomputers in
Civil Engineering, Orlando, 1991.

Kunnath, S.K., Reinhorn, A.M. and Panahshahi, N.,"**Computational Modeling of RC
Buildings with Inelastic Floors,**" ACI Spring Convention, Boston, 1991.

Kunnath, S.K., Reinhorn, A.M. and Mander, J.B., "**Seismic Response and
Damageability of Gravity-Load-Designed Buildings,**" Proceedings, Ninth European
Conference on Earthquake Engineering, Moscow, 1990.

Kunnath, S.K. and Reinhorn, A.M., "**Inelastic Response of RC Beam-Columns under
Biaxial Excitations,**" Proceedings, Fourth U.S. National Conference on Earthquake
Engineering, Palm Springs, 1990.

Reinhorn, A.M., Mander, J.B., Bracci, J.M. and Kunnath, S.K., "**A
Post-Earthquake Damage Evaluation Strategy for RC Buildings,**" Proceedings, Fourth
U.S. National Conference on Earthquake Engineering, Palm Springs, 1990.

Panahshahi, N., Kunnath, S.K. and Reinhorn, A.M., "**Inelastic Modeling of RC
Buildings with Flexible Floor Diaphragms,**" Proceedings, Fourth U.S. National
Conference on Earthquake Engineering, Palm Springs, 1990.

Reinhorn, A.M., Mander, J.B. and Kunnath, S.K., "**Damage Based Design and
Evaluation of Structural Systems: Future Approach,**" Developments in Seismic
Design, (A.Scarlat, editor) Israel. 1990.

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