nonlinear analysis of reinforced concrete three-dimensional

CUREe-Kajima Research Project Final Project Report

Nonlinear Analysis of Reinforced Concrete Three-Dimensional Structures

Dr. Takashi Miyashita Dr. Norio Suzuki Mr. Hiroshi Morikawa Mr. Masaaki Okano Mr. Makoto Maruta Mr. Motomi Takahashi

By

,,.. ........... , \ \ \ \

Prof. Graham H. Powell Prof. Filip C. Filippou Mr. Vipul Prakash Mr. Scott Campbell

............ . ,

Report No. CK 92-02 February 1992

California Universities for Research in Earthquake Engineering ( CUREe)

...... ...... ..... ...... ...... . , ,_ ....

t

' .J

Kajima Corporation

CUREe (California Universities for Research in Earthquake Engineering)

• California Institute of Technology • Stanford University • University of California, Berkeley • University of California, Davis

\ • University of California, Irvine • University of California, Los Angel~s • University of California, San Diego /

• University of Southern California

Kajima Corporation I

( • Kajima Institute of Construction Technology • Information Processing Center

I I

• Structural Department, Architectural Design Division • Civil Engineering Design Division • Kobori Research Complex

CUREe-KAJIMA RESEARCH PROJECT

NONLINEAR ANALYSIS OF

REINFORCED CONCRETE THREE-DIMENSIONAL STRUCTURES

CUREe-Kajima Team

consisting of

Takashi Miyashita Norio Suzuki

Hiroshi Morikawa Masaaki Okano Makoto Maruta

Motomi Takahashi (Kajima Corporation)

Graham H. Powell Filip C. Filippou

Vipul Prakash (University of California, Berkeley)

January 15, 1990- July 15, 1991

SUMMARY

This project addresses the development of three closely related computer programs and several advanced

elements for modeling the nonlinear dynamic behavior of reinforced concrete high-rise structures and their

members.

The first computer program deals with the nonlinear static and dynamic analysis of two-dimensional

structures and is called DRAIN-2DX. The second deals with the nonlinear static and dynamic analysis of

three dimensional structures and is called DRAIN-3D and the third program deals with the nonlinear static

and dynamic analysis of three dimensional buildings called DRAIN-BUILDING.

The project also addresses the development of suitable element libraries for these three programs. Several

elements are already complete and some are in the final stages of development. Most of these elements are two-dimensional and thus presently work only with DRAIN-2DX. Three-dimensional beam-column ele

ments have also been developed by the CUREe-Kajima team, and are presently awaiting implementation

in the three-dimensional program DRAIN-BUll..DING. The Kajima beam-column element accounts for

bending, shear and bond-slip deformations and is presently implemented in the in-house computer program.

The UC Berkeley beam-column element presently accounts for bending deformations, only, and is

implemented in a stand alone version. A two dimensional beam-column joint element based on an extension

of the fiber concept is in the final stages of development and will be implemented in DRAIN-2DX.

Finally, the project also addresses issues related to the pre- and post-processing of the results. A set of

interactive tools are proposed to facilitate the data input and the evaluation of the response of the nonlinear

analysis. These tools are, presently, in the final stages of development and should be released shortly.

In order to facilitate the future addition of elements by other researchers and to guide engineers in the

practical use of the program a major part of the total effort was directed towards documenting the modeling

and analysis procedures used in this project.

1. OVERVIEW

Program Architecture

The three developed computer programs consist of a "base" program with an extendable element library.

The programs emphasize the analysis of building structures, rather than general finite element systems, but

are otherwise very flexible. In particular, the programs are not only limited to regular building frames, but

are based on a "construction set" approach, in which the analyst has a wide variety of elements available,

which can be combined together in many ways to form a model for the 2D and 3D analysis of structures.

1

Base Program

The salient features of the base programs are summarized below:

• The program architecture is a "base" program to which an element library is added. The base program

handles everything that is not element specific, particularly the data management and the solution

strategy. It also incorporates a well defined and well documented procedure for adding elements to

t~e library. This approach was pioneered for linear analysis by the SAP program (Wilson 1984), and

for nonlinear analysis by DRAIN-2D (Kanaan and Powelll973).

• The program is able to model a variety of structural types and forms, including girders, columns,

walls, infill panels, girder-to-column connections, frame-to-panel connections, etc. The program

allows a great deal of modelling flexibility, and makes it easy to add new elements to the element

library.

• The program features a simple and well documented procedure for adding new elements to the

element library.

• Non-linear static, as well as dynamic, analyses can be performed. The original DRAIN-20 could

only perform non-linear dynamic analyses.

• Static and dynamic loads can be applied in any sequence.

• Static loads can be applied on elements as well as nodes (e.g., distributed loads along beam element

lengths). The structure is assumed to remain linear elastic under gravity loads.

Mode shapes and frequencies can be calculated. and linear response spectrum analyses can be

performed.

• Dynamic analysis can be carried out for ground accelerations (all suppons moving in phase), ground

displacements (suppons may move out of phase), specified external forces (e.g. wind) and for initial

velocities (corresponding to initial impulse). This option is presently not implemented and will

become available at a later version of the program.

• It is possible to perform both linear and nonlinear analyses on the same model, since linear analysis

is often used to gain insight into the structural response. It is also possible to perform both static and

dynamic analyses, since a great deal can be learned about a structure from its nonlinear static response.

• It is possible to apply nonproportional and cyclic static loads, and to combine static and dynamic

loads. It is also possible to apply static and dynamic loads in any sequence, for example to allow

dynamic loading with subsequent static loading to investigate stiffness deterioration of the structure.

2

• It is possible to consider both material and geometric nonlinearity. Note, however, that for the majority

of civil engineering structures, particularly reinforced concrete structures, geometric nonlinearity

can be accounted for adequately by a P-~ type of theory. It is not necessary to perform a true large

displacement analysis.

The solution strategy is based on the "event-to-event" method. Limit points in the static load to

collapse analysis can be passed using a displacement control option.

• The adopted solution strategy is reliable and automatic. It does not require the analyst to have a deep

knowledge of the solution strategy, and no "coaxing" is necessary to obtain the solution. The computer

program warns the user if it appears that significant inaccuracy has developed.

More sophisticated dynamic step-by-step solution strategies may be specified. In particular, (a) the

time step can be varied automatically (this is valuable for contact problems), and (b) corrections can

be applied to compensate for errors in force equilibrium and energy balance.

• Energy balance computation can be performed, and detailed logs of energy and equilibrium errors

can be obtained. Energy breakdowns by element group are possible.

• The structure state can be saved permanently at the end of any analysis. A new an~ysis can then

begin from any previously saved state.

• Cross sections can be specified through a structure, and the resultant normal, shear and overturning

effects on these sections can be computed.

• Generalized displacements can be specified as a weighted combination of up to 8 displacements, to

compute effects such as interstory drift, shear distortions etc.

• The output data contains information which can be used to make rational damage assessments (Powell

et al. 1988). The analysis results also include an energy breakdown. Output information includes the

input energy, the kinetic energy, and the absorbed elastic, hysteretic and viscous energies in different

parts of the structure.

• Post-processing files can be produced, and an interactive post-processing program has been devel

oped to permit tabulation and graphic representation of results from these files. It is also possible

for users to extend the post-processing capabilities to suit specific needs.

• A reorganization of the program input to use SEPARATORS, and to allow for comments in the input

file. The node numbers need not be in sequence. More generation options are provided for ease of

program input.

• Suppon for compound nodes (the main node with a few subnodes).

• . Suppon for nodal rigid link slaving for DRAIN-2DX and DRAIN-3D, and rigid diaphragm slaving

for DRAIN-BUll . .DING.

3

• Support for modelling the building with help of FLOOR and INTERFLOOR subassemblies of

elements (element group templates) and out-of-core hypermatrix solution for DRAIN-BUll.DING.

• Support for controlled event overshoot for each element. This increases the speed of execution and

allows support of elements which are non-linear throughout their behavior an~ thus, do not have

well defined events.

Element Library

Existing DRAIN elements have been modified and implemented in the new program family. These existing

elements are empirical in nature. New elements in the DRAIN library have been developed by the

CUREe-Kajima team based mostly on "physical" rather than "empirical" models.

For frame-type elements, "physical" models are based on "fiber" concepts, in which a beam or column

cross section is modelled using a number of fibers (not necessarily a large number), each with a specified

uniaxial stress-strain or force-extension relationship. Interaction between axial force and bending moment

is also accounted for directly, without the need to define yield surfaces and flow rules. The Kajima team

extended the basic beam-column element to account for shear and bond-slip deformations and numerous

comparisons of analytical with experimental results demonstrate the validity of this model.

Pre- and Postprocessing

In order to facilitate the preparation of input data and the evaluation of the analysis results separate pre

and post-processorprograms were developed. These will later be integrated with the execution of the main

program for a seamless nonlinear analysis. The pre- and post-processorprogram are interactive and hardware

independent. On PC's they work under Microsoft Windows 3.0 and on Unix workstations under

X-Windows. For this development we have used a software package that also supports other windows

environments such those by Apple, HP and Sun. We intend to deliver a PC version of the pre- and post

processor only. The users will be responsible for compiling and relinking the code with the appropriate

graphics library.

2. SUMMARY OF RESEARCH ACCOMPLISHMENTS

• DRAIN-2DX Extension (complete)

• Basic 3D Program Extension (by December 1991)

• New DRAIN Building Program Development (by December 1991)

• Rigid Diaphragm Option (complete)

4

• 2D Fiber Beam-Column Element for bending and axial force

{UC standalone version complete)

• 3D Fiber Beam-Column Element for bending and axial force which accounts for shear and

bond-slip deformations

{Kajima version implemented in-house)

• Extensive comparisons of analytical with experimental beam-column test results for cases

where shear and bond-slip deformations have a pronounced effect on the hysteretic response

• Interactive Input Data Preparation and Graphics (by December 1991)

• Interactive Post Processing Capabilities {by December 1991)

• Documentation of Element Modelling Procedures (by end of Phase II)

• User Documentation and Example Analyses (by end of Phase II)

3. CONCLUSIONS

~, · The final product of this research project is a flexible analytical platform capable of performing three

dimensional· static and dynamic analysis of reinforced concrete structures with a variety of structural

systems. Several element models were also developed in this project by UC Berkeley and Kajima

Corporation researchers. The developed program has improved interactive capabilities for pre- and post

processing of the static and dynamic analysis. Extensive documentation of the modeling and the analysis

~; , procedures are provided to facilitate the addition of elements by other researchers of CUREe and the use

"' of the program by Kajima engineers.

5

CUREe-KAJIMA RESEARCH PROJECT

NONLINEAR ANALYSIS OF

REINFORCED CONCRETE THREE-DIMENSIONAL STRUCTURES

Professors Graham H. Powell and Filip C. Filippou

Graduate Students Vipul Prakash and Scott Campbell

Department of Civil Engineering

University of California, Berkeley

January 15, 1990- July 15, 1991

SUMMARY

This project addresses the development of three closely related computer programs: the frrst program deals

with the nonlinear static and dynamic analysis of two-dimensional structures and is called DRAIN-2DX.

The second deals with the nonlinear static and dynamic analysis of three dimensional structures and is

called DRAIN-3D and the third program deals with the nonlinear static and dynamic analysis of three

dimensional buildings called DRAIN-BUll..DING.

The project also addresses the development of suitable element libraries for these three programs. Several

elements are already complete and some are in the final stages of development. Most of these elements are

two-dimensional and thus presently work only with DRAIN-2DX. A few three-dimensional elements have

also been developed in stand-alone versions and have not yet been incorporated in the three-dimensional

program versions.

Finally, the project also addresses issues related to the pre- and post-processing of the results. A set of

interactive tools are proposed to facilitate the data input and the evaluation of the response of the nonlinear

analysis. These tools are, presently, in the final stages of development and should be released shonly.

In order to facilitate the future addition of elements by other researchers and to guide engineers in the

practical use of the program a major part of the total effort was directed towards documenting the modeling

and analysis procedures used in this project.

1. INTRODUCTION

It has been possible for several years to perform inelastic dynamic analysis of 3D structures. However,

practical applications have been relatively few outside of the nuclear and offshore industries, and the task

has been one which requires special skills.

The basic need is a nonlinear structural analysis program which can consider a variety of structural types;

is applicable to 3D structures of quite general shape; allows RIC columns, girders, wall, panels and con-

1

nections to be modelled; and is neither over-simplified nor excessively complex for routine use in structural

engineering design. Because computer software is rarely static, it is also important that the program be

designed to allow continued development over time.

2. RESEARCH ACCOMPLISHMENTS

2.1 Program Architecture

Three related programs have been developed in the course of this project. The frrst program deals with the

nonlinear static and dynamic analysis of two-dimensional structures and is called DRAIN-2DX. The second

deals with the nonlinear static and dynamic analysis of three dimensional structures and is called DRAIN-3D

and the third program deals with the nonlinear static and dynamic analysis of three dimensional buildings

called DRAIN-BUILDING.

The three progr:uns consist of a "base" program with an extendable element library. The programs emphasize

the analysis of building structures, rather than general finite element systems, but are otherwise very flexible.

In particular, the programs are not only limited to regular building frames, but are based on a "construction

set" approach, in which the analyst has a wide variety of elements available, which can be combined together

in many ways to form a model for the 2D and 3D analysis of structures.

2.2 Base Program

The starting point of the development was the program DRAIN-2DX (Allahabadi 1987; Powell and

Allahabadi 1988). This program was revised to allow its extension to three-dimensional structures and to

comply with the requirements of the pre- and post-processing tools that were developed in the course of

the project. The most important revision of program DRAIN-2DX concerns the allocation of memory. This

allocation was considerably improved so as to permit the program to run on a variety of platforms including

2

personal computers with a limit of 640 Kbytes in memory. At present the program has been tested on

personal computers and workstations using three different operating systems (Unix, DOS and OS/2) and

three different Fortran compilers.

The salient features of program DRAIN-2DX are summarized below:

• The program architecture is a "base" program to which an element library is added. The base program

handles everything that is not element specific, particularly the data management and the solution

strategy. It also incorporates a well defined and well documented procedure for adding elements to

the library. This approach was pioneered for linear analysis by the SAP program (Wilson 1984), and

for nonlinear analysis by DRAIN-20 (Kanaan and Powell1973).

• The program is able to model a variety of structural types and forms, including girders, columns,

walls, inflll panels, girder-to-column connections, frame-to-panel connections, etc. The program

allows a great deal of modelling flexibility, and makes it easy to add new elements to the element

library.

• The program features a simple and well documented procedure for adding new elements to the

element library.

• Non-linear static, as well as dynamic, analyses can be performed. The original DRAIN-20 could

only perform non-linear dynamic analyses.

• Static and dynamic loads can be applied in any sequence.

• Static loads can be applied on elements as well as nodes (e.g., distributed loads along beam element

lengths). The structure is assumed to remain linear elastic under gravity loads.

• Mode shapes and frequencies can be calculated, and linear response spectrum analyses can be

performed.

3

• Dynamic analysis can be carried out for ground accelerations (all supports moving in phase), ground

displacements (supports may move out of phase), specified external forces (e.g. wind) and for initial

velocities (corresponding to initial impulse). This option is presently not implemented and will

become available at a later version of the program.

• It is possible to perform both linear and nonlinear analyses on the same model, since linear analysis

is often used to gain insight into the structural response. It is also possible to perform both static and

dynamic analyses, since a great deal can be learned about a structure from its nonlinear static response.

• It is possible to apply nonproportional and cyclic static loads, and to combine static and dynamic

loads. It is also possible to apply static and dynamic loads in any sequence, for example to allow

dynamic loading with subsequent static loading to investigate stiffness deterioration of the structure.

• It is possible to consider both material and geometric nonlinearity. Note, however, that for the majority

of civil engineering structures, particularly reinforced concrete structures, geometric nonlinearity

can be accounted for adequately by a P-A type of theory. It is not necessary to perform a true large

displacement analysis.

• The solution strategy is based on the "event-to-event" method. Limit points in the static load to

collapse analysis can be passed using a displacement control option.

• The adopted solution strategy is reliable and automatic. It does not require the analyst to have a deep

knowledge of the solution strategy, and no "coaxing" is necessary to obtain the solution. The computer

program warns the user if it appears that significant inaccuracy has developed.

• More sophisticated dynamic step-by-step solution strategies may be specified. In particular, (a) the

time step can be varied automatically (this is valuable for contact problems), and (b) corrections can

be applied to compensate for errors in force equilibrium and energy balance.

• Energy balance computation can be performed, and detailed logs of energy and equilibrium errors

can be obtained. Energy breakdowns by element group are possible.

4

• The structure state can be saved permanently at the end of any analysis. A new analysis can then

begin from any previously saved state.

• Cross sections can be specified through a structure, and the resultant normal, shear and overturning

effects on these sections can be computed.

• Generalized displacements can be specified as a weighted combination of up to 8 displacements, to

compute effects such as interstory drift, shear distortions etc.

• The output data contains information which can be used to make rational damage assessments (Powell

et al. 1988). The analysis results also include an energy breakdown. Output information includes the

input energy, the kinetic energy, and the absorbed elastic, hysteretic and viscous energies in different

parts of the structure.

• Post-processing files can be produced, and an interactive post-processing program has been devel

oped to permit tabulation and graphic representation of results from these files. It is also possible

for users to extend the post-processing capabilities to suit specific needs.

The following enhancements will be implemented in the new release of the three DRAIN programs and

target user guides for these programs have already been prepared and distributed to KAJIMA.

• A reorganization of the program input to use SEPARATORS, and to allow for comments in the input

file. The node numbers need not be in sequence. More generation options are provided for ease of

program input.

• Support for compound nodes (the main node with a few subnodes).

• Support for nodal rigid link slaving forDRAIN-2DX and DRAIN-3D, and rigid diaphragm slaving

for DRAIN-BUaDING.

• Support for modelling the building with help of FLOOR and INTERFLOOR subassemblies of

elements (element group templates) and out-of-core hypermatrix solution for DRAIN-BUaDING.

5

• Support for controlled event overshoot for each element. This increases the speed of execution and

allows support of elements which are non-linear throughout their behavior and, thus, do not have

well defined events.

DRAIN-2DX and DRAIN-3D programs with the above features have been written, but not yet tested. Work

is continuing on writing the program DRAIN-BUILDING, which is about half done, and on modification

of the elements for supporting event overshoot. The implementation and testing of all the three programs

will be done during Phase II of the CUREe-Kajima project. Other features which may come to mind later

may also be incorporated at some stage. The writing and implementation of dynamic analyses for ground

displacements, specified external forces and for initial velocities (corresponding to initial impulse) is

planned after the programs have been finalized, as these analyses options are expected to largely use the

same code as for ground acceleration analyses.

2.3 Element Library

When creating an analysis model, an analyst should not need to perform a great deal of preliminary

computation. Ideally, the analyst should be required to provide data on only the structure geometry and the

material properties, and should not be required to precompute properties such as moment-curvature rela

tionships or hysteresis loops. This is an ideal, however, and must be tempered with reality. In principle, it

is possible to model a structure as a mesh of solid, nonlinear 3D elements, leaving nothing to the judgement

of the engineer. This is clearly impractical, however (and not necessarily more accurate). Hence, special

purpose structural elements will inevitably be required, and they will be to some extent empirical. The

important goal is to make the elements as rational as possible, with small numbers of empirical parameters

which can be easily calibrated.

To meet the goal of developing rational elements several new elements were developed in this project, in

conjunction also with similar work under a project sponsored by the National Science Foundation on Precast

Seismic Structural Systems (PRESSS). These elements are: a fiber beam-column element, a fiber beam

column joint element, a panel element and improved versioiiS of gap and link elements. Several old elements

6

(such as the 2D plastic hinge beam-column element) of DRAIN-2D have been modified and included in

the element library, which presently consists of some 13 elements. Many of these elements have been

checked, but more work is certainly required. We hope to complete this work in the Phase II of the

CUREe-Kajima project.A list of elements which includes a short description of the theory and an input

data section from the manual is given in Appendix A.

2.3.1 Rational element models

New elements in the DRAIN library are based mostly on "physical" rather than "empirical" models. A

physical model is one in which the element is conceived as an assemblage of bars, fibers, springs, hinges,

etc., each of which has relatively simple behavior. These components then interact to create the complex

behavior of the complete element. In contrast, the behavior of an empirical, or "phenomenological", model

is defined in terms of empirically determined functions and rules (Banon et al. 1981; Meyer et al. 1983;

Saiidi 1982). One advantage of a physical model is that it can readily be made "logically complete", meaning

that no matter what the current state of an element, or how it arrived at that state, its subsequent behavior

is always defined. With empirical models, it can be difficult to define sufficient rules to ensure logical

completeness, with the result that in some situations the rules either fail to define the subsequent behavior,

or define behavior which is unreasonable.

For frame-type elements, "physical" models are based on "fiber" concepts, in which a beam or column

cross section is modelled using a number of fibers (not necessarily a large number), each with a specified

uniaxial stress-strain or force-extension relationship (Kaba and Mahin 1984; Zeris and Mahin 1988). The

cross section properties then follow by summation of the fiber properties, and the need to predetermine

moment-curvature or similar relationships for a complete cross section is avoided. Interaction between

axial force and bending moment is also accounted for directly, without the need to define yield surfaces

and flow rules. Where data is available in experimental form, the usual approach will be to create a physical

model and calibrate it against the empirical data. However, the computer program is able to accept strictly

empirical models.

7

Fiber beam-column element

Fig. 1 shows a simplified reinforced concrete column section and a fiber representation of the section. In

the fiber representation the section is divided into a number of steel and concrete longitudinal fibers, and

each is assigned a uniaxial stress-strain relationship. These relationships are empirical, based on observed

behavior of steel and concrete. Note, in particular, that different stress-strain curves are assumed for confined

and unconfined concrete. In order to assign the confined curve it is necessary to account for the amount of

confinement and its effect on the stress-strain curve. A truly rational model would use a inultiaxial model

for concrete and consider the confining steel directly. However, such a model would be much more complex.

A major advantage of the fiber representation is that it allows P-M interaction to be considered without

postulating multiaxial yield surfaces and flow rules. Figure 2a shows that a two-fiber model with simple

yielding fibers provides a P-M interaction surface that is a reasonable approximation of that for a steel I

section. A three-fiber model (not shown) gives a hexagonal yield surface. Figure 2b shows that a model

with two yielding steel fibers combined with two yielding and cracking concrete fibers gives a P-M

interaction surface of reinforced concrete type (the four-fiber interaction surface is actually somewhat more

complex than that shown). As the number of fibers is increased, the interaction surfaces more closely

approximate those of actual cross sections. Hardening and softening behavior is also captured, without the

need for complex hardening rules.

The fiber procedure can also be applied to hinges. In order to obtain the force-moment-strain-curvature

behavior for a section, the fibers are assigned stress-strain curves. The force-moment-extension/rotation

behavior of a lumped hinge can be obtained by assigning force-extension relationships to zero-length fibers

in a hinge. This is more empirical, since the hinge seeks to capture inelastic behavior along the element

length as well as over the cross section, but the model can be useful.

Models of both hinges and cross sections have also been developed with only small numbers of fibers (as

few as 4 or 5 for a three dimensional concrete element). Since the number of fibers is small, the fiber

stress-strain curves are not those of the basic steel and concrete materials. Instead, they are empirically

determined curves which combine steel and concrete properties into one fiber, and which also account for

8

the small number of fibers. Although these curves must be determined empirically, elements based on small

numbers of fibers are computationally much less costly than more rational elements with large numbers

of fibers. The issue of computational cost is addressed later in this paper.

Extension of fiber concept to shear deformation

The controlling mode of inelastic deformation for a frame member is ideally flexural, since this provides

the greatest ductility. Some frame members may, however, be controlled by shear, particularly in structures

designed to old design codes. Inelastic shear deformation is less ductile than flexural deformation, and it

could be argued that shear failure will immediately be followed by structural collapse. However, a structure

may be able to redistribute load in such a way that local shear failure does not lead to overall collapse.

Also, inelastic shear deformation is not entirely brittle, and there may be sufficient ductility to avoid major

damage. It may be important, therefore, to model inelastic shear deformation, and the interaction of shear

forces with axial forces and bending moments. It is possible to extend the fiber modelling concept to include

shear deformation. An outline of the procedure is as follows.

Figure 3a shows a column section, and Figure 3b shows its fiber representation. Figure 3b also shows a

side view of a beam-column "slice" of infinitesimal length dx. This slice is a cuboid of dimension b by d

by dx. Its behavior under axial force and bending moment is defined by the properties assigned to the

longitudinal steel and concrete fibers, as already considered. In addition, the slice may deform in shear.

Shear resistance is provided by the concrete and by transverse shear reinforcement In Figure 3b the shear

reinforcement is shown as a transverse fiber. The fiber area is the shear reinforcement area per unit length

multipled by dx. Figure 3c indicates the way in which shear deformation is usually modelled in beam

elements: the shear force produces shear deformation in the slice, based on the shear stiffness of the material.

Since concrete has a large shear stiffness, the amount of shear deformation of this type is small and can

usually be ignored. Figure 3d shows that shear deformation can also be caused by diagonal cracking. In

this figure, the cracks are shown as opening with zero sliding parallel to the cracks, corresponding to perfect

aggregate interlock. In fact there will be sliding as well as crack opening, but the essential behavior is

similar to that shown. In particular, the cracking produces not only shear deformation but also longitudinal

9

and transverse extension. There is thus interaction between shear cracking and axial effects (axial force

and bending moment). Shear cracking also deforms the transverse reinforcement in tension, so that the

reinforcement resists shear. By adding cracking degrees of freedom to the slice, it is possible to account

for shear cracking, to include the shear reinforcement fiber in the model, and to account for interaction

among shear force, axial force and bending moment The theory is presently under development by the

authors. It is planned to extend it to incude torsional shear cracking in three dimensional members, by

allowing a spiral pattern of cracks.

Connection Deformation

In frame analysis it is usual to assume that the beam-to-column connections are rigid, in the sense that the

beams and columns remain at right angles to each other as the frame deforms. It is well known, however,

that there can be significant deformation in these connections. In steel frames this deformation is caused

by high shear stresses in the "panel zone" of the connection. In concrete frames it is caused by shear cracking

and by bond slip in the joint region. In some cases the connection may be the weakest part of the frame,

and large deformations can be present. In order to develop a rational model for inelastic frame analysis it

may be necessary to model these deformations.

Connection models which are similar in concept to a plastic hinge are the simplest. In these models the

beam elements to the left and right of the connection are rigidly connected to each other, and similarly the

column elements above and below the connection. However, the beams and columns are not joined rigidly

but are connected by a deformable connection element. Any moments which are transferred from the beams

to the columns thus pass through this element, causing it to deform. Inelastic connection behavior is modelled

by assigning inelastic properties to the connection element. This type of element can be useful, but since

its inelastic properties must be assigned empirically, it has the same weaknesses as other empirical elements.

Reinforced concrete connections are complex, and it is desirable to capture the underlying behavior in the

analysis model.

10

(3) Subject both analysis models to the same loadings, in the form of imposed end forces and/or dis

placements. The load magnitudes should be those expected to act on the element in the complete

structure.

( 4) Compare the results. If they agree within engineering accuracy, the empirical properties assigned in

Step 2 are "correct". If not, revise these properties and repeat. That is, calibrate the empirical model

against the rational model. It might be possible to automate the process using formal system iden

tification techniques.

(5) Repeat the process for a representative selection of members.

(6) Analyze the complete structure using the calibrated empirical models for the elements.

(7) From the analysis results, confirm that the loadings on the elements are similar to those used in Step

3, and hence that the calibration is correct.

With this approach, the overall computational cost can be minimized, and the engineer can have confidence

in the analysis results. In the future, as computers get faster, it will be possible to use elaborate models

directly in the analysis of the complete structure.

This procedure addresses an important preprocessin~ problem. Problems of computational cost can also

arise in postprocessin~. During a nonlinear analysis it is usual to save histories of node displacements and

element response. These saved results are then used for postprocessing. For a large structure the amount

of data to be saved can be immense, straining the capacity of even very large disks. This is especially true

for dynamic analysis. Also, the element data to be saved must be specified before the analysis starts, and

the engineer can not always be sure that all needed element response data has been saved.

A procedure which can be used to solve this problem is as follows.

(1) During the analysis, save time- or load-histories of node displacements, but not of element responses.

The amount of data to be saved is thus greatly reduced.

12

(2) Identify the elements which are likely to have the most damage, by examining the maximum (en

velope) values of element forces and deformations. These envelopes are computed during the

analysis.

(3) Choose one of these elements. Extract, from the histories of node displacements, a history of the

displacements at the nodes to which the element connects.

(4) Set up an analysis model containing only this single element, and analyze it for these node dis

placements. During the analysis, calculate and output damage measures, and other pertinent results.

(5) Repeat for other potentially critical elements.

In Step 4 this process repeats calculations already performed during the main analysis. However, there can

be a cost saving because histories of element response no longer need to be saved, and because the single

element calculations can be performed on low cost microcomputers. Also, decisions do not need to be made

on which element responses to save during the main analysis. The postprocessing analysis recreates all of

the detailed time history information on the element state, and can process this information in ways which

may not have been foreseen during the main analysis.

2.4 Nonlinear Solution Strategy

The ideal solution strategy for nonlinear structural analysis is one which is simple to specify, reliable and

computationally efficient. Unfortunately, no such strategy exists, and trade-offs are inevitable (Bergan et

al. 1978; Riks 1979; Crisfield 1981; Powell et al. 1984).

For analysis under static load, the "event-to-event" strategy has advantages in terms of simplicity and

reliability, but tends to be more expensive computationally (Simons and Powell 1982). In this strategy, the

"exact" behavior path of the model is traced out in the numerical scheme, from stiffness change to stiffness

change, each change being an "event", and the structure stiffness is modified at each event. In contrast,

iterative methods can be less expensive because they modify the stiffness less often, but tend to be unreliable

13

(Bergan et al. 1978). The event-to-event strategy is reliable precisely because it traces out the exact behavior

of the model, and hence the computed behavior does not depart from the equilibrium path. Iterative strategies

can depart substantially from the equilibrium path, and frequently have trouble finding their way back.

This can be especially true for analyses of frame models, which develop localized nonlinearities and are

often poorly behaved for iterative solutions. For small models, with simple elements which have few

stiffness changes, the computational effort for the event-to-event strategy can be modest. However, for

large models, with complex elements which have many stiffness changes, the number of events is larger,

and the cost of modifying the stiffness is greater for each event. The cost thus tends to grow rapidly.

Fortunately, the speed of computers is continually increasing, so that analyses which were of mainframe ~

scale a few years ago are now of microcomputer scale. Hence, simplicity and reliability are likely to be the

most important criteria. This suggests the use of a strategy which has at least an event-to-event flavor.

For analysis under dynamic load, the choice of strategy is affected by the need to perform the analysis for

many small time steps. Because of this, the difference in computational cost between event-to-event and

iterative solutions tends to be less (Bergan et al. 1978). A more important consideration, however, is the

ability to vary the time step automatically during the analysis, so that large time steps are used when possible

and small steps only when necessary (Golafshani 1982).

An event-to-event type of strategy has been used successfully in both the original DRAIN-2D program and

the DRAIN-2DX extension, and has proven to be both reliable and economical (Kanaan and Powell1973;

Allahabadi and Powel11988). It also has advantages for the computation of energy balances. In DRAIN-2DX

the strategy is combined with automatic time step variation for dynamic analysis, and also allows for

iteration. This basic strategy underlies the new DRAIN family of programs. It is presently been extended

to allow for event overshoot, thus permiting the use of nonlinear force-deformation relations in the elements.

2.5 Pre- and Postprocessing

In order to facilitate the preparation of input data and the evaluation of the analysis results separate pre

and post-processor programs were developed. These will later be integrated with the execution of the main

14

program for a seamless nonlinear analysis. The pre- and post -processor program are interactive and hardware

independent. On PC's they work under Microsoft Windows 3.0 and on Unix workstations under

X-Windows. For this development we have used a software package that also supports other windows

environments such those by Apple, HP and Sun. We intend to deliver a PC version of the pre- and post

processor only. The users will be responsible for compiling and relinking the code with the appropriate

graphics library.

3. SUMMARY OF RESEARCH ACCOMPLISHMENTS

• DRAIN-2DX Extension (complete)

• 2D Fiber Beam-Column Element (complete)

• Basic 3D Extension (by December 1991)

• New DRAIN Building Program Development (by December 1991)

• Rigid Dia:>hragm Option (complete)

• Interactive Input Data Preparation and Graphics (by December 1991)

• Interactive Post Processing Capabilities (by December 1991)

• Documentation of Element Modelling Procedures (by end of Phase ll)

• User Documentation and Example Analyses (by end of Phase ll)

4. CONCLUSIONS

The final product of this research project is a flexible analytical platform capable of performing three

dimensional static and dynamic analysis of reinforced concrete structures with a variety of structural

systems. Several element models were also developed in this project by U.C. Berkeley and Kajima

15

Corporation researchers (see companion report by Kajima). The developed program has improved inter

active capabilities for pre- and post-processing of the static and dynamic analysis. Extensive documentation

of the modeling and the analysis procedures are provided to facilitate the addition of elements by other

researchers of CUREe and the use of the program by Kajima engineers.

S. ACKNOWLEDGEMENTS

This work .was performed under a CUREe-Kajima project. This support is gratefully acknowledged. Our

counterparts in the Kajima corporation who contributed to the success of this work are Dr. Takashi Miy

ashita, Dr. Norio Suzuki and Mr. Masaaki Okano. We would like to thank them for their commitment and

patience during the development of the DRAIN family of programs. The opinions of this report are those

of the authors and do not reflect the views of CUREe-Kajima.

6. REFERENCES

Allahabadi, R. (1987). "DRAIN-2DX, Seismic Response and Damage Assessment for 2D Structures",

PhD. Dissertation, University of California, Berkeley.

Allahabadi, R. and Powell, G.H. (1988). "DRAIN-2DX, User Guide", Earthquake Engineering Research

Center, Repon No. EERC 88-06, University of California, Berkeley.

Ban on, H., Biggs, J.M. and Irvine, M.H. (1981 ). "Seismic Damage in Reinforced Concrete Frames" ,Journal

of the Structural Division, ASCE, Vol. 107, No. ST9.

Bergan, P.G. et al. (1978). "Solution Techniques for Nonlinear Finite Element Problems", International

Journal for Numerical Methods in Engineering, Vol. 12, pp. 1677-1696.

Crisfield, M.A. (1981). "A Fast Incremental/Iterative Solution Procedure that Handles 'Snap-Through'",

Computers and Structures, Vol. 13, pp. 55-62.

Golafshani, A. (1982). "A Program for Inelastic Seismic Response of Structures", PhD. Dissertation,

University of California, Berkeley.

16

Kaba, S.A. and Mahin, S.A. (1984). "Refmed Modeling of Reinforced Concrete Columns for Seismic

Analysis", Eanhquake Engineering Research Center, Repon No. EERC 84-03, University of Cali

fornia, Berkeley.

Kanaan, A.E. and Powell, G.H. (1973). "General Purpose Computer Program for Inelastic Dynamic

Response of Plane Structures", Eanhquake Engineering Research Center, Repon No. EERC 73-06,

University of California, Berkeley.

Meyer, C., Roufaiel, M.S. and Arzoumanidis, S.G. (1983). "Analysis of Damaged Concrete Frames for

Cyclic Loads", Eanhquake Engineering and Structural Dynamics, Vol. 11, pp. 207-228.

Powell, G.H. et al. (1984). "WIPS-Computer Code for Whip and Impact Analysis of Piping Systems-Part

B-Theory Manual", Lawrence Livermore National Laboratory, Livermore, California.

Powell, G.H. and Allahabadi, R. (1988). "Seismic Damage Prediction by Deterministic Methods: Concepts

and Procedures", Earthquake Engineering and Structural Dynamics, Vol. 16, pp. 719-734.

Riks, E. (1979). "An Incremental Approach to the Solution of Snapping and Buckling Problems", Inter

national Journal of Solids and Structures, Vol. 15, pp. 529-551.

Saiidi, M. (1982). "Hysteresis Models for Reinforced Concrete", Journal of the Structural Division, ASCE,

Vol. 108, No. ST5.

Simons, J.W. and Powell, G.H. (1982). "Solution Strategies for Statically Loaded Nonlinear Structures",

Earthquake Engineering Research Center, Report No. EERC 82-22, University of California,

Berkeley.

Zeris, C.A. and Mahin, S.A. (1988). "Analysis of Reinforced Concrete Beam-Columns under Uniaxial

Excitations", Journal of Structural Engineering, ASCE, Vol. 114, No.4.

17

• •

Steel

E

Concrete outside confinement

:v. :I 'l. ~ /. v. rl: '/ ~ /. v.; '/.

'//. v:; 'l_

'/ ~ ~ 'l '/ v.; ./. 'l.

cr

Concrete inside confinement

FIG. 1 FffiER REPRESENTATION OF A SECTION

p p

M

E

M

(a) Two Steel Fibers (b) Two Steel and Two Concrete Fibers

FIG. 2 P-M INTERACTION SURF ACES FOR VERY SIMPLE SECTIONS

18

~r ..

Ia \. Ill

(a) Section

d

(b) Fiber Representation

..

Steel Fiber

Concrete Fiber

Shear Reinforcement Fiber

(c) Conventional Shear Deformation

(d) Shear Deformation caused by Diagonal Cracks

FIG. 3 EXTENSION OF FffiER MODEL TO INCLUDE SHEAR DEFORMATION

High bond stress, __ _...A'lv·-·-

Crack opening at column face'-. /

1/

'//// -... v ::. V/////. ~ "' '////~ .r ::. i/__/////_ ~ rL. '/. '///. ~ V/////. -,.

/ '/// "'

/

Diagonal cracking in jOint region

') l<e~

FIG. 4 SOURCES OF DEFORMATION IN

BEAM-COLUMN CONNECTION

19

APPENDIX A

ELEMENT INFORMATION

20

DRAIN-2DX USER GUIDE

ELEMENT THEORY

TRUSS ELEMENT <TYPE 01)

EOl.l GENERAL CHARACTERISTICS

Truss elements may be oriented arbitrarily .in the XY plane, but can transmit axial load

only. Two alternative modes of inelastic behavior may be specified, namely (a) yielding in

both tension and compression (Fig. EOl.la) and (b) yielding in tension but elastic buckling in

compression (Fig. EOl.lb). Strain hardening effects are included by dividing each element

into two parallel components, one elastic and one inelastic (Fig. E01.2). Element loads and

second order effects can be included.

E01.2 ELEMENT DEFORMATIONS

The only element deformation is axial extension, v. The displacement transformation

relating increments of deformation and displacement (Fig. E01.3) is

] { dr1

} dv [ -x -Y X y dr2 (EOl.l) = L drs L L L

dr4

or

dv = adr (E01.2)

where X, Y and L are the projections and element length in the undeformed state.

The measure of inelastic deformation is the extension beyond yield of the inelastic com-

ponent of the element.

E01.3 STATIC ELASTO-PLASTIC STIFFNESS

The static elasto-plastic stiffness in terms of deformation is

dS = Et A dv L

21

(E01.3)

-2-

or

dS = kep dv (E01.4)

in which Et = tangent modulus in current state, A = cross section area, and L = undeformed

length. Hence the static stiffness in terms of nodal displacements is

E01.4 GEOMETRIC STIFFNESS

The geometric stiffness in terms of rigid body rotation (Fig. E01.4) is

s dM8 = L de

(E01.5)

(E01.6)

in which S = current axial force and L = undeformed length. The transformation relating

rigid body rotation and nodal displacement is

[ -Y X Y -X ]

dB = T L L L dr (E01.7)

or

dB = a6 dr (E01.8)

Hence the geometric stiffness is

K aT S g = g L ag (E01.9)

E01.5 DYNAMIC STIFFNESS

If stiffness dependent (pK) damping is specified, a viscous damping component is added

in parallel with the elasto-plastic component. The axial force in this damping component is

(EOl.lO)

in which v = rate of change of extension and E = elastic modulus.

22

-3-

A damping geometric stiffness is not considered (i.e., the geometric stiffness is based

only on the static axial force, S , not on S + Sd ).

E01.6 RESISTING FORCE

The element resisting force is used to calculate equilibrium unbalance. This resisting

force is

(EOl.ll)

where the term containing~ applies only when second order effects are included. Note that

since the value of S used in Eqn. (EOl.ll) is at the end of a step, it will generally be different

from that used for K 6 , which is at the beginning of the step. Hence, the second order effect

can cause an equilibrium unbalance even if K ep is constant.

E01.7 ELEMENT LOADS

Static loads applied along the lengths of truss elements may be taken into account by

specifying end clamping forces as shown in Fig. E01.5. These forces are those which must

act on the element ends to prevent end displacements. The fixed end forces for any element

contribute to the static loads (gravity load cases only) on the nodes to which the element con-

nects.

E01.8 ACCUMULATED PLASTIC DEFORMATIONS

The computed response includes both maximum and "accumulated" plastic deforma

tions. The accumulated values are defined by Fig. E01.6.

23

-4-

DRAIN-ANAL USER GUIDE

INPUT DATA SECTION C2.01

TRUSS ELEMENT (TYPE 01)

See Fig. E01.1 through E01.6 for element behavior and properties.

C2.01(a). Control Information

One line.

Columns Notes Variable

1-5(1)

C2.01(b). Stiffness Types

One line for each stiffness type.

Columns

1-5(1)

6-15(R)

16-25(R)

26-35(R)

36-45(R)

46-55(R)

60(1)

Notes Variable

Data

No. of stiffness types (max. 40). See Section C2.01(b).

Data

Stiffness type number, in sequence beginning with 1.

Youngs modulus.

Strain hardening ratio, as a proportion of Youngs modulus.

Cross section area.

Yield stress in tension.

Yield stress or elastic buckling stress in compression. See Fig. E01.1

Code for compression behavior. (a) 1: Element buckles elastically in compression. (b) 0: Element yields in compression without

buckling.

24

-5-

C2.01(c). Element Generation Commands

One line for each generation command. The :first element can be assigned any number.

Subsequent elements must be defined in numerical sequence. Lines for the first and last ele-

ments must be included

Columns

1-5(1)

6-10{1)

11-15(1)

16-20(1)

21-25(1)

Notes Variable

C5

Data

Element number, or number of first element in a sequentially numbered series of elements to be generated by this command.

Node number at element at end i.

Node number at element at endj.

Node number increment for element generation. Default= 1.

Stiffness type number.

25

-6-


INPUT DATA SECTION D2(b)(ii).Ol

ELEMENT LOAD DATA FOR TRUSS ELEMENT (TYPE 01)

D2(b)(ii).Ol(a). Load Sets

NLOD lines (see section D2(bXi)), one line per element load set. See Fig. E01.5.


1-5(1)

6-10(1)

11-20{R)

21-30(R)

31-40(R)

41-50CR)

Data

Load set number, in sequence beginning with 1.

Coordinate code. (a) 0: Forces are in local (element) coordinates. (b) 1: Forces are in global (structure) coordi

nates.

Clamping force Pi.

Clamping force Vi.

Clamping force Pi.

Clamping force Vi.

D2(b)(ii).Ol(b). Loaded Elements and Load Set Scale Factors

As many lines as needed. Terminate with a blank line.

Columns

1-5(1)

6-10(1)

11-15(1)

16-20(1)

21-30(R)

31-45(1-R)

46-60(1-R)

61-75{1-R)

Notes Variable Data

No. of :first element in series.

No. of last element in series. Default= single element.

Element no. increment. Default = 1.

Load set number.

Load set scale factor.

Optional second load set no. and scale factor.

Optional third load set no. and scale factor.

Optional fourth load set no. and scale factor.

26

"

-7-

DRAIN-POST USER GUIDE

OUTPUT ITEMS FOR POSTPROCESSING

TRUSS ELEMENT (TYPE 01)

Item Description

1 Axial force, tension positive.

2 Total axial extension.

3 Accumulated positive plastic extension.

4 Accumulated negative plastic extension.

5 Node number at end I.

6 Node number at end J.

7 Yield code ( 1: yielded; 0: not yielded ).

27

- 10 -

s

v

(a) YIELD IN TENSION AND COMPRESSION

s

(b) YIELD IN TENSION. BUCKLING IN COMPRESSION

FIG. E01.1 TRUSS ELEMENT BEHAVIOR

28

--

s

_-:::J ----

- 11 -

CBOTH COMPONEN_!~ ""--_....._.,~-----,.,.-·~-----lJ ELASTO-PLASTIC IJ COMPONENT

FIG. E01.2 DECOMPOSITION OF BILINEAR RELATIONSHIP INTO TWO COMPONENTS

X

dS,dv ~

(a) (b)

RG. E01.3 DEFORMATIONS AND DISPLACEMENTS

29

- 12 -

~s

FIG. E01.4 TERMS FOR ROTATIONAL GEOMETRIC STIFFNESS

. p j

~ \Vj

Pj -Pi

(a) CODE c 0 (b) CODE c 1

RG. E01.5 END CLAMPING AND IN mAL FORCES

30

FORCE OR MOMENT

- 13 -

ACCUMULATED POSITIVE DEFORMATION a SUM OF POSITIVE YIELD EXCURSIONS

\..M~XIMUM POSITIVE DEFORMATION

EXTENSION OR ROTATION

c SUM OF NEGATIVE YIELD EXCURSIONS

NOTE THAT MAXIMUM NEGATIVE DEFORMATION IS ZERO, ALTHOUGH ACCUMULATED NEGATIVE DEFORMATION JS NOT ZERO

FIG. E01.6 PROCEDURE FOR COMPUTATION OF ACCUMULATED PLASTIC DEFORMATIONS

31

DBAIN-2DX USER GUIDE

ELEMENT THEORY

BEAM-COLUMN ELEMENT (TYPE 02)

E02.1 GENERAL CHARACTERISTICS

Beam column elements can: be oriented arbitrarily in the XY plane. The elements pos

sess :flexural and axial stiffness. Elements of variable cross section can be considered by spec

ifying appropriate :flexural stiffness coefficients. Flexural shear deformations and the effects

of eccentric end connections can be taken into account. Yielding can take place only in con

centrated plastic hinges at the element ends. Strain hardening is approximated by assuming

that the element consists of elastic and inelastic components in parallel. The hinges in the

inelastic component yield under constant moment, but the moment in the elastic component

can eontin9-e to increase.

With this strain hardening model, if the bending moment on the element is constant,

and if the element is of uniform strength, then the moment-rotation relationship for the ele

ment will have the same shape as its moment-curvature relationship (Fig. E02.la). This fol

lows because curvature and rotation in this ease are directly proportional. If, however, the

bending moment or strength vary, then the curvatures and rotations are no longer propor

tional, and the moment-rotation and moment-curvature variations may be quite different

(Fig. E02.1b). With the parallel component model, a moment-rotation relationship is, in

effect, being specified. Care must be taken in relating this to the moment-curvature relation

ship.

The yield moments can be specified to be different at the two element ends, and for pos

itive and negative bending. The interaction between axial force and moment in producing

yield can be taken into account approximately.

Static loads applied along any element length can be taken into account by specifying

fixed end force values. Second order effects can be approximated by including a simple P-

32

-2-

delta equlibrium correction and geometric stiffness, based on the element axial force.

E02.2 ELEMENT DEFORMATIONS

A beam-column element has three modes of deformation, namely axial extension, flexu-

ral rotation at end i, and flexural rotation at end j. The displacement transformation relat-

ing increments of deformation and displacement (Fig. E02.2) is

-X -Y X y dr1

L L 0

L L 0 dr2

{ dv1

} dv2 = -Y X 1

y -X 0

dr3 (E02.1) L2 L2 L2 L2 dr4 dv3 -Y X

0 y -X

1 dr5 L2 L2 L2 L2

dr6

or

dv = adr (E02.2)

Where X. Y and L are the projections and element length in the undeformed state.

A plastic hinge forms when the moment in the inelastic component of the element

reaches its yield moment. A hinge is then introduced into this component, the elastic compo-

nent remaining unchanged. The measure of flexural plastic deformation is the plastic hinge

rotation.

For any increments of total flexural rotation, dv2 and dv3 , the corresponding incre-

ments of plastic hinge rotation, dv p2 and dv p3, are given by

{dvp2} =[A B]{dv2 } dvp3 C D dv3

(E02.3)

in which A, B, C and Dare given in Table E02.1. Unloading occurs at a hinge when the incre-

ment in hinge rotation is opposite in sign to the bending moment.

Inelastic axial deformation is assumed not to occur. Hence, only an approximate proce-

dure for considering interaction effects is included, as explained in the following section. This

procedure is not theoretically sound, but may be reasonable for practical applications.

33

-3-

E02.3 INTERACTION SURFACES

This section applies to the inelastic component only. Yield interaction surfaces of three

types can be specified for the ends of this component, as follows.

(1) Beam type (shape code = 1, Fig. E02.3a). This type of surface should be specified where

axial forces are small or interaction can be ignored. Yielding is affected by bending

moment only.

(2) Steel column type (shape code = 2, Fig. E02.3b). This type of surface is intended for use

with steel columns.

(3) Concrete column type (shape code = 3, Fig. E02.3c). This type of surface is intended for

use with concrete columns.

For any combination of axial force and bending moment within a yield surface, the

cross section is assumed to be elastic. If the force-moment combination lies on or outside the

surface, a plastic hinge is introduced. Combinations outside the yield surface are permitted

only temporarily, being compensated for by applying corrective loads in the succeeding step.

This procedure is not strictly correct because the axial and flexural deformations interact

after yield, and it is therefore wrong to assume that only the flexural stiffness changes

whereas the axial stiffness remains unchanged.

If a force-moment combination goes from the elastic range to beyond the yield surface in

any time or load step, an equilibrium correction is made as shown in Fig. E02.4a. Because

the axial stiffness is assumed to remain unchanged, in subsequent steps the force-moment

combination at a plastic hinge will generally move away from the yield surface within any

time step, as shown in Fig. E02.4b. An equilibrium correction, as shown, is therefore made.

The axial force in an element with a column-type interaction surface can, in reality,

never exceed the yield value for zero moment. However, because of the computational proce

dure which is used, axial forces in excess of yield can be computed. For axial forces in excess

of yield, the yield moments are assumed to be zero. The printed results from th~ program

34

-4-

should be examined carefully and interpreted with caution. If axial forces approaching or

exceeding yield are computed for a column, the results are probably incorrect, and severe col-

umn damage is probably implied.


The element is considered as the sum of an elastic component and an inelastic compo-

nent. The element actions and deformations are shown in Fig. E02.2a. The axial stifthess is

constant, and is given by

EA. dS1 =- dv1

L (E02.4)

in which E = elastic modulus, and A = effective uniform cross sectional area. The flexural

stifthess in the elastic range is given by

(E02.5)

in which I = reference moment of inertia; and ku,kiJ, k ii are coefficients which depend on the

cross section variation. For a uniform element, I = actual moment of inertia, ku = k .ii = 4 and

ku = 2. The coefficients must be specified by the program user, and may, if desired, account

for such effects as shear deformations and nonrigid end connections, as well as cross section

variations.

After one or more hinges form, the coefficients for the inelastic component change to k~,

ku and k~, as follows

k~ = k" (1- A) - ku C (E02.6)

k~ = ku (1-D) - k" B (E02.7)

k~ = kii (1-D)- kv· B (E02.8)

in which A, B, C and D are defined in Table E02.1.

35

-5-

The elasto-plastic stiffness in terms of node displacements is

(E02.9)

where Kep is the sum of the stiffness for the elastic and inelastic components.

E02.5 GEOMETRIC STIFFNESS

The geometric stiffness which is used is the same as for the truss bar element. This is

not the exact geometric stiffness for a beam column element, but is sufficiently accurate for

taking into account second order effects in typical building frames.

E02..6 DYNAMIC STIFFNESS

As for the truss bar element, if pk damping is specified, a viscous damping element is

added in parallel with the elastic component. This component contributes both axial and flex-

ural stiffness during dynamic analysis, and develops both axial and flexural damping resis-

tance. As for the truss bar, the geometric stiffness is based on the elastic-plastic axial force

·only.

E02.. 7 RESISTING FORCE

The element resisting force is

(E02.10)

where s and sd are the static and viscous damping actions, respectively, sl is the static

axial force, and a8 is the same as for the truss bar element. As for the truss bar, the geomet-

ric stiffness is based on S at the beginning of a step, and the resisting force on the value at

the end of the step, so that second order effects can produce equilibrium unbalances even for

an otherwise linear structure.

E02..8 ELEMENT LOADS

Static loads applied along the lengths of beam column elements can be taken into

account by specifying end clamping forces as shown in Fig. E02.5. These forces are those

which must act on the element ends to prevent end displacement.

36

-6-

The fixed end forces for any element contribute to the static loads on the nodes to which

the element connects. Frequently, the live load reduction factor permitted for a column in a

building will exceed that for the beams it supports, because columns support tributary loads

from several floors. Hence, if the full live load fixed end shears for each beam are applied at

the structure nodes, the accumulated loads on the the columns may be unnecessarily large.

This can be taken into account by means of live load reduction factors for the fixed end forces,

which are used as follows.

For initialization of the element shear and axial forces, the full specified fixed end

forces are used. However, for computation of the static loads on the nodes connected to the

element, the fixed end shear and axial forces due to live load (but not the moments) are first

multiplied by the specified reduction factor. The forces producing axial loads in the columns

can therefore be reduced to account for differences in the permissible live load reductions for

beams and columns, yet the shear forces computed for the beams will still be correct.

E02.9 SHEAR DEFORMATIONS

If desired, effective flexural shear areas can be specified. The program then modifies the

flexural stiffness to account for the additional shear deformations. The fixed end forces are

not changed. Hence if shear deformations are important, the specified fixed end force pat-

terns should take these deformations into account.

· E02.10 END ECCENTRICITY

Plastic hinges in frames and coupled frame-shear-wall structures will form near the

joint faces rather than at the theoretical joint centerlines. This effect can be approximated by

assuming rigid and infinitely strong connecting links between the nodes (which are located at

the joint centerlines) and the element ends, as shown in Fig. E02.6. The displacement trans-

formation relating the node displacements, {dr,} , with those at the element ends is

dr1 1 0 -Yt 0 0 0 dr1n dr2 0 1 xi 0 0 0 dr2n dra 0 0 1 0 0 0 dran (E02.11) = dr4 0 0 0 1 0 -Yi dr4n dr5 0 0 0 0 1 Xi dr5n drs 0 0 0 0 0 1 dr&n

37·

-7-

This transformation has been incorporated into the calculation of the element stiffnesses and

deformations. If end eccentricities are specified, the stiffness coefficients in Eqn. E02.5 must

apply to that part of the element between the joint faces, ignoring the joint region. Similarly,

the fixed end forces are those applying at the joint faces. The end eccentricity effects are

taken into account in transferring the fixed end forces to the nodes (i.e the moment loads are

augmented by couples created by the fixed end shears and axial forces). Any specified live

load reduction factors are applied to the fixed end shear and axial forces before they are

transferred from the joint forces to the nodes.

For second order effects with end ecentricities, an approximate theory is currently used.

This assumes that second order effects are produced by a truss bar extending directly from

node to node, and that the axial force in this bar is the axial force in the element. The reason

for this is that it is not correct to form the geometric stiffness and resisting forces at the joint

faces then :imply transform to the nodes.

38·

- 21-

TABLE EOl.l

COEFFICIENTS FOR PLASTIC HINGE ROTATIONS

Yield Condition A B c D

Elastic ends 0 0 0 0

Plastic binge at end i only 1 k;j

0 0 k;;

Plastic binge at end j only 0 0 k;j

kjj 1

Plastic binges at both ends i and j 1 0 0 1

Coefficients k;;. k;i• and kii are defined by Eq. E02.5 ..

39·

-9-



BEAM-COLUMN ELEMENT <TYPE 02)

See Fig. E02.1 through Fig. E02.6 for element behavior and properties.


One line.


1-5(1)

6-10(1)

11-15(1)

Data

No. ofstiffuess types (max. 40). See section C2.02(b).

No. of end eccentricity types (max 15). See section C2.02(c).

No. of yield surfaces for cross sections (max. 40) See section C2.02(d).

40·



Columns

1-5U)

6-15(R)

16-25(R)

26-35(R)

26-45(R)

46-50(R)

51-55(R)

56-60(R)

61-70(R)

71-SO(R)

Notes Variable

C2.02(c). End Eccentricities

-10-

Data


Young's modulus.

Strain hardening ratio, as a proportion of Young's modulus.

Cross sectional area.

Moment of inertia.

Flexural stiffness factor kii.

Flexural stiffness factor k ii·

Flexural stiffness factor kv.

Shear area. Leave blank if shear deformations are to be ignored, or if shear deformations have already been taken into account in computing the flexural stiffness factors.

Poisson's ratio (used for computing shear modulus, and used only if shear area is nonzero).

One line for each end eccentricity. Omit if there are no end eccentricities. See Fig.

E02.6 for explanation. All eccentricities are measured from the node to the element end.

Columns

1-5(1)

6-15(R)

16-25(R)

26-35(R)

26-45(R)

Notes Variable Data

End eccentricity number, in sequence beginning with 1.

Xi = X eccentricity at end i.

Xi = X eccentricity at end j.

Yi = Y eccentricity at end i.

Y i = Y eccentricity at end j.

41·

-11-

C2.02(d). Cross Section Yield Surfaces

One card for each yield surface. See Fig. E02.3 for explanation.

Columns

1-5(1)

10(1)

ll-20(R)

21-30(R)

31-40(R)

41-SO(R)

51-55(R)

56-60(R)

61-65(R)

66-70(R)

Notes Variable Data

Yield surface number, in sequence beginning with 1.

Yield surface shape code, as follows. (a) 1: beam type, without P-M interaction. (b) 2: steel I-bea1.a type. (c) 3: reinforced concrete column type.

Positive yield moment, My+ (counter clockwise).

Negative yield moment, M :r- (clockwise).

Compression yield force, P yc· Leave blank if shape code= 1.

Tension yield force, P yt· Leave blank if shape code = 1.

M coordinate of balance point A, as a proportion of My+· Leave blank if shape code = 1.

P coordinate of balance point A, as a proportion of P yc· Leave blank if shape code = 1.

M coordinate of balance point B, as a proportion of M :r-· Leave blank if shape code = 1.

P coordinate of balance point B, as a proportion of P yc· Leave blank if shape code = 1.

42·

-12-

C2.02(e). Element Generation Commands

One line for each generation command. The first element can be assigned any number.



Columns

1-5(1)

6-10(1)

11-15(1)

16-20(1)

21-25(1)

26-300)

31-35(1)

36-40(1)

Notes Variable

C5

Data


Node number at element at end i.

Node number at element at end j.



End eccentricity number. Default = no end eccentricity.

Yield surface number at end i

Yield surface number at endj.

43

-13-


INPUT DATA SECTION D2(b)(ii).02

ELEMENT LOAD DATA FOR BEAM-COLUMN ELEMENT <TYPE 02)

D2(b)(ii).02(a). Load Sets

NLOD lines (see Section D2(bXi)), one line per element load set. See Fig. E02.5.


1-5(1)

6-10(1)

ll-20(R)

21-30(R)

31-40(R)

41-SO(R)

51-60(R)

61-70(R)

71-SO(R)

Data

Load set number, in sequence beginning with 1.

Coordinate code. (a) 0: Forces are in local (element)

coordinates. (b) 1: Forces are in global (structure)

coordinates.

Live load reduction factor.

Clamping force P;.

Clamping force Vi.

Clamping moment M;.

Clamping force Pi·

Clamping force Vi.

Clamping moment M i·

44

-14-

D2(b)(ii).02(b). Loaded Elements and Load Set Seale Factors

As many as lines needed. Terminate with a blank line.

Columns Notes Variable Data

1-5(1)

6-10(1)

11-15(1)

16-20(1)

21-30(R)

31-45(1-R)

46-60(1-R)

61-75(1-R)

No. of first element in series.

No. of last element in series. Default= single element.

Element no. increment. Default = 1.

Load set number.

Load set scale factor.

Optional second load set no. and scale factor.

Optional third load set no. and scale factor.

Optional fourth load set no. and scale factor.

45

- 15-

DRAIN-POST USER GUIDE OUTPUT ITEMS FOR POSTPROCESSING

BEAM-COLUMN ELEMENT (TYPE 02)

Item Description

1 Bending moment at end I.

2 Bending moment at end J.

3 Shear force at end I.

4 Shear force at end J.

5 Axial force at end L

6 Axial force at end J.

7 Current plastic hinge rotation at end I.

8 Current plastic hinge rotation at end J.

9 Accumulated positive plastic hinge rotation at end I.

10 Accumulated positive plastic hinge rotation at end J.

11 Accumulated negative plastic hinge rotation at end I.

12 Accumulated negative plastic hinge rotation at end J.

13 Yield code at end I (1: hinge; 0: no hinge).

14 Yield code at end J (1: hinge; 0: no hinge).



46

M

- 29 -

MOMENT,M

r_ ....... -I ,

{b)

~ / ~

"',8

(a)

--------

CURVATURE, o/

M 8 M ("~)·

M

"',8 (c)

FIG. E02.1 MOMENT -CURVATURE AND MOMENTROTATION RELATIONSHIP

I X

'"I ds1 , dv1 ds5

, dv5

dr6

:J.-/ dr2

ds2

, dv2 J f

( (.

{a) dr5 {b)

RG. E02.2 DEFORMATIONS AND DISPLACEMENTS 47

t~r, ~

dr4

B

My-

- 30 -p

(a) SHAPE CODE = 1

p

(b) SHAPE CODE= 2

My-

(c) SHAPE CODE • 3

M

M

A

M

FIG. E02.3 YIELD INTERACTION SURFACES 48

p

Mi

Pi~ \vi

- 31 -

EQUILIBRIUM UNBALANCE

t- •I t+~t

M (a)

EQUILIBRIUM p · UNBALANCE

(b) M

FIG. E02.4 EQUILIBRIUM CORRECTION FOR YIELD SURFACE OVERSHOOT

Pi

Mi

" --f Vi

Mj

r:_ __ Pj

fvj

(a) CODE • 0 (b) CODE • 1

FIG. E02.5 END CLAMPING AND INITIAL FORCES

49

- 32 -

FIG. E02.6 END ECCENTRICITIES

50


ELEMENT THEORY

SIMPLE CONNECTION ELEMENT (TYPE 04)


The element connects two nodes which must have identical coordinates (ie. it is a zero

length element). It can connect either the rotational displacements of the nodes or the trans

lational displacements. Positive actions (moments or forces) and deformations are shown in

Fig. E04.1. For a translational connection the element can connect horizontal displacements

or vertical displacements, but not inclined displacements. The element can be specified to

behave elastically or inelastically, as shown in Fig. E04.2. Complex modes of behavior can be

obtained by placing two or more elements in parallel.

One application is to allow for angle changes at beam-column connections, for example

(a) panel zon£: deformations in steel frames, and (b) crack opening and closing in precast con

crete frames. In this case the element connects rotational displacements. A second applica

tion is to model panel-to-frame connections in frames with structural cladding or infill pan

els. In this case the element connects translational displacements.

E04.2 ELEMENT DEFORMATION

The element has two degrees of freedom, providing one deformation mode and one rigid

body mode. The deformation is the relative rotation or translation between the connected

nodes, as follows:

(E04.1)

or

dq = a dr (E04.2)

in which dq = increment of element deformation (8, bz or 6y, Fig. E04.1); and drit dri =

51

-2-

increments of rotation, X translation or Y translation of the connected nodes.

E04.3 STATIC TANGENT STIFFNESS

The tangent action-deformation relationship is

dQ = kt dq (E04.3)

where dQ = increment of element action (moment or force) and kt = connection tangent stiff

ness. The positive sign convention is shown in Fig. E04.1. Hence, in terms of node displace

ments the static tangent stiffness, K t. is given by

(E04.4)

E~40THERPROPERTIES

As for other elements, pK damping has the effect of adding a viscous damping element

in parallel with the elasto-plastic element. The damping stiffness is based on the initial stiff

ness k1 (Fig. E04.2). Since k1 may be large, care should be taken in assigning the value of p,

to avoid excessive viscous damping. It is probably wise to include viscous damping in the

beam and column elements only, and to set p = 0 for connection elements.

There is no provision for second order effects or for element loads.

52

-3-



SIMPLE TRANSLATIONUROTATIONAL CONNECTION ELEMENT (TYPE 04)

See Figs. E04.1 and E04.2 for element behs:vior and properties.


One line.


1-5(1)

C2.04(b). Property 'JYpes

One line for each property type.


1-5(1)

6-15CR)

16-25(R)

26-35(R)

36-45(R)

46-50(1)

51-55(1)

Data

No. of property types (max. 40).

Data

Property type number, in sequence beginning with 1.

Initial stiffness (for rotation, moment per radian).

Strain hardening stiffness, as a proportion of initial stiffness.

Positive yield moment or force.

Negative yield moment or force.

Direction code. (a) 1: X translation. (b) 2: Y translation. (c) 3: Rotation.

Elasticity code. (a) 0: Unload inelastically (Fig. E04.1(a)). (b) 1: Unload elastically (Fig. E04.1(b)). (c) 2: Unload inelastically with a gap

(Fig. E04.1(c)).

53

-4-




ments must be included.

Columns

1-5(1)

6-10{1)

11-15(1)

16-20(1)

21-25(1)

Notes Variable

C5

Data


Node number at element end I.

Node number at element end J.


Property type number.

54

-5-



SIMPLE TRANSLATIONIJROTATIONAL CONNECTION ELEMENT (TYPE 04)

Item Description

1 Static force or moment.

2 Viscous force or moment.

3 Total deformation.

4 Accumulated positive plastic deformation (sum of all positive excursions with yield code = 1).

5 Accumulated negative plastic deformation (sum of all negative excursions with yield code = 1).



8 Yield code for element ( 0: not yielded; 1: yielded; 2: gap open).

55

IYf ~

Dr. Deformation, e = rw - rei

(a) Node Displacement (b) Rotational Connection

J lronslational

Spring /_ ::7"::

Tra nslational Spring

I F F - I J -

Notes:

Deformation, o = rxJ - rxl Deformation, o = ryJ - ry1

(c) X Translational Connection (d) Y Translational Connection

(1) Nodes I and J should have identical coordinates. (2) One element provides only one type of connection. For example, a rotational connection

does not provide any X or Y translational connection. It may be necassary to use zero displacement commands or to specify addiional connection elements to avoid an unstable modeL

FIG. E04.1 CONNECT ION T'I'PES

56

F or M

(a) Inelastic Unloading

(Elasticity Code = 0 )

F or M

k1 =initial stiffness k2 = kl x ( strain bardenting ratio )

F or M

B or8

5

(b) Elastic Unloading (c) Inelastic Unloading with Gap ( Elasticity Code = 1 ) ( Elasticity Code = 2 )

F16. E04-.2 ELEMENT BEHAVIOR

57

9

B or8


ELEMENT THEORY

GAP-FRICTION JOINT ELEMENT (TYPE 05)


E05.1.1 Gap Behavior

Consider, first, a gap/bearing element with zero friction. Such an element consists of a

spring with zero length, placed normal to the joint surface. A finite stiffness is assigned to

the element in compression. This stiffness will typically be large. However, avoid assigning

astronomically large values, since (a) they are probably unrealistic, and (b) they can lead to

numerical sensitivity problems. A zero stiffness is assigned in tension, corresponding to a

gap across the joint.

The force-deformation relationship is as shown in Fig. E05.1. For a horizontal joint, the

element provides this relationship between vertical force and vertical deformation. For a ver

tical joint, the relationship is between horizontal force and deformation. For an inclined joint

the gap/bearing direction is normal to the joint (Fig. E05.2). An element can connect up to

four nodes, as shown. Nodes I and J may be assigned the same node number, and similarly

nodes K and L, to connect three or two nodes. The node numbers must be specified carefully

to ensure that the tension and compression directions are correctly defined. The element

should have zero length. Typically, lines I.J and K-L should be parallel to the joint direction, .

but this is not essential.

The force-deformation relationship allows for nonlinear behavior in compression, with

the joint bearing surfaces yielding as the normal compressive force increases. The element

has options to unload elastically or inelastically, as shown.

Compressive deformation is assumed to be positive. An element may be preloaded, for

example to represent gravity and/or posttensioning effects. Separation occurs when any

58

-2-

added tension force exceeds the preload. Alternatively an element can be given an initial gap.

H event calculations are not specified, substantial unbalances can occur when a .gap

closes, especially if the time step is long or the bearing stiffness is high. When gap elements

are used, the element stiffnesses should generally be made as low as possible, the variable

time step option should be chosen, event calculations should be specified, and the results

should be examined to ensure that there is an energy balance and that oscillation or diver

gence of results does not occur following gap closure.

E05.1.2 Opening of a Wide Joint

H a wide joint between, say, two structural panels develops a gap, one side tilts relative

to the other. It will be natural to place a gap element at each end of the joint. H a gap opens,

the assumption is then that the joint pivots about the end point, as shown in Fig. E5.03a. In

an actual structure, joint opening is likely to take place progressively, rather than suddenly,

with significant distortion of the joint plane. Since the element assumes rigid connections

between nodes I,J and K,L, this distortion is not modelled. The error in assuming a rigid

joint plane can be partially corrected either by moving the assumed pivot points or by mod

elling the joint with several gap elements. The pivot points can be moved by specifying two

gap elements located within the joint rather than at the comers, as shown in Fig, E05.3b. A

less sudden joint opening can be obtained by specifying several gap elements along the joint,

as shown in Fig. E05.3c. In this case the elements must be made relatively flexible in com

pression, otherwise tilting will occur essentially about one comer, and all gap elements will

open at essentially the same time.

E05.1.3 Combination With Friction

A complete gap-friction element combines a gap element with a friction element, and

adjusts the friction element so that its strength at any time is equal to the compressive force

on the gap element multiplied by a coefficient of friction. The friction strength becomes zero

if gap opening occurs.

59

-3-

Frictional slip under varying bearing force is a complex process. The procedure used to

deternrine the state of an element at the end of any time or load step is not exact, but is

believed to be reasonable. The state of the gap element is found first. Two friction strengths

are then used, based on the new bearing force and specified upper and lower values of the

friction coefficient. If the friction element is locked (ie., not slipping) at the beginning of the

step, its state is changed to slipping at the end of the step if the friction force exceeds the

upper friction strength. The friction force is also set equal to the upper friction strength (pro-

ducing a temporary unbalanced load). If the friction element is slipping at the beginning of

the step, its state is changed to locked if the lower friction strength at the end of the step

exceeds the current friction force. If the element continues to slip, the friction force says con-

stant, except that if it exceeds the upper friction strength it is set equal to this strength.

With this procedure, the friction force for a slipping element can lie anywhere between the

upper and lower friction strengths. If desired, the range between the two strengths can be

specified to be small (nearly equal upper and lower friction coeffcients). However, this will

lead to more stiffness change events.


The element combines a gap element with a friction element. The element has up to

eight displacement degrees of freedom. The gap element has one extensional mode of defor-

mation, positive in compression. The friction element has one shear mode of deformation,

positive when rigid link IJ slides towards end L of rigid link KL. The gap deformation, q8 , is

given by

rl r2 ra

a1s _ b2s a2s b1c a 1c b2c ~c r4 05.1)

Ll L2 - L2 - L1 - Ll L2 -> L2 rs

rs r7 rs

60

-4-

where s = sin 8, c = cos 9, r 1 through r 4 are the X displacements at nodes I through K, and r5

through r8 are the Y displacements. This can be written as

(E05.2)

Hence, the gap stiffness matrix is

(E05.3)

where kgt is the tangent stiffness of the element ( k~o k2 , k3 , k4 , or zero, Fig. E05.1). The the

ory for a two-node element is obtained by setting b1 = b2 = 0, and a three-node element by

setting either b1 = 0 or b2 = 0.

The stiffness of the friction element is obtained in essentially the same way, except that

the displacements are parallel, not normal, to the joint surface.

E05.3 OTHER PROPERTIES

There is no provision for second order effects or for element loads. Also, regardless of

what value is specified for the stiffness dependent (,BK) damping coefficient, ,8, this value is

assumed to be zero (i.e. no viscous damping).

61

-5-



GAP FRICTION JOINT ELEMENT (TYPE 05)



One line.


1-5(1)

6-10(1)

C2.05(b). Gap Property Types



1-5(1)

6-10(1)

ll-20(R)

21-30(R)

31-40(R)

41-50(R)

51-60(R)

61-70(R)

Data

No. of gap property types (max. 20). See section C2.05(b).

No. of friction property types (max. 20). May be zero. See section C2.05(c).

Data


Unloading code (0 = inelastic, 1 = elastic).

Displacement limit U1.

Displacement limit U2.

Stiffness k1•

Stiffness k3 •

Unloading stiffness k4 • Default= k1 •

62

-6-

C2.05(c). Friction Property 'JYpes

One line for each property type. Omit if there are no friction properties.

Columns Notes Variable Data

1-5(1)

6-15(R)

16-25(R)

26-35(R)


Upper friction coefficient.

Lower friction coefficient. Must be < upper coefficient.

Shear stiffness (i.e., stiffness when not slipping).

63

-7-

C2.05(d). Element Generation Commands

One line for each generation command. The :first element can be assigned any number.

Subsequent elements must be defined in numerical sequence. Lines for the :first and last ele-


Columns

1-5a>

6-10{1)

11-15(1)

16-20(1)

21-25(1)

26-30(1)

31-35a>

36-40(1)

41-50(R)

51-60(R)

61-70(R)

71-SO(R)

Notes Variable

C5

Data

Element number, or number of :first element in a sequentially numbered series of elements to be generated by this command.

Node number at element point I.

Node number at element point J.

Node number at element point K.

Node number at element point L.


Gap property type number.

Friction property type number (if zero, no friction).

Initial bearing force(+ value) or initial gap(- value).

Joint angle (degrees, counterclockwise from global X axis).

Location ratio al/L1.

Location ratio a2/L2.

64

-8-



GAP FRICTION JOINT ELEMENT (TYPE 05)

Item Description

1 Bearing force.

2 Bearing deformation (negative = gap opening).

3 Accumulated plastic deformation of gap element.

4 Friction force.

5 Friction deformation (current total slip).

6 Accumulated positive slip.

7 Accumulated negative slip.

8 Node number at point I.

9 Node number at point J.

10 Node number at point K

11 Node number at point L.

12 Line number for gap element..

13 Slip code (0 =locked; 1 =slipping).

65

Bearing Force

Inelastic Unloading

Corrpressive Deformation

FIG. EOS.1 BEARING BEHAVIOR

. Fl6. EOS.2

66

/ Joint Direction ps l -9o" < B < 90° l

Rigid

For 2-node element put: Node I = Node J Node K = Node L

b1 = b2 = o .

ElEMENT GfOMET~Y

panel eleJTBnt

_.J--- gap eleJTBnts

(a) 2 gap elements per joinL Tilting it about panel comers.

(b) 2 gap elements per joint, moved inwards to change pivot points.

(c) Several gap elements per joinL Elements must be soft in compression to get progressive opening ofjoinL

Fl <S • f 0 5. 3 PAN £ L Tl L T I N G

67

DRAIN-2DX USER GUIDE

ELEMENT THEORY

STRUCTURAL PANEL ELEMENT <TYPE 06)


In analyses of buildings with structural panels it will often be reasonable to idealize

each panel as a single elastic element in which the overall extensional, flexural, and shear

stiffnesses of the panel are modeled. This element provides this type of idealization.

Fig. E06.1 shows a large panel with an opening. In the vertical direction an effective

centroidal axis can be found, such that an axial force applied along this axis produces no

bending (and, correspondingly, a bending moment produces no axial deformation). A similar

effective centroidal axis can be found in the horizontal direction. The extensional and flexu

ral stiffnesses of the panel must be specified as effective EA and EI values along these axes,

where E = Youngs modulus, A = effective cross section area, and I = effective cross section

moment of inertia. In addition, the shear stiffness of the panel must be specified.

A panel is idealized as shown in Fig. E06.2, with four nodes and eight degrees of dis

placement freedom. These provide for five deformation modes, as shown in Fig. E06.3, plus

three rigid body modes.

As noted, the five deformation modes are assumed to be uncoupled, with stiffuesses for

vertical extension (effective vertical EA), vertical bending (effective vertical El), horizontal

extension (effective horizontal EA), horizontal bending (effective horizontal El), and shear.

The shear stiffness is defined in terms of shear strain and shear force per unit edge length

(effective Gt, where G = shear modulus and t = effective panel thickness). These stiffnesses

must be determined by experiment or by separate calculations, taking into account openings,

stiffening ribs, thickness variations, etc.

Note that the element contributes no rotational stiffness to the nodes. Hence, it may be

necessary to restrain the rotational displacements.

68

-2-

The mass of each panel must be lumped at its nodes. This permits a reasonable repre-

sentation of the translational inertia (both vertical and horizontal) of the panel, but overesti-

mates its rotational inertia. This is an inherent error of this panel model. If it is believed that

the rotational inertia will substantially affect the dynamic response, each panel should be

divided into several elements to provide a more accurate representation of the mass distribu-

tion in the panel. Note, however, that the panel edges do not remain straight (see Fig. E06.3 -

in effect the element is a plane stress finite element with one-point shear quadrature).

Hence, a panel modelled with several elements may be too :flexible.

E06.2 STATIC ELASTIC STIFFNESS

E06.2.1 Deformations and Actions

. The displacement degrees of freedom are r 1 through r 8 as shown in Fig. E06.2. The

deformations are q1 through q5 as shown in Fig. E06.3.

The stifihess matrix in terms of deformations is

(E06.1)

in which Au = effective area for vertical extension; lu = effective moment of inertia for verti-

cal bending; A4 = effective area for horizontal extension; I 4 = effective moment of inertia for

horizontal bending; t = effective thickness for shear racking; E = Young's modulus; G = shear

modulus; h =panel height; and w = panel width.

The extensional and bending modes are uncoupled because of the way in which they are

defined. The racking mode is assumed to be uncoupled from the other modes.

69

-3-

E06.2.2 Stiffness Matrix

The element actions, Q, are shown in Fig. E06.3. The basic stiffness relationship is

Ql Ql Q2 Q2

Qa =kd Qa (E06.2)

Q4 Q4 Q5 Q5

where kd is given by Eq. E06.1.· The vectors Q and q are conjugate (that is, 0.5QT q =strain

energy).

A transformation between nodal displacements, r, and element deformations, q, can be

set up in the form

q=ar (E06.3)

Hence, the (8 x 8) element stiffness matrix, K, is given by

(E06.4)


The element is elastic, with no nonlinear behavior. As for other elements, pK damping has

the effect of adding a viscous damping element in parallel with the elastic element.

There are no provisions for second order effects or element loads.

76

-4-



STRUCTURAL PANEL ELEMENT (TYPE 06)



One line.


1-5(1)



Columns

1-5(1)

6-15(R)

16-25(R)

26-35(R)

36-45(R)

46-55(R)

56-65(R)

66-75(R)

Notes Variable

Data

No. of stiffness types (max. 40). See section C2.06(b).

Data


Effective EA for vertical extension (i.e., effective EA of horizontal section).

Effective EI for vertical bending (i.e., effective EI of horizontal section).

Effective EA for horizontal extension (i.e., effective EA ofvertical section).

Effective EI for horizontal bending (i.e., effective EI of vertical section).

Effective Gt for shear racking.

Distance from panel centerline to effective vertical centroidal axis, plus or minus, as a proportion of panel width (i.e. range is -0.5 to +0.5, - to left, + to right). Default = 0.

Distance from panel midheight to effective horizontal centroidal axis, plus or minus, as a proportion of panel height (i.e. range is -0.5 to +0.5, - down , + up). Default= 0.

71

-5-





Columns

1-5(1)

6-10(1)

11-15(1)

16-20(1)

21-25(1)

26-30(1)

31-35(1)

Notes Variable

C5

Data


Node number at element point I (top left).

Node number at element point J (top right).

Node number at element point K (bottom left).

Node number at element point L (bottom right).



•r'

72

-6-



STRUCTURAL PANEL ELEMENT (TYPE 06)

Item Description

1 Vertical axial force (tension + ).

2 Vertical bending moment (tension at right edge + ).

3 Horizontal axial force (tension+).

4 Horizontal bending moment (tension at bottom edge + ).

5 Shear force per unit edge length (to right at top+).

6-10 As for 1-5, but viscous damping forces and moments.

11 Vertical extension.

12 Rotation of top edge relative to bottom (counterclockwise + ).

13 Horizontal extension.

14 Rotation of right edge relative to left (counterclockwise + ).

15 Shear strain.

16 Rigid body rotation (counterclockwise + ).

17 Node number at point I.

18 Node number at point J.

19 Node number at point K

20 Node number at point L.

73

j w :-~--.._;·~~--- Effective

/ centroidal

-!-----d-~----- _ . axes h

Fl6. E06. I rAN E L EFFECTIVE AXES

I J ' I ' r-----------j-----'

K I L

FIG. E06.2 I

NODES ANJ> DISPLACE-MENT l>OF S

---------1

_________ I

-_, \ IE31 / ------ "\ D/ : --l :--t -------- ) ~ // / i 1 1 I I \

I L I l._ \ -J --- --~ ~ ---- ---

-r----- \

' \ I I

I I ... __ I -- -

FIG. E06. 3 I>EFORNATtO N MODE'S

74


ELEMENT THEORY

LINK ELEMENT (TYPE 09)


The link element is a uniaxial element with finite length and arbitrary orientation. An

element can be specified to act in tension (tension force and extension are positive) or in com-

pression (compression force and shortening are positive). A tension element has :finite stiff-

ness in tension and goes slack in compression. A compression element has finite stiffness in

compression and a gap opens in tension.

The force-deformation relationship is as shown in Fig. E09.1. Either one of two unload-

ing paths, namely elastic or inelastic, may be specified. An element can be preloaded to a

specified positive force if desired, or alternatively can be prestrained to a specified negative

deformation. 'rhe element can thus function as (a) a cable prestressed in tension, (b) a cable

with initial slack, (c) a bearing element prestressed in compression, or (d) a bearing element

with an initial gap. Complex modes of behavior can be obtained by placing two or more ele-

ments in parallel.

E09.2 STATIC TANGENT STIFFNESS

A link element has four displacement degrees of freedom and one deformation degree of

freedom, as shown in Fig. E09.2. For a tension element the relationship between nodal dis-

placement and element deformation is

{

dr1}

dq = < - cose - sine + cose +sine > :~: dr4

(E09.1)

or

dq=a dr (E09.2)

75

-2-

For a compression element the signs in matrix a are changed. The static tangent stiffness

matrix is thus

(E09.3)

where kt is the element tangent stiffness (i.e., ktt k2 , k8 , k4 or zero).


As for other elements, fJK damping has the effect of adding a viscous damping element

in parallel with the elasto-plastic element. The damping stiffness is based on stiffness k 1• It

may be wise to include viscous damping in beam, column and panel elements only, and to set

fJ = 0 for link elements.

There is no provision for second order effects or for element loads.

76

-3-



LINK ELEMENT <TYPE 09)

See Fig. E09.1 and Fig. E09.2 for element behavior and properties.


One line.


1-5(1)

C2.09(b). Property Types


Columns

1-5(1)

10(1)

11-20(1)

21-30(R)

31-40(R)

41-50(R)

51-60(R)

61-70(R)

Notes Variable

Data

No. ofproperty types (max. 40). See section C2.09(b).

Data


Property code. (a) + 1: Acts in tension, unloads inelastically. (b) +2: Acts in tension, unloads elastically. (c) -1: Acts in compression, unloads inelastically. (b) -2: Acts in compression, unloads elastically.

Displacement limit u1•

Displacement limit u2•

Stiffness k1•

Stiffness k2 •

Stiffness k3 •

Unloading stiffness k4 •

Default = k1•

77

-4-


One line for each generation command the first element can be assigned any number.



Columns

1-5U)

6-10(1)

11-15(1)

16-20(1)

21-250)

26-35(R)

Notes Variable

C5

Data


Node number at element end I.

Node number at element end J.

Node number increment for element generation. Default=!.

Property type number.

Initial force or deformation. (a) < 0.0 Initial deformation (slack if tension ele

ment, gap if compression element). (b) > 0.0 Initial force (tension if tension element,

compression if compression element).

78

-5-



LINK ELEMENT (TYPE 09)

Item Description

1 Static force.

2 Viscous force.

3 Deformation.

4 Accumulated inelastic deformation (sum of all positive excursions on lines 2 and 3 if element is inelastic).



7 Line number (0,1,2,3 or 4).

79

Conpression or Tension

Force

Initial state may be prestressed

or hove initial gop/slack.

Gop Opening or Slack

Length nust be non-zero

-r,

FIG. E09'. l

Elastic Unloading

Inelastic Unloading/Reloading

Ll NIC. BEHAVIOR

Shortening or Extension

Link element. Deformation = axial shortening

or axial extension

F'l 6. E'OCJ. 2 E" LE ME NT G Eo METe V

80

Kajima - CUREe Project

Nonlinear Analysis of Reinforced Concrete Three Dimensional Structures

Final Project Heport

Takashi Miyashita N orio Suzuki Hiroshi Morikawa Masaaki Okano Makoto Maruta Motomi Takahashi

(Kajima Corporation)

Graham I-I. Powell Filip C. Fili ppou (CUREe Team)

August 1991

ABSTRACT

The goal of this research is development of a 3-dimensional nonlinear

analysis program for reinforced concrete buildings that considers effects, such

as the biaxial bending moment-axial force interaction, shear deformation in

columns and bond slip in beam-column joits.

The nonlinear analysis methods of reinforced concrete frames have been

investigated by many researchers. The fiber model includes more of the

phenomena involved in reinforced concrete behavior, especially the biaxial

bending moment-axial force interaction in columns. The columns of reinforced

concrete frames in Japan usually have a small shear span ratio. Therefore the

analysis model has to consider the shear deformation of columns. The slippage

of a reinforcing bar from the beam-column joints is too large to neglect at

high stress levels. Such phenomena have to be considered during the analysis.

The improved fiber model that consider:; the shear deformation in columns

and the bond slip in beam-column joints was developed. This method was applied

to the experimental models of columns and beam-column joints which were

performed by Kajima Corporation, Tohoku University, etc.

The conclusions from these analyses are

(1) The fiber model is successful for the analysis of column and

beam subjected to cyclic loading.

(2) The analytical results, especially with regard to deformation

of a column, are in good agreement with the experimental values.

(3) The improved fiber model can easily take into account the effect

of the bond slip of connections.

(4) The analysis of column subjected to high compressive axial

stress ( a = 0. 7F ) underestimate the bending strength, c c

so the stress-strain relationship of concrete at the strain

softning zone should be modified.

TABLE OF CONTENTS

1. Introduction

2. 2- Dimensional Analysis by The Fiber Model

2-1. General

2-2. Analytical Method

2-3. The Fiber Model Considering Shear Deformation

2-4. The Fiber Model Considering Slippage of Reinforcing Bar

2-5. Analytical Studies of Columns and Beams

3. 3 - Dimensional Analysis by The Fiber Model

3-1. General

3-2. Analytical Method

3-3. Analytical Studies of Columns Subjected to Biaxial Bending

and Axial Forces

4. Conclusions

page

1

4

7

10

17

20

69

69

73

89

1. INTRODUCTION

The latest reinforced concrete high-rise buildings tend to have an

irregular shaped plan and elevation. Therefore an accurate analysis method for

such buildings is necessary to predict more realistic behavior when they are

subjected to strong earthquake motions. The columns subjected to biaxial

bending moment and axial force especially show very complicated behavior. The

goal of this research is the development of a 3-dimensional nonlinear analysis

program for reinforced concrete frames.

Nonlinear analysis methods for reinforced concrete frames have been

investigated by many researchers.

A member model based on plasticity theory, which includes the interaction

between the biaxial bending moment and the axial force, has been introduced [1-

1], [1-2], [1-3]. A multi-spring model [1-4], [1-5] has been proposed, which

can simulate both the varying axial force and the biaxial bending interactions,

and consider the stiffness degrading behavior of reinforced concrete columns.

An extensive study of the fiber model approach to structural dynamic analysis

was performed by Mark, et. al [1-6] and the method has been developed by Mahin,

et. al [1-7].

The plasticity theory model can simulate the strength of the bending

moment of members, but the hysteresis loop der:l ved by the analysis does not

always agree with the experimental results. The multi-spring model is not clear

about the definition of spring length. The spring length usually depends on the

magnitude of the bending moment but in the analysis the spring length are held

constant. The fiber model can take into consideration more of the phenomena

involved in the behavior of reinforced concrete members. But, the fiber model

is apt to overestimate the stiffness of a short ·column in which the shear

deformation cannot be neglected. Because the columns of reinforced concrete

frames in Japan usually have a small shear span ratio, the analytical method

- 1-

has to consider the shear deformation of columns. The slippage of a reinforcing

bar from the beam-column joints is too large to neglect at high stress levels.

The above mentioned phenomena, for example the shear deformation of a

column and the slippage of a reinforcing bar, have to be considered during the

analysis. This paper presents a three-dimensional nonlinear analytical method

for reinforced concrete frames using a fiber model that takes into account the

shear deformation of columns and the slippage of the reinforcing bar from the

beam-column joints, and describes the analytical results of columns and beam

column joints.comparing with the experimental results.

-2-

REFERENCES

[ 1-1 ]

[1-2]

[1-3]

[ 1-4]

[1-5]

[1-6]

[1-7]

H.Takizawa and H.Aoyama, "Biaxial Effects in Modeling Earthquake

Response of RIC Structures," Earthquake Engineering and Structural

Dynamics, Vol. 4, 1976.

G. H. Powell and P. F. S. Chen, 11 3D Beam-Column Element with Generalized

Plastic Hinges, 11 ASCE.,EM .. Vol. 112, No. 7, July, 1986.

E. Fukuzawa, Y. Isozaki and M. Takahashi, "Elastic-Plastic Analysis of

Reinforced Concrete Frame in Consideration of Fluctuation of Axial

Forces on Columns, 11 Transactions of Architectural Institute of Japan,

Journal of Structural and Construction Engineering, No. 372, February,

1987.

S.S.Lai, G.T.Will and S.Otani, "Model for Inelastic Biaxial Bending of

Concrete Members, 11 ASCE.,SE. Vol. 110, No. 11, November, 1984.

H. Aoyama, S. Otani and LI. Kang-Ning, "Nonlinear Earthquake Response of

Reinforced Concrete Structures Including Axial Force-Bending Moment

Interaction," Proc. of Annual Meeting of the Architectural Institute

of Japan, Kanto, October, 1988.

Mark, K.M.S. and Rosset, J.M., 11 Nonlinear Dynamic Response of

Reinforced Concrete Frames," Publication R76-38, Department of Civil

Engineering, M.I.r.

S.A.Kaba and S.A.Mahin, "Refined Modeling of Reinforced Concrete

Columns for Seismic Analysis, 11 UCB/EERC-84/03, April, 1984.

-3-

2. 2 - DIMENSIONAL ANALYSIS BY THE FIBER MODEL

2-1. GENERAL

In the earthquake resistant design of reinforced concrete structures, it

is important to understand the inelastic behavior of each constituent member in

3-D frames subjected to multi-directional earthquake motions [2-1]. The

importance of member-by-member analysis is emphasized for this purpose and

various inelastic member models have been developed, to take into account the

biaxial bending and varying axial force.

With the rapid progress in computer capabilities, the member-by-member

analysis has been made possible. More effort .is still necessary, with regard to

cost and computational time. In the earthquake response- analysis of regular

shaped structures, 2-D frame models subjeeted to one-component earthquake

excitation have generally been used, because the dynamic behavior of those

structures do not include torsional vibration components. In order to predict

the inelastic behavior of 2-D frames, pertinent inelastic member models have to

be developed, which can represent the mechanical properties of each member

element such as columns, beams, and walls. Various inelastic member models have

been developed by many researchers. In this report, those models except for a

wall will be reviewed.

One-component model was proposed by Giberson [2-2]. It has two

concentrated inelastic springs at both ends of an elastic line element [Fig.2-

1]. This model has been widely adopted for the analysis of reinforced concrete

frames, because it can incorporate arbitrary moment-rotation relationships in

the end springs. Takeda, et al [2-3] proposed a nonlinear characteristics rule

for cyclic bending behavior of a reinforced concrete element including

cracking, yielding and stiffness degradation.

-4-

Multi-component parallel model was proposed by Clough [ 2-4]. It consists

of two or more components in parallel, superposition of which should represent

the overall elemental nonlinear behavior [Fig.2-2]. For example, in order to

represent a bi-linear skeleton curve, two parallel components are necessary,

one of which is elastic and the other elastic-perfectly plastic. A tri-linear

model needs three or four parallel components.

The above-mentioned models can conside1· only uni-directional bending. A

shear or axial resistance property is usually treated independently, without

considering interaction between multi-component stresses. Therefore, these

models can not represent the effect of a varying axial force.

Fukuzawa, et al [2-5] proposed a plasticity model, which consists of two

similar surfaces representing cracking and yielding in a bending moment-axial

force plane of reinforced concrete columns [Fig.2-3]. Rules for movement and

expansion of these two surfaces are based on plasticity theory. This model can

consider the effect of varying axial force, but can not represent the complex

behavior of reinforced concrete members subjeeted to cyclic loadings.

Kaba and Mahin [2-6] proposed a multi-slice fiber model which can consider

varying axial force and bending [Fig.2-4]. This model consists of several

slices along the member axis. The inelastic moment-curvature relationships are

calculated at the slice located points and the member stiffness is evaluated

assuming linear variation in flexibility between slices. Each slice consists of

several fiber springs representing uni-axial material properties of concrete

and steel. Therefore, the analytical results show good agreement with the

overall experimental data.

Handou, et al [2-7] investigated the influence of a varying axial force in

columns with the static analysis of a 2-dimensional reinforced concrete frame,

using the fiber model. This model can be applied similarly to the earthquake

response analysis of reinforced concrete frames.

- 5-

The fiber model has a tendency to overestimate the stiffness of members at

a small displacement level, especially for a short column, because it neglects

shear deformation. Furthermore, the fiber model cannot take into account the

increasing deformation due to the slippage of the reinforcing bar from beam

column joints.

This chapter presents a fiber model that includes the effect of the

inelastic shear deformation and the slippage· of' the reinforcing bar. And this

model was applied to column and beam elements.

-6-

2-2. ANALYTICAL METHOD

This section describes. the theory of the two-dimensional fiber model

representing the behavior of reinforced concrete members. Usually, the fiber

model is divided into several segments along the axis of the member, and has

slices at the ends of each segment. The fiber model proposed in this section

has only one segment [Fig.2-5], in order to reduce computational requirements.

The slices at the ends of each member are further divided into concrete and

steel fibers [Fig.2-6]. The strains in these fibers are calculated from the

centroidal strain and section curvature by assuming that plane sections remain

plane.

where,

b.£. 1

A£. 1

strain increment of the i-th fiber

(2-1)

A£0

strain increment at centroidal axis

Arp ; slice curvature increment

X. ; distance from centroidal axis 1

If the strains of each fiber are determined, the stresses are derived from

the cyclic stress-strain material relationships, which will be presented later.

A6· =E .. A£. 1 1 1

(2-2)

where, A6. 1

stress increment of the i-th fiber

A£. 1

strain increment of the i-th fiber

E. 1 i tangent stiffness of the i-th fiber

-7-

To obtain good analytical results, exact stress-strain curves should be

· adopted. The steel material properties are represented by the bi-linear curve

[Fig.2-7] or the Ramberg-Osgood curve· [Fig.2-8]. The bi-linear model has only

two possible stiffness. One is the elastic stiffness and the other is the post-

yield stiffness. In this model the post-yield unloading stiffness is the

elastic one. The Hamberg-Osgood model in this report consists of a linear part

and a curvi-linear part. The former is used for the pre-yield and post-yield

loading stiffness and the latter is used for the post-yield unloading

stiffness. The concrete model shown in [Fig.2-9] is assumed to be incapable of

taking any tension. For the compressive stress, the bi-linear curve is adopted

if the stress is less than the compressive strength. Zero stiffness is assumed

after the compressive yield. Unloading proceeds using the degrading stiffness

and reloading to the envelope curve again proceeds using the same stiffness.

If the stiffness of the constituent fibers is determined, the slice

stiffness can be easily calculated.

(2-3)

where, area of the i-th fiber

By assuming linear variation of the bending moment along the member

[Fig.2-10], the axial force and the bending moment at the length z are ;

{ ~N(z)} = [l ~M(z) 1

0

1- z /L

where,

· 0 J{~:;} = [T(z)]{~:;} 1-z /L ~M. ~M.

J J

(2-4)

incremental bending moment of i-end

~Mj ineremental bending moment of j-end

L ; member length

-8-

Therefore, the (3X3) element flexibility matrix [ F] is given by

L

[ F ] = f 0

[ T ( z ) ]T [ f ( z ) ][ 'I' ( z ) ] dz (2-5)

In the multi-slice fiber model [ 2-6], the member stiffness is evaluated

assuming linear variation in flexibility between slices [Fig.2-12]. This model

can represent the behavior of reinforced collcrete members, if the slices are

located at suitable points along the member axis. The computational efforts

increase as more slices are used. Handou, et al [2-7] adopted a two-slice fiber

model which has slices only at both ends of members, and assumed linear

~ variation in flexibility. At all analytical steps, this model has the elastic

flexibility at the center of the longitudinal axis [Fig.2-13].

In this report, two types of fiber models were adopted with regard to the

flexibility distribution. One is the two-slice fiber model. It assumes that the

distribution of the flexibility along the member is a parabolic shape [Fig.2-

11]. The elastic flexibility can be located at any point along the member axis,

according to the balance between the i-end sliee flexibility and the j-end one.

Therefore, the flexibility of any section ean be calculated by parabolic

interpolation. The other method is the multi-slice fiber model, and it is

assumed that the distribution of the flexibility between slices is linear.

- 9 -

2-3. THE FIBER MODEL CONSIDERING SHEAR DEFORMATION

(1) Introduction

A fiber model has been considered to be one of the appropriate analytical

models to simulate the nonlinear response of a reinforced concrete column.

However, most fiber models consider only bending deflection and neglect the

effect of shear deformation. Those fiber models have been acceptable because

they have been applied to columns which had a large shear span ratio. Because

the shear span ratio of the columns in Japanese reinforced concrete buildings

is small, the effect of shear deformation should be considered in a analytical

model. In this section, the necessity of taking the effect of shear deformation

into account is shown, and a model considering shear deformation is proposed.

(2) Necessity of Considering the Effect of Shear Deformation

[Fig.2-14] shows an experimental test specimen of a reinforced concrete

column. This specimen is a half -scale model of a column in a 25-s tor ied

reinforced concrete building constructed by Kajima Corporation in Japan. The

clear span and depth of the original column is 200(cm) and 80(cm),

respectively. If the point of contraflexure was located at the middle of the

span, the shear span ratio could be as small as 1.25. [Fig.2-15] shows the

crack pattern observed after the experiment. Many diagonal cracks due to shear

stress occurred during the loading. The first shear crack was observed at a

shear stress of 26(kgf/cm2), which was about 60(%) of the maximum shear force.

These diagonal cracks suggest the necessity of considering the shear

deformation in an analytical model. This test specimen was analyzed by a fiber

model in which only the bending deflection was considered. The details of this

-10-

fiber model will be presented later. [Fig.2-16] shows the comparison between

the observed load-deformation relationship and the analyzed one. The fiber

model which considered only the bending deflection apparently overestimated the

stiffness. If the effect of shear deformation was considered in the analytical

model, the analytical result would be closer to the observed one.

(3) A Model Considering the Effect of Shear Deformation

(a) Basic Assumptions

A simple way to consider the effect of shear deformation is to introduce

a spring which expresses the relationship between the shear force (stress) and

shear deformation (strain). Hereinafter, such a spring will be referred to as a

"shear spring". One of the features of a fiber model is that it only needs the

dimensions of the column and uniaxial stress-strain relationships of the

materials to predict the restoring force characteristics for bending

deflection. The restoring force characteristics of the shear spring should be

predicted by using the same information used for the fiber model. A truss

analogy model is proposed here to predict the characteristics of a shear

spring. Hereinafter, this model will be referred to as a "shear model". The

basic assumptions for this shear model are as follows.

QD No interaction is considered between the restoring force

characteristics for bending and shear.

GD The volume of the longitudinal bars is infinite.

CD Shear reinforcement is distributed uniformly.

~ Stress and strain of shear reinforcement and concrete are

uniform.

~ Shear cracks distribute uniformly.

-11-

(b) Backbone Curve of Shear Force-Deformation Relationship

The backbone curve of the shear force-shear strain relationship is assumed

to be as shown in [Fig.2-17]. This curve contains three characteristic points,

which are expressed as "C1 11, "C2 11

, and "Y". The points and curves between them

are defined as following.

CD Point 11C 111

This point represents where the first shear cracks occur. It is

assumed that shear cracks occur when the principle tensile stress

reaches the tensile strength of concrete. Shear stress does not

distribute uniformly in a section as shown in [Fig.2-18]. In a

rectangular section, the maximum shear stress is 1.5 times the average

shear stress. Thus, the stress at point 11 C1 11 is defined as follows.

r C1 = V ft ( ft + 6 0 ) I 1.5

e = 0.5 tan-1 ( 2 Tel I 60)

(2-6)

(2-7)

ft, 60

, 0 stand for the tensile strength of concrete, axial stress, and

the angle of the cracks. The total sectional area is used as the

effective shear sectional area.

A=B·D (2-8)

11 8 11 and 11 011 stand for the width and depth of the column.

-12-

® Curve between the origin and point "C1"

This is a linear region. Therefore, the stiffness can be calculated

as follows.

K=G·A (2-9)

@ Point "C2"

At this point, a truss mechanism is assumed to be completed. The

stress at this point is assumed to be the average shear stress when

the tensile principal stress reaches the tensile strength of concrete.

(2-10)

@ Curve between points "C2" and "Y"

A truss mechanism is assumed to be completed at point "C2". This

mechanism consists of the shear reinforcements and concrete struts

with an angle of 8 as shown in [Fig.2-19]. "Strut" means that

concrete is assumed to be a material which can resist only a

compressive stress with an angle of e and that shear stiffness and

tensile principle stress are zero. In this region, the sectional area

which is effective for shear is assumed to be as follows.

(2-11)

jt stands for the distance between the longitudinal reinforcements. In

this model the equilibrium conditions are as follows.

-13-

6 L = 6 1 cos2 8

6 = 6 sin2 8 t 1

r = 6 1 sin 0 cos 0

(2-12)

(JL' (Jt' r stand for the normal stresses in the longitudinal direction

and transverse direction and shear stress. In order to deduce the

compatibility conditions, deformation is separated into two modes (

See [Fig.2-21] ). In one mode, called "Mode A", only the concrete

struts are deformed. In the other mode, called "Mode B", only the

shear reinforcements are deformed. Compatibility conditions can be

expressed as follows.

For "Mode A"

0.5 YA sin 20 (2-13)

El' YA stand for the strain of concrete struts and shear strain of

Mode A.

For "Mode B"

y 8 cot 8 (2-14)

.Et' y8 stand for the strain of shear reinforcements and shear strain

of Mode B. Uniaxial stress-strain relationships of the materials are

assumed as follows.

Shear reinforcement Stress-strain curve is expressed as a bi-linear

type.

6 = E8 E ( E < EY )

(J = (J y ( € > Ey)

-14-

(2-15)

Concrete strut : Compressive strength is reduced as a function of the

tensile strain in the direction perpendicular to the struts. The

function proposed by Collins and Vecchio [2-8] is used here.

a 6 = f3 fc { 1 - (( Eo- E ) I Eo ) }

a = Ec I (fc I E0 ) (2-16)

f3 = 1 I { 0.8 + 0.34 C E2 I e:0 )}

e:2 stands for the strain perpendicular to the strut. From Eq. (2-10) to

Eq.(2-16), the following equations can be deduced.

E 1 sin2 0 eos2 0 6Q =A 6y

1 + E 1 sin4 0 I ( Pw E8 )

(2-17)

E5 , E1 , Pw stand for the tangent moduli of the shear reinforcements

and concrete struts and the ratio of shear reinforcement.

@Point "Y"

At this point, shear reinforcements yield. The force can be

calculated as follows.

(2-18)

6wy stands for the ratio of shear reinforcement.

® After the point "Y"

After the yielding of shear reinforcements, the relationship

between the incremental stress and shear force can be deduced by

exchanging the elastic moduli of the shear reinforcements with the

moduli after yielding.

-15-

(b) Hysteresis Curve

It is not easy to evolve the truss model fot· cyclic loading, because shear

cracks do not intersect at a right angle. Furthermore, the derived hysteresis

curve would not be realistic because some phenomena such as the slip of

reinforcement, etc., are not considered. In this shear model, the hysteresis

curve is derived empirically. One feature of the hysteresis curve of the shear

force - shear deformation relationship is that it shows the "pinching effect"

and the area of the loop is small. In this model, the hysteresis curve is

assumed to be an origin-oriented-type before point "C2 11 • After point "C2", a

tri-linear curve without a hysteresis loop is assumed as shown in [Fig.2-17].

-16-

2-4. THE FIBER MODEL CONSIDERING SLIPPAGE OF REINFORCING BAR

The slippage of a reinforcing bar from the beam-column joints is too large

deformation to be neglected at high stress levels. The hysteretic response of

reinforcing bar anchorages under cyclic excitations is analyzed using F.E.M.

The bonds are idealized as a set of link elements connecting the reinforcing

bar and the concrete element. For bond stress vs. relative displacement between

the reinforcing bar and concrete element, a slip-type hysteresis loop is

assumed. The hysteresis loop of reinforcing bar anchorages can be obtained from

this analysis. Then, the relationship between the stress of the reinforcing bar

and the slippage is idealized as a tri-linear hysteresis loop.

(1) Analysis of the slippage of reinforcing bar by F.E.M.

The analytical model is considered to be composed of concrete, a

reinforcing bar and a bond between the concrete and reinforcing bar as shown in

[Fig.2-22]. Concrete elements are idealized elastic material in order to

simplify the analysis. A slip-type tri-linear .loop proposed by Morita and Sumi

[ 2-9] is assumed for the bond, and the deterioration of bond stress under

cyclic excitations is assumed to take the maximum value * 0.9. The reinforcing

bar is represented by a rod element possessing only axial stiffness and the

stress-strain relationship is assumed to be a bi-linear loop. The analysis is

performed on the beam-column joint model which was done by Kajima Corporation

[Fig.2-23].

The analytical model takes out the beam-column joint [Fig.2-24] and the

bending moments act upon the front of the beam-column joint. [Fig.2-24] shows

the mesh layout and boundary conditions. The bond property is shown in [Fig.2-

25]. The reinforcing bar is assumed to be linear (CASE-A) to make clear the

slippage and bi-linear (CASE-B).

-17-

(2) Results of analysis

The obtained results for the relationships between the external force and

the slippage at the reinforcing bar are shown in [Fig.2-26] (CASE-A) and

[Fig.2-27] (CASE-B). [Fig.2-28] shows the experimental results. The analytical

results don't agree with the experimental results for high external forces,

because in the experiment the reinforcing bar yielded at a lower external force

than the analytical result. [Fig.2·-29] and [Fig.2-30] show the analyzed

relationships between the bond stress and the displacement at typical points.

[Fig.2-31] shows the stress distribution of the reinforcing bar and [Fig.2-32]

shows the stress distribution of the bond.

(3) Assumed hysteresis loop for the slippage

For the analysis by the fiber model, it is necessary to clearly define the

hysteresis loop between the slippage and the stress of the reinforcing bar. The

hysteresis loop is proposed through the analyzed results as shown in [Fig.2-

33]. The skeleton curve is composed of three straight lines as shown in [Fig.2-

34 ].

Point "A" in [Fig.2-34] corresponds to the appearance of bond plasticity.

Point "B" represents the yield point which corr·esponds to the occurrence of the

plasticity of a bond at all points. The stiffness for unloading from the

skeleton curve takes the elastic stiffness (Ke) at all times. During reloading

when the stress of the reinforcing bar is lower than oy, the loading from point

"C" has a stiffness terminating at the maximum deformation. When the stress of

the reinforcing bar is higher than qy and the slippage is lower than ou/4' the

stiffness is usually zero. The loading point "D" moves towards point "E" which

is 0.9 * maximum stress experienced.

-18-

(4) The fiber model considering the slippage of the reinforcing bar

An analysis method that considers bond slip in connections is introduced

as.follows.

[Fig.2-35] shows the analysis method. F'al't; 8 and 8' usually experience a

bending crack, so the analytical length of the beam is assumed to be e. Part C

and C' consider only shear deformation. In the analysis more than four slice

elements are put in a member. When the reinforcing bar has the tensile stress

at part 8 and B', the stiffness of the reinforcing bar is replaced by the

stiffness which is introduced from the idealized hysteresis loop of bond slip

as shown in [Fig.2-36]. Except for this treatment, a member is analyzed using

the general equations of the fiber model. In this analysis the yielding of the

reinforcing bar occur at both ends of part A.

-19-

2-5. ANALYTICAL STUDIES OF COLUMNS AND BEAMS

(1) Analysis of a long reinforced concrete column subjected to an axial force

and a one-directional shear force

a. Objective

With the analysis of reinforced concJ'ete members, it is important to

investigate the effect of the axial force. In this section, long columns

subjected to axial force and cyclic shear force are analyzed by the fiber model

and the analytical results are compared with the experimental ones.

b. Objects of Analysis

Ogawa, et al [2-10] performed the experiment on columns subjected to

cyclic shear force. The specimen as shown in [Fig. 2-37] is a cantilever long

column type. The adopted constant axial forces are of two types ( N= 12. Otonf,

N:24.0tonf ).

c. Analytical Model

The analytical model is shown in [Fig.2-38]. One-component model, multi

component parallel model and two-slice fiber model were used for the analysis.

In the one-component model and the multi-component parallel model, Takeda's

rule [2-3] was used for the bending moment - rotation relationship. In the

fiber model, the division into fibers is shovm in [Fig.2-39], and the Hamberg

Osgood model was used for the steel stress-strain relationship. The material

properties are shown in [Table 2-1].

-20-

d. Results of Analysis

The results of the analysis are shown in [Fig.2-40] and [Fig.2-41). Those

figures show that the fiber model can predict the behavior of the reinforced

concrete members very well.

(2) Analysis of an HiRC column tested by Kajima Corporation

a. ObJective

The columns of reinforced concrete high-rise buildings have a small shear

span ratio in Japan. Therefore it is important to investigate the behavior of

short columns. In this section, an HiRC column is analyzed by the fiber model

and the analytical result is compared with the experimental one.

b. ObJect of Analysis

Bessho, et al [2-11] performed an experiment on an HiRC column subjected

to cyclic shear force. The specimen is shmm in [Fig.2-42]. It consists of

high-strength concrete ( Fc=420kgf/cm2 ) , core rebars, and spiral hoops. The

shear span ratio of this specimen is 1.29.

c. Analytical Model

The analytical model is shown in [Fig.2-43], and the two-slice fiber model

was used for the analysis [Fig.2-44]. The material properties are shown in

[Table 2-2]. The steel stress-strain relationship is a bi-linear type.

d. Result of Analysis

The result of analysis is shown in [Fig.2-45]. It shows that the fiber

model can not accurately predict the behavior of a short column subjected to

cyclic shear force, because of the effects of shear deformation and bond slip.

-21-

(3) Analysis of a Column Subjected to Pure Bending Moment

a. Objective

It is important to investigate whether the fiber model can predict the

moment-curvature relationship of a column well or not. In this section, a test

specimen of a reinforced concrete column subjected to a pure bending moment is

analyzed by the developed fiber model and the analytical result is compared

with the experimental one.


[Fig. 2-46] shows the test specimen of the column in the experiment

performed by Takiguchi [2-12]. This specimen was subjected to an eccentric

axial load as shown in [Fig.2-47]. It was subjected to a pure bending moment

and a varying axial load. The properties of the materials are as shown in

[Table 2-3].

c. Analytical Model

A section was divided into fibers as sh01-m in [Fig.2-48]. The uniaxial

stress-strain relationship of concrete and steel were modeled as shown in

[Fig.2-49]. A confinement effect is considered for the strain softening branch

in compression of core concrete. As the eecentric distance was 50(cm), the

relationship between the bending moment "M" and axial force "N" can be

expressed as M = 50N. In the analysis, this relation was taken into account.

d. Result of Analysis

[Fig.2-50] shows the comparison between the calculated and observed

moment-curvature relationship. It shows that; the fiber model can predict the

behavior of the column subjected to pure bending very well.

-22-

(4) Analysis of a Beam

a. Objective

In order to verify the analytical model considering the effect of shear

and slippage of reinforcements, test data from experiments in which those

effects are measured are needed. From one ex per· iment on a beam-column joint,

such data was obtained. In this section this specimen is analyzed and compared

with the obtained results.

b. Object of Analysis '

[Fig. 2-23] shows a test specimen of a beam-column joint performed by

Kajima Institute of Construction Technology. In this test, the distribution of

curvature along the beam and the slippage of the longitudinal reinforcements of

a beam were measured as shown in [Fig.2-51]. From these data, the contribution

of bending, shear and slippage to the deformation of a beam was estimated. A

beam in this specimen was analyzed as a cantilever element. The properties of

the material were as shown in [Table 2-4].

c. Analytical Model

[Fig.2-52] shows the fiber model idealization of this specimen. The

specimen was modeled as a cantilever beam with one fiber slice at the critical

section. It was assumed that the curvature distributes linearly along the

length of the beam. The stress-strain relationship of materials was assumed to

be as shown in [Fig.2-49]. The restoring force characteristics of shear springs

were defined by using the method mentioned in Chapter 2-3. A slice to

calculate the slippage of longitudinal reinforcements was set at the location

of the bar of the column. The relationship between the slippage and the pull-

out load was as mentioned in Chapter 2-4.

-23-

d. Result of analysis

[Fig.2-53] shows the relationship between the slippage of a reinforcement

and the shear force of a beam. The analytical result shows good agreement with

the observed one. [Fig.2-54] shows the load-deformation curve of the beam. In

this figure, the contributions of bending deflection, shear and slippage are

shown. The comparison between the calculated and obtained results demomstrates

that a model considering the effect of shear deformation and slippage can

simulate the behavior of the beam very well.

(5) Analysis of Reinforced Concrete Short Columns Subjected to Axial Force

and One-Directional Shear Force

a. Objective

Fiber models considering the effect of shear deformation and slippage of

reinforcements were developed. In this section, columns with a small shear span

ratio are analyzed and the developed model is checked.


[Fig.2-55] shows the experimental test specimens of reinforced concrete

column performed by Kajima Institute of Construction Technology. These

specimens were half-scale models of a column in a 25-storied reinforced

concrete building. The shear span ratio wa:> 1 .25. Four tests were performed

under constant axial force as shown in [Fig.2-56]. The parameter of this

experiment was the level of axial force as shown in [Table 2-5]. The properties

of the materials were as shown in [Table 2-6].

c. Analytical Model

[Fig. 2-57] shows the fiber model idealization of these specimens. The

specimen was modeled as a cantilever column with one fiber slice at the

-24-

critical section. It was assumed that the curvature distributes linearly along

the length of the column. Stress-strain relationships of materials were assumed

as shown in [Fig.2-49]. The restoring force characteristics of shear springs

were defined by using the method mentioned in Chapter 2-3. A slice to calculate

the slippage of longitudinal reinforcements was set at the location of the bar

of a beam. The relationship between the slippage and the pull-out load was as

mentioned in Chapter 2-4.

d. Case of Analysis

For each of the four test specimens, three type of analyses, namely (1)

Model-1 ; an analysis considering oniy bending deflection, (2) Model-2 ; an

analysis considering bending deflection and shear deformation and (3) Model-3 ;

an analysis considering bending deflection, shear deformation and slippage of

the longitudinal reinforcements, were performed. In each analysis, an

incremental axial force was loaded up to the applied value in the test, and

then, forced displacement was applied to the free end with an increment of

0.005(cm) under a constant axial force.

e. Results of Analysis

[Fig.2-58] to [Fig.2-69] shows the comparison between the calculated and

observed load-displacement relationship. Although Model-1 could predict the

maximum strength, except for specimen C-4, it overestimated the stiffness.

Model-2 could predict the load at the shear crack. The calculated stiffness was

smaller than the stiffness calculated by Model-1, but it was still larger than

the observed one. By considering the effect of slippage of the reinforcements,

-25-

the analytical results by Model-3 agreed well with the observed ones. The

analysis underestimated the strength of specimen C-4 which was subjected to

high compressive axial stress. In the analysis, cover concrete was crushed and

a rapid reduction of strength occurred. This suggests that the stress-strain

relationship of the strain softening branch of concrete should be modified.

-26-

~--------- L --------~

[ Fig.2-l ] One- component model

1------L

[ Fig.2-2] Multi- component parallel model

N Initi~l Yield Surface h=O Initial Crack Surface

f=O

Moment Formula

[ Fig.2-3] Plasticity model

slices to be located by user

[ Fig.2-4] Fiber 1nodel

-27-

slice ( thickness is zero )

/ 7

R.C. member Two- slice fiber model

[ Fig.2-5 ] 2- Dimensional fiber model

fl N ( axial force )

/1M ( bending moment)

[ Fig.2-6 ] Section slice

-28-

[ Fig.2-7] Steel stress- strain relationship (Bi-linear type) (J

* E - E fi =--·

£ O- E r

a*= a-a. ao- Or

E

[ Fig.2-8] Steel stress- strain relationship (Ramberg- Osgood type) (J

Ea = o

Et E4=-

./ £ a/•o

E

0. 2 Fe

Fe

[ Fig.2-9] Concrete stress- strain relationship

-29-

I fe L ( elastic flexibility )

[ Fig.2-10] Bending moment distribution [ Fig.2-11] Flexibility distribution (parabolic variation)

(a) Actual flexibility variatiol)

I 2 3 567

4

(b) Flexibility variation using 7 slices

I 2 s 6

(c) Superior flexibility v.ariation using only 6 slices

[ Fig.2-12] Flexibility distribution ( 1nulti-slice fiber model)

f se

t--t I I o I

f Sill I :

.i

~· Plber A • f

j_ {SUI

[ Fig.2-13] Flexibility distribution (linear variation)

-30-

uni.t : mm

-81 z _f ,.----7<""--~

9" . I 4oo

~001 687.5 1375 687.5 120d

L _ _l.Q"""O __ ...J-..31.5 . .._ __ 1 O.QO. ___ L3]5c..~l.____,_7....._00......_---t t== ----·--~-L!?.9.

[ Fig.2-14] Test Spechnen of a Short Column

[ Fig.2-15] Crack of a Short Column

-31-

100.~--------------~-----------------------------,

----------

50.

-c: 0

w (.) 0: 0 LL

0: <t w :c en 0.

-50.

---- ANALYSIS

----------· EXPER I HENT

-100.~--------------~--------------~--------------~ -0.5 0.0 0.5 1.0

DISPLACEMENT lcml

[ Fig.2-16] Comparison between the Observed Load-Deformation Relationship and Calculated One by a Fiber Model

-32-

Q

I I I I I I . I

'f

[ Fig.2-17] Assurned Shear Force- Shear Strain Relationship

h

~max.

[ Fig.2-18] Distribution of Shear Stress in a Rectangular Section

-33-

SHEAR REINFORCEMENT M

N N(i/, , , , , , , , , ,

Q' , , , , , , , ,

CONCRETE STRUT

[ Fig.2-19] Truss Mechanis1n

a t

a t

[ Fig.2-20] Equilibriu1n Condition

r-cot8

1 + e t

[ Fig.2-21 ] Mode of Shear Deformation

-34-

y

Reinforcing bar

~

0 X

[ Fig.2-22] Concrete, Reinforcing Bar and Bond Element

JO o[, ___ ~2 QQ._O '2 o =o o:::._ ____ -_____ IJool

0 CJ M

[ Fig.2-23] Experiinental Model of Bean1- Column Joint

-35-

AI ... .... AI

r-<' I"' ['<I "'' "' "' 'I\ 4:C 41

\ 2D 41

\\Bond

I

Concrete Ei t)

0 cc

-

2D 41

4D ~1

1... '~• 1-11 kl ~· HI 1... l..J ,, "'IT 'If "' '<I " 'II '11 'II

80cm

[ Fig.2-24] Analytical Model

r (kgf/cm2)

Su

0.005tf/cm3 S(cm)

Sd

[ Fig.2-25] Idealized Material Property of Bond

-36-

f.:! (tf)

i i 100 i i r ---------- -r-- -- --- -------- - -- --r --- --- -- --- -~

l l 50 l l r ---- -- ----------r-- ---- - ----- ---- --- ----------!------ ---- --- ---~

0 0 i2 o( crti)

[ Fig.2-26] Analytical Result of Slippage of Reinforcing Bar (CASE -A)

o o.~ o(Fm)

[ Fig.2-27] Analytical Result of Slippage of Reinforcing Bar (CASE-B)

o(cm)

[ Fig.2-28] Experimental Result of Slippage of Reinforcing Bar

-37-

j~~~--------·--------------.------------- -------------- --- ... . . . . . . : : . . . . . . . . : : . . 0 0 0 0 0 0 0 0

: : T : : . 1 1 .(kgf/cm ) l 0----------------------- ---1-------------------------- ------------------------""0""------------------------. 0

: 75 i : • 0 • : : : • 0 0 • 0 0

: : : • 0 0 0 0 • 0 0 • 0 0 0 0 0 0 0 0 • 0 0 0

: ! : 0 0 0 0 0 0 • 0 0

0 :

o<crrD

1-------------------75

0

t __________________________ L ___________________________________________________ 1 __________________________ :

[ Fig.2-29] Relationship between Bond Stress and Bond Slip

r- ------- ----- ------------T-------------------r---- --------------- ·--------- ·1· ------------------------ -! ! ! ~~m 1 1 : : : : 1 I I I o I t I I I 1 I

!___________________ _,_______ --··---- ---· -···· ··-----····-- - - ; ---------- ________ _]

: : 50 0 0 0 0 0 0

: :

. !

-O.Or ~005

-------·------------------·-- - .. ---------- ------ ------------·

. : I I 0 • •

..... -------- .. ------------------------------- .. ----- .. --- --------------- ------- .. ----------------------- .. ---- .......

[ Fig.2-30] Relationship between Bond Stress and Bond Slip

-38-

[ Fig.2-31] Distribution of Reinforcing Bar Stress

Q=15tf

QL_ ______ ~2~0cLm------~4~0c-In------~60~c-m------~80cm

[ Fig.2-32] Distribution of Bond Stress

-39-

r ................................................. T ..................................... '"(j ....... ................................................... :-····-······-·--·--·-···--!

l ! (tf/cm2) l l I I • 1

i .. -................... 1 .......................... ············ .......... : .................. ..1 i i 2.5 : i

I i l : :

[ Fig.2-33] Relationship between Stress of Reinforcing Bar and Slippage

(J (tf/cm2

(Jy

o(cm)

[ Fig.2-34] Idealized Relationship between Stress of Reinforcing Bar and Slippage

-40-

Column R . fi B c 1

~L em orcmg ar o umn

j ~ I ---Slice 1,0

_j_/ Slice 2,0 I

(C) ~ I.-Slice 1 Slice 2 (;;(C')

(B) 1 Beam (A)' ~

I I

- .fl .fo .f2-

E:: e

[ Fig.2-35] Analytical Model

Ee

Es C:( a/.ft or olf2)

[ Fig.2-36] Stress- Strain Relationship of Bond Slip

-41-

T 85cm

1

V Axial force

<J-fshear rorce

~

400X400XI200

D b X D = 17 em X 17 em

reinforcement 4-D10

hoop 1.65~@100

a/D=5

[ Fig.2-37] Specimen of a Long Column

~ l:l U (0 .. 004cm) ~Enforced displacement

2

+----17cm -~

1 17cm

1 [ Fig.2-38] Analytical model [ Fig.2-39] Division into fibers

[Table 2-1] Material properties

Concrete Fc=250kg/cm2 E 1 =2.0X105kg/cm2

Steel o-Y=3240kg/cm2 E 1 =2.1X106kg/cm2

E 2 =E1 I 100

-42-

[ N=l2. Oton ] Q (ton)

Multi- component

parallel model

One- component

model

Fibe1· model

Ex peri men t

Q (ton)

Q (ton)

I - - -- -- - - ,- - ---- - - 1 ~ - -- - .LO ~.D.- -2 • 0 I I 1 I I I I I I I I I

' ' ' ' ' ' -1----------~ I I I I I I I I I I

3.0

:--------~---~----~-------~~----1.0 -~--------1 1 I I I I I I

-J', 0 -2'. 0 -1!, 0 I

~-----i ~~~~~~~~-<~!------- 1------~

I I

1-------1 I I

i 1:0 2!0 3:0 I I 1 I

' ----1 o--~--------~--P.l_&r.!1c~DJ ____ J • I I I 1

I I I o I I

I I I t t t

1--------l--------•--------1------- -2.0--J ________ J _______ ~l--------·

(em)

& (em)

[ Fig.2-40] Shear force- horizontal displacement relationships

( N = 12.0 ton )

-43-

Multi- component

parallel model

One- component

model

Fiber model

gxperiment I I I I I I I 1---------.---1 I I I I I I I

L--------•-------

Q (ton)

6 (em)

I

: Jp I

11 1 DISP.(rnm) 1

I I I -1------•--------r-------4--------4 ; I : : I I I 1 I I

-z .. -- --! - ... - - ... -- ... L-- ...... - ... - .J-- ......... -- ... -~


( N = 24.0 ton )

-44-

Shear force

10

[ Fig.2-42] Specimen of a HiRC Column

,J_N ll U ( 0. 004cm )

......,___,_ Enforced displacement

2 I +---- 31 em ----1

80cm <D

l

Q

¢6 (SR30)

Fc=420~m2

1 31cm

l [ Fig.2-43] Analytical model [ Fig.2-44] Division into fibers

[Table 2-2] Material properties

Concrete Fc=459kg/cm2 E 1 =2.46Xl05kg/cm2

E0 = 0.003 Eu = O.OOG(unconfined) Eu = 0.015(confined)

Steel o-Y=4000kg/cm2 E 1 =2.1X10r.kg/cm2

E2 =E 1 I 100

-45-

Experiment

Q (ton) ;-----------~----------1 I I I I . I

~----------~-------60.0--1 I I I I I I I 1-----------1-----------I I I" I I I I I

1-----------.!.-1 I

I i-----1 I I I ,. ----1=:..=.--~~ I I I I I I •-----------r------1 I I I I I I I

L----------~----------

-15 ,;,w'

;' I

60

I

!l ,, { ,.

-r;o

1/100 t.-:oo 1/tOO

VIOO

I

10

-----R(rad)

VIO

IS 20 c::::::,._ 8 ( ...... )

·"

I I

I I

I I

,/ ,/

/

JO

8 • I' j1

N--+~..-.v +P (15511)

r CJ- 2

' H - 1100


( N = 155.0 ton)

-46-

(em)

[Table 2-3 ] Properties of Materials

Concrete Compressive Strength Young's Modulus

Fc{kgf/cm 2) Ec {kgf/cm2

)

248 ?

Steel Yield Strength Young's Modulus

a y{kgf/cm 2) Es{kgf/cm 2

)

DlO 3117 2. 1 X 10 8

6r/J 2746 2.1Xl0 8

Weld

unit mm

[ Fig.2-46] Test Speciinen of a Colmnn Subjected to Pure Bending

[ Fig.2-4 7] Loading Apparatus for Pure Bending

-47-

t'T'\ l't'\

r i'\ '-1./ ./

[ Fig.2-48] Fiber Model Idealization

a· I

1 EuL I

----~---------------------· e o

(a) Concrete

(b) Steel

a = E c/ ( f c/ E. o)

EuL= Ec

E UL = E c E 0 / E T

[ Fig.2-49] Stress- Strain Relationship of Materials

-48-

40.0

-e 0

-40.0 -4.0

,.

CURVATURE (x 1/1000 1/cm)

--- - ---

ANALYSIS

EXPERIMENT

10.0

[ Fig.2-50] Comparison between the Calculated Mornent- Curvature Relationship and Observed One

-49-

20 lll!HISIIrerncnt or ~Iii I -n-'. 11nr~n

"'";"'"";"§:=-±· -

- ~

~-== ·~ OM rneasurernentofbeom bendil 1g deformation

I r---"' 11t -

~~ '-'---

I JSO l JSO

·-I

OM~ I 2 3 4 li 6

~I= cy~ 0[)2 f~ ' +-- -- -- - - . -- -

~I= 01)3 . OD4 I§ 1

r 0~~ .r;;M UH9J 10 1 II 1 12 J :~ -12002oo1 400 I 4QQ I 4QQ I

15( ·~ 50

~ .o-Displocement Tr F t -2s plate • rl: f- • Strain Gaee

·~

ansducer

[ Fig.2-51] Location of the Displace1nent Transducers

r.. ~

-""" -'"" ~ '-'


-50-

-<:: 0

II)

(.)

~

0 lL

~

"' II)

.&: ([)

[Table 2-4] Properties of Materials


Fc(kgf/cm 2) Ec (kgf/cm 2

)

552 2.13Xl0 8


a y(hf/cm 2) Es(kd/cm2

)

041 4200 2.1Xl0 8

016 3640 2.1Xl0 8

150. Ur-----------------.

100.

50.

,. I

I

ANALYSIS -------- EXPERIMENT

o.w---------------+-------------~ 0.0 0.1 0.2

Slip (cml

[ Fig.2-53] Comparison between the Calculated Load- Slippage Relationship and Observed One

-51-

150.~----------------------------------------------~

c: BENDING DEFLECTION + SLIPPAGE 0

w 100. (..) a: 0 lL

a: <( w :c (/)

50.

--- ANALYSIS

----------· EXPERIMENT

0.~--------------~----------------~--------------~ 0.0 1.0 2.0 3.0

OISPLACEM~NT (cml

[ Fig.2-54] Comparison between the Calculated Load-Deformation Relationship and Observed One

-52-

400(b)

0..

+

0.. I

® ~ .~

• F-~ ~ c:: 'll - -..... Cf> I ..,.

rr ~ iQ I

·1111 I !I I -·

~I ~;::::p

8 './)

..... "' .. I , . ..,.

.Jii I

I" I ·-

..

~Q.

!;J;:::;ll;l .A

1

[r

·gl

•

=i Specimen C2-- C4

I I

!riJI~ Long! tudinai Reinforcement Shear Reinforcement : 4-9¢

~T -.. i<: N

12-D22

[ Fig.2-55] Test Speciinens of Short Colmnns

[ Fig.2-56] Loading Apparatus

-53-

0.. I

Cl

[Table 2-5 ] Parameter in Test of Short Columns

Specimen Axial Force (tonf) Axial Stress (kgf/cm') C-1 -87 (Tension) -54.4 C-2 40 25.0 C-3 174 108.8 C-4 348 217. 5


Concrete Specimen Compressive Strength Young·s Modulus

Fc(kgf/cm2) Ec (kgf/ em')

C-1 270 3.09X10 5

C-2 302 3.50Xl0 5

C-3 324 2. 89X 105

C-4 296 3.17Xl0 5

Steel Yield Strength Young· s Modulus

a y(kgf/cm 2) Es(kgf/cm1

)

D22 3880t 2.1 x to• 9¢ 3190 2.1Xl08

......,., 1 ..... I~

) c

~ (

1.-. ....., 1- ~

[ Fig.2-57) Fiber Model Idealization

-54-

100.~------------~~--------------------------~

50.

c: 0 -----

w (.) a: 0 LL

a: <: w I en 0.

-50.

--- ANALYSIS

----------· EXPER I HENT

-100.~--------------+---------------+-------------~ -0.5 0.0 0.5 1.0

DISPLACEMENT lcml

[ Fig.2-58] Cmnparison between the Calculated Load- Deformation Relationship and Observed One ( C-1, Model-l)

-55-

-c 0

w u a: 0 lL

a: <t w :r: en

100.~--------------~------------------------------.

50.

0.

' ' I I

-50.

' I

I ' I

---- ----

---ANALYSIS

----------· EXPEA I MENT

-100.~---------------r---------------+--------------__, -0 . 5 0 . 0 0 . 5 1. 0

DISPLACEMENT lcml

[ Fig.2-59] Comparison between the Caleulated Load- Deformation Relationship and Observed One ( C-1, Model-2)

-56-

c: 0

w (_) a: Cl LL.

a: <( w I en

100.~--~----------~-------------------------------.

50.

0.

-50.

---ANAL YS l S

----------· EXPER l MENT

-100.~--------------~----------0.5 0.0 0.5

DISPLACEMENT (cml 1.0

[ Fig.2-60] Comparison between the Calculated Load- Deformation

Relationship and Observed One ( C-1, Model-3)

-57-

100.~--------------~-------------------------------.

---------

50.

c: 0

UJ C,) a: 0 LL

a: <t UJ :r: en 0.

-50.

---ANALYSIS

----------·EXPERIMENT

-100.~--------------~--------------~----------------; -0 . 5 0. 0 0. 5 1. 0

DISPLACEMENT !cml

[ Fig.2-61] Comparison between the Calculated Load- Deformation Relationship and Observed One ( C-2, Model-l)

-58-

----------------------------

-c: 0

w u a: 0 I.J...

a: <t w :c en

100.~--------------~------------------------------.

50.

0. 0 -----------

-50.

---ANALYSIS


-IOO.w---------------~---------------+---------------4 -0. 5 0. 0 0. 5 1. 0

DISPLACEMENT (cml

[ Fig.2-62] Comparison between the Calculated Load- Deformation Relationship and Observed One ( C-2, Model-2)

-59-

c 0

lJ.J (.) cr: 0 lL.

cr: <( lJ.J :c en

100.~------------~~----------------------------~

50.

0.

-50.

--- ANALYSIS


-100.~------------+---------------+------------~ -0 . 5 0 . 0 0 . 5 I. 0

DISPLACEMENT !cml

[ Fig.2-63] Comparison between the Calculated Load- Defonnation Relationship and Observed One ( C-2, Model-3 )

-60-

100.~--------------~----------------------------.

/

50.

c: 0

w (.) a: 0 lL..

a: < w :c en 0.

I

/

/

I /

I

I

-50.

---- ANALYSIS


-lOO.rn---------------+---------------~------------~ -0.5 0.0 0.5 1.0

DISPLACEMENT !cml

[ Fig.2-64] Comparison between the Calculated Load- Deformation Relationship and Obs~rved One ( C-3, Model-l)

-61-

...... c: 0

w (.) a: 0 LL

a: <( w :c en

100.~--------------.-----------------------------~

50 .

0.

-50. I

I I

I I

----

I

---- ANALYSIS


-100.~---------------+--------------~--------------~ -0.5 0.0 0.5 1.0

DISPLACEMENT (cml

[ Fig.2-65] Cmnparison between the Calculated Load- Deformation Relationship and Observed One ( C-3, Model-2)

-62-

...... c: 0

w (.) a: 0 IJ._

a: <(

w :r: rn

100.~--------------~-----------------------------.

50 .

0.

-50. I

I I

I I

I

I I

I

/

/ I

I

I

---ANALYSIS


-lOO.rn---------------4---------------+-------------~ -0. 5 0. 0 0. 5 1. 0

DISPLACEMENT lcml


-63-

c:: 0

w u a: 0 LL

a: <t w :::r:: en

100.~--------------~-----------------------------~

50.

0.

-50.

I I

I

'

/ /

/-, / .....

I I I I I

I I

I

/

~,----//

/ I

I

------------------------------

----- ANALYSIS


-100.~---------------+--------------~--------------~ -0. 5 0. 0 0. 5 1. 0

DISPLACEMENT !cml

[ Fig.2-67] Cmnparison between the Calculated Load- Deformation Relationship and Observed One ( C-4, Model-l)

-64-

..... c: 0

w (.) a: a lL

a: <( w :r: en

100.~--------------~----------------------------,

50 .

0.

-50.

I

I

,-'1 / ,,

/ ' / I I 1 I

I

----- ANALYSIS


-100.~--------------~------------~~----------~ -0. 5 0. 0 0. 5 1. 0

DISPLACEMENT lcml


-65-

100.~--------------~----------------------------~

/

I

/

I

.... - ...........

-- -----

---ANALYSIS


-1 00. U+-------__,f--------0.5 0.0 0.5

DISPL.ACEMENT lcml 1.0

[ Fig.2-69] Comparison between the Caleulated Load- Deformation Relationship and Observed One ( C-4, Model-3)

-66-

REF'ERENCES

[2-1]

[2-2]

[2-3]

[2-4]

[2-5]

[2-6]

[2-7]

A.Shibata, "State-of-the-Art Report: Inelastic Response of 3-D

Structures and Multi-Directional Seismic Forces on Structural

Components," Proc. of the 9th World Conference on Earthquake

Engineering, Japan, 1988.

M.F .Giberson, "Two Nonlinear Beams with Definition of Ductility,"

Proc. of ASCE., Vol. 95, No. ST2, 1969, pp. 137-157.

T.Takeda, M.A.Sozen and N.N.Nielsen, "Reinforced Concrete Response to

Simulated Earthquakes," Proc. of ASCE., Vol. 96, No. ST12, Dec. 1970,

pp. 2557-2573.

R.W.Clough, K.L.Benuska and E.L.Wilson, "Inelastic Earthquake Response

of Tall Buildings, II Proc. of 3rd World Conference on Earthquake

Engineering, New Zealand, 1965.

E.Fukuzawa, Y.Isozaki and K.F'ujisaki, "Elastic-plastic Earthquake

Response Analysis of Reinforced Concrete Frame in Consideration of

Fluctuation of Axial Forces on Columns," Proc. of the 9th World

Conference on Earthquake Engineering, Japan, 1988.

S.A.Kaba and S.A.Mahin, "Refined Modeling of Reinforced Concrete

Columns for Seismic Analysis," Ear·thquake 'Engineering Research

Center, University of California, Berkeley Report, No. UCB/EERC-84/03,

April, 1984.

M.Handou, A.Shibata and J .Shibuya, "Earthquake Response Analysis of

R.C. Columns with Varying Axial Forces by means of Fiber Model,"

Summaries of Technical Papers of Annual Meeting, Arch. Inst. of Japan,

1987. 10.

-67-

[2-8]

[2-9]

[2-10]

[ 2-11 J

[2-12]

F. Vecchio and M.P. Collins, "The Mod Hied Compression-Field Theory for

Reinforced Concrete Elements Subjected to Shear," ACI Journal, No.

83-22, pp. 219-231, 1986.

S.Morita and T.Sumi, "Local Bond Stress Slip Relationship under

Replaced Loading," Transactions of 1\rchitectural Institute of Japan,

Journal of Structural and Construction Engineering, No. 229, March,

1975.

J.Ogawa, M.Hoshi and Y.Abe, "Experimental Study on ·Force-Displacement

Relation of Reinforced Concrete Columns," Summaries of Technical

Papers of Annual Meeting, Arch. Inst. of Japan, 1979.9.

S.Bessho, M.Fukushima and H.Hatamoto, "Columns and Beam-Column joints

of a 30-Story Reinforced Concrete High-Rise Building," Annual Report

of Kaj ima Institute of Construction Technology, Kaj ima Corporation,

Vol. 34.

K.Takiguchi, S.Kokusho and K.Okada, "Experiments on Reinforced

Concrete Columns Subjected to Bi-axial Bending Moments," Transactions

of Architectural Institute of Japan, Journal of Structural and

Construction Engineering, No. 239, May, 1975.

-68-

3. 3-DIMENSIONAL ANALYSIS BY FIBER MODEL

3-1. GENERAL

A column in a building is subjected. to bi-directional lateral force and/or

varying axial force during an earthquake. A fiber model has been considered as

one of the appropriate analytical models to simulate the nonlinear response of

a reinforced concrete column under such a complicated loading process. In this

chapter, the outline of a 3-dimensional (3-D) fiber model and its application

will be mentioned.

3-2. ANALYTICAL METHOD

(1) Theory of 3-Dimensional Fiber Model

It is easy to expand the 2-D fiber model, which was mentioned in the

previous chapter, to a 3-D fiber model. In the 3-D fiber model, each section is

divided into 2-dimensional meshes as shown in [Fig.3-1]. Each mesh is called a

"fiber" and such a section is called a "slice".

In order to calculate the deformation of a element, a finite number of

slices are set along the length of an element. In the k-th slice, the curvature

around y and z axis ( ¢y' ¢z ) and axial strain ( Eo ) at the center of the

section are related to the moments around y and z axis ( MY, Mz ) and axial

force ( ~ ) by the following equation.

( 3-1)

-69-

If a shape function was assumed, the rotation angle at both ends of the

element ( ryi' rzi' ryj, rzj ) can be calculated from the curvatures and strains

at the slices as follows.

(3-2)

In this report, it is assumed that the curvatures and strain distribute

linearly between the slices as shown in [Fig.3-1]. The moments, axial forces,

curvature and strain are related as following.

(3-3)

The moments and axial forces at each slice are calculated from the moments

and axial forces at both ends as following.

{ M 1 M 1 N 1 ••• M M N }T = [A] { M . M. N. M . M. N. }T y z yn zn n y1 z1 1 YJ ZJ J (3-4)

When the element is subjected to axial force

Mzi, Myj, Mzj ) and shear force ( Qyi, Qzi' Qyj' Qzj

the following relations.

Ni, Nj ) , moments ( Myi'

at the ends, there exists

{ N. Q . Q . M . M . N. Q . Q . M . M . }T I Yl Zl Yl Zl J YJ ZJ YJ ZJ

= [ T ]T { N. M . M . N. M . M . }T 1 yt Zl J YJ ZJ

(3-5)

When the element is subjected to axial for·ce, moments and shear force at

the ends, those forces can be related to the displacement at both ends as

follows.

-70-

{ Ni Qyi Qzi Myi Mzi Nj Qyj Qzj Myj Mzj }T = [ T JT ( [ N] [ D ]·1 [A] )·1 [ ~]

{u. v. w. e. e. u. v. w. e. e .}T (3-6) 1 1 1 y1 7.1 J J J YJ ZJ

[ T ]T ( [ N] [ D ]-1 [A] )·1 [ T] is the 3-D stiffness matrix of the element and

it can be used to make a stiffness matrix of the total structure.

(2) Constitutive Equations of Materials

In a fiber model, uniaxial stress-strain relationships of concrete and

steel are needed. In this model, they are assumed to be as shown in [Fig.2-49].

In the stress-strain relationship of concretE~, strain softening in compression

and tension stiffening are considered. If the tangent stiffness of many fibers

was negative, the calculation may become difficult. Therefore, the tangent

stiffness in these regions was assumed to be zero and the unbalanced force are

released at the next calculation step. The confinement effect from the shear

reinforcements is taken into account in the calculation of the stiffness in the

strain softening region by using Suzuki's proposal [3-1].

The stress-strain model of steel was based on the proposal by Filippou [3-

2]. In this model Baushinger effect is considered.

(3) Effect of Shear Deformation

The two shear springs, which were mentioned in chapter 2-3, are inserted

in the y and z direction. Although there may be some interaction between the

bi-directional shear forces and deformations, those two springs are assumed to

be independent in this model.

-71-

(4) Effect of Slippage of Longitudinal Reinforcements

As mentioned in chapter 2-4, one slice is set at the end of the element to

estimate the rotation due to the slippage of longitudinal reinforcements. As

this slice is divided into 2-dimensionals, the effect of bi-directional bending

is automatically taken into account.

-72-

3-3. ANALYTICAL STUDIES OF COLUMNS SUBJECTED TO BIAXIAL BENDING AND

AXIAL FORCES

(1) Analysis of a Column Subjected to a Pure Bi-directional Bending Moment

a. Objective

Test specimens of a reinforced concrete column subjected to a pure bi

directional bending moment are analyzed by the developed fiber model and the

analytical results are compared with the experimental one.

b. ObJect of Analysis

Test specimens and the loading apparatus are the same as those shown in

section 2-5.(3). In the experiments with foUJ• specimens, a constant bending

moment is held in one direction and a cyclic forced curvature was applied in

the orthogonal direction as shown in [Fig.3-2]. The parameter was the level of

the constant moment. One specimen was loaded diagonal forced curvature.

c. Analytical Model

Each section was divided into fibers as shown in [Fig.3-3]. The uni-axial

stress-strain relationships of concrete and steel were modeled as shown in

[Fig.2-49]. As the eccentric distance was 50(cm) in each direction, the

relationship between the bending moment "M" and axial force "N" can be

expressed as M = 50N. In the analysis, this relation was taken into account.

d. Results of Analysis

[Fig.3-4] to [Fig.3-8] show the comparison between the calculated moment

curvature relationships and observed ones. When the constant moment is small,

the calculated results agreed well with the observed ones. When the constant

moment is large, the analysis underestimated the ductility. The calculated and

- 73-

observed results agreed well up to the maximum strength. For diagonal loading,

the analysis could predict the behavior of the specimen very well.

(2) Analysis of Columns Subjected to Bidi1·ectional Shear/Bending

a. ObJective

Test specimens subjected to bi-directj ona.l shear force are analyzed to

verify the developed fiber model.

b. ObJects of Analysis

[Fig.3-9] shows the test specimen performed by Aoyama et. al [3-3]. This

column was subjected to a constant axial force of 40(kgf/cm2 ) and bi

directional forced deformation. The orbit of the displacement of the top of the

column was rectangular as shown in [Fig.3-10]. The properties of the materials

were as shown in [Table 3-1].

c. Analytical Model

The section was divided into fibers as shown in [Fig.3-11]. Stress-strain

relationships of materials were assumed to be as shown in [Fig.2-49]. The

restoring force characteristics of the shear springs were defined by using the

method mentioned in Chapter 2-3. A slice to calculate the slippage of

longitudinal re~nforcements was set at the location of the bar of a base. The

relationship between the slippage and the pull-out load was as mentioned in

Chapter 2-4.

-74-

d. Case of Analysis

For each of the four test specimens, three type of analyses, (1) Model-1 ;

an analysis considering only bending deflection, (2) Model-2 ; an analysis

considering bending deflection and shear d•~formation and (3) Model-3 an

analysis considering bending deflection, shear deformation and slippage of the

longitudinal reinforcements, were performed. In each analysis, an incremental

axial force was loaded to the applied value in the test, and then, forced

displacement was applied to the free end with an increment of 0.005(cm) under

constant axial force.

e. Results of Analysis

[Fig.3-12] to [Fig.3-14] shows the comparison between the calculated and

observed load-deformation relationship in eaeh principal direction. As the

column had a large shear span ratio of 3.0, the effect of shear deformation was

not so large. Therefore, the results calculated by Model-1 and Model-2 were

very similar and overestimated the stiffness. On the other hand, the model in

which the effect of slippage and shear deformation were considered could

predict the behavior of the column well. Namely, in a element with a large

shear span ratio, the effect of slippage of the reinforcements are important.

-75-

y

Curvature

[ Fig.3-l ] 3- Dirnensional Fiber Model

-76-

M2 B-R-1.0 B-0-1.0

-Moy 8-R-0.75 B-0-0.75

s-R-o:5 B-0-0.5

B-R-0.25 B-0-0.25

B-R-0.0 B-0-0.0 I M1

Moy M2

[ Fig.3-2] Loading Pattern in the Test of Bidirectional Bending


-77-

40.0.-----------~------------------~

'\

, , , , - I = 0 I ..... I a:: 0 .... I

f-:z; w ::e 0 ::e

I

ANALYSIS

-------- EXPERIMENT

-40.0+-------------~------------------~ - 4• O CURVATURE (x 1/1000 1/cm) B. O

[ Fig.3-4] Comparison between the Calculated Mo1nent- Curvature Relationship and Observed One ( B-R-0.25)

-78-

~0.0,-----------~----------~

-= 0

...... = 0 .....

-----------1

f I

I I

, -.. -

I

I

I I I I I I I I I I I I

I

I

:--- ANALYSIS I

-~O.O ~--------------~:-----_--_-_--__ E_X_P_E_R_IM_E_N~T -~. 0 CURVATURE (x 1/1000 1/cm) ~. 0

[ Fig.3-5 ] Comparison between the Calculated Moment- Curvature Relationship and Observed One ( B-R-0.5)

-79- "

30.0

-IS ()

...... c 0 ....

I I I I f

I . -

I /

I I

' , I 1

I

ANALYSIS

EXPERIMENT

-30.0 +-------------~------------~ -4. 0 CURVATURE (x 1/1000 1/cm) 4. 0

[ Fig.3-6] Con1parison between the Caleulated M01nent- Curvature Relationship and Observed One ( B-R-0.75)

-80-

i , .. I

30.0

·;'- ... ~-, -= I 0

~ ..... /I

I s:: 0 I .....

I !-::z: ------,-t<:l ::e 0 ::e I I

I I I

I I I

I I I

I I I , . I r

l jl

/ : lv ANALYSIS

_20 _0 +---------~--EX_P_ER_I_ME_N~T -2.0 2.0

CURVATURE ( x 1/1000 1/cm}

[ Fig.3-7] Comparison between the Calculated Mmnent- Curvature Relationship and Observed One ( B-R-1.0)

-81-

30.0 .---------~~---------------------.

-s (.)

...... r::: 0 .... -

I I I I

I I I I I

I I

I I

--- ,--------,---- ---- ------- ----------------------

I I

1 I

I I

I I I

ANALYSIS

-------- EXPERIMENT

-30.0+-----------~---------------------4 -2.0 4.0

CURVATURE (x 1/1000 1/cm}

[ Fig.3-8] Co~parison between the Calctilated Moment- Curvature Relationship and Observed One ( B-R-1-1)

-82-

0 0 \0

0 0 M

p

4D 13

¢6@50

(Unit: mm)

l N=l6tonf 25 ISO 25

II 1-t 1-- ID 1-

~ --

~ Ps=l.27%

~ pw=0.56% 1-

----

I) ' r--.. T t.=: I, 200 ,

[ Fig.3-9] Test Speci1nen of a Column Subjected to Bidirectional Shear and Bending Moment

0-Y(cm)

-2.0 0-X(cm)

-2.0 14~------~=}--____ _

5

[ Fig.3-10] Orbit of the Forced Deforn1ation

-83-



Fc(kgf/cm 2) Ec (kgf/cm 2

)

162 1.9X10 5


u y(kgf/cm 2) Es (kgf/cm 2

)

D13 3973 2.1Xl0 5

6¢ 2516 2.1Xl0 5

r

~

[ Fig.3-ll ] Fiber Model Idealization

-84-

~ 6.~---------------------------------------.---------------------------------------,

>j' 4-. w (_) a: a I.J... 2. 0 a: <t: w :c (F) 0.0

-2.

-4-.

-----..: / /

' ' ' ' ' ' ---ANALYSIS


-6.~----------~--------~----------~-----------~---------+--------~

c: 0

>-I

w (_) a: a I.J...

a: <t: w :c en

-3.0 -2.0 -1.0 0.0 1.0 2.0 3.0

6.

4-.

2.

0.

-2.

-4-.

-6. -3.0 -2.0

D I SPLACEMENT-X ( c m I

----1 ~;:;...::.-=----,,

.,.---------------

~ ' - ----- -- ..,....~-5.:::"'~- - - - : _.... :

-1.0

' ' ' ' ' 0. 0 1. 0

---ANALYSIS


2.0 DISPLACEMENT-¥ (cml

3.0

[ Fig.3-12] Comparison between the Calculated Load- Defonnation Relationship and Observed One (Model-l)

-85-

-c: 6. 0

X 4-. I w (.) a: Cl lL. 2. a: <( w :r: U1 0.

-2.

-4-.

-6. -3.0

- 6. c: 0

>- 4-. I w (.) a: Cl lL. 2. a: <( w :r:

-2.0 -1.0

---ANALYSIS


0.0 1.0 2.0 DISPLACEMENT-X lcml

---"'-==-=-~---------..,\ I I \ . I

I

3.0

U1 0. -7---------------

-2.

---ANALYSIS -4-.


-6. -3.0 -2.0 -I. 0 0. 0 I. 0 2.0

DISPLACEMENT-'!' lcml

[ Fig.3-13] C01nparison between the Calculated Load- Deformation Relationship and Observed One ( Model-2)

-86-

3.0

- 6. c: C>

X 4-. I lJJ (.) a: CJ LL 2. a: <{ lJJ I o. en

-2.

---ANALYSIS -------


m------~2.~o-----~~.~o~---~o~.~o~--~~,~.o~---~2~.on-------~3 .. o DISPLACEMENT-X lcml

c: 6.~----------------~--------,-·------------------------~ 0

>-1

lJJ (.) a: CJ LL

a: <{ lJJ I en

4-.

2.

0.

-4-. --------- ---· -;~-::-.:-~;;: ~--:.: i--I I I

'

- __ , --- ' \_~=~--~~

I I

I

-7---------------

---- ANALYSIS


-6.~--------------+-------------~--~----------~~.~----------~~----------~~~----------~ -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0

DISPLACEMENT-Y lcml

[ Fig.3-14] Comparison between the Calculated Load- Deformation Relationship and Observed One ( Model-3)

-87-

REFERENCES

[3-1]

[3-2]

- [3-3)

K.Suzuki, T.Nakatsuka and M.Sugata, "Mechanism of Confinement and

Strength - Deformantion Characteristics of Confined Concrete with

Rectangular Lateral Reinforcement," Proc. of the Japan Concrete

Institute, Vol. 11, No. 2, pp. 449-454, 1989.

F.C.Filippou, E.P.Popov and V.V.Bertero, "Effects of Bond

Deterioration on Hysteretic Behavior of Reinforced Concrete Joints,"

Earthquake Engineering Research Center, University of California,

Berkeley Report, No. UCB/EERC-83/19, August, 1983.

H.Minamino, M. Yoshimura, S.Fujii and ll.Aoyama, "Study on Reinforced

Concrete Columns Subjected to Bi-aKial Bending," Summaries of

Technical Papers of Annual Meeting, Arch. Inst. of Japan, pp. 1293-

1294, 1973.

-88-

4. CONCLUSIONS

The analysis code of three-dimensional reinforced concrete frames is

developed using a fiber model which can take into consideration the biaxial

bending moment - axial force interaction in columns. Through the analysis of an

experimental model of columns and beam-column connections, it is recognized

that the fiber model has to consider the shear deformation in columns and bond

slip in connections. The improved fiber model considering such phenomena, is

applied to the columns and the beam-column connections which were tested by

Kajima Corporation, Tohoku University, etc.

The conclusions from these analyses are

(1) The fiber model is successful for the analysis of column and

beam subjected to cyclic loading.

(2) The analytical results, especially deformation of the column is

in good agreement with the experimental ones.

(3) The improved fiber model can easily consider the bond slip of

connections.

(4) The analysis of column subjected to high compressive axial

stress ( cJ = 0. 7F ) underestimate the bending strength, c c

so the stress-strain relationship of concrete at the strain

softning zone should be modified.

-89-

Shedule for the Presentation in 1991 and 1992

Domestic Conference

1. Annual Meeting of Architectural Institute of Japan

( September 1991, in Sendai )

2. Annual Meeting of Architectural Institute of Japan

( 1992 )

3. Japan Concrete Institute Conference

( 1992 )

nonlinear analysis of reinforced concrete three-dimensional

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