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
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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
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
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-
DRAIN-ANAL USER GUIDE
INPUT DATA SECTION C2.02
BEAM-COLUMN ELEMENT <TYPE 02)
See Fig. E02.1 through Fig. E02.6 for element behavior and properties.
C2.02(a). Control Information
One line.
Columns Notes Variable
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·
C2.02(b). Stiffness Types
One line for each stiffness type.
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
Stiffness type number, in sequence beginning with 1.
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.
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.
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)
26-300)
31-35(1)
36-40(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 end j.
Node number increment for element generation. Default= 1.
Stiffness type number.
End eccentricity number. Default = no end eccentricity.
Yield surface number at end i
Yield surface number at endj.
43
-13-
DRAIN-ANAL USER GUIDE
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.
Columns Notes Variable
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).
15 Node number at end I.
16 Node number at end J.
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
DBAIN-2DX USER GUIDE
ELEMENT THEORY
SIMPLE CONNECTION ELEMENT (TYPE 04)
E04.1 GENERAL CHARACTERISTICS
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-
DRAIN-ANAL USER GUIDE
INPUT DATA SECTION C2.04
SIMPLE TRANSLATIONUROTATIONAL CONNECTION ELEMENT (TYPE 04)
See Figs. E04.1 and E04.2 for element behs:vior and properties.
C2.04(a). Control Information
One line.
Columns Notes Variable
1-5(1)
C2.04(b). Property 'JYpes
One line for each property type.
Columns Notes Variable
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-
C2.04(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 end I.
Node number at element end J.
Node number increment for element generation. Default= 1.
Property type number.
54
-5-
DRAIN-POST USER GUIDE
OUTPUT ITEMS FOR POSTPROCESSING
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).
6 Node number at end I.
7 Node number at end J.
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
DBAIN-2DX USER GUIDE
ELEMENT THEORY
GAP-FRICTION JOINT ELEMENT (TYPE 05)
E05.1 GENERAL CHARACTERISTICS
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.
E05.2 STATIC ELASTO-PLASTIC STIFFNESS
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-
DRAIN-ANAL USER GUIDE
INPUT DATA SECTION C2.05
GAP FRICTION JOINT ELEMENT (TYPE 05)
See Fig. E05.1 through E05.3 for element behavior and properties.
C2.05(a). Control Information
One line.
Columns Notes Variable
1-5(1)
6-10(1)
C2.05(b). Gap Property Types
One line for each property type.
Columns Notes Variable
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
Property type number, in sequence beginning with 1.
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)
Property type number, in sequence beginning with 1.
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-
ments must be included
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.
Node number increment for element generation. Default= 1.
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-
DRAIN-POST USER GUIDE
OUTPUT ITEMS FOR POSTPROCESSING
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)
E06.1 GENERAL CHARACTERISTICS
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
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)
E06.3 OTHER PROPERTIES
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-
DRAIN-ANAL USER GUIDE
INPUT DATA SECTION C2.06
STRUCTURAL PANEL ELEMENT (TYPE 06)
See Fig. E06.1 through E06.3 for element behavior and properties.
C2.06(a). Control Information
One line.
Columns Notes Variable
1-5(1)
C2.06(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)
56-65(R)
66-75(R)
Notes Variable
Data
No. of stiffness types (max. 40). See section C2.06(b).
Data
Stiffness type number, in sequence beginning with 1.
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-
C2.06(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)
26-30(1)
31-35(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 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).
Node number increment for element generation. Default= 1.
Stiffness type number.
•r'
72
-6-
DRAIN-POST USER GUIDE
OUTPUT ITEMS FOR POSTPROCESSING
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
DBAIN-2DX USER GUIDE
ELEMENT THEORY
LINK ELEMENT (TYPE 09)
E09.1 GENERAL CHARACTERISTICS
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).
E09.3 OTHER PROPERTIES
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-
DRAIN-ANAL USER GUIDE
INPUT DATA SECTION C2.09
LINK ELEMENT <TYPE 09)
See Fig. E09.1 and Fig. E09.2 for element behavior and properties.
C2.09(a). Control Information
One line.
Columns Notes Variable
1-5(1)
C2.09(b). Property Types
One line for each property type.
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 type number, in sequence beginning with 1.
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-
C2.09(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
1-5U)
6-10(1)
11-15(1)
16-20(1)
21-250)
26-35(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 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-
DRAIN-POST USER GUIDE
OUTPUT ITEMS FOR POSTPROCESSING
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).
5 Node number at end I.
6 Node number at end J.
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
: : 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
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