Research Collection
Conference Paper
User Elements Developed for the Nonlinear Dynamic Analysis ofReinforced Concrete Structures
Author(s): Wenk, Thomas; Linde, Peter; Bachmann, Hugo
Publication Date: 1993
Permanent Link: https://doi.org/10.3929/ethz-a-007215071
Rights / License: In Copyright - Non-Commercial Use Permitted
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ETH Library
ABAQUS Users' Conference, June 23-25, 1993, Aachen, Germany
User Elements Developed for the Nonlinear Dynamic Analysis of
Reinforced Concrete Structures
Thomas Wenk, Peter Linde and Hugo Bachmann
Institute of Structural Engineering
Swiss Federal Institute of Technology (ETH)
CH-8093 Zurich, Switzerland
ABSTRACT
A library of ABAQUS user elements has been developed for the seismic analysis of reinforced
concrete structures. The library includes elements for the modelling of the hysteretic behaviour of
reinforced concrete beams, columns and walls.
As numerical example the user elements were used for the seismic analysis of a six-storey
reinforced concrete building with different configurations of the load-bearing system. The results of
the analyses are presented in the form of a video animation showing the dynamic behaviour of the
structures during the earthquake.
INTRODUCTION
For the numerical analysis of reinforced concrete structures under seismic action a software tool is
necessary which properly simulates the hysteretic behaviour of the plastic hinges in walls and
frames. A thorough evaluation of existing finite element programs for nonlinear dynamics led to the
development of ABAQUS user elements for the modelling of the plastic hinge zones (Wenk,
Bachmann, 1991). These elements, treating the cyclic behaviour of plastic hinge zones in reinforced
concrete beams, columns, and structural walls are presented in this paper.
The user elements were developed for the verification of the seismic behaviour of reinforced
concrete buildings designed according to the capacity design method (Paulay, Bachmann, and
Moser, 1990). Briefly explained, this method focuses on the establishment of clearly defined plastic
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hinge zones, the proper detailing of these, and the protection of the remaining elastic parts of the
building against yielding.
To show the applicability of the user element, nonlinear dynamic analyses of a series of two-
dimensional frame wall buildings were carried out (Bachmann, Wenk, and Linde 1992). The
analysis results of one of these buildings is presented here as numerical example.
MODELLING OF BEAMS
In reinforced concrete frames designed according to the capacity method, plastic hinges will form
only at predetermined and specially detailed locations. To model these plastic hinge zones a two
node beam element with nonlinear hysteretic flexural behaviour has been developed as ABAQUS
user element U1 (fig. 1). The element length Lp is taken equal to the ductile detailed length of the
beam in the structure. The remainder of the beams is modelled by linear beam elements (type B23).
A linear moment gradient is assumed over the length of the plastic hinge element. The bending
stiffness EI of the element is a function of the average moment Mi (fig. 1) and is kept constant over
the element length. A simplified hysteretic model with an asymmetric bilinear skeleton curve is
used for the moment vs. curvature relation as shown in fig. 1 (right). The moment curvature relation
is completely defined by four design parameters of the cross section: elastic bending stiffness EIel,
positive yield moment My+, negative yield moment My
-, and yielding stiffness EIpl.
The often observed phenomena in cyclic behaviour of reinforced concrete sections such as
strength degradation, pinching and bond slip of reinforcement are avoided in a capacity designed
structure by appropriate constructive measures. These phenomena are consequently not included in
the hysteretic model. The influence of concrete cracking is taken into account from the beginning
by reducing the elastic flexural stiffness to 40% of the stiffness of the uncracked section.
The axial behaviour of user element U1 is assumed to be linear. Shear deformations are
neglected. A summary of the input properties required in the *UEL PROPERTY option is shown in
table 2. The internal state variables utilised in the formulation of the user element U1 are given in
table 3.
MODELLING OF COLUMNS
Plastic hinges in columns are avoided in general by the capacity design concept, since it is more
difficult to obtain a ductile behaviour in compression members. However, at the foundation level
the formation of plastic hinges in the columns can usually not be prevented. In addition plastic
column hinges are often provided at the top floor, where axial forces in the columns are small.
The modelling of the column hinges is similar to the modelling of the beam hinges. Over the
plastic hinge length Lp the column is discretised by a nonlinear user element U2 as in fig. 1 (left).
For the rest of the column the linear beam element B23 is used. To account for the influence of the
axial force, the skeleton curve of the moment curvature relation is expanded or shrunk as a function
of the current axial force Ni, as shown in fig. 2. Inside the skeleton curve for flexural behaviour, the
same hysteretic model as for user element U1 (fig. 1) is employed for user element U2. The axial
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behaviour of U2 is always kept elastic and shear deformations are neglected. The additional input
properties of user element U2 compared to U1 are summarised in table 4.
MODELLING OF STRUCTURAL WALLS
The numerical modelling of structural walls is carried out with a macro model, consisting of four
nonlinear springs connected by rigid beams, as seen in fig. 3 (left). The corresponding user element
U3 with its four nodes and ten degrees of freedom is shown in fig. 3 (right).
The two outer vertical springs Kf model the flexural behaviour of the entire wall cross section,
and follow hysteretic rules seen in fig. 4, showing spring force vs. spring displacement. The main
features of the hysteretic rules consist of the skeleton curve, and of unloading and reloading curves.
The skeleton curve is made up of an elastic compressive stiffness Kel, a cracked tensile stiffness Kcr
and a yielding stiffness Ky, the latter two of which are taken as fractions of the compressive elastic
stiffness.
The unloading rule in the tensile region (fig. 2) is parallel to the stiffness Ku indicated in the
figure. The unloading in the compressive region occurs towards a point !clFy on the elastic
compressive branch, denoting the point where flexural cracks are closing. This point determines the
fatness of the hysteresis loops, and its force level was found to be roughly equal to the effective
axial force acting on the wall section in order to get reasonable flexural hysteretic behaviour. The
reloading always occurs towards the maximum displacement reached. A more detailed discussion
on the hysteretic rules is provided in (Linde, 1993).
The central vertical spring Kc models the axial behaviour together with the flexural springs, and
is active only in compression. The horizontal spring Ks models the shear behaviour. Since the walls
studied here behaved mainly elastically in shear, although some minor shear cracking may occur, a
bilinear origin oriented hysteretic model as shown in fig. 4 is considered sufficient (Linde, 1993).
A complete description of the input properties as well as the internal state variables of the wall
user element U3 is presented in tables 5 and 6.
NUMERICAL EXAMPLE
Description of six-storey building designs
As numerical examples four different designs of the load bearing structure of a six-storey reinforced
concrete building were analysed (fig. 6):
Design F: consists of a moment resisting frame designed for gravity load and masses tributary to
one transverse bay width of 6.40 m. Plastic hinges are allowed to form in the beams at column
faces, and in the columns at the foundation and roof only.
Design W: consists of a structural wall combined with gravity load columns designed for gravity
load tributary to one bay as in design F, however for masses tributary to two transverse bays with a
width of 6.40 m each.
Design FW1: is a combination of design W and design F. The structural wall of design W is
combined in its plane with a moment-resisting frame as in design F. The same gravity load and
masses as in design W are assumed. 687
Design FW2: is equal to design FW1 except that the wall itself is designed for masses tributary
to one bay only, but in the time history analysis masses tributary to two bays are considered.
In this paper results of design FW2 only are given. For a complete overview of the results of all
four designs the reader is referred to (Bachmann, Wenk, and Linde 1992).
Finite element discretisation
Each of the three described ABAQUS user elements were used for the discretisation of design
FW2. In the wall a plastic hinge was modelled at the base over a height equal to the wall length.
The plastic hinge was discretised by two wall user elements U3. The rest of the first storey, as well
as the remaining storeys, were discretised by one user element U3 each, allowing for cracking
behaviour. In a similar manner the beam and column hinges were modelled by user element U1 or
U2 over a length equal to the beam height or column width, respectively. The remaining elastic
portions of beams and columns were modelled by elements of type B23.
Ground motion input
The analysed building is located in the highest seismic zone (3b) specified by the Swiss earthquake
code SIA 160 with a maximum ground motion acceleration of 16 % g (SIA, 1991). An artificially
generated ground motion compatible to the SIA code elastic design spectrum of the zone (3b) for
medium stiff ground was used for the time history analysis (fig. 7). The strong motion duration is
approximately 7 s, the total duration of the ground motion is 10 s, and the total analysis time is 12 s
in increments of 0.01 s. Instead of performing a calculation in absolute coordinates with the ground
acceleration applied to the boundary nodes of the model, a calculation in relative coordinates was
carried out with the ground acceleration applied as GRAV-load to all mass elements. The ground
motion was applied horizontally in the plane of the frames. The dynamic analysis was preceded by
an elastic static gravity load step.
Discussion of results
The horizontal roof displacement history of design FW2 is plotted in fig. 8. A small lateral
displacement due to the static gravity preload is visible at time zero. A maximum displacement of
90 mm corresponding to 0.35 % of the building height is reached at the time of about 10 s.
Typical moment-curvature behaviour of beam hinges (user element U1) at interior column faces
and at the wall face are shown in figs. 9 and 10, respectively. The curvature ductility demand,
defined as the ratio of the maximum curvature "u (fig. 1) reached during the time history analysis
and the yield curvature "y, is about 2 for the beam hinge at the interior column in fig. 9, and 4 for
the beam hinge at the wall in fig. 11. The time history of the hysteresis rule number of the beam
hinge element of fig. 10 is shown in fig. 11. The hysteresis rule numbers of user element U1 are
explained in fig. 12. An integer number between -3 and +3 is assigned to each characteristic branch
of the hysteresis model.
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For the wall hinge the moment-curvature and base shear-lateral displacement behaviour are
shown in figs. 13 and 14. The values are taken from the hinge element closest to the base. The
maximum value of fig. 13 corresponds to a rotational ductility demand of 1.7.
In fig. 15 the distribution of plastic deformations is shown by small dials indicating the
maximum rotational ductility demand during the 12 s time history analysis. A maximum rotational
ductility demand of 1.7 is obtained in the lower wall hinge element, as mentioned above. The
second user element U3 for the wall hinge did not reach yielding (fig. 15). The highest ductility
demand (7.2) occurred in the beam hinges next to the wall.
SUMMARY AND CONCLUSIONS
The development of ABAQUS user elements modelling the hysteretic behaviour of plastic hinges in
reinforced concrete beams, columns, and walls was described. As a numerical example the seismic
analysis of a six-storey building modelled by these user elements and general ABAQUS elements
was presented.
The example presented served as a first check on the reliability of the developed user elements.
Although no comparison basis, such as experimental data, was available, the results appear
reasonable. Especially, the main features of the hysteretic behaviour of reinforced concrete sections
in the plastic range could be reproduced satisfactorily in the ABAQUS calculation. It is planned to
expand the library of user elements for the analysis of three-dimensional reinforced concrete
structures.
REFERENCES
Bachmann, H., Wenk, T., and Linde, P., Nonlinear Seismic Analysis of Hybrid Reinforced
Concrete Frame Wall Buildings, Workshop on Nonlinear Seismic Analysis and Design of
Reinforced Concrete Buildings, Fajfar, P. and Krawinkler, H. editors, Elsevier Applied Science,
London, 1992.
Linde, P., Numerical Modelling and Capacity Design of Earthquake-Resistant Concrete Walls,
Swiss Federal Institute of Technology (ETH), Zurich, 1993.
Paulay, T., Bachmann, H., and Moser, K., Erdbebenbemessung von Stahlbetonhochbauten,
Birkhäuser Verlag, Basel-Boston, 1990
SIA Standard 160, Actions on Structures, Edition 1989 (in English), Swiss Society of Engineers
and Architects, Zurich 1991
Wenk, T., Bachmann, H., Ductility demand of 3-D reinforced concrete frames under seismic
excitation. Proceedings of the European conference on structural dynamics Eurodyn'90, A.A.
Balkema, Rotterdam, 1991, Vol. 1, pp. 537-41.
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TABLES
Type
Description ABAQUS Input Specification
U1 2-node beam element for
beam plastic hinges
*USER ELEMENT, TYPE=U1, NODES=2,
COORDINATES=3, PROPERTIES=5
VARIABLES=11
U2 2-node beam element for
column plastic hinges
*USER ELEMENT, TYPE=U2, NODES=2,
COORDINATES=3, PROPERTIES=10,
VARIABLES=11
U3 4-node macro element for
wall plastic hinges
*USER ELEMENT, TYPE=U3, NODES=4,
COORDINATES=3, PROPERTIES=8,
VARIABLES=45
Table 1. Definition of user elements U1, U2 and U3
*UEL PROPERTY
Parameter Number
Description
PROPS(1) Axial stiffness EA
PROPS(2) Elastic flexural stiffness EIel
PROPS(3) Ratio of elastic flexural stiffness vs. plastic
bending stiffness EIel / EIpl
PROPS(4) Positive yield moment My+
PROPS(5) Negative yield moment My-
Table 2. Description of input properties of user element U1
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State Variable
Number
Description
SVARS(1) Hysteresis rule number
SVARS(2) Plastic curvature
SVARS(3) Total curvature
SVARS(4) Maximum positive curvature
SVARS(5) Maximum negative curvature
SVARS(6) Rotational ductility demand
SVARS(7)-(11) Stress resultants
Table 3. Description of internal state variables of user element U1 and U2
*UEL PROPERTY
Parameter Number
Description
PROPS(1)-(5) Same as user element U1
PROPS(6) Maximum axial force in tension
PROPS(7) Maximum axial force in compression
PROPS(8) Maximum positive yield moment max My+
PROPS(9) Maximum negative yield moment min My-
PROPS(10) Axial force corresponding to max My+ and min My
-
Table 4. Description of input properties of user element U2
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*UEL PROPERTY
Parameter Number
Description
PROPS(1) Cross sectional area of entire wall section
PROPS(2) Moment of inertia (about strong axis) of entire wall section
PROPS(3) Young's modulus of uncracked concrete
PROPS(4) Cracking factor for stiffness of flexural springs in tension,
equal to ratio of compressive to tensile stiffness
PROPS(5) Yielding factor, equal to ratio of yielded to compressive stiffness
PROPS(6) Bending moment at flexural yielding of cross section with zero
axial load
PROPS(7) Shear force at the onset of shear cracking
PROPS(8) Cracking factor in shear, equal to ratio of cracked to uncracked
shear stiffness
Table 5. Description of input properties of user element U3
State Variable
Number
Description
SVARS(1) Hysteresis rule number for left vertical spring
SVARS(2) Force in left vertical spring
SVARS(3) Deformation in left vertical spring
SVARS(4)-(8) Stiffness change parameters of left vertical spring
SVARS(9) Initial yield level of left vertical spring
SVARS(10) Instantaneous ductility of left vertical spring
SVARS(11)-(20) Same as SVARS(1)-(10) for right vertical spring
SVARS(21-27) Same as SVARS(1)-(7) for horizontal spring
SVARS(31-33) Same as SVARS(1)-(3) for center vertical spring
SVARS(34)-(45) Output quantities
Table 6. Description of internal state variables of user element U3
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FIGURES
Mi
My+
My-
EIpl
EIel
EIpl
EIel!
!y
Ml MiMr
LP/2
LP
EI constant
!u
Figure 1. User element U1 for beam plastic hinges (left), hysteretic rules of
bending behaviour of U1 (right)
Mi
My+(N)
My-(N)
EIpl
EIel
!
!yNiN
My
Figure 2. Yield moment-axial force relation for column plastic hinge U2 (left),
hysteretic rules of bending behaviour of U2 (right)
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u5
u2
u3
u4
u6
u7u8
u9u10
u1
u4
u3
u2u1
u8
u9u10u5
u6
u7
Kf KfKs
Kc
rigid
Figure 3. Macro model simulating structural wall behavior (left),
corresponding user element U3 (right)
!
KyFy
Kcr
KuKu
Kel
-aclFy
F
!y
Figure 4. Hysteretic rules for flexural springs Kf in fig. 3 of wall model
Vc
!s
V
!sc-!sc
-Vc
Kcr
Ku
Kel
Figure 5. Hysteretic rules for shear spring Ks in fig. 3 of wall model
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