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User Elements Developed for theNonlinear Dynamic Analysis ofReinforced Concrete Structures

Conference Paper

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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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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695

696

697

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