User Elements Developed for the Nonlinear Dynamic Analysis of Reinforced Concrete Structures

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1 Research Collection Conference Paper User Elements Developed for the Nonlinear Dynamic Analysis of Reinforced Concrete Structures Author(s): Wenk, Thomas; Linde, Peter; Bachmann, Hugo Publication Date: 1993 Permanent Link: Rights / License: In Copyright - Non-Commercial Use Permitted This page was generated automatically upon download from the ETH Zurich Research Collection. For more information please consult the Terms of use. ETH Library

2 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 685

3 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 twodimensional 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 L p 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 M i (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 EI el, positive yield moment M y +, negative yield moment M y -, and yielding stiffness EI pl. 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 L p 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 N i, 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 686

4 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 K f 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 K el, a cracked tensile stiffness K cr and a yielding stiffness K y, 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 K u indicated in the figure. The unloading in the compressive region occurs towards a point! cl F y 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 K c models the axial behaviour together with the flexural springs, and is active only in compression. The horizontal spring K s 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

5 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. 688

6 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, Linde, P., Numerical Modelling and Capacity Design of Earthquake-Resistant Concrete Walls, Swiss Federal Institute of Technology (ETH), Zurich, 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

7 TABLES Type Description ABAQUS Input Specification U1 U2 U3 2-node beam element for beam plastic hinges 2-node beam element for column plastic hinges 4-node macro element for wall plastic hinges *USER ELEMENT, TYPE=U1, NODES=2, COORDINATES=3, PROPERTIES=5 VARIABLES=11 *USER ELEMENT, TYPE=U2, NODES=2, COORDINATES=3, PROPERTIES=10, VARIABLES=11 *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) PROPS(2) PROPS(3) Axial stiffness EA Elastic flexural stiffness EI el Ratio of elastic flexural stiffness vs. plastic bending stiffness EI el / EI pl PROPS(4) Positive yield moment M y + PROPS(5) Negative yield moment M y - Table 2. Description of input properties of user element U1 690

8 State Variable Number SVARS(1) SVARS(2) SVARS(3) SVARS(4) SVARS(5) SVARS(6) SVARS(7)-(11) Description Hysteresis rule number Plastic curvature Total curvature Maximum positive curvature Maximum negative curvature Rotational ductility demand Stress resultants Table 3. Description of internal state variables of user element U1 and U2 *UEL PROPERTY Parameter Number PROPS(1)-(5) PROPS(6) PROPS(7) Description Same as user element U1 Maximum axial force in tension Maximum axial force in compression PROPS(8) Maximum positive yield moment max M y + PROPS(9) Maximum negative yield moment min M y - PROPS(10) Axial force corresponding to max M y + and min M y - Table 4. Description of input properties of user element U2 691

9 *UEL PROPERTY Parameter Number PROPS(1) PROPS(2) PROPS(3) PROPS(4) PROPS(5) PROPS(6) PROPS(7) PROPS(8) Description Cross sectional area of entire wall section Moment of inertia (about strong axis) of entire wall section Young's modulus of uncracked concrete Cracking factor for stiffness of flexural springs in tension, equal to ratio of compressive to tensile stiffness Yielding factor, equal to ratio of yielded to compressive stiffness Bending moment at flexural yielding of cross section with zero axial load Shear force at the onset of shear cracking 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 SVARS(1) SVARS(2) SVARS(3) SVARS(4)-(8) SVARS(9) SVARS(10) SVARS(11)-(20) SVARS(21-27) SVARS(31-33) SVARS(34)-(45) Description Hysteresis rule number for left vertical spring Force in left vertical spring Deformation in left vertical spring Stiffness change parameters of left vertical spring Initial yield level of left vertical spring Instantaneous ductility of left vertical spring Same as SVARS(1)-(10) for right vertical spring Same as SVARS(1)-(7) for horizontal spring Same as SVARS(1)-(3) for center vertical spring Output quantities Table 6. Description of internal state variables of user element U3 692

10 FIGURES M i M l M i M r M y + EI pl EI el L P /2 L P EI constant! y! u EI el! M - y EI pl Figure 1. User element U1 for beam plastic hinges (left), hysteretic rules of bending behaviour of U1 (right) M y M i M y +(N) EI pl Ni N! y EI el! M y - (N) Figure 2. Yield moment-axial force relation for column plastic hinge U2 (left), hysteretic rules of bending behaviour of U2 (right) 693

11 u 6 u 9 u 6 u 9 u 7 u 5 rigid u 10 u 8 u 5 u 10 u 7 u 8 K f u 2 u 1 K s K c K f u 4 u 3 u 2 u 4 u 1 u 3 Figure 3. Macro model simulating structural wall behavior (left), corresponding user element U3 (right) F F y K cr K y Ku K u! y! K el -a cl F y Figure 4. Hysteretic rules for flexural springs Kf in fig. 3 of wall model V K cr V c K u -! sc! sc! s K el -V c Figure 5. Hysteretic rules for shear spring Ks in fig. 3 of wall model 694

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