Nonlinear Dynamic Analysis of Reinforced Concrete Building Structures

SHUNSUKE OTANI

Department of Civil Engineering, University of Toronto, Toronto, Ontario, Canada M5S 1A4 Received January 10，1979

Revised manuscript accepted March 3，1980

This state-of-the-art paper discusses nonlinear dynamic analysis of reinforced concrete building structures. Nonlinear analysis of a reinforced concrete building is difficult (a) because inelastic deformation is not confined at critical sections, but spreads throughout the structure; and (b) because stiffness of the reinforced concrete is dependent on a strain history.

The paper reviews the behaviour of reinforced concrete members and their subassemblies observed during laboratory tests. Then different hysteresis and analytical models of reinforced concrete members are reviewed, and their application to the simulation of building model behaviour is discussed.

________________ Cet article traite de l’etat actuel des connaissances au sujet d’une analyse dynamique

non1ineare de structures de batiments en beton arme. L’analyse nonlineaire d’un batiment en beton arme est difficile (a) puisque la deformation inelastique n’est pas limitee a des sections critiques, mais s’etend a travers la structure et (b) parce que la rigidite du beton arme depend d’une histoire de deformation unitaire.

Cet article fait la revue du comportement des membrures en beton arme et leurs sous-assemblages obseves au cours d’essais en laboratoire. Ensuite．differents modeles analytiques et hystesis de membrures en beton arme sont examines et leur application a la simulation du comportement de modele de batiment est discutee. Canadian Journal of Civil Engineering, Vol. 7、pp. 333－344 (1980) [Traduit par la revue]

Introduction

Dynamic response of a structure can be caused by different loading conditions such as: (a) earthquake ground motion; (b) wind pressure; (c) wave action; (d) blast; (e) machine vibration; and (f) traffic movement. Among these, inelastic response is mainly caused by earthquake motions and accidental blasts. Consequently, more research on nonlinear structural behaviour has been carried out in relation to earthquake problems. This paper describes the research development in earthquake engineering.

Note that dynamic problems are different from static one in the following points: (a) inertial force; (b) damping; (c) strain rate effect; and (d) oscillation (stress reversals). These need to be clarified in order to analyze a structure under dynamic loading.

Dynamic characteristics up to failure cannot be identified solely through a dynamic test or a real structure for the following reasons: (a) difficult to understand the behaviour due to complex interactions of various parameters; (b) expensive to build a structure, as a specimen, for destructive testing; and (c) capacity of loading devices insufficient to cause failure. Consequently, dynamic tests of real buildings are rather aimed toward obtaining data (a) to confirm the validity of mathematical modelling techniques for a linearly elastic structure; and (b) to obtain damping characteristics of different types of structures. A specifically designed laboratory test becomes inevitable in order to complement the weakness of full scale tests and to study the effect of individual parameters. Damping

Any mechanical system possesses some energy-dissipating mechanisms, for example: (a) inelastic hysteretic

energy dissipation; (b) radiation of kinetic energy through foundation; (c) kinetic friction; (d) viscosity in materials; and (e) aerodynamic effect. Such capacity or energy dissipation is vaguely termed “damping,” and is most often assumed to be of viscous type because of its mathematical simplicity.

Damping capacity is often determined by the band width of the response curve during a sinusoidal steady-state test. Figure 1 shows such acceleration response curves for a reinforced concrete building at different excitation levels（Jennings and Kuroiwa 1968). Note the shift of resonant frequencies and the change in amplitudes of damping with increase of excitation level despite low response amplitudes.

FIG. 1. Observed acceleration amplitudes from steady-state test (Jennings and Kuroiwa 1968).

Damping capacity is not a unique value of a structure, but it depends on the level of excitation. The

state-of-the-art does not provide a method to determine the damping capacity based on the material properties and geometrical characteristics of a structure.

Strain Rate Effect

It is technically difficult to test structural members under dynamic conditions in a laboratory. Before force-deformation relations already obtained from thousands of static tests can be studied for use in a dynamic analysis, the effect of strain rate on the force-deformation relation needs to be examined.

The speed of loading is known to influence the stiffness and strength of various materials (Cowell 1965, 1966). Some member test results are available (Mahin and Bertero l972). Important findings from these investigations are as follows: (a) high strain rates increased the initial yield resistance, but caused small differences in either stiffness or resistance in subsequent cycles at the same displacement amplitudes; (b) strain rate effect on resistance diminished with increased deformation in a strain-hardening range; and (c) no substantial changes were observed in ductility and overall energy absorption capacity.

Note that strain rate (velocity) during an oscillation is highest at low stress levels, and that the rate gradually decreases toward a peak strain. Cracking and yielding of a reinforced concrete member reduce the stiffness, elongating the period of oscillation. Furthermore, such damage is normally caused by the lower modes of vibration having long periods. Therefore, the strain rate is small in the case of earthquake response, and its effect on the response is small.

Consequently, the static hysteretic behaviour observed can be utilized in a nonlinear dynamic analysis of reinforced concrete structures.

Stiffness Properties of Reinforced Concrete Members It is not feasible to analyze an entire structure using microscopic material models. It is more important to

study the behaviour of isolated members and their subassemblies (beam-column, slab-column, and slab-wall connections) so that their analytical models can be developed for use in the analysis of a complete structure.

A typical force-deflection curve of a cantilever column is shown in Fig. 2 (Otani et al. 1979). Note the following observations: (a) tensile cracking of concrete and yielding of longitudinal reinforcement reduced the stiffness; (b) when a deflection reversal was repeated at the same newly attained maximum amplitude (for example, cycles 3 and 4) the loading stiffness in the second cycle was lower than that in the first cycle, although the resistances at the peak displacement were almost identical; and (c) average stiffness (peak-to-peak) of a complete cycle decreased with a maximum displacement amplitude. For example, the peak-to-peak stiffness of cycle 5, after large amplitude displacement reversals, was significantly reduced from that of cycle 2 at a comparable displacement amplitude. Therefore, the hysteretic behaviour of the reinforced concrete is sensitive to loading history. Let us study some typical stiffness characteristics of reinforced concrete.

FIG. 2. Hysteretic characteristics of reinforced concrete member (Otani et al. 1979).

Flexural Characteristics

The flexural deformation index (average curvature) is obtained from longitudinal strain measurements at two levels assuming that a plane section remains plane. This flexural deformation index does not represent the flexural deformation in a strict sense because a plane section does not remain plane in a region where an extensive shear deformation occurs. However, the index is useful for understanding flexural deformation characteristics qualitatively.

A typical moment-flexural deformation index curve obtained from a simply supported beam test (Celebi and Penzien 1973) is shown.in Fig. 3. Note that the stiffness during loading gradually decreases with load, forming a fat hysteresis loop, and absorbing a large amount of hysteretic energy. The hysteresis loops remain almost identical even after several load reversals at the same displacement amplitude beyond yielding. Consequently, vibration energy can be efficiently dissipated through flexural hysteresis loops without a reduction in resistance. Many hysteretic models, as discussed later, are currently available to represent the nexural behaviour.

FIG. 3. Flexural deformation characteristics (Celebi and Penzien 1973). The increase in axial force decreases the flexural ductility of a reinforced concrete member, but increases

force levels corresponding to (a) tensile cracking of concrete; and (b) tensile yielding of longitudinal reinforcement. Shear Characteristics

Similar to the flexural deformation index, a shear deformation index is defined from strain measurements in the two diagonal directions. Again, this index does not represent the true shear deformation because the interference of shear and flexure exists.

A typical lateral load-shear deformation index curve (Celebi and Penzien 1973) is shown in Fig. 4. Unlike what occurs in flexure, the stiffness during loading gradually increases with load, exhibiting a “pinching” in the curve. The hysteretic energy dissipation is smaller. The hysteresis loop decays with the number of load reversals, resulting in a smaller resistance at the same peak displacement in each repeated loading cycle. Although the curve shows a “yielding” phenomenon, it is important to recognize that the shear force of the member was limited by flexural yielding at the critical section rather than by yielding in shear. This yielding clearly indicates the interaction of shear and bending.

FIG. 4. Shear deformation characteristics (Celebi and Penzien 1973).

The pinching in the force-deformation curve is obviously less desirable. The shear span to effective depth ratio is the most significant parameter. Decreasing the shear span to depth ratio causes a more pronounced pinching in the curve, and a faster degradation of the hysteretic energy-dissipating capacity. Considerable improvements in delaying and reducing the degrading effects can be accomplished by using closely spaced ties. Existence of axial force tends to retard the decrease in stiffness and resistance with cycles. However, it is hard to eliminate this undesirable effect when high shear stress exists. Consequently, it becomes important to include this degrading behaviour in a behavioural model for a short, deep reinforced concrete member. The current state of knowledge is not sufficient to define the stiffness degrading parameters on the basis of the member geometry and material properties. Bar Slip and Bond Deterioration

When a structural element is framed into another element, some deformation is initiated within the other element. Consider a beam-column subassembly. Bertero and Popov (1977) reported a significant rotation at a beam end caused by the slippage (pullout) of the beam's main longitudinal reinforcement within the beam-column joint (Fig. 5). The general shape of the moment-bar slip rotation curve is similar to that shown in Fig. 4, demonstrating a pronounced pinching of a hysteresis loop. The contribution of bar slip to total deformation cannot be neglected, especially in a stiff member (short or deep).

FIG. 5. Rotation due to bar slip (Bertero and Popov 1977).

Biaxial Lateral Load Reversals

During an earthquake, columns of a framed structure must resist lateral forces simultaneously in longitudinal and transverse directions. Recent tests at the University of Toronto (Otani et al. 1979) on reinforced concrete columns showed that the columns under lateral load reversals in two perpendicular directions exhibited the decay of resistance and stiffness at a faster rate than those under uniaxial lateral load reversals. This topic needs further study.

Hysteretic Models for Reinforced Concrete Nonlinear dynamic analysis of a reinforced concrete structure requires two types of mathematical modelling:

(a) modelling for the distribution of stiffness along a member; and (b) modelling for the force-deformation relationship under stress reversals.

A hysteresis model must be able to provide the stiffness and resistance under any displacement history. At the same time, the basic characteristics need to be defined by the member geometry and material properties. The current state of knowledge is sufficient to define flexural hysteresis models. However, it is not sufficient to determine the degree of stiffness degradation due to the deterioration of shear-resisting and rebar-concrete bond mechanisms. Bilinear Model

The elastic-perfectly plastic hysteretic model was used by many investigators because the model was simple. The maximum displacement of an elasto-plastic simple system was found (Veletsos and Newmark 1960) to be practically the same as that of an elastic system having the same initial period of vibration as long as the period was longer than 0.5 s.

A finite positive slope was assigned to the postyield stiffness to account for the strain-hardening characteristic, and the model was called a bilinear model. The bilinear model does not represent the degradation of loading and unloading stiffnesses with increasing displacement amplitude reversals (Fig. 6), and the model is not suited for a refined nonlinear analysis of a reinforced concrete structure.

FIG. 6. Bilinear hysteresis model.

Clough 's Degrading Stiffness Model

A qualitative model for the reinforced concrete was developed by Clough (1966), who incorporated the stiffness degradation in the elasto-plastic model : the response point during loading moved toward the previous maximum response point. The unloading slope remained parallel to the initial elastic slope. This small modification improved the capability to simulate the flexural behaviour of the reinforced concrete. Compared with the elasto-plastic model, less energy is absorbed per cycle beyond yielding by Clough's degrading model.

From the response analysis of a series of single-degree-of-freedom systems, Clough (1966) concluded that (a) the degrading stiffness model did not cause any significant change in the ductility demand of long-period structures (period longer than 0.6 s) compared with the elasto-plastic model; on the other hand, (b) the degrading stiffness model required significantly larger ductility from short-period structures than the corresponding elasto-plastic systems; and (c) the response waveform of a degrading stiffness model was distinctly different from that of an ordinary elasto-plastic model.

The model is relatively simple, and has been used extensively in nonlinear analysis with the inclusion of strain-hardening characteristics (Fig. 7).

FIG. 7. Clough’s degrading stiffness model.

Takeda's Degrading Stiffness Model

A more refined and sophisticated hysteresis model was developed by Takeda et al. (l970) on the basis of experimental observation. This model included stiffness changes at flexural cracking and yielding, and also strain-hardening characteristics. The unloading stiffness was reduced by an exponential function of the previous maximum deformation. Takeda et al. also prepared a set of rules for load reversals within the outermost hysteresis loop. These are major improvements over the Clough (1966) model.

Failure or extensive damage caused by shear or bond deterioration was not considered in the model. The Takeda model, similar to the Clough model, simulates dominantly flexural behaviour (Fig. 8). Simplified Takeda hysteresis models were proposed by Otani and Sozen (1972) and by Powell (1975), using a bilinear backbone curve.

FIG. 8. Takeda’s degrading stiffness models.

To test the goodness of the Takeda model, cantilever columns tested on the University of Illinois earthquake

simulator were analyzed (Takeda et al.1970). Calculated waveforms were favourably compared with the observed waveform as shown in Fig. 9.

FIG. 9. Takeda model applied to column analysis (Takeda et al. 1970)

Takeda-Takayanagi Models The amplitude of the exterior column axial load varies greatly due to the earthquake overturning moment,

and changes its moment-carrying capacity. Takayanagi and Schnobrich (1976) incorporated the effect of axial force variation in the Takeda model by preparing various backbone curves at different axial load levels (Fig. 10a).

(a) axial force variation (b) pinching and strength decay FIG. 10. Takeda-Takayanagi models (Takayanagi and Schnobrich 1976):

A pinching action and strength decay are inevitable in a short and deep member due to bar slip and

deterioration in shear resistance. Takayanagi and Schnobrich (1976) introduced a pinching action and strength decay in the Takeda model (Fig. 10b). Whenever a response point was located in the positive rotation-negative moment range or the negative rotation-positive moment range, the pinching was introduced. After the moment exceeded the yield level, a strength decay was incorporated. The values of guideline for strength decay and pinching stiffness were not related to the member geometry and material properties. Degrading Trilinear Hysteresis Model

A model that simulates dominantly flexural stiffness characteristics was developed in Japan (Fukada 1969). The backbone curve is a trilinear shape with stiffness changes at cracking and yielding. Up to yielding, the model behaves in the same way as the bilinear model. Once deformation exceeds the yield point, the model behaves as a perfectly plastic system. Upon unloading, the unloading point is treated as a new “yield” point, and unloading stiffnesses corresponding to pre- and postcracking are reduced proportionately so that the behaviour becomes of the bilinear type in a range between the positive and the negative yield points (Fig. 11).

FIG. 11. Degrading trilinear model.

The degrading trilinear model can easily include strain-hardening characteristics. The hysteresis energy

dissipation per cycle beyond the initial yielding is proportional to the displacement, and the equivalent viscous damping factor becomes constant. The fatness of a hysteresis loop is sensitive to the choice of a cracking point. Comments

Many other hysteresis models have been proposed and used in the past. Figure 12 shows attained ductility factors of single-degree-of-freedom systems with any of four flexural hysteresis models: bilinear; Clough; Takeda; and degrading trilinear models. The four models have the same backbone curve except the cracking point. The four models show similar variations of attained ductility factors with periods, but attained ductility factors show a wide scatter from one model to another, especially in a short-period range.

Fig. 12. Effect of hysteresis models on earthquake response. Further research is necessary to study the effect of hysteretic characteristics on earthquake response, and

develop a simple standard hysteresis model for general use.

Reinforced Concrete Member Model Inelastic deformation of a reinforced concrete member does not concentrate in a critical location, but rather

spreads along the member (Fig. 13). Various member models have been proposed to represent the distribution of stiffness within a reinforced concrete member. The effect of gravity load on the beam behaviour and the contribution of slabs to the structural stiffness will not be discussed.

FIG. 13. Deformation of beam under gravity and earthquake loads.

One-component Model

An elasto-plastic frame structure was analyzed by placing a rigid plastic spring at the location where yielding is expected. The part of a member between the two rigid plastic springs remains perfectly elastic. All

inelastic deformation is assumed to occur in these springs (Fig. 14). This one-component model was generalized by Giberson (1967).

FIG. 14. One-component member model. A major advantage of the model is that inelastic member-end deformation depends solely on the moment

acting at the end so that any moment-rotation hysteretic model can be assigned to the spring. This fact is also a weakness of the model because the member-end rotation should be dependent on the curvature distribution along the member, hence dependent on moments at both member ends. Consider two cases of moment distribution along a member AB with corresponding curvature distributions as shown in Fig. 15. The inelastic rotations at the A end are given by the shaded areas. For the same moments at the A end, case II causes larger inelastic rotation at the A end. Consequently, this simple model does not simulate actual member behaviour. Furthermore, it is not rational to lump all inelastic deformations at member ends.

FIG. 15. Inelastic rotation of beam: (a) moment; (b) curvature and inelastic rotation. The stiffness of an inelastic spring is normally defined by assuming an asymmetric moment distribution

along a member with the infection point at midspan. The usage of the initial location of the inflection point in evaluating spring properties was suggested by Suko and Adams (1971). However, once yielding is developed at one member end, the moment at the other end must increase to resist a higher stress, moving the inflection point toward the member centre. At the same time, a large concentrated rotation starts to occur near the critical section. Despite rational criticisms against this simple model, the performance of the one-component model is expected to be reasonably good for a relatively low-rise frame structure, in which the inflection point of a column locates reasonably close to midheight.

A special-purpose computer program, SAKE (Otani 1974), for a regular rectangular reinforced concrete frame structure and recent modifications (Powell 1975) to general-purpose computer program DRAIN 2D (Kanaan and Powell 1973) used the one-component model.

Multi-component Model In an effort to analyze frame structures well into the inelastic range under earthquake excitation, an

interesting model was proposed by Clough et al. (1965). A frame member was divided into two imaginary parallel elements: an elasto-plastic element to represent a yielding phenomenon, and a fully elastic element to represent strain-hardening behaviour. When the member-end moment reaches the yield level, a plastic hinge is placed at the end of the elasto-plastic element. A member-end rotation depends on both member-end moments. Aoyama and Sugano (1968) adapted the two-component model, creating the multicomponent model (Fig.16), using four parallel beams to account for flexural cracking, different yield levels at two member ends, and strain-hardening. The deformation compatibility of the imaginary components is satisfied only at their ends.

FIG. 16. Multi-component member model. The multi-component model appears to have merit; rotation at one end of a member depends on both

member-end moments. In other words, the moment distribution along a member can be approximately reflected in the analysis. However, the stiffness of the multi-parallel components must be evaluated under a certain assumed moment distribution. Therefore, the stiffness parameters are valid only under such a moment distribution, and are bound to be approximate when the moment distribution becomes drastically different.

Giberson (1967) discussed the advantages and disadvantages of the one-component and the two-component models, and concluded that the one-component model was more versatile than the two-component model because the two-component model was restricted to the bilinear-type hysteresis characteristics. This two-component model was used in a general-purpose computer program DRAIN 2D (Kanaan and Powell 1973) for plane structure. The interaction of the bending moment and the axial force was easily incorporated by simply changing the yield value of the elasto-plastic component depending on the existing axial force. Connected Two-cantilever Model

When a frame is analyzed under lateral loads only, the member moment distributes linearly. From the similarity of moment distribution, the member can be considered to consist of two imaginary cantilevers, free at the point of contraflexure and fixed at the member end, and connected at the inflection point, satisfying the continuity of displacement and rotation (Otani and Sozen 1972).

The flexibility relation of a member was formulated by assuming: (a) the inflection point did not shift much during a short time increment; (b) free-end rotation and displacement were proportional to the beam length and the square of the beam length, respectively; and (c) instantaneous stiffness for shear-rotation and shear-displacement curves of a unit length reference cantilever could be defined by hysteretic models.

The weakness of this method is that the member flexibility matrix is a function of the location of the inflection point, which tends to shift rapidly when the sign of a member-end moment changes. This causes a numerical problem. Consequently, this method cannot be recommended for a general dynamic analysis. However, the method is useful for incremental static load analysis of a structure. Discrete Element Model

In order to overcome difficult problems of variable stiffness distribution along a member, the member can be

subdivided into short line segments along the length, with each short segment assigned a nonlinear hysteretic characteristic. The nonlinear stiffness can be assigned within a segment, or at the connection of two adjacent segments.

Wen and Janssen (1965) presented a method for dynamic analysis of a plane frame consisting of elasto-plastic segments. Consequently, the mass and flexibility of a member were lumped at the connecting points on a tributary basis. Powell (1975) suggested using a degrading stiffness hysteresis model for rigid inelastic connecting springs (Fig. 17a). Shorter segments were recommended in a region of high moment, and longer segments in a low-moment region.

FIG. 17. Discrete element model: (a) lumped inelastic stiffness; (b) distributed inelastic stiffness.

An alternative method is to divide a member into short segments, each segment with a uniform flexural

rigidity that varies with a stress history of the segment (Fig. 17b). Local concentration of inelastic action can be easily handled by arranging shorter segments at the location of high concentration of inelastic deformation (Takayanagi and Schnobrich 1976).

These methods are useful when more accurate results are required, or in the analysis of walls. More computational effort is required compared with the other simple models. Distributed Flexibility Model

Once cracks develop in a member, the stiffness becomes nonuniform along the member length. Instead of dividing a member into short segments, Takizawa (1973) developed a model that assumed a prescribed distribution pattern of cross-sectional flexural flexibility along member length. A parabolic distribution with an elastic flexibility at the infection point was used (Fig. 18). The flexura1 flexibility at the member ends was given by a hysteretic model dependent on a stress history.

FIG. 18. Distributed flexibility model.

This is an interesting concept in analyzing an inelastic member. However, the parabolic flexibility distribution may not describe the actual concentration of deformation at critical sections (normally at member ends) due to flexural yielding and deformation attributable to slippage of longitudinal reinforcement within a beam-column connection. The usage of inelastic springs at locations of concentrated deformation in conjunction with this model may be a useful solution. Summary

Various member models are reviewed, and their advantages and disadvantages are discussed. These models have been developed specifically for earthquake response. Development of a simple model for simultaneous gravity and earthquake situations is desired.

Reliability of Analytical Models Earthquake simulator tests provide interesting opportunities to examine the goodness of different analytical

models in simulating the observed response of small- to medium-scale highly inelastic model structures. This section reviews the reliability of different analytical models in relation to the capability to simulate the observed behaviour.

These test structures were designed to behave dominantly in flexure, being prevented as much as possible from failing in shear or anchorage because the two types of failure are not desirable in real construction and are avoided in a design process. Three-storey One-bay Frames (I)

Small-scale three-storey one-bay reinforced concrete frames were tested on the University of Illinois earthquake simulator, and were analyzed using the connected two-cantilever member model (Otani and Sozen 1972). The stiffness properties of individual members were calculated on the basis of the geometry and material properties. The Takeda model was used to represent the force-deflection of each cantilever model. Member-end rotation due to bar slip was approximated by the simplified bilinear Takeda model, which did not simulate the pinching behaviour. The damping matrix was assumed to be proportional to the instantaneous stiffness matrix.

FIG. 19. Connected two-cantilever model applied to three-storey frame analysis (Otani and Sozen 1972): (a) measured; (b) calculated (h=0.0); (c) calculated (h=0.02)

The model structure was subjected to a base motion simulating the El Centro (NS) 1940 accelerogram. The

first-floor displacement was measured to be as much as four times the yield displacement calculated under static lateral loads. The analytical models with and without viscous damping favourably simulated the large-amplitude oscillations at 1.0, 2.0, and 5 s from the beginning of the motion (Fig. 19). The analytical models, however, failed to simulate the medium- and low-amplitude oscillations. Note that the frequencies at the medium- to low-amplitude oscillations are higher for the analytical model, which indicates that the test structure was more flexible at low stress levels than the analytical model. In order to reproduce lower amplitude oscillations of the observed response waveforms, the pinching behaviour needs to be incorporated in a hysteretic model. Three-storey One-bay Frames (II)

Another set of three-storey one-bay small-scale reinforced concrete frame structures was tested on the University of Illinois earthquake simulator (Otani 1976). The base motion is significantly more intense than a design earthquake motion.

A member was represented by the one-component model with two inelastic rotational springs at each member end: one for the flexural deformation and the other for the member-end rotation due to bar slip (Otani 1974). Takeda models with trilinear and bilinear backbone curves were assigned to the two inelastic springs. Two types of damping were used in the analysis: (a) a damping matrix proportional to the constant mass matrix; and (b) a damping matrix proportional to an instantaneous stiffness matrix. The first mode damping factor was 5% of critical at the initial elastic stage. Observed and calculated third-level displacement waveforms are compared in Fig. 20 (Otani 1976). The comparison is fair for large-amplitude oscillations, and poor at low-amplitude oscillations. Again in this analysis, the pinching characteristic was not incorporated. A fair agreement between the computed and the observed may be attributable to the fact that the yielding was developed at most member ends at the large-amplitude oscillations, and that the inflection point tended to be near the midpoint of each member in such a low-rise frame structure.

FIG. 20. One-component model applied to three-story frame analysis (Otani, 1976):

(a) measured; (b) calculated (mass proportional damping); (c) calculated (stiffness proportional damping).

Two-storey One-bay Frame

A two-storey one-bay medium-scale frame structure with slabs was tested on the University of California earthquake simulator (Hidalgo and Clough 1974). The structure was analyzed using the two-component model. In an effort to improve the correlation, the elastic stiffness of the two parallel components was degraded as a function of the first-mode-response amplitude history. The observed and the calculated second-floor displacement waveforms are satisfactorily compared in Fig. 21. However, the parameters controlling stiffness degradation could not be determined from the theory.

FIG. 21. Two-component model applied to two-story frame analysis (Hidalgo and Clough 1974).

Ten-storey Coupled Shear Walls

Ten-storey coupled shear walls were tested on the University of Illinois earthquake simulator (Aristizaba1-Ochoa and Sozen 1976). Takayanagi and Schnobrich (l976) divided a wall into short segments of uniform stiffness, and represented connecting beams by the one-component model. The Takeda-Takayanagi model with changing axial force was assigned to a wall element, and the Takeda-Takayanagi model with pinching action and strength decay was used in a beam. It was judged that the usage of two-dimensional plane stress elements for the walls was less desirable because such an approach might cost more computational effort without any compensating increase in accuracy.

(a) displacement at level 10, in inches (1 in. = 25.4 mm) (b)Acceleration at level 10, g.

FIG. 22. Analysis of ten-story coupled shear wall (Takayanagi and Schnobrich 1976): The comparison of the measured and calculated displacement and acceleration is excellent, as shown in Fig.

22. It is necessary to include the effects of inelastic axial rigidity of the wall section and pinching action and strength decay of the connecting beams to reproduce the maximum displacement response and the elongation

of the period. Some stiffness parameters for the walls and connecting beams were defined on the basis of static tests of connecting beam-wall assemblies.

Summary The favourable comparison of the measured and the calculated response waveforms encourages the use of

correct analytical and hysteretic models. It is desirable in developing a mathematical model that all parameters of the proposed model should be evaluated on the basis of the geometry of a structure and the properties of materials.

Three-dimensional Building Analysis

The development of analytical methods has made it feasible to discuss nonlinear behaviour of reinforced concrete plane structures with a certain confidence. However, we have not yet reached a point to discuss, with any confidence, the nonlinear behaviour of three-dimensional building structures. Columns of a framed structure must resist lateral forces in two horizontal directions. The stiffness is reduced significantly under biaxial lateral load reversals (Otani et al. 1979).

The first nonlinear analysis of a frame structure under horizontal biaxial ground motion was made by Nigam (1967) with elasto-plastic columns. Multi-storey frames with rigid beams and floor slabs under horizontal biaxial ground motion were studied by Pecknold (1974). Prager's kinematic hardening theory as modified by Ziegler (1959) was used. The basic effect of biaxial inelastic interaction is to produce a softer structure. Pecknold (1974) confirmed Nigam's finding that: (a) horizontal biaxial ground motion increased ductility demand for stiff structures (initial natural period less than 0.3 s); and (b) horizontal biaxial ground motion had little effect on the ductility demand of flexible structures. Neither study included the stiffness degradation property.

Aktan et al. (1973) used the finite element technique to include the stiffness degradation through the degradation of material properties. The biaxial ground motion was found to cause 20-200% larger response than the uniaxial ground motions from columns for which the calculated deflection under uniaxial ground motion exceeded approximately twice the yield deflection.

Takizawa and Aoyama (1976) extended the one-dimensional degrading trilinear hysteretic model into a two-dimensional model on the basis of plasticity theories (Ziegler 1959). The proposed model was judged to predict the significant trends of the biaxial behaviour of reinforced concrete test columns. The effect of biaxial response interaction was reported to be significant for degrading stiffness models, and not so important for nondegrading type hysteretic models.

A brief review of nonlinear analysis of three-dimensional structures indicates the necessity of further study in this area.

Summary

The behaviour of reinforced concrete buildings, especially under earthquake motion, was briefly reviewed. When a structure can be idealized as plane structures, the current state-of-the-art provides useful and reliable analytical methods.

However, more research is required to understand the effect of slabs, gravity loads, and biaxial ground motion on nonlinear behaviour of a three-dimensional reinforced concrete structure.

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