If the structure has a symmetric plan and torsional response is expected to be small, the three-dimensional model may be reduced into a two-dimensional plane frame model in each principal direction as shown in Fig. 7.5. The model connects all the plane frames in one principal direction by assuming the iden- tical horizontal displacement in a floor.
In the example of Fig. 7.1, where the two outer frames with wall are sup- posed to be identical, these two frames may be modeled as one frame with doubled stiffness, strength and weight. Axial deformation in the beam member model is neglected in this case. Vertical displacement and rotation is consi- dered at each node and the horizontal displacement is identical at each floor.
In this case, JVdof = 2 x iVn 4- JV/} where Nf is number of stories, therefore iVdof = 2 x 8 x 8 + 8 = 136. The number of degrees of freedom can be reduced about one-fourth compared to the three-dimensional model.
In the plane frame model, two-dimensional member models are used for beams and columns as shown in Fig. 7.6. Usually axial deformation is ne- glected in beam but considered in column using one-component model shown in Fig. 7.6(a). The axial deformation in the beam may also be included into
Fig. 7.5. Two-dimensional frame model for building structures.
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(a) One-component model (b) Multi-Spring model
Fig. 7.6. Beam and column models in two-dimensional frame model.
the plane frame model by considering independent horizontal displacement at each node and using the multispring model, as shown in Fig. 7.6(b). Detailed description of the member models are given in Sec. 7.3.
7.2.3. Multimass Model
In the early age of application of nonlinear response analysis to the practi- cal design of highrise buildings in Japan, equivalent multimass model shown in Fig. 7.7 has been most frequently used. The reduction of the frame model to a multimass model is based on the static push-over analysis by the frame model as above or more simplified model. The characteristic of the nonlinear spring, which is the force-deformation relation, is idealized based on story-shear vs. interstory drift relations obtained from the push-over analysis. Simplified method based on inelastic story stiffness of columns and beams are also avail- able to determine the relations. Especially for the analysis of highrise buildings, not only shear spring but also rotational spring must be located in every story to simulate an overall flexural deformation of the building due to axial defor- mation of columns or bending deformation of wall, as shown in Fig. 7.8. The degrees-of-freedom of the model is the number of story Nf. The model is simple and the required amount of calculation and storage can be very much efficiently reduced from the frame model.
If the properties of the nonlinear springs are determined properly, the res- ponses calculated by the multimass model would be in fair agreement with the
Fig. 7.7. Multimass system.
Fig. 7.8. Shear and rotational springs.
responses calculated by the frame model, on condition that the first mode res- ponses are dominant and within moderately inelastic displacement range. In case that the higher mode response is dominant or the response is in well plastic range, calculated responses could be different from those by the frame model.
Another disadvantage of this model is that the responses of the members such as inelastic deformations or internal forces cannot be calculated. The inelastic deformation of the members may be estimated from the push-over analysis, for example, at the corresponding interstory displacement. On the other hand, the member forces in the columns and walls including the effects of the higher mode responses cannot be evaluated directly from the push-over analysis. The effect of the higher mode responses on the moment and shear forces in columns and walls, which is called dynamic magnilcation, should be taken into account in design of these members in addition to the static calculation by the push-over analysis.7,4
However, the multimass model is still useful in practical design, because the responses of highrise buildings to design earthquake motion are relatively small in Japan and the buildings are designed to ensure the beam-yielding mechanism so that the first mode response would be dominant.
7.2.4. Soil-Structure Model
In the recent revision of Japanese Building Standard, a new procedure was adopted for the verification of seismic performance in addition to traditional design requirement. One of the innovative features in seismic design procedure is that safety performance is to be verified by the limit states criteria defined using inelastic displacement response and deformation capacity of the struc- ture. Another is that the standard design earthquake is specified as the elastic
Fig. 7.9. Soil-structure model for response analysis.
response spectrum at the engineering bedrock. Amplification by the surface soil should be calculated for each construction site. Simple method is available for this calculation, while soil-structure model, shown in Fig. 7.9, may be used as a sophisticated design tool.
The soil-structure model is not being used frequently even for the special design procedure for highrise buildings, because the rocking deformation is relatively small and the design motion is defined at the base of the structure.
The model will be useful in the future to estimate (a) amplification of input earthquake by the surface soil, (b) input energy loss due to deformation or viscosity of soil, and (c) action of piles induced by the response of soil shear deformation.
7.3. M e m b e r M o d e l s