Xt

Figure 2.13. Damped free vibration of a single-degree-of-freedom system.

accelerations, forces, and overturning moments. Each predominant mode is analyzed separately and the results are combined statistically to compute the multimode response.

Buildings with symmetrical shape, stiffness, and mass distribution and with vertical continuity and uniformity behave in a fairly predictable manner whereas when buildings are eccentric or have areas of discontinuity or irregularity, the behavioral characteristics are very complex. The predominant response of the building may be skewed from the apparent principal axes of the building. The torsional response as well as the coupling or interaction of the two translational directions of response must be considered. This is similar to the Mohr's circle representation of principal stresses.

Thus, three-dimensional methods of analysis are required as each mode shape is defined in three dimensions by the longitudinal and transverse displacement and the rotation about a vertical axis. Thus, building irregularities complicate not only the method of dynamic analysis, but also the methods used to combine modes.

For a building that is regular and essentially symmetrical, a two-dimensional model is generally sufficient. Note that when the floor plan aspect ratio (length-to-width) of the building is large, torsion response may be predominant, thus requiring a 3-D analysis in an otherwise symmetrical and regular building.

For moderate- to-high-rise buildings, the effects of higher modes may be significant. For a fairly uniform building, the dynamic characteristics can be approximated using the general modal relationship shown in Table 2.1. The fundamental period of vibration may be estimated by using code formulas, and the periods for the second through fifth modes

(a) Fundamental mode of a multimass system

(b) Equivalent single-mass system

Figure 2.14. Representation of a multimass system by a single-mass system.

(a) Fundamental mode of a multimass system

(b) Equivalent single-mass system

Figure 2.14. Representation of a multimass system by a single-mass system.

TABLE 2.1 General Modal Relationships

Mode

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