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2.13.9 Damping

Damping is found to increase with the increasing amplitude of vibration. It arises from the dissipation of energy during vibration. The mechanisms that contribute to energy dissipation are material damping, friction at interfaces between components, and energy dissipation due to the foundation interacting with soil, among others. Material damping arises from the friction at bolted connections and the frictional interaction between structural and nonstructural elements such as partitions and cladding.

The amount of damping in a building can never be predicted precisely, and design values are generally derived based on dynamic measurements of structures of a corresponding type. Damping can be measured based on the rate of decay of free vibration following an impact, by spectral methods based on analysis of response to wind loading, or by force excitation by mechanical vibrator at varying frequency to establish the shape of the steady state resonance curve. However, these methods may not be easily carried out if several modes of vibration close in frequency are presented.

Table 2.10 gives the values of modal damping that are appropriate for use when amplitudes are low. Higher values are appropriate at larger amplitudes where local yielding may develop, for example, in seismic analysis.

2.13.10 Numerical Analysis

Many less complex dynamic problems can be solved without much difficulty by hand methods. More complex problems, such as determination of natural frequencies of complex structures, calculation of response due to time-varying loads, and response spectrum analysis to determine seismic forces, may require numerical analysis. Finite element method has been shown to be a versatile technique for this purpose.

The global equations of an undamped force-vibration motion, in matrix form, may be written as

where n n n

;=i ;=i ;=i are the global stiffness, mass, and force matrices, respectively. [k,-j, [m,-j, and {fi} are the stiffness, mass, and force of the ith element, respectively. The elements are assembled using the direct stiffness method to obtain the global equations such that intermediate continuity of displacements is satisfied at common nodes and, in addition, interelement continuity of acceleration is also satisfied.

Equation 2.398 is the matrix equations discretized in space. To obtain the solution of the equation, discretization in time is also necessary. The general method used is called direct integration. There are two methods for direct integration: implicit or explicit. The first, and simplest, is an explicit method known as the central difference method (Biggs 1964). The second, more sophisticated but more versatile, is an implicit method known as the Newmark method (Newmark 1959). Other integration methods are also available in Bathe (1982).

TABLE 2.10 Typical Structural Damping Values

Structural type

Damping value, X (%)

Unclad welded steel structures

0 0

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