Structures are subjected to loading that is mostly time dependent in a weak or strong fashion. Response histories under weakly time-dependent loading may be calculated by using the quasi-static analysis procedure.

For moderately or strongly time dependent loading, calculation of response quantities requires a full dynamic analysis procedure as it is presented in the previous chapter assuming that the structure is deterministic and the loading history is fully determined or known,i.e., it obeys a specific rule or a definite function of time such as constant, linear, non-linear, harmonic, etc.

time functions with known properties. Under such a structural and loading case, the corresponding analysis type is called as the deterministic dynamic analysis since all necessary parameters of the analysis can be uniquely determined or known. However, the difficulty in the structural dynamic analysis is to determine the loading functions and their properties correctly, such as frequencies, durations, amplitudes, and phases, in practice. Due to lack of sufficient knowledge of dynamic excitations in nature, we possess limited information on loading parameters which is usually obtained from recorded data or observations of occurrences, such as earthquakes and sea waves, which occur in arbitrary fashions. Other examples can be wind loading on high-rise buildings and towers, and traffic loading on bridges and viaducts, which do not follow specific rules. Earthquakes occur periodically in seismic areas with unknown information and sea waves occur continuously with random fluctuation of the sea surface. The only information that we have is based on experiences of past occurrences from which we can predict information of the structural response in a probabilistic manner. When the excitation loading varies arbitrarily in time, the corresponding response will also be arbitrary in time. Such a response process deals with the random vibration and its characteristic properties can be determined by using statistical and probabilistic methods.