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3.1 Dynamic Forces and Structural Models

### 3.1.1 Characteristics of Dynamic Forces

The subject presented herein deals with the response of structures subjected to dynamic forces or loads, whose magnitude varies with time. Generally, most of the forces applied to a structure involve, in some manner, time variation; static force may be viewed as a special dynamic case when the force applied is slow enough without causing structural vibration. As structural materials, construction methods, and computer technology rapidly advance, constructed facilities of building structures and nonbuilding structures become taller and slender. Therefore, dynamic behavior of such structures must be included in their design.

The dynamic forces acting on a structure can be categorized in different ways according to (1) the original sources causing vibration, (2) the characteristics of vibration, whether periodic, nonperiodic, or random, or (3) the definite function of time as deterministic or nondeterministic. Rotating machinery, blast, wind, and earthquake are in the first category. Dynamic force due to unbalanced machinery varied repeatedly in magnitude with time is called periodic force. Earthquake, wind, and blast, however, do not have any periodicity and are hence called nonperiodic or random forces. A deterministic force is one where its time function can be specified in regular or irregular variation; for instance, the time variation of rotating machinery can be represented by a mathematical function, blast and impulse may be specified by mathematical curves or lines, and an earthquake may be specified by accelerograms in magnitude with time intervals. These forces may be classified as deterministic. On the contrary, nondeterministic forces cannot be specified as definite functions of time because of the inherent uncertainty in their magnitude and in their variation with time. These types of load should be described through a statistical approach. Wind is in this category and earthquake is also non-deterministic because the magnitude and frequency distribution of any future earthquake cannot be predicted with certainty but can be estimated only in a probabilistic sense. The classification of loading is shown in Figure 3.1. This chapter deals with both deterministic and nondeterministic loadings or forces for which the response analysis methods are presented. Stochastic analysis is illustrated with seismic response.

### 3.1.2 Mathematical Models of Structural Systems

Analytical accuracy and computational efficiency of dynamics problems depends on several key features: structural modeling, material property idealization, loading assumptions, and numerical techniques. This chapter covers three well-known models:

• Lumped-mass system

• Continuous-mass system

• Finite element system

In fact, the lumped-mass system and the finite element system are similar in modeling and therefore are sometimes classified into one group known as the discrete system.

r Static

Nondeterministic (random)

Periodic

Loading <

L Dynamic <

Nonperiodic

Nondeterministic (random)

FIGURE 3.1 Loading classification.

3.2 Response Analysis of Single d.o.f. Systems

The degree of freedom (d.o.f.) of a structure may be first explained from the nature of loading as statically applied load or dynamic excitation to a structure. In general, d.o.f. represents the independent movement of structural nodes for a static case, but the independent movements of lumped masses for a dynamic case. The number of structural nodes can be more than the number of lumped masses. Furthermore, each lumped mass may have more than one independent motion. For instance, the plane frame shown in Figure 3.2 should have three d.o.f. (x1, x2, x3) for a static case. However, the lumped-mass model may have one (x3), two (x3 and x4), or three (x3, x4, and x5) d.o.f. corresponding to the response analysis of lateral motion (x3) only, lateral and vertical motions (x3, x4) only, or lateral, vertical, and rocking (x3, x4, X5) motions. Therefore, the number of masses and the dynamic d.o.f. are determined by the structural analyst based on the structural configuration and the interest of the analytical results. Note that x1 and x2 in Figure 3.2b are not dynamic d.o.f., but they must be given in order to allow the structural joints to rotate during vibration.

3.2.2 Undamped and Damped Free Vibration 3.2.2.1 Undamped with Initial Conditions

Consider the spring-mass model shown in Figure 3.3. This model, which consists of a mass of weight, W, suspended by means of a spring with stiffness, K, is idealized from the accompanying unsymmetrical rigid frame where L and I signify the member's length and its cross-sectional moment of inertia, respectively. The spring stiffness, K, is defined as the force necessary to stretch or compress the spring

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