## Dynamic Analysis Theory

A good portion of the loads that occur in buildings can be considered static, requiring static analysis only. Although almost all loads except dead loads are transient, meaning that they change with time, it is customary to treat them as static. For example, lateral loads imposed by transient wind pulses are usually treated as static loads and even in earthquake design, one of the acceptable methods of design, particularly for buildings with regular configuration, is to use an equivalent static force procedure. Under these circumstances, the analysis of a structure reduces to a single solution for a given set of static loads. Although the equivalent static method is a recognized method, most building codes typically mandate dynamic analysis for certain types of buildings such as those with irregular configurations (see ASCE 7-03, Table 9.5.2.5.1). It is therefore necessary, particularly in seismic design, to have a thorough understanding of dynamic analysis concept.

Consider a building subjected to lateral wind loads. Although wind loads are dynamic, in typical design practice, except in the case of slender buildings, wind loads are considered as equivalent static loads. The variation of wind velocity with time is taken into account by including a gust factor in the determination of wind loads. Therefore, for a given set of wind loads, there is but one unique solution.

Now consider the same building, instead of being buffeted by wind, subjected to ground motions due to an earthquake. The input shaking causes the foundation of the building to oscillate back and forth in a more or less horizontal plane. The building would follow the movement of the ground without experiencing lateral loads if the ground oscillation took place very slowly over a long period of time. The building would simply ride to the new displaced position. On the other hand, when the ground moves suddenly as in an earthquake, building mass, which has inertia, attempts to prevent the displacement of the structure.

Therefore, lateral forces are exerted on the mass in order to bring it along with the foundation. This dynamic action maybe visualized as a group of horizontal forces applied to the structure in proportion to its mass, and to the height of the mass above the ground.

These earthquake forces are considered dynamic, because they vary with time. Since the load is time-varying, the response of the structure, including deflections, axial and shear forces, and bending moments is also time-dependent. Therefore, instead of a single solution, a separate solution is required to capture the response of the building at each instant of time for the entire duration of an earthquake. Because the resulting inertia forces are a function of building accelerations, which are themselves related to the inertia forces, it is necessary to formulate the dynamic problem in terms of differential equations.

### 2.6.1. Single-Degree-of-Freedom Systems

Consider a portal frame, shown in Fig. 2.65, consisting of an infinitely stiff beam supported by flexible columns that have negligible mass as compared to that of the beam. For horizontal motions, the structure can be visualized as a spring-supported mass, as shown in Fig. 2.66a, or as a weight W suspended from a spring, as shown in Fig. 2 66b. Under the action of gravity force on W, the spring will extend by a certain amount x. If the spring is very stiff, x is small, and vice versa. The extension x can be related to the stiffness of the spring k by the relation

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