ABSTRACT
This chapter presents a brief account of the basics of dynamic behaviour of structures; the representation of earthquake ground motion by response spectra; and the principal methods of seismic structural analysis.Dynamic analysis is normally a two-stage process: we first estimate the dynamic properties of the structure (natural frequencies and mode shapes) by analysing it in the absence of external loads, and then use these properties in the determination of earthquake response.Earthquakes often induce non-linear response in structures. However, most practical seismic design continues to be based on linear analysis. The effect of non-linearity is generally to reduce the seismic demands on the structure, and this is normally accounted for by a simple modification to the linear analysis procedure. A fuller account of this basic theory can be found in Clough and Penzien (1993) or Craig (1981). 3.2 Basic dynamics
This section outlines the key properties of structures that govern their dynamic response, and introduces the main concepts of dynamic behaviour with reference to single-degree-of-freedom (SDOF) systems. 3.2.1 Dynamic properties of structures
For linear dynamic analysis, a structure can be defined by three key properties: its stiffness, mass and damping. For non-linear analysis, estimates of the yield load and the post-yield behaviour are also required. This section will concentrate on the linear properties, with non-linearity introduced later on.First, consider how mass and stiffness combine to give oscillatory behaviour. The mass, m, of a structure, measured in kg, should not be confused with its weight, mg, which is a force measured in N. Stiffness, k, is the constant of proportionality between force and displacement, measured in
N/m. If a structure is displaced from its equilibrium position then a restoring force is generated equal to stiffness × displacement. This force accelerates the structure back towards its equilibrium position. As it accelerates, the structure acquires momentum (equal to mass × velocity), which causes it to overshoot. The restoring force then reverses sign and the process is repeated in the opposite direction, so that the structure oscillates about its equilibrium position. The behaviour can also be considered in terms of energy – vibrations involve repeated transfer of strain energy into kinetic energy as the structure oscillates around its unstrained position.In addition to the above, all structures gradually dissipate energy as they move, through a variety of internal mechanisms that are normally grouped together and known as damping. Without damping, a structure, once set in motion, would continue to vibrate indefinitely. There are many different mechanisms of damping in structures. However, analysis methods are based on the assumption of linear viscous damping, in which a viscous dashpot generates a retarding force proportional to the velocity difference across it. The damping coefficient, c, is the constant of proportionality between force and velocity, measured in Ns/m. Whereas it should be possible to calculate values of m and k with some confidence, c is a rather nebulous quantity that is difficult to estimate. It is far more convenient to convert it to a dimensionless parameter ξ, called the damping ratio:
ξ= c km2 (3.1)
ξ can be estimated based on experience of similar structures. In civil engineering it generally takes a value in the range 0.01 to 0.1, and an assumed value of 0.05 is widely used in earthquake engineering.In reality, all structures have distributed mass, stiffness and damping. However, in most cases it is possible to obtain reasonably accurate estimates of the dynamic behaviour using lumped parameter models, in which the structure is modelled as a number of discrete masses connected by light spring elements representing the structural stiffness and dashpots representing damping. Each possible displacement of the structure is known as a degree of freedom. Obviously a real structure with distributed mass and stiffness has an infinite number of degrees of freedom, but in lumped-parameter idealisations we are concerned only with the possible displacements of the lumped masses. For a complex structure the finite element method may be used to create a model with many degrees of freedom, giving a very accurate representation of the mass and stiffness distributions. However, the damping is still represented by the approximate global parameter, ξ.
