Advanced Methods of Structural Analysis

514 14 Dynamics of Elastic Systems

t

y

T

y 0

A

T

t

y

T T

t

y

T T

ab c

Fig. 14.1 Types of oscillatory motions

may be divided into the following groups: disturbing forces, positional (restoring)

forces, resisting forces, and forces of the mixed character.

1.Disturbing forcesmay be of the following types:

(a)Immovable periodical loads are produced by stationary units and mechanisms

with moving parts. These loads have a periodical, but not necessary a har-

monic character and generally do not depend on the elastic properties of the

structure.

(b)Impact (impulsive) loads are produced by falling weights or collision of bod-

ies. Impulsive loads are characterizedby very short duration of their action

and depend on the elastic properties of the structure, which is subjected to

such loads.

(c)Moving loads act on the structures through wheels of a moving train or truck.

The availability of the rail joins on the railway bridge or irregularities of the

deck on the car bridge lead to appearance of inertial forces. These type of

loads should be distinguished from moving one, which has been studied in

the sections “Influence lines” because unit moving loadP D 1 had been

considered without dynamical effects.

(d)Seismic loads arise due to earthquakes. The reason of the seismic load on

the structure is the acceleration of the supports caused by acceleration of the

ground. This type of disturbance is called kinematical. The acceleration of

supports leads to the acceleration of the separate parts of the structure, and

as a result inertial forces act on these parts. Seismic forces, which arise in

the members of the structure, depend on the type and the amount of ground

acceleration, distribution of the mass within the members of the structure and

their elastic properties.

2.Restoring forcesdepend on the displacement of the structure, arise due to devi-
ation of system from a static equilibrium position, and tend to return the system
to its initial position. Restoring properties of a system are described by its elas-
tic characteristicPDP.y/,wherePis a static force, which is applied to the
structure. CharacteristicPymay be linear or nonlinear. Some types of char-
acteristicsPyare presented in Table14.1; in all casesyis the displacement
at the point ofP.
3.Resisting forces. The forces of inelastic resistance (friction or damping forces)
depends on the velocityof motion,RDR./, and always act in the opposite
direction of velocity. These forces are a result of internal friction in the material
of a structure and/or in the connections of a system.