Advanced Methods of Structural Analysis

518 14 Dynamics of Elastic Systems

of freedom will be 2 and 6, respectively. Forgently slopingarches, the horizontal
displacements of the masses may be neglected; in this case, the arches should be
considered as structures having one and three degrees of freedom in thevertical
direction.
All cases shown in Figs.14.2–14.4 present the design diagrams for systems with
lumped (concentrated) parameters. Since masses are concentrated, the configura-
tion of a structure is defined by displacement of each mass as a function of time,
i.e.,yDy.t/and behavior of such structures is described byordinarydifferen-
tial equations. It is worth discussing the term “concentrated parameters” for cases
14.2f (pontoon bridge) and 14.4c (two-story frame). In both cases, the mass, in fact,
is distributed along the correspondence members. However, the stiffness of these
members is infinite, and the position of these members is defined by onlyoneco-
ordinate. For the structure given in Fig.14.2f, such coordinate may be the vertical
displacement of the pontoon or the angle of inclination of the span structure, and for
the two-story frame (Fig.14.4c), it may be the horizontal displacements of the each
cross bar.
The structures with distributed parameters are more difficult for analysis. The
simplest structure is a beam with a distributed massm. In this case, the configuration
of a system is determined by displacement of each elementary mass as a function of
time. However, since masses are distributed, then the displacement of any point is a
function of a timetand locationxof the point, i.e.,yDy.x; t/,sobehaviorofthe
structures is described bypartialdifferential equations.
Figure14.5presents the four possible design diagrams for dynamical analysis of
the beam subjected to a moving concentratedload. Parameters that are taken into
account (mass of moving loadMand distributed mass of a beamm)areshown
by bold and thick solid lines. The scheme (a) does not take into account the mass
of the beam and mass of the load; therefore, the inertial forces are absent. This
case corresponds to static loading, and parameteronly means that forcePmay
be located at any point. Just this case of loading is assumed for construction of
influence lines. Case (b) takes into account only the mass of the moving load. Case
(c) corresponds to the motion of the massless load along the beam with distributed
massm. Case (d) takes into account the mass of the load and mass of the beam. The
difficulty in solving these dynamical problems increases from case (b) to case (d).

Fig. 14.5 (a–d)Design
diagrams for beam subjected
to moving load

a

u

P

M u

b

u

m

c P

u

m

M

d

It is possible to obtain a combination of the members with concentrated and
distributed parameters. Figure14.6shows a frame with a massless strut BF.mD0/,