Page 66 - Dynamic Loading and Design of Structures
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Figure 2.16 Time response of an elastoplastic SDOF system under maintained load F(t)=F1.
In the final stage, we observe that we have harmonic vibration about a neutral position which
is given by . Redefining a new time , we have the response
as
(2.51)
Figure 2.16 plots the dynamic displacement y(t) for the case described above, while Figure
2.17 is a nomograph for the ductility ratio µ of the SDOF elastoplastic oscillator which is
defined as the ratio ym/yel for a load of magnitude F1 and duration td. We finally observe that
in order for the elastoplastic SDOF system to behave elastically (i.e.µ≤1), the maximum
spring resistance R must have at least twice the value of the magnitude of the applied load F .
m
1
2.3 MULTIPLE DEGREE-OF-FREEDOM SYSTEMS
The definition of a Multiple Degree-of-Freedom (MDOF) system is one which requires a
second order, ordinary differential equation to describe the motion of each independent DOF.
A DOF is an active translation or rotation component of motion at a given nodal point of the
structure in question. In three dimensions, we have a total of six DOF per node, namely three
displacements and three rotations, while on the x-y plane there is a total of three DOF, namely
two displacements and one rotation. As a simple example, we have the two DOF system of
Figure 2.18(a) with the following coupled equations of motion:
