Dynamic Analysis of the Structure
From an experiment it can be shown that the
loading deformation graph, when drawn with unloading v/s deformation, will not
retrace the earlier path, when it is reloading we get another path, This system
is called inelastic system. It is expected that after an earth quake there has
been some cracks, ground movements, and deformation etc.
This force is a function of deformation s
and velocity ds/dt, mathematically f = f (s, ds/dt).
Force due to acceleration of mass mass
m=m.d2s/dt2=fa,
Now the external forces cause deformation
s, velocity ds/dt, and acceleration m.d2s/dt2.
Then for an inelastic system equation of
motion is
Where c is called damping coefficient. The
three forces acting are the resisting force, damping force, and force make
acceleration. These forces are due to external force F (t).
D Alembert's principle
Like static equilibrium there is also
dynamic equilibrium. A moving body is in dynamic equilibrium.
Static Analysis
After solving the equation of motion, we
can start to find bending moments and shearing forces by static analysis. At
each end of a beam or column the displacement and rotation is known. Now the bending
moments and shears can be calculated through the element stiffness properties
and stresses is obtained from element forces.
In a nonlinear system the two analysis cannot
be done separately. The initial stiffness property is needed to start the
dynamic analysis.
Solution of Differential Equation
The differential equation of motion for a
linear single degree of freedom system is
Md2s/dt2 + cds/dt +ks
= p (t)
This is a 2nd order differential
equation. The displacement and initial velocity is zero at time t=0, or the
structure is at rest before the excitation. The complete solution is the sum of
complimentary solution and particular solution. Since it is a 2nd
order equation there are two constants of integration. It is appeared in the
complementary solution and can be find from initial condition.
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