Structural dynamics

Challenge 2 :-

Structural dynamics

Ans 1:Differentiate Static analysis and Dynamic analysis with proper representing pictures

AIM-:1. Differentiate Static analysis and Dynamic analysis

Ans.-

Static Analysis	Dynamic Analysis
static analysis time invariant analysis of forces like Stress, strain and displacement etc.	Dynamic analysis is time dependent analysis
The results are a consequence of the applied constraints and loads, for example, bearings, gravity, forces, etc.	Inertia and damping forces taken into account
Static loads applied on structure slowy	Dynamic Loads applied on the structures Suddenly
Example- Dead Load , Live load	Example- Earthquake load , Wind Load
Static analysis done with minimum three degree of freedom	Dynamic analysis of structure done with single degree of freedom
Static analysis done where the structure deformations behave linear	Dynamic analysis done where structure behaves non-linear deformations

.In dynamic analysis we assume the mass is concentrated at a point with single degree of freedom as see in below picture

Ans 2:

Explain the following with relevant force-displacement graph

1. Elastic behavior
2. Inelastic behavior
3. Plastic behavior
4. Non-linear Inelastic behavior

AIM-

Elastic behavior

For the linear system relation between static force and displacement is linear

That is

fs=ku where K is lateral stiffness of the structure

and u is Displacement

the loading and unloading curves are identical such the system is said to be linear elastic

\

Conside the frame a with shown parameters L,h,E and second moment of area about the axis of bending

when the flexural rigidity of Beam is infinty

k=∑12EIch3∑12EIch3 for columns

=24EIch324EIch3

if stiffness of the beam is Zero of no stiffness

EIb=0

=3EIch3=3EIch3

Inelastic Behaviour

Determined by experiments, the force–deformation relation for a structural steel component
undergoing cyclic deformations expected during earthquakes.
The initial loading curve is nonlinear at the larger amplitudes of deformation, and the un-
loading and reloading curves differ from the initial loading branch; such a system is said
to be inelastic. This implies that the force–deformation relation is path dependent, i.e., it
depends on whether the deformation is increasing or decreasing.

Plastic Behaviour of structure

Plastic behiviour known as permanent deformation after that member can not attains its original shape or position of deformation or displacement

Non-Linear Inelastic behaviour

Force displacement relation Becomes Non-linear and inelastic at large deformations

For fs corresponding to the deformation is not single valued and depends on history of deformations and whether deformation is increasing or decreasing thus the resisting force can be expressed as

fs=fs(u,v)

Ans.3:Explain Mass, Stiffness & Damping components in the equation of motion

AIM-:Explain Mass, Stiffness & Damping components in the equation of motion

Single degree of freedom system can be viewed as combination of three pure independent system

The stifffness component- ; the frame without damping or mass , the external forces fs on the stiffness componenet is relaed to displacement by u by fs=ku if the system is linearly elastic

The Damping component -: the frame with its damping property but no stiffness or mass the external force on the damping component is related to velocity by fd=cv

The mass component = The roof mass without the stiffness or damping of the frame the external force fi on the mass component is related to acceleration by fi=mu

m..umu..+fd+fs = pt rightarrow mddot u++cdot u`+ku = pt

Ans 4: Provide relationship between Natural period (Tn) & Natural frequency (f) and provide their definitions

AIM-Provide relationship between Natural period (Tn) & Natural frequency (f) and provide their definitions

Natural period (tn)-The time required for the undamped system to complete one cycle of free vibration natural period of vibration of the system, which we denote as Tn in units of sec It is related to the natural circular frequency of vibration, ωn in units of radians per second

Tn=2π/ωn

A system executes 1/Tn cycles in 1 sec. This natural cyclic frequency of vibration is denoted by

fn=1/Tn

Ans 5: Explain in detail about Response spectrum & its Graph

Aim- Explain in detail about Response spectrum & its Graph

Response spectrum analysis is a method to estimate the structural response to short, nondeterministic, transient dynamic events. Examples of such events are earthquakes and shocks. Since the exact time history of the load is not known, it is difficult to perform a time-dependent analysis.

A response spectrum is a function of frequency or period, showing the peak response of a simple harmonic oscillator that is subjected to a transient event. The response spectrum is a function of the natural frequency of the oscillator and of its damping. Thus, it is not a direct representation of the frequency content of the excitation (as in a Fourier transform), but rather of the effect that the signal has on a postulated system with a single degree of freedom (SDOF)

Example The El Centro Earthquake

One of the most studied earthquake recordings is that of the "El Centro" earthquake on May 18, 1940. Recorded signals (with some filtering) are shown below