Structural dynamics

1. Static Analysis and Dynamic Analysis

Static Analysis :

• Static analysis can only be performed if the system being simulated does not depend on time,and if the loads being applied are constant.
• Static analysis don't consider inertia.
• Static refers to the force that has constant size,Position and Direction on or with in a structure.
• There is no necessity to count inertia and damping forces in case of static analysis , but the elastic force should be accounted.

Dynamic Analysis :

• In dynamic analysis ,the system itself load application or both might change with time
• The inertial force developed by the system due to accelaration are taken into account. 
• Dynamic refers to a force on a structure that changes size, position or direction.
• In dynamic , two additional forces must be accounted such as inertia and damping force in equilibrium equation .
• It can be disregarded in static analysis.

2. 

a. Elastic behavior

• Elasticity , ability of a deformed material body to return to its original shape and size when the forces causing the deformations are removed
• A body with this ability is said to behave elastically
• The theory of elasticity is used to design safe and stable man-made structures such as skyscrappers and overbridges to make life convenient.
• 

b. Inelastic behavior

• Deformation material that does not disappear on the removal of force produced by it
• Inelastic material will experience no elongation or deflection and up to its ultimate strength is reached will just rupture suddenly usually through shear break
• Example: wood, fabric etc.

c. Plastic behavior

• For stresses beyond the elastic limit,a material exhibits plastic behavior.
• This means the material deforms irreversibly and does not return to its original shape and size, even when the load is removed
• When stress is gradually increased beyond the elastic limit, the material undergoes plastic deformation
• Plastic wrap is an example of plasticity.

d. Non-linear Inelastic behavior

• Materials have elastic behvior  usually only upto a certain point after that deformation are plastic and the relation in this is not linear, that is non linear elastic behavior
• Fore displacement relationship becomes non-linear and inelastic at large deformations.

3.

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

Stiffness component

The frame without damping or mass , the external forces fs on the stiffness component is related to displacement, if the system is linearly elastic

fs=ku

Damping component

The frame with its damping property but no stiffness or mass the external force on the damping component is related to velocity

fd=cu

mu + fd +fs = p(t) --> mu + cu + ku = p(t)

where p(t) is the external force

4.

Natural period: 

The time taken for undamped system to complete one cycle of free vibration is called as the natural period of vibration and it is denoted by the formula Tn = 2π/ωn and the unit is seconds.

Natural frequency: 

The natural frequency is defined as the number of cycles i.e, one full cycle of free vibration completed in one second. It is denoted in radians/sec.The relation between and natural period and natural frequency is that frequency is inversely propotional to the period of vibration.

fn=1/Tn.

The natural Vibration properties `wn, Tn , and fn` depend only on the mass and stiffness of the structure. 

                                                                    `k = m w^2`        

5.

Response Spectrum

• It is a central concept in earthquake engineering, response spectrum is a practical means of characterizing ground motions and their effects on structures of different Tn 
• It provides a convenient means to summarize the peak response of all possible linear SDF systems to a particular component of ground motion
• It also provides a particular approach to apply the knowledge of structural dynamics to the design structures and development of lateral force requirements in building codes 
• A plot of the peak value of a response quantity (u,v and a) as a function of the natural vibration period Tn of the system,is called the response spectrum for that quantity
• Each such plot is for SDF systems having a fixed damping ratio ζ  and several such plots for different values ζ are included to cover the range of damping values encountered in actual structures
• Three types of plot are generally plotted with u and Tn as deformation response spectra and a similar plot is done with u as relative velocity spectrum and with u as acceleration response spectrum.
• Each plot is done with fixed damping ratio and several such plots with different damping ratio are created for the range of damping values encountered in the structures.