Structural dynamics

1.Diffrence between Static analysis and Dynamic analysis :

             Dynamic analysis is the testing and evaluation of an application during runtime. Static analysis is the testing and evaluation of an application by examining the code without executing the application. Many software defects that cause memory and threading errors can be detected both dynamically and statically.

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 :

• If, after a load has been applied and then quickly removed, a material returns rapidly to its original shape, it is said to be behaving elastically.
• Elasticity is a crucial characteristic of building materials,if it were not buildings would suffer continuous deformation under load and ultimately would collapse.
• When we stretch a slingshot, the force we applied allowed it to stretch, but when we removed the force, the deformation reverted to its original shape. This property is known as elasticity.
• 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 :

• The term, “inelastic” is used to define the material response in relation to the stress-strain diagram that is non-linear and that retains a permanent strain or returns to an unstrained state on complete unloading.
• Inelastic behavior is possible in beams that are made of ductile materials, and as such can be loaded beyond the elastic limit or proportional limit of the material. This implies that the ultimate load carrying capability of a ductile beam is higher than its maximum elastic load.
• 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 :

• The term 'plasticity' refers to the degree to which as material has the characteristic of being easily shaped, moulded or formed.
• After stretched it stays stretched.Most materials have an amount of force or pressure for which they deform elastically. If more force or pressure is applied, then they have plastic deformation.
• 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 behaviour usually only up to a certain load, which is called the 'yield point'. After that, the deformations are plastic. Rubber, for instance, has an elastic stress–strain curve, but the relation is not linear − in such cases nonlinear elastic models are required.
• Material nonlinearity involves the nonlinear behavior of a material based on a current deformation, deformation history, rate of deformation, temperature, pressure, and so on. Examples of nonlinear material models are large strain (visco) elasto-plasticity and hyperelasticity (rubber and plastic materials).
• A non-linear story will present events in a different order from how they occurred in the setting. For example, a story might open to a scene from the future before flashing back to explain that scene
• 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.

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

The equation of motion in single degree of freedom system consists of three indepenedent system components acting on the frame.

Mass component: 

The mass componenet in equation of motion considers only the mass of the frame and stiffness and damping are considered to neglected. The force is related with mass component with the equation of fi=mu¨.

Stiffness component: 

The rigidity of the material  to resist deformation is called as the stiffness and in this component only stiffness is considered in the frame and other mass and damping components are neglected. The force is related with stiffness componet with the equation of fs=ku.

Damping component: 

Damping is the dissipiation of energy due to vibrations over time and distances. It occurs due to the friction between structural and non stuctural elements. The force is related with damping component with the equation of fd=cu˙ 

Hence the equation of motion can be written as 

mu¨+ fd + fs = p(t) that is 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.

5.Response spectrum:

          A response spectrum is a graphical plot of the frequency of an oscillator and its damping. The response spectrum plot represents the peak or steady-state response (velocity, displacement, or acceleration) of a series of oscillators of varying natural frequencies.

           It is a concept of characterizing the ground motion and their effects on structures of different time period Tn. A plot of the peak value of a response activity (u,v,a) as a function of natural vibration period Tn of the system is called the response spectra. The peak response of all linear single degree of freedom systems can be computed by response spectra plot. 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.