Structural dynamics

1. Differentiate Static analysis and Dynamic analysis with proper representing pictures

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 within 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, the 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. Explain the following with relevant force-displacement graph

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

1. 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.
• From graph it is clearly mensioned that upto certain point from 'O' the graph is linear and shows elastic properties.

2. 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.,
• In graph inelastic behavior has been shown in the upper shape of the curve upto point "a".

3. 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
• In graph the plastic behavior occurs during unloading after point "a"

2. 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.
• Non-linear elastic behavior occurs after the line from "o" tends to bend towards point"a".

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

Stiffness component

• The frame without damping or mass.
• The external force 'fs' on the stiffness component is related to displacement u by fs=ku
• If the system is linearly elastic

Damping component

• The frame with its damping property with no stiffness or mass .
• The external force on the damping component is related to velocity by fd=cv

Mass component

• The roof mass without stiffness or damping of the frame.
• The external force fi on the mass component is related to the accelaration

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

Natural Period Of Vibration (Tn)

• It is the time required for the undamped system to complete one cycle of free vibration.
• Units in seconds
• It is realated to the natural circular frequency of vibration wn , in units of radians per second

Natural Cyclic Frequency (fn)

• The number of cycles (or one complete revolution) executed by system in 1 sec
• units in hertz

5. Explain in detail about Response spectrum & its Graph.

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.
• The response spectrum can be of 3 types , the deformation response spectrum is a plot of peak u, against Tn for fixed ζ . A similar plot for peak ú, is the relative velocity response spectrum, and for peak ü^T ,is the accelaration response spectrum.