Structural dynamics

Q1

STATIC ANALYSIS	DYNAMIC ANALYSIS
Load is variable with time and neglected due to slow applying of landlord loan	Load is variable with time due to fast applying of load and vibration motion motion motion
Motion of the structure can be disregarded	Motion can not be disregarded where there there
There is no necessity to count inertia and damping forces in case of static analysis but the elastic force should be account inertia	Inertia and damping forces should be accounted for the calculation of dynamic analysis there where there
There is no inertia effect on static analysis	Inertia effect is considered in dynamic analysis

Fig 1: Pictorial representation of static and dynamic analysis

Q2)

1. Elastic behavior

It is the property of the material up to which the material can retain its shape. From the graph mentioned below, it is clear that up to a certain point from “O” the graph is linear and shows its elastic properties

2. Inelastic Behaviour

It is the property of the material in which the material can’t retain its previous shape after a certain deformation. From the graph mentioned below, it has been shown in the upper shape of the curve up to point “a”.

3. Plastic Behaviour:

It occurs when the material completely lost its ductility. In the graph, the plastic behavior occurs during unloading after point “a”

4. Non-Linear Inelastic Behaviour

This occurs after the line from “o” tends to bend towards point “a”

Fig 2: Force-Displacement Graph (Ref: Dynamics of structures Theory and applications to earthquake engineering, A.K.Chopra )

Q3)

Equation of motion:External force

An alternative point of view for structural engineering wherein principle of statics can be used to develop the equation of motion

`P-fs-fD=m u ̈ → m u ̈+fD+fs=P(t)→m u ̈+cu ̇+ku=P(t)`

Stiffness component

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

Damping component

The frame with its damping property but no stiffness or mass.The fs on the damping component is related to velocity by ,fd=CV.

Mass component

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

Q4)

Relationship between Natural period of vibration (Tn) and natural cyclic frequency(fn) of the system is

fn=1/Tn`

Natural period of vibration (Tn)

• The time required for the undamped system to complete one cycle of free vibration (unit in seconds)
• It is related to the natural circular frequency of vibration,ὠn ,in units of radians per second.

Tn=(2π/ωn)

Natural cyclic frequency (fn)

• Natural cyclic frequency of vibrationof the systemis the number of cycles (or one complete revolution )executed by the system in one second
• unit-Hertz

`u_0``fn=(1/Tn)`

Q5)

• It’s a central concept in earthquake engineering and provides a convenient means to summarize the peak response of all possible linear SDF systems to a particular component of ground motion
• It provides a practical approach to applying the knowledge of structural dynamics to the design of structures and the development of lateral force requirements in building codes.
• A plot of the peak value of response quantity as a function of the natural vibration period Tn of the system, or a related parameter such as circular frequency wn or cyclic frequency fn, is called the response spectrum for that quantity
• Each such plot is for SDF systems having a fixed damping ratio of `zeta` and several such plots for different values of `zeta` are included to cover the range of damping values encountered in actual structures.
• A variety of RS can be defined depending on the response quantity plotted,Consider the peak response

• The deformation response spectrum is a plot `u_0` against Tn for fixed ( damping ratio). A similar plot for `u_0` is the relative velocity response spectrum and for `(〖u^t〗_0 ) ̈` is the acceleration response spectrum.

Fig 2: Ground acceleration (b) deformation response of three SDF systems and deformation response system(Ref: Dynamics of structures Theory and applications to earthquake engineering,A.K.Chopra )