Structural dynamics

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

Static analysis	Dynamic analysis
Static Loads are the loads which varies very slowly (the acceleration of load is very very less than the natural frequency of the structure)	Dynamic Loads are the loads which changes with time fairly quickly in comparison to the structure's natural frequency.
Load is not variable with time and neglected due to slow applying of load.	Load is variable with time due to fast applying of load and vibration.
Motion of the structure can be disregarded.	Motion can not be disregarded
There is no necessity to count Inertia and damping forces in the case of static analysis, but the elastic force should be accounted for.	Inertia and damping forces should be accounted for in the calculation of dynamic analysis.
There is no inertia effect on static analysis.	Inertia effect is considered in dynamic analysis.

2.Explain the following with relevant force-displacement graph (40 marks)

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

The following with the relevant force-displacement graph

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

Elastic behavior

• The ability of a deformed material body to return to its original shape and size when the forces causing the deformation are removed.
• A body with this ability is said to behave (or respond) elastically. From the force-displacement graph at point O to a certain point, it shows a linear path, hence the body shows elastic properties at a certain limit with the forces applied.

Inelastic behavior

• The deformation of a material that does not disappear on the removal of the force that produces it.
• From the force-displacement graph, inelastic behavior corresponds to the nonlinear path.
• Inelasticity is experienced by the body as soon as the path goes out of the linear curve.

Plastic behavior

• The behavior of plastic structures beyond the bending collapse load is influenced by the changes in the geometry of a structure during the process of plastic deformation.
• In the present paper, an investigation of the load-deflection relationship for simply supported rectangular reinforced concrete plates is presented

Non-linear Inelastic behavior

• Materials have elastic behavior usually only up to a certain load, which is called the ‘yield point’.
• After that, the deformations are plastic.
• From the force-displacement graph, this behavior occurs when the line from point “O” tends to bend to point “a”.

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

Mass:

• Newton's second law, which states that the force F acting on a body is equal to the mass m of the body multiplied by the acceleration a of its Centre of mass
• F = ma, is the basic equation of motion in classical mechanics.
• Mass is always constant for a body. One way to calculate mass: Mass = volume × density. Weight is the measure of the gravitational force acting on a mass.

Stiffness:

• Stiffness relates to how a component bends under load while still returning to its original shape once the load is removed.
• Since the component dimensions are unchanged after load is removed, stiffness is associated with elastic deformation. A material can have high strength and low stiffness.

Damping:

• Damping is a reduction in vibration over a period of time. Damping will eventually bring a vibrating object to rest.
• A damping mechanism suppresses vibrations of the structure.

In this section the governing equation for the idealized one-story frame is formulated based on an alternative viewpoint.

Under the action of external force p(t) the state of the system is described by

• Displacement u(t)
• Velocity `u^.`(t)
• Acceleration`u^(..)`(t) the system as the combination of three pure components:

• (1) Stiffness component: The frame without damping or mass
• (2) Damping component: The frame with its damping property but no stiffness or mass
• (3) Mass component: The roof mass without the stiffness or damping of the frame.

• The external force fs ;on the Stiffness component is related to the displacement u by Eq. fs = Ku

• If the system is linearly elastic, the External force fd on the Damping component is related to the velocity `u^.` by Eq. fd =`cu^.`

• External force fi on the mass component is related to the acceleration by fi = m`u^.`

• The external force P(t) applied to the complete system may therefore be visualized as distributed among the three components of the structure, and fs + fd + fi must equal the applied force P(t) leading to
• Eq. m`u^.`+`cu^.`+ku = P(t)

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

Period of vibration:

• Time require to complete one complete vibration cycle to structure is know the period of vibration.
• Formula for period

`T = (2π)/(√(k ÷ m))`

Frequency of vibration:

• An object's natural frequency is the frequency or rate that it vibrates naturally when disturbed.
• Objects can possess more than one natural frequency and we typically use harmonic oscillators as a tool for modeling the natural frequency of a particular object.

When calculating the natural frequency, we use the following formula:

`f = ω /(2π)`

We know that,

`ω = √(k / m)`

therefore,

`f = √(k /m) /( 2π)`

The relation between period and frequency is as follow:

`T = 1/f`

unit of period is second s.

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

• The typical earthquake ground motion response spectrum represents an envelope of the peak responses of many single-degree-of-freedom systems with different periods.
• The acceleration response spectrum of a ground motion is a relationship between the natural period of vibration of a single-degree-of-freedom system and the maximum absolute acceleration that it experiences under the ground motion.
• Similarly, a displacement response spectrum typically represents the peak displacement, relative to the ground, of many single-degree-of-freedom systems with different periods.
• Hence, the construction of a response spectrum involves the analysis of many different sdof systems.
• The value of each point on the spectrum is the peak response of a single degree of freedom system of a given period.

RESPONSE SPECTRUM

• A central concept in earthquake engineering, response spectrum is a practical means of characterizing ground motions and their effects on structures of different Time 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 practical approach to apply the knowledge of structural dynamics to the design of 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 and several such plots for different values of 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. A similar plot for peak ú, is the relative velocity response spectrum, and for peak üT, is the acceleration response spectrum.

• Pseudo acceleration for an SDF system with natural frequency ɷn is related to its peak deformation D = u0, due to earthquake ground motion:

• The quantity A has units of acceleration and is related to the peak value of base shear Vb0 (just another name for the peak value of the equivalent static force fs0):