Structural dynamics

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

Answer:

Static Analysis

• Load is not variable with time and neglected due to slow applying of load.
• Static forces does not change in size, position or in direction.
• Incase of building structures is static, the analysis is not sufficient for high ends.
• Static analysis doesn't considered inertia force along with static force.

Dynamic Analysis

• The displacement obtained by dynamic analysis are less
• Dynamic forces changes in either size, position or in direction.
• Load is variable with time due to fast applying of load and vibrations.
• Motion cannot be disregarded.

The equation of equilibrium for a structure under static loading is f=ku, u is the displacement.

The equation of equilibrium for a structure under dynamic loading is mu¨(t)+cu˙(t)+ku(t)=p(t). u¨ is the acceleration and u˙ is the velocity.

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2. Explain the following with relevant force-displacement graph

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

Answer:

1. Elastic behavior:

• Elasticity, ability of a deformed material body to return to its original shape and size when the forces causing the deformation are removed.
• For a linear elastic system the relationship between the lateral force fs and resulting deformation u is linear that is, fs=ku; where k - lateral stiffness of the system, its unit are force/length.
• In linear deformation k is constant
• This deformation would be linear at smaller displacement. But would become non linear at larger deformations.
• Elastic behavior is the property of the material upto which the material can retain it's shape. From the aforementioned graph it is clearly mentioned that upto a certain point from "O" the graph is linear and shown its elastic properties.

2. Inelastic behavior:

• For extreme loadings, the structure is allowed to experience a limited amount of deformation beyond the elastic limit.
• This deformation is called “inelastic” since in contrast to elastic deformation, when the loading is removed, the structure does not return to its original position.
• Inelastic behavior is the property of the material when the material can't retain it's previous shape after a certain deformation. In the mentioned 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 behavior occurs when the material completely lost it's ductility. In the above mentioned graph plastic behavior occurs during unloading after point "a".

4. Non-linear Inelastic behavior:

• Force displacement relationship becomes non-linear and inelastic at large deformations
• Force fs corresponding to deformation is not single valued and depends on history of deformations and whether deformations is increasing or decreasing. Thus the resisting force can be expressed as fs=fs(u,v).
• Non-linear inelastic behavior occurs after the line from "O" tends to bend towards point"a".

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

Answer:

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.

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 u̇(t), and acceleration ü(t); the system as the combination of three pure components:(1) the stiffness component: the frame without damping or mass (2) the damping component: the frame with its damping property but no stiffness or mass and (3) the mass component: the roof mass without the stiffness or damping of the frame.

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

Answer:

Natural period:

• The natural vibration of the system (Tn) is the time required for the undamped system to complete one cycle of free vibration 
•  It is Related to the natural circular frequency of vivatarion ωn,unit of radiation/second

                                             Tn = 2π/ωn or ωn = 2π/Tn

Natural frequency (f):

• The number of cycle executed by system in one second with unit in hertz. where fn = 1/Tn
• Natural vivaration properties ωn, Tn  and F depend only one mass and stiffness of the structure.
•  Tn, ωn,Fn depend only on mass and stiffness of the structure
• ωn = sqrt(k/m), The stiffer of single degree of freedom having the same mass will have higher natural frequency and the shorter natural period.
• Natural frequency and natural time period is inversely proportional to each other .

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5. Explain in detail about Response spectrum & its Graph 

Answer:

• It is a central concept in earthquake engineering, response spectrum is a practical means of characterising 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 
• It also provides a particular approach to apply the knowledge of structural dynamics to the design structures
• 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
• 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.

Pseudo-Acceleration Response Spectrum:-

• It is for an single degree of freedom system with natural frequancy ωn 

                                A = (ωn^2) * D = ((2π/Tn)^2) * D.

• The quantity a has unit of acceleration and is related to the peak value of base shear vbo

                                 Vbo = fso = mA (or)

                                  Vbo = (A/g)*ω.

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