Structural dynamics

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

Static analysis can only be performed if the system being simulated does not depend on time and if the loads being applied are constant. In dynamic analysis, the system itself, the load application or both might change with time.

In dynamic analysis, the inertial loads developed by the system due to acceleration are taken into account. Static analysis do not consider inertia.

When loads or displacements are applied very slowly to a structure, inertia force which are equal to mass and acceleration are negligible and may be disregarded in the equation of force equilibrium. So, we can neglate this variation of load or displacement with time when the load is applied slowly. This corresponds to the static responce of the structure where force does not varry with time. This variation of force with respect to time is is neglated is assumed to be constant in static analysis.

By contrast the loads are applied quickly inertia force cannot be disregarded in the equilibrium equation and the structure responce dynamically to those exitations further more damping force may also be developed and should be considered in the equilibrium of motion equation.

2. Explain the following with relevant force-displacement graph.

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

Elastic behaviour:

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.

Inelastic behaviour:

Inelastic behavior is the property of the material when the material can't retain it's previous shape after a certain deformation. In the aforementioned graph inelastic behavior has been shown in the upper shape of the curve upto point "a".

Plastic behaviour:

Plastic behavior occurs when the material completely lost it's ductility. In the above mentioned graph plastic behavior occurs during unloading after point "a".

Non-linear inelastic behaviour:

Non-linear inelastic 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 the stiffness or damping of the frame. The external force fi on the mass component is related to the acceleration by fi = mű

mü + f_D + f_s = p(t) → mü + cὺ + ku = p(t)

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

Natural period of vibration (Tn):

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

T_n = 2π/ω_n or ω_n = 2π/T_n

Natural cyclic frequency (f):

The number of cycles (or one complete revolution) executed by system in 1 second (unit in hertz).

f_n = 1/T_n

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

• It is a central concept in earthquake engineering, responce spectrum is a parctical means of characterizing ground motions and their effects on structures of different T_n
• 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 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 T_n of the system, is called the response spectrum for that quantity.
• Each such plot is for SDF systems having a fixed damping ratio ᶗ and several such plots for different values of ᶗ are included to cover the range of damping values encountered in actual structures.
• The responce spectrum can be of 3 types, The deformation responce spectrum is a plot of peak u, against T_n for fixed ᶗ. A similar plot for peak ὺ, is the relative velocity response spectrum, and for peak ü^T, is the acceleration responce spectrum.