Structural dynamics

Question 1) Differentiate Static analysis and Dynamic analysis with proper representing pictures (30 marks)

Static analysis :

it is an analysis of a structure considering the static loads like dead and live ,which are does not vary with respect of time. in this analysis we did not consider sudden applied .

In static analysis momenet of structure and intertial forces will not be considered. we only considered stiffness matrix alone.

Dynamic Analysis:

this analysis is quite differ from strcuture analysis.this analysis considers dynamic loads (such as wind load ,seismic loads) acting on the structure which are varies with respect to the time.these loads are generated from the nature and it produces suddenly. which does not follow the symetrical pattren.this analysis include stiffness matrix,mass matrix,damping matrix.in this analysis we consider the motion of structure.

so.. the main difference is acting the loads with respect to time acting on the structure.

Question 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

Elastic behavior :

the elastic property defines the deformation occurs when the load is applied then after removal load it will be set right to its original shape or position .For linear system force can be expressed as

extrnal force = fs

displacement =u

lateral displacement = k which is constant and its unit are force/length.

• The deformation is linear at smaller displacement. But it will become non linear at larger deformations.

Inelastic behavior :

this is quiet oppoiste to the elastic natrure .this behavior is having incapable of regain its original shape or position after removal of load.is called inelastic behavior.

• For extreme loadings, the structure is allowed to experience a limited amount of deformation beyond the elastic limit.

Plastic behavior :

the irriversable natrure of the material having deformation even the load is also removed. if the stress is beyond its elastic limit, the material undergoes plastic deformation..

• the above graph is showing the it occurs beyond point 'a' after unloading as represented in the curve.

Non lenear inelastic behavior :

the large defrmations occurs at displacement and force shows the non-lenear and inelastic of the material, Force corresponding to displacement is not single valued in non linear behaviour and depends on the history of deformations of the material. For non linear system 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".

Question 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 of the component is related to the displcement u by fs = Ku, if the system is learly elastic.

Damping component :

the frame with its damping property but no stiffness or mass.the external force fs on the damping of the component is related to the 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 acceleration by fi=mu¨
• mu¨+ fd + fs = p(t) that is mu¨ + cu˙ + ku = p(t)

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

Natural period:

The time required for undamped system to complete one cycle of free vibration is known as the natural period of vibration and

• it is denoted by the formula Tn = 2π/ωn
• the unit is seconds.

Natural frequency:

The natural frequency is defined as the number of cycles are one full cycle of free vibration completed in one second. It is denoted in radians/sec or hertz.

The relation between and natural period and natural frequency is that frequency is inversely propotional to the period of vibration.

fn=1/Tn.

The natural vibration properties w, T, and f, depend only on the mass and stiffness of the structure

k=mw

The stiffer of two SDF systems having the same mass will have the shorter natural period (less time to complete one cycle) and the higher natural frequency Similarly, the heavier (more mass) of two structures having the same stiffness will have the longer natural period and the lower natural frequency.

`omega` = `root`k/m.

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

Response spectrum :

the method of finding motions of earth under the ground which effects the structure with in a different time periods is known as response spectrum Tn.

it provides the summurization of the peak response of the all lenear SDF to the ground motions.

it can be divide 3 types : Deformation response spectrum ,Relative velocity spectrum ,Pseudo acceleration response spectrum.

Pseudo acceleration response spectrum:

Pseudo acceleration for an single degree of freedom system with natural frequency ωn is related to its peak deformation D = uo,

due to earthquake ground motion: A = (ωn^2) * D = ((2π/Tn)^2) * D.

The D here represents relative deformation and not total deformation hence, the acceleration obatained is relative so, it is called as pseudo acceleration. The quantity A has units of acceleration and it is related with base shear Vbo. Vbo = fso = mA (or) Vbo = (A/g)*ω. In this formula A/g is base shear coefficient which is acceleration divided by gravitational force. So multiplying this with total mass of the structure provides total base shear or equivalent static force which can be used for analysis of the structure.

The above image shows the graphical difference between deformation spectrum and pseudo acceleration spectrum.

• Pseudo-acceleration would be calculated by multiplying deformation with natural circular frequency of vibration at any instant of time.