Structural dynamics

Question 1) Differentiate State analysis and Dynamic analysis with proper representating 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 moment of stucture and intertial forces will not be considered.We only considered stiffness matrix alone.

Dynamic Analysis:

This analysis is quite differ from structure 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 pattern ,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.

1. 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

external 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 ono linear at larger deformations.

2.Inelastic behavior:

Inearstic behaviour of materials is defined as the inability of the material to return to its original shape when the load is removed and tend to be in the deformed shape.The material undergone deformation acts opposite to elastic behaviour hence,called as inelastic behaviour.

3.Plastic behavior

The property of the material which has the deformed shape and permanent strain in it when the loading is removed is called the plastic behaviour of structure.The plastric behaviour of the structure occures beyond the yield point of the structure and the deformation is irreversible even when the loading is tremoved.It occurs beyond point 'a' after unloading as represented in the curve.

4.Non-linear Inelastic behaviour:

When large determinartions occur at the matewrial, the force displacement relationship of the material becomes non linear.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).The bend from O to a represents the non linear.

The graph represents the behavourial of the above mentioned systems and the graph obtained is based on the cycle loading and unloading.

3.Explain mass,stiffness & Damping components in the equation of motion(30 marks)

The equation of motion in single degree of freedom system consists of three independent system components acting on the frame.

Mass component:

The mass component in equation of motion considers only the mass of the frame and stiffness and damping are considered to neglected.The force is related with mass component with the equation of fi=mu.

Stiffness component:

The rigidity of the material to resist deformation is called as the stiffness and in this component only stiffness is considered in the frame and other mass and damping components are neglected.The Force is related with stiffness component with equation of fs=KU.

Damping component:

Damping component is the disspiation of energy due to vibrations over time and distances.It occurs due to the friction between structural and non structural elements.The force is related with damping component with the equation of fd=cu.

Hence the equation of motion can be written as

mu¨+fd+fs=p(t)that is mu¨+cu+ku=P(t) Where p(t) is ther external force.

4.Provide relationship between Natural period(Tn) & Natural frequency (f) and provide their definations(25 marks).

Natural period:

The time taken for undamped system to complete one cycle of free vibration is called as the natural period of vibration and it is denoted by the formula Tn=2π/ωn and the unit is seconds.

Natural Frequency: The natural frequerncy is defined as the number of cycles i.e, one full cycle of free vibration completed in one second. It is denoted in radians/sec.

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

fn=1/tn.

5.Explain in detail about Response spectrum & its Graph (25 marks)

Response spectrum:

It is a concept of characterizing the ground motion and their effects on structures of different time period Tn .Aplot of the peak value of a response activity (u,v,a) as a function of natural vibration period Tn of the system is called the response sperctra.The peak response of all linear single degree of freedom systems can be computed by response spectra plot.Three types of plot are generally plotted with u and Tn as deformation response spectra and a similar plot is done with U* as relative velocity spectrum and with u** as acceleration response spectrum.Each plot is done with fixed damping ratio and several such plots with different damping ratio are created for the range of damping values encountered in the structures.

Deformation response spectrum:

The above image contains the plot with deformation with respect to time .The 1st plot (a) represents the ground acceleration recorded during a earthquake.The second plot (b) represents the plots of deformation respone of three independent SDF systems.In each plots of b has been done with fixed damping ratio nas 2% and variation in natural time period as 0.5s,1s,2s and the corresponding deformations are obtained with respect to time.Finally a plot of recorded dimensions and Tn is done to record the peak responses.So based on the time period of vibration the deformation of the system can be obtained which can be multiplied with stiffness to obtain equivalent static force which can be further used for the analysis of the structure.

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 griound 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 accelaration.The quantity A has units of accelaration and it is related with base shear Vbo.

Vbo=fso=mA(or)Vbo=(AS/g)*ω.

In this formula A/g is base shear coefficient which is accelaration 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 represents the graphical difference between deformation spectra and pseudo acceleration spectra.Pseudo acceleration can be calculated by multiplying deformation with natural circular frequency of vibration at any instant of time.