Structural dynamics

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

Static Analysis: Static analysis of structures considers static loads acting on the structure such as dead,live load which does not vary with time and are not considered to be sudden. In static analysis stiffness matrix alone is considered and the inertial forces are not considered.Motion of the structure is not considered.

Dynamic Analysis: Dynamic analysis of structures considers dynamic loads acting on the structure such as wind,earthquake which varies with time. These loads are sudden and does not follow a constant pattern and vary with time. In dynamic analysis along with stiffness matrix, mass matrix and damping matrix is also included.Motion of the structure is considered.

So the basic difference between static and dynamic analysis is that the loads considered in static analysis does not vary with time whereas, the loads considered in dynamic analysis vary with time.

2. Explain the following with relevant force-displacement graph

• Elastic behavior
• Inelastic behavior
• Plastic behavior
• Non-linear Inelastic behavior

Elastic behaviour: Elastic behaviour can be defined as the ability of a component to return to its original shape when an application of external loading is removed. The explanation is that any component under loading like compression or tension loses its original shape and tends act according to the load and when the load is removed the component restores to its original shape when the loading is removed.For linear system force can be expressed as fs=ku where k is lateral stiffness constant.

Inelastic behaviour: Inealstic 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.

Plastic behaviour: 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 plastic behaviour of the structure occurs beyond the yield point of the structure and the deformation is irreversible even when the loading is removed. It occurs beyond point 'a' after unloading as represented in the curve.

Non-linear Inelastic behaviour: When large deformations occur at the material, 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 cyclic loading and unloading.

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

ANSWER-->

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 stiffness or damping of the frame.
• The external force fi on the mass component is related to the accelaration

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

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Natural Period Of Vibration (Tn)

• It is the time required for the undamped system to complete one cycle of free vibration
• Units in seconds
• It is realated to the natural circular frequency of vibration wn , in units of radians per second

Natural Cyclic Frequency (fn)

• The number of cycles (or one complete revolution) executed by system in 1 sec
• units in hertz

5. Explain in detail about Response spectrum & its Graph

Response spectrum: It is a concept of characterizing the ground motion and their effects on structures of different time period Tn. A plot 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 spectra. 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 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 as 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 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 reprsents 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.