Structural dynamics

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

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 (40 marks)

• 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.

3. Mass, Stiffness & Damping components in the equation of motion are as follows;

• The Single Degree of Freedom System as a combination of 3 pure independent systems such as the stiffness, the damping and the mass component.
• The 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. 
• The 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 = ců, where fd is damping force, c is the viscous damping coefficient and ů is the velocity.
• The 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

        Then, mű + fd + fs = p(t) --à mű + ců+ ku = p(t)

4. Natural period (Tn) & Natural frequency (f) are as follows;

• Natural period (Tn) – is the vibration of the system (Tn) is the time required for the undamped system to complete one cycle of free vibration (Unit is in seconds). It is related to the natural circular frequency of vibration, ωn, in units of radians per second. With, Tn = 2π/ωn or ωn = 2π/Tn
• Natural frequency (f) - number of cycle (or one complete revolution) executed by system in one second with units in hertz. Where, f=1/Tn
• Natural vibration properties ωn, Tn and f depends only on mass and stiffness of the structure.

5. Response spectrum;

• It’s the central concept in earthquake engineering, provides a convenient means to summarize the peak response of all possible linear SDF systems to a particular component of ground motion.
• It provides a practical approach to apply the knowledge of structural dynamics to the design structures and development of lateral force requirements in building codes.
• A plot of the peak value of a response quantity as a function of the natural vibration period Tn of the system, or a related parameter such as circular frequency ωn or cyclic frequency fn, is called the response spectrum for that quantity. 
• Each such plot is for Single Degree of Freedom systems having a fixed damping ratio Շ, and several such plots for different values of Շ are included to cover the range of damping values encountered in actual structures. 
• A variety of response spectra can be defines depending on the response quantity is plotted. Consider the following peak responses.
• • The deformation response spectrum is a plot of u_o against T_n for fixed Շ (damping ration). A similar plot for ů_o is the relative velocity response spectrum, and for ű^t_o is the acceleration 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.