Structural dynamics
1.) Structural analysis is mainly concerned with finding out the behavior of a physical structure when subjected to force. This action can be in the form of load due to the weight of things such as people, furniture, wind, snow, etc. or some other kind of excitation such as an earthquake, shaking of the ground due to a blast nearby, etc. In essence all these loads are dynamic, including the self-weight of the structure because at some point in time these loads were not there. The distinction is made between the dynamic and the static analysis on the basis of whether the applied action has enough acceleration in comparison to the structure's natural frequency. If a load is applied sufficiently slowly, the inertia forces (Newton's first law of motion) can be ignored and the analysis can be simplified as static analysis. Structural dynamics, therefore, is a type of structural analysis which covers the behavior of structures subjected to dynamic (actions having high acceleration) loading. Dynamic loads include people, wind, waves, traffic, earthquakes, and blasts. Any structure can be subjected to dynamic loading. Dynamic analysis can be used to find dynamic displacements, time history, and modal analysis.
A dynamic analysis is also related to the inertia forces developed by a structure when it is excited by means of dynamic loads applied suddenly (e.g., wind blasts, explosion, earthquake).A static load is one which varies very slowly. A dynamic load is one which changes with time fairly quickly in comparison to the structure's natural frequency. If it changes slowly, the structure's response may be determined with static analysis, but if it varies quickly (relative to the structure's ability to respond), the response must be determined with a dynamic analysis. Dynamic analysis for simple structures can be carried out manually, but for complex structures finite element analysis can be used to calculate the mode shapes and frequencies.
2.)
Elastic Behavior:-Elastic behavior is the property of the material up to which the material can retain it's shape. From the aforementioned graph it is clearly mentioned that up to a certain point from "O" the graph is linear and shown its elastic properties.
Inelastic Behavior:-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 up to point "a".
Plastic Behavior:-Plastic behavior occurs when the material completely lost it's ductility. In the aforementioned graph plastic behavior occurs during unloading after point "a".
Non-linear Inelastic Behavior:-Non-linear inelastic behavior occurs after the line from "O" tends to bend towards point "a".
3.) 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 the equation of fs=k u.
Damping component: Damping is the dissipation 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 f d=cu˙
Hence the equation of motion can be written as
mu¨+ f d + fs = p(t) that is mu¨ + cu˙ + k u = p(t) where p(t) is the external force.
4.)
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 T n = 2π/ωn and the unit is seconds.
Natural frequency: The natural frequency is defined as the number of cycles , 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 proportional to the period of vibration.
f n=1/Tn.
5.)
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 response 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 T n 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 = u o, 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 obtained 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 = f s o = mA (or) V b o = (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.