Structural dynamics
Differentiate Static analysis and Dynamic analysis with proper representing pictures
1.STATIC ANALYSIS
Static analysis can only be performed if the system being simulated does not depend on time, and if the loads being applied oare constant.
Static analysis don't consider inertia
static refers to the force that has constant size,position and direction on or with in a structure
There is no necessity to count inersa and damping forces in case of static analysis, but the elastic forces should be accou jjnjuigt6re4wseyxtfrcgvubopop[]ntable.
Static analysis of a structure considers static load acting on the structure such loads acting on the structure such as dead load, love load which does not vary with time and are not considered to be sudden. In static analysis stiffness matric is alone considered and the inersia forces are not considered.
Dynamic Analysis
Dynamic analysis of a structure considers dynamic loads acting on the structure such as wind load, earthquake which varies with time these loads are with time. These loads are sudden and does not follow a constant pattern are sudden and does not follow a constant pattern and vary with time. In dynamic analysis along with stiffness matrix, mass dynamic analysis along with stiffness matrix is also included. Motion of the structure is considered.
So the base different between static and dynamic and dynamic analysis does not vary with
time whereas, the loads considered in dynamic analysis vary with time.
2.
Elastic behavior
Inelastic behavior
plastic behavior
Non linear inelastic behavior
Elastic behavior: 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 shapes and tends to the load and when the load is removed the component restores to its original shape when the loading is removed the component restores to its shpe when the loading is removed. For linear system force can be expresses as fs=ku where k is lateral stffiness constant.
Inelastic behaviour:
Inelastic brhaviour of materials is defined as the inability of the material to return to its original shape when the load is remomed it tends to be in deformation shape. the mateial goes deformation hence it is called as inelastic behaviour
Plastic behaviour:
The property of the material which has the deformed shape and permanent strain in it when the load is removed it is called as plastic behaviour of structure' The plastic behaviour of the structure occurs beyond point a after unloading as represesented in the curve.
Non-linear inelastic behaviour behaviour: when large deformation occurs at the material, the force displacement relationship of the material becomes non linear. Forces corresponding to displacement is not single valued in non lenear behaviour and depends on the history of deformations of the material. For non linear system forces can be expressed as fs=fs(u,v). The bend from O to arepresents the non linear.
The graph represents the behaviourial of the above mentioned systems and the garph obtained is based on the cyclic loading and unloading
3. Explain Mass, Stiffness & Damping components in the equation of motion
The equation of motion in single degree of freedom system consists of three independent system components acting on the frame.
Mass component: The mass componenet in equation 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.
stifness component
The rigidity of the material to resist deformation is called as the component only stiffness is considered in the frame and other mass and damping components are neglected. The force is related with stifness component with the equation of fs=ks
Damping component:
Damping is the dissipiation of energy due to vibrations over time and distances. It occurs due to the friction between structual and non structural elements. The force is related with damping components with 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 the external forces.
4. Provide relationship between Natural period (Tn) & Natural frequency (f) and provide their definitions
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 frequency 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
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.