Structural dynamics

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

Ans.

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.

- Displacement is static

- No velocity and acceleration

- Solution time taken less

- Output: stress and displacement

- One degree of fredom and how to rotate are neccessary to known

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.

- Displacement is function of time and frequency

- Velocity and acceleration developed due to variation in displacement

- Dmping force and inertia force due to velocity and acceleration

- Volution time taken more

- Output: stress, displacement, velocity and acceleration with respect to time and frequency

- 3 degree of fredom and how to rotate are neccessary to known

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)

1. Elastic behavior
2. Inelastic behavior
3. Plastic behavior
4. Non-linear Inelastic behavior

Ans:

1) 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 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.

- Elasticity is the ability of a body to resist any permanent change to it when stress is applied. When stress application ceases, the body regains its original shape and size.

- Different materials show different elastic behavior. The study of the elastic behavior of a material is of much importance.

- When it is linear elastic system the deformation will be less and when non linear system deformation is varying

2) Inelastic behavior

- 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.

- Deformation of a material that does not disappear on removal of the force that produced it.

3) Plastic behavior

-For stresses beyond the elastic limit, a material exhibits plastic behavior. This means the material deforms irreversibly and does not return to its original shape and size, even when the load is removed.

-When stress is gradually increased beyond the elastic limit, the material undergoes plastic deformation.

-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.

4) Non-linear Inelastic behavior

- Force displacement relationship become non linear and inelastic at large deformation

-Materials have elastic behavior usually only up to a certain load, which is called the 'yield point'. After that, the deformations are plastic.

- Rubber, for instance, has an elastic stress–strain curve, but the relation is not linear − in such cases nonlinear elastic models are required.

- 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 (30 marks)

Ans.

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

Mass component:

- The mass componenet in equation of motion considers only the mass of the frame and stiffness and damping are considered to neglected.

-Newton's second law, which states that the force F acting on a body is equal to the mass m of the body multiplied by the acceleration a of its Centre of mass, F = ma, is the basic equation of motion in classical mechanics.

-Mass is always constant for a body. One way to calculate mass: Mass = volume × density. Weight is the measure of the gravitational force acting on a mass.

- 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.

-Damping is a reduction in vibration over a period of time. Damping will eventually bring a vibrating object to rest.

- A damping mechanism suppresses vibrations of the structure.

- The force is related with stiffness componet with the equation of fs=ku.

Damping component:

- Damping is the dissipiation of energy due to vibrations over time and distances.Damping is a reduction in vibration over a period of time. Damping will eventually bring a vibrating object to rest.

-A damping mechanism suppresses vibrations of the structure. It occurs due to the friction between structural and non stuctural 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 the external force.

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

Ans.

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.

- The natural period of a building is simply a time taken by it to undergo one complete cycle of oscillation. The natural period is one of the properties or building which is controlled by mass and stiffness.

- The time period of a building is its natural period of oscillation when subjected to ground shaking in the event of an earthquake.

- The building is modelled to be under harmonic oscillation and the time period is calculated in a similar way as in harmonic oscillation

T = 2π√(k/m)

- where k is the total stiffness and m is the total mass.

- k is calculated as the sum of the lateral stiffness of all stiffness providing and moment resisting vertical members (columns, moment resisting frames, sheer walls, etc.) m is calculated as the sum of masses of all members on the Storey.

- [ Note: The above method is only applicable for single Storey buildings and for multiple Storey buildings, these are vector quantities depending on the mode of vibration. A detailed vector analysis (modal analysis) is done to calculated time periods for each vibration mode, which requires detailed explanation.]

- IS 1893 part I lists empirical formulae for approximate estimation of time period of multi-Storey buildings given in section 7.6 on page 24.

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 (25 marks)

Ans.

Response spectrum:

- The typical earthquake ground motion response spectrum represents an envelope of the peak responses of many single-degree-of-freedom (SDOF) systems with different periods Tn.

- The acceleration response spectrum of a ground motion is a relationship between the natural period of vibration of a sdof system and the maximum absolute acceleration that it experiences under the ground motion. Similarly, a displacement response spectrum typically represents the peak displacement, relative to the ground, of many sdof systems with different periods.

- It is also provide a practical approach to apply the knowledge of structural dynamics to the design of structures and development of lateral froces requirements in building code

- Hence, the construction of a response spectrum involves the analysis of many different sdof systems. The value of each point on the spectrum is the peak response of a single degree of freedom system of a given period.

Pesponse 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.