Structural dynamics

1.Difference between static and dynamin analysis:

Static Analysis	Dynamic Analysis
Load is not variable with time and neglected due to slow applying of load.	Load is variable with time due to fast applying of load and vibrations.
Motion of the structure can be disregarded.	Motion cannot be disregarded.
Static analysis does not considered inertia force along with static force.	Dynamic analysis considered inertia force along with dynamic force.
The displacement obtained by staic analysis is more.	The dispalcement obtained by dynamic analysis is less.
Static forced does not change in size, position or in direction.	Whereas in dynamic forces changes in either size, position or in direction.
Incase of building structures is static, the analysis is not sufficient for high ends.	Incase of building structures is dynamic, the analysis is sufficient for high ends.
In static force analysis, inerta force and inertia torque caused by acceleration are neglected.	In dynamic force analysis, inerta force and inertia torque caused by acceleration are not neglected.

1. 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.
2. The equation of equilbrium for a structure under the static loading is f = ku, where u is the displacement.
3. Unlike staic loading's which remain constant or act in such a manner so that there is no substantial variation in structural response with time, dynamic loading's vary substantially with time, thus the structural response needs to be calculated at every time instant.

The image is represented by static and dynamic analysis:

2.

i.Under cyclic deformation due to earthquake, the initial loading curve is non-linear with larger amplitute of formation.

ii.After initial loading there are umloading and reloading curves. Both differ from initial loading branch.

iii.We see that at certain point in cyclic loading it becomes non linear until the point (a).

iv.Then there is unloading then load moves in negative direction and then there is reloading then it does not follow initial unloading path. It will follow a new path.

v.This implies the fs corresponding anin built direction (lateral deformation) is not a single value. It depends on the history of deformation.

i.Force-dispalcement graph on Elastic behaviour:

i.Elasticity,ability of a deformed material body to return to its original shape and size when the forces causing the deformation are removed.

ii.For a linear elastic system the relationship between the lateral force fs and resulting deformation u is linear and i.e fs = ku, where k is the stiffness of the system; its unit are force/length.

iii.In linear deformation k is constant.

iv.This deformation would be linear at smaller dispalcement but it would become non linear at larger deformations.

v.Elastic behaviour is the property at the material upto which the material can retain its shape. From the above mentioned graph it is clearly mentioned that upto a certain point from 'O' the graph is linear and shown its elastic properties.

ii.Force-dispalcement graph on inelastic behaviour:

i.For extreme loadings, the structure is allowed to experience a limited amount of deformation beyond the elastic limit.

ii.This deformation is called ''inelastic'' since in contrast to elastic deformation, when the loading is removed, the strucuture does not return to its original position.

iii.Inelastic behavoiur is the property of the material when the material cannot retain its previous shape after a certain deformation. In the above mentioned graph inelastic behaviour has been shown in the upper shape of the curve upto point 'a'.

iii.Force-dispalcement graph on Plastic behaviour:

i.For stresses beyond the elastic limit, amaterial exhibits plastic behaviour.

ii.This means the material deforms irreversibly and does not return to its original shpe and size, even when the load is removed.

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

iv.Plastic behaviour occurs when the material completely lost its ductility. In the above mentioned graph plastic behaviour occurs during unloading after point 'O'.

iv.Force-dispalcement graph on Non-linear inelastic behaviour:

i.Force dispalcement relationshipbecomes non-linear and inelastic at large deformations.

ii.Force fs corresponding to deformation is not single valued and depends on history of deformations whether deformations is increasing or decreasing. Thus the resisting force can be expressed as fs=fs(u,v).

iii.Non-linear inelastic behaviour occurs after the line from 'O' tends to bend towards point 'a'.

3.The single degree of freedom system can be viewed as a combination of three pure independent systems.

i.Stiffness system

ii.Damping system

iii.Mass system

1.Stiffness:

Stiffness relates to how a component bends under load while still returning to its original shape once the load is removed. Since the component dimensions are unchanged after load is removed, stiffness is associated with elastic deformation. A material can have high strength and low stiffness.

Stiffness component:

i.The term stiffness is used in earthquake engineering to designate how well a building will endure large lateral displacements imposed by ground shaking.

ii.Stiffness is a measure of how much force is required to displace a building by a certain amount. The frame without damping or mass.

iii.The external force fs on the stiffness component is related to displacment u by fs=ku, if the system is linearly elastic

2.Damping:

Damping is a reduction in vibration over a period of time. Damping will eventually bring a vibration object to rest. A damping mechanism supresses vibratons of the structure.

Damping Coefficient:

i.Damping is the dissipation of vibratory energy in solid mediums and structures over time and distance.

ii.Similar to the absorption of sound in air, damping occurs whenever there is any type of friction that diminishes movement and disperses the energy.

iii.The frame with its damping property but no stiffness or mass.

iv.The external force on the damping component is related to velocity by fd = cv.

3.Mass:

i.Newton's second law, which states that the force F acting on 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.

ii.Mass is always constant for a body. One way to calculate mass; Mass= Volume x Density. Weight is the measure of the gravitational force acting on a mass.

Mass component:

i.The vertical irregularities in structures are stiffness irregularity, mass irregularity, vertical geometric irregularity, discotinuity in capacity A, centre of mass.

ii.The centre of mass is a position defined as the average position of all the parts of the system, weighed according to their masses.

iii.The roof mass without the stiffness or damping of the frame.

iv.The external force fi on the mass component is related to acceleration by fi=mu''

mu''+ fd + fs = p(t) i.e mu''+ cu' + ku = p(t)

Now, the external force p(t) is applied to the complete system. They may therfore visualized among the 3 components of the structure i.e inertial, stiffness and damping forces.

4.Natural Period (Tn):

i.The natural vibration of the system (Tn) is the time required for the undamped system to complete one cycle of free vibration (unit is seconds).

ii.It is related to the natural circular frequency of vibration, ωn, in units of radians per seconds.

Tn = 2Π/ωn or ωn = 2Π/Tn

Natural Frequency (f):

i.The number of cycle (or one complete revolution) executed by system in one second with units in hertz. where fn = 1/Tn

ii.Natural vibration properties ωn, Tn and f depends only on mass and stiffness of the structure.

- Tn,ωn, fn (Natural vibration properties) depends only on two quantities i.e mass and stiffness.

- ωn = √(k/m), the stiffer of single degree of freedom having the same mass will have higher natural frequency and the shorter natural period.

- Natural frequency and natural time period is inversely proportional to each other.

- Similarly, the heavier of two structures or one with more mass and if both structure have the same stiffness. The one with the heavier mass will have the lower frequency and lower frequency and longer natural period.

- ωn, n is natural (indicates to emphasize the natural properties of system and when it is allowed freely without any natural excitation).

- It is important to remember that earthquake last for few seconds. The ground motion subsides freely vibrates the structure based on the velocity, displacement impacted to it. Because, the system is linear these vibration properties are independent of displacement and velocity.

- These properties are related to its mass ans stiffness.

- The natural vibration properties ωn, Tn and fn depends only on the mass and stiffness of the structure i.e k=mw2.

- The stiffer of 2 single degree of freedom system having the same mass will have the shorter natural period (less time to complete one cycle) and higher natural frequency.

- Similarly the heavier the mass of two structures having the same stiffness will have the longer natural period and the lower natural frequency i.e ωn = √(k/m).

- Frequency and time periods is inversely proportional to the number of vibrations per second.

f=1/t or t=1/f

5.Response Spectrum:

i.A central concept in earthquake engineering, response spectrum is a practical means of characterizing ground motion and their effects on structure of different Tn.

ii.It provides a convenient means to summarize the peak response of all possible linear single degree of freedom systems to a particular component of ground motion.

iii.It also provides a practical approach to apply the knowledge of structural dynamics to the design structures and development of lateral force requirements in building codes.

iv.A plot of the peak value of a response quantity (u,v and a) 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 of that quantity.

v.Each such plot is for single degree of freedom systems having a fixed damping ratio G, and several such plots for different values of G are included to cover the range of dampung values encountered in actual structures.

vi.A variety of response spectra can be defines depending on the response quantity is plotted. Consider the following peak responses.

vii.The response spectrum can be of two types. The deformation response spectrum is a plot of peak u, against Tn for fixed G.

viii.A similar plot for peak u', is relative velocity reaponse spectrum, for the peak u'', is the acceleration response spectrum.

Deformation Response Spectrum:

i.Here is the particular ground motion i.e L-centro earthquake.

ii.The response developed for L-centro earthquake as shown.

iii.Now the time variation of the deformation induced in the ground motion in two different single degree of freedom is presented.

iv.Each single degree of freedom as its own period of vibrations i.e Tn = 1 sec and Tn = 2 sec and the damping ratio kept constant.

v.Each of the single degree of freedom system subjected this earthquake i.e the displacement history that we obtain.

i.First peak deformation we obtain is 2.67 inch fot Tn = 0.5sec and G =2%

ii.Second peak deformation we obtain is 5.97 inch fot Tn = 1sec and G =2%

iii.Third peak deformation we obtain is 7.47 inch fot Tn = 2sec and G =2%

iv.In X-axis it shows the natural period of vibration and Y-axis peak dispalcement.

v.Now, to determine the peak deformation value, repeat the process again and compute dispalcement for natural period vibration.

vi.For a range of single degree of freedom system with different period of vibration keep the damping ratio as constant.

vii.For getting the peak period of vibration i.e displacement from response spectrum curve. That deformation can be multiplied by stiffness to get equivalent static force.

viii.From there we can do simple structural analysis to obtain the bending moment and shear force in the structure.

Pseudo-Acceleration Response Spectrum:

i.Pseudo acceleration for an single degree of freedom system with natural frequency ωn is related to its peak defprmation D= uo, due to earthquake ground motion A=(ωn^2)*D = ((2Π/Tn)^2)*D.

ii.The quantity A has units of acceleration and is related to the peak value of base shear Vbo (just another name for the peak value of the equivalent static force fso) Vbo = fs = mA or Vbo = (A/g)*ω.

iii.A(t) is called pseudo acceleration and can be easily obtained by multiplying deformation at each instant of time with ωn^2 = ((2Π/Tn)^2).

iv.Most codes world wide are force based values related to peak acceleration are mentioned in the code and those values can be easily multiply by mass of the structure to obtain the equivalent static force.

v.When equivalent static force experssed in the form of deformation.

vi.The stiffness of the system as to be calculated and it can be leborious task.