Structural dynamics

1. Differentiate Static analysis and Dynamic analysis with proper representing pictures

Ans)

Static: Static load are the loads, the variation of load with respective time is very slowly

Dynamic: the dynamic loads are the loads, the variation of load with respective time is fairly quick in comparison to the structure's natural frequency.

Static	Dynamic
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 of the structure cannot be disregarded.
Static analysis doesn't consider inertia force along with static force.	Dynamic analysis considered inertia force along with dynamic force
The displacement obtained by static analysis is more.	The displacement obtained by dynamic analysis are less
In static force analysis, inertia force and inertia torque caused by acceleration are neglected	dynamic force analysis, inertia force and inertia torque caused by acceleration are not neglected.
The equation of equilibrium for a structure under static loading ie: f=ku, where u is the displacement.	The equation of equilibrium for a structure under dynamic loading ie: mu¨(t)+cu˙(t)+ku(t)=p(t). where, u¨ is the acceleration and u˙ is the velocity

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

• Under the cyclic deformation due to the earthquake, the intial loading curve is non - linear with the larger amplitude of formation.
• After the intial loading there are unloading and reloading curves
• At certain loading point in cylic loading it becomes nonlinear when until the point a
• There is unloading when load moves in negative direction and there is reloading then it does not follow initial unloading path. It will follow a new path.

Elastic behaviour

• Elasticity, ability of a deformed material when the body is return to its original position and size when the forces causes the deformation are it removed.
• For a linear elastic system, the relationship between the lateral force fs and resulting deformation u is linear that is, fs=ku;
• where k is the lateral stiffness of the system;
• its unit are force/length.
• In linear deformation k is constant
• This deformation would be linear at smaller displacement. But would become nonlinear at larger deformations.
• Elastic behaviour is a type property of the material can retain its shape. From the deformed graph it is clearly mentioned that a certain point from "O" the graph is linear and shown its elastic properties.

Inelastic behaviour

• The deformation of a material that does not disappear when the force were removed
• From the force-displacement graph, inelastic behaviour corresponds to the nonlinear path. Inelasticity is experienced by the body as soon as the path goes out of the linear curve.

Plastic Behaviour:

• For stresses beyond the elastic limit, a material exhibits plastic behaviour.
• This means the material deforms irreversible 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.
• Plastic behaviour occurs when the material completely lost its ductility. In the abovementioned graph plastic behaviour occurs during unloading after point "a".

Non-Linear inelastic behaviour:

• Force displacement relationship becomes non-linear and inelastic at large deformations
• Force fs corresponding to deformation is not single value and it depends on history of deformations and whether deformations is increasing or decreasing. Thus the resisting force can be expressed as: fs=fs(u,v).
• Non-linear inelastic behaviour occurs after the line from "O" tends to bend towards point "a".

3. Explain Mass, Stiffness & Damping components in the equation of motion.

Ans)

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

1. The stiffness component
2. The damping component
3. The mass component
4. The stiffness component:

The stiffness component:

• The stiffness component is used in earthquake engineering to designate how well a building will endure large lateral displacements and imposed by ground shaking.
• It measures how much force is required to displace a building by a certain amount of deformation
• The external force fs on the stiffness component is related to displacement u by fs=ku
• If the system is in early elastic

The damping Component:

• Damping is the kind of vibratory energy in solid mediums and structures over time and distance.
• Similar to the absorption of sound in air, damping occurs whenever there is any type of friction that diminishes movement and disperses the energy.
• The frame with its damping property but there is no stiffness or mass.
• The external force on the damping component is related to velocity by fd=cv.

The mass component

• The vertical irregularities in a structures are Stiffness irregularity, Mass irregular, Vertical geometric irregularity, Discontinuity in capacity.
• The centre of mass is a position defined as the average position of all the parts of the system, weighted according to their masses.
• The roof mass without the stiffness or damping of the frame.
• The external force f1 on the mass component is related to acceleration by fi=mu¨

mu¨+ fd + fs = p(t) that is mu¨ + cu˙ + ku = p(t)

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

Ans)

Natural period (Tn):-

• 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).
• 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):-

• The number of cycle (or one complete revolution) executed by system in one second with units in hertz. Where, fn=1/Tn
• 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 that is mass and stiffness.
• - ωn = sqrt(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.
• - Similar, the heavier of 2 structures (or) one with more mass and if both structures 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 and stiffness,
• - The natural vibration properties ωn, Tn and fn depends only on the mass and stiffness of the structure. That is k=mw2.
• - The stiffer of 2 single degree of freedom system having the same mass will have the shorter natural period (less time to comlete one cycle) and higher natural frequency.
• - Similarly, the heavier mass of two structures having the same stiffness will have the longer natural period and the lower natural frequency. That is ωn = sqrt(k/m).

5. Explain in detail about Response spectrum & its Graph

Ans)

• The response spectrum is characterizing ground motion and their effects on structures.
• 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.
• 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.
• 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 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 response spectrum can be of 2 types. The deformation response spectrum is a plot of peak u, against Tn for fixed Շ.
• A similar plot for peak u˙, is the relative velocity response spectrum, for the peak u¨, is the acceleration response spectrum.

Deformation Response Spectrum: -

• Here is the particular ground motion that is L-Centro earthquake.
• The response developed for L-Centro earthquake as shown.
• Now, the time variation of the deformation induced in the ground motion in 2 different single degree of freedom system is presented.
• Each single degree of freedom as its own period of vibrations that is Tn=1sec and Tn=2sec. And the damping ratio kept constant.
• Each of the single degree freedom system subjected this earthquake. That is the displacement history that we obtain.
• First Peak deformation we obtain is 2.67 inch for Tn=0.5sec and Շ = 2%
• Second Peak deformation we obtain is 5.97 inch for Tn=1sec and Շ = 2%
• Third Peak deformation we obtain is 7.47 inch for Tn=2sec and Շ = 2%
• In x-axis it shows the natural period of vibration and y-axis peak displacement.
• Now, to determine the peak deformation value, repeat the process again and compute displacement for natural period vibration.
• For a range of single degree of freedom system with different period of vibration keep the damping ratio as constant.
• For getting the peak period of vibration that is displacement from response spectrum curve. That deformation can be multiplied by stiffness to get equivalent static force.
• From there we can do simple structural analysis to obtain the bending moment and shear force in the structure.

Pseudo-Acceleration Response Spectrum: -

• Pseudo acceleration for an single degree od freedom system with naturak frequency ωn is related to its peak deformation D = uo, due to earthquake ground motion: A = (ωn^2) * D = ((2π/Tn)^2) * D.
• 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 = fso = mA (or) Vbo = (A/g)*ω.