Structural dynamics

CHALLENGE 2:

Structural dynamics

QUESTION NUMBER 1:

Differentiate Static analysis and Dynamic analysis by properly representing pictures

ANSWER:

The nature of loading governs the kind of analysis to be performed on the structure. The loads are divided into two basic categories

• Static Loads are the loads, the variation of load with respective time is very slow (the acceleration of load is very very less than the natural frequency of the structure)
• Dynamic Loads are the loads, the variation of load with respective time is fairly quick in comparison to the structure's natural frequency.

Static Analysis	Dynamic Analysis
1. static analysis time-invariant analysis of forces like Stress, strain, displacement, etc.	1. Dynamic analysis is a time-dependent analysis
2. The results are a consequence of the applied constraints and loads, for example, bearings, gravity, forces, etc.	2. Inertia and damping forces are taken into account
3.Static loads applied on structure slowy	3.Dynamic Loads applied on the structures Suddenly
4. Example- Dead Load, Live load	4. Example- Earthquake load, Wind Load
5. Static analysis was done with a minimum of three degrees of freedom.	5. Dynamic analysis of structure done with a single degree of freedom
6. Static analysis is done where the structure deformations behave linearly.	6. Dynamic analysis is done where the structure behaves in non-linear deformations

In the dynamic analysis, we assume the mass is concentrated at a point with a single degree of freedom as seen in below picture,

QUESTION NUMBER 2:

Explain the following with a relevant force-displacement graph

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

ANSWER:

1. Under cyclic deformation due to earthquake, the initial loading curve is non-linear with a larger amplitude of formation.
2. After initial loading there are unloading and reloading curves. Both differ from the initial loading branch.
3. We see that at a certain point in cyclic loading, it become-linear until point (a).
4. Then there is unloading then the load moves in the negative direction and then there is reloading then it does not follow the initial unloading path. It will follow a new path.
5. This implies the fs corresponding to an inbuilt direction (lateral deformation) is not a single value. It depends on the history of deformation

Force-displacement graph of Elastic behavior:-

1. Elasticity is the ability of a deformed material body to return to its original shape and size when the forces causing the deformation are removed.
2. 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 is force/length.
3. In linear deformation k is constant.
4. This deformation would be linear at a smaller displacement. But would become nonlinear at larger deformations.
5. Elastic behavior is the property of the material up to which the material can retain its shape. From the aforementioned graph, it is clearly mentioned that up to a certain point from "O," the graph is linear and shows its elastic properties.

Force-displacement graph of Inelastic behavior:-

1. For extreme loadings, the structure is allowed to experience a limited amount of deformation beyond the elastic limit.
2. This deformation is called “inelastic” since in contrast to elastic deformation when the loading is removed, the structure does not return to its original position.
3. Inelastic behavior is the property of the material when the material can't retain its 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".

Force-displacement graph of Plastic behavior:-

1. For stresses beyond the elastic limit, a material exhibits plastic behavior.
2. This means the material deforms irreversibly and does not return to its original shape and size, even when the load is removed.
3. When stress is gradually increased beyond the elastic limit, the material undergoes plastic deformation.
4. Plastic behavior occurs when the material completely lost its ductility. In the above-mentioned graph, plastic behavior occurs during unloading after point "a".

Force-displacement graph of Non-linear Inelastic behavior:-

1. Force displacement relationship becomes non-linear and inelastic at large deformations
2. Force fs corresponding to deformation is no single-valued and depends on the history of deformations and whether deformation is increasing or decreasing. Thus the resisting force can be expressed as fs=fs(u,v).
3. Non-linear inelastic behavior occurs after the line from "O" tends to bend towards point"a".

QUESTION NUMBER 3:

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

ANSWER:

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

• The stiffness component
• The damping component
• The mass component

The stiffness component:-

• The term stiffness is used in earthquake engineering to designate how well a building will endure large lateral displacements imposed by ground shaking.
• Stiffness is a measure of how much force is required to displace a building by a certain amount. The frame is without damping or mass.
• The external force fs on the stiffness component is related to displacement u by fs=ku
• If the system is linearly elastic.

The Damping component:-

• Damping is the dissipation 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 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 structures are Stiffness irregularity, Mass irregularity, Vertical geometric irregularity, and Discontinuity in capacity. A. Centre of Mass.
• The center 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)

Now, the external force p(t) is applied to the complete system. They may therefore be visualized among the 3 components of the structure. That is inertial, stiffness, and damping force

QUESTION NUMBER 4:

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

ANSWER:

Natural frequency (f):

Natural frequency, also known as eigenfrequency, is the frequency at which a system tends to oscillate in the absence of any driving force. Also referred to as wn

Natural period (Tn):

The period of one complete oscillation of a body or system.

Relationship:

• Both refer to quantities which are mass and stiffness.
• Natural frequency is inversely proportional to the natural period.
• The following formulas are taken from the book, Dynamics of structure,
• 

QUESTION NUMBER 5:

Explain in detail about Response spectrum & its Graph

ANSWER:

1. A central concept in earthquake engineering, the response spectrum is a practical means of characterizing ground motion and its effects on structures of different Tn.
2. 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.
3. It also provides a practical approach to applying the knowledge of structural dynamics to the design structures and development of lateral force requirements in building codes.
4. 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.
5. 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.
6. A variety of response spectra can be defined depending on the response quantity plotted. Consider the following peak responses.
7. The response spectrum can be of 2 types. The deformation response spectrum is a plot of peak u, against Tn for fixed Շ.
8. A similar plot for peak u˙ is the relative velocity response spectrum, and for the peak u¨, is the acceleration response spectrum.

Deformation Response Spectrum:-

1. Here is the particular ground motion that is an L-Centro earthquake.
2. The response developed for the r L-centro earthquake is shown.
3. Now, the time variation of the deformation induced in the ground motion in 2 different single degrees of freedom systems is presented.
4. Every single degree of freedom has as its own period of vibrations that is Tn=1sec and Tn=2sec. And the damping ratio was kept constant.
5. Each of the single-degree freedom systems subjected to this earthquake. That is the displacement history that we obtain.

1. First Peak deformation we obtain is 2.67 inch for Tn=0.5sec and Շ = 2%
2. Second Peak deformation we obtain is 5.97 inch for Tn=1sec and Շ = 2%
3. Third Peak deformation we obtain is 7.47 inch for Tn=2sec and Շ = 2%
4. In x-axis it shows the natural period of vibration and y-axis peak displacement.
5. Now, to determine the peak deformation value, repeat the process again and compute displacement for natural period vibration.
6. For a range of single degree of freedom system with different period of vibration keep the damping ratio as constant.
7. For getting the peak period of vibration that is displacement from response spectrum curve. That deformation can be multiplied by stiffness to get an equivalent static force.
8. From there we can do a simple structural analysis to obtain the bending moment and shear force in the structure.

Pseudo-Acceleration Response Spectrum:-

• Pseudo acceleration for a 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 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.

• 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).

• Most codes worldwide are force-based values related to peak acceleration are mentioned in the code and those values can be easily multiplied by the mass of the structure to obtain the equivalent static force.
• When equivalent static force is expressed in the form of deformation.
• The stiffness of the system is to be calculated and it can be a laborious task.

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