Structural dynamics

QUESTIONS

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

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

3. Explain Mass, Stiffness & Damping components in the equation of motion (30 marks)

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

5. Explain in detail about Response spectrum & its Graph (25 marks)

ANSWERS

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

ANS :

The difference b/w static and dynamics analysis of structure lies in the way they approach analyzing the behavior and performance of a structural system.

STATIC ANALYSIS

• Static analysis focuses on the equilibrium state of a structure under a specific set of loads, without considering the effects of time or motion.
• It determines the internal forces, stresses, and deformations that arise in the structural members due to the applied loads.
• This analysis is typically used in the preliminary design stage to ensure the structure's stability and adequacy under service conditions.

DYNAMIC ANALYSIS

• Dynamic analysis, on the other hand, considers the time-dependent behavior of a structure subjected to dynamic loads, such as earthquakes, wind, or impact.
• It takes into account the inertial and damping effects of the structure, as well as the changing nature of the applied loads over time.
• Dynamic analysis is used to evaluate the structure's response, including displacements, velocities, accelerations, and internal forces, under these dynamic conditions.
• This analysis is crucial for the design of structures that are subjected to significant dynamic loads, such as high-rise buildings, bridges, and industrial facilities.

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 -

Elastic behavior -

1. Elasticity, ability of a deformed material body to return to its original shape and size when the forces causing the deformations are removed
2. A body with this ability is said to behave elastically
3. The theory of elasticity is used to design safe and stable man-made structures such as skyscrapers and overbridges to make life convenient
4. From graph it is clearly mentioned that up to certain point from 'O' the graph is linear and shows elastic properties

Inelastic behavior -

1. Deformation material that does not disappear on the removal of force produced by it
2. Inelastic material will experience no elongation or deflection and up to its ultimate strength is reached will just rupture suddenly usually through shear break
3. Example: soil under loading and unloading, RCC at first cracking, wood, fabric etc.
4. In graph inelastic behavior has been shown in the upper shape of the curve up to point "a"

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 wrap is an example of plasticity
5. In graph the plastic behavior occurs during unloading after point "a"

Non-linear Inelastic behavior -

1. Materials have elastic behavior usually only up to a certain point after that deformation are plastic and the relation in this is not linear, that is nonlinear elastic behavior
2. Non-linear elastic behavior occurs after the line from "o" tends to bend towards point"

3. Explain Mass, Stiffness & Damping components in the equation of motion (30 marks)

ANS -

Mass component -

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

The force is related with mass component with the equation of f_i = m*d²u/dt²

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 component are neglected.

The force is related with stiffness component with the equation of f_s =k*u

Damping component -

Damping is the dissipation of energy due to vibrations over time and distance. It occurs due to the friction between structural and non structural element.

The force is related with damping component with the equation of f_d =c*du/dt

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

T_n = 2*pi/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 proportional to the period of vibration.

F_n =1/T_n

5. Explain in detail about Response spectrum & its Graph (25 marks)

ANS -

Response spectrum

Response spectrum analysis is a method to estimate the structural response to short, nondeterministic, transient dynamic events. Examples of such events are earthquakes and shocks. Since the exact time history of the load is not known, it is difficult to perform a time-dependent analysis. Due to the short length of the event, it cannot be considered as an stationary process, so a random response approach is not applicable either.

The response spectrum method is based on a special type of mode superposition. The idea is to provide an input that gives a limit to how much an eigenmode having a certain natural frequency and damping can be excited by an event of this type.

• A response spectrum is a function of frequency or period, showing the peak response of a simple harmonic oscillator that is subjected to a transient event. The response spectrum is a function of the natural frequency of the oscillator and of its damping.
• It is a central concept in earthquake engineering, response spectrum is a practical means of characterizing ground motions and their effects on structural of different T_n .
• It provides a convenient means to summarize the peak response of all possible linear SFD systems to a particular component of ground motion.
• It also provides a particular approach to apply the knowledge of structural dynamics to the design structures and development of lateral force requirement in building code.
• A plot of the peak value of a response quantity (u, v, a) as a function of the natural vibration period T_n of the system is called the response spectrum for that quantity.