Structural dynamics

1Q: Differentiate Static analysis and Dynamic analysis with proper representing pictures

In structural dynamics, static analysis and dynamic analysis refer to two different approaches for analyzing the behavior of structures subjected to loads or external forces.

STATIC ANALYSIS: Static analysis in structural dynamics involves studying the response of a structure under static loads. It assumes that the load or external force is constant and does not change over time. Static analysis is often used to calculate the deformation, stresses, and strains in a structure and to determine its stability under static loads. It is particularly useful for analyzing structures that experience long-term loading or for verifying the design of structures before they are constructed.

DYNAMIC ANALYSIS: Dynamic analysis, on the other hand, involves studying the response of a structure under dynamic loads that vary with time. This includes analyzing the response of a structure to wind, earthquakes, vibrations, and other types of transient loads. Dynamic analysis takes into account the time-varying behavior of the load and the corresponding response of the structure. It is particularly useful for analyzing structures that experience short-term, high-intensity loads, such as earthquake or wind loads.

In summary, static analysis in structural dynamics focuses on the response of a structure under static loads, while dynamic analysis considers the response of a structure under dynamic loads that vary with time. Both approaches have their own strengths and weaknesses, and are often used together to provide a more comprehensive analysis of the behavior of structures.

2Q: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 refers to the ability of a material to return to its original shape after deformation when a force is applied and then removed. This behavior is shown on a force-displacement graph by a linear relationship between force and displacement, as shown in the graph below.
• Inelastic behavior refers to the behavior of a material when it does not return to its original shape after the applied force is removed. Inelastic behavior can be further classified into plastic and non-linear inelastic behavior.
• Plastic behavior occurs when the material undergoes permanent deformation after the applied force is removed. This behavior is shown on a force-displacement graph by a non-linear relationship between force and displacement, as shown in the graph below.
• Non-linear inelastic behavior occurs when the material undergoes permanent deformation and the relationship between force and displacement is not linear. This behavior is shown on a force-displacement graph by a non-linear relationship between force and displacement, as shown in the graph below.

In summary, elastic behavior is characterized by a linear relationship between force and displacement, while inelastic behavior is characterized by non-linear relationships between force and displacement. Plastic behavior is a type of inelastic behavior in which the material undergoes permanent deformation, while non-linear inelastic behavior is another type of inelastic behavior in which the relationship between force and displacement is not linear.

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

ANS:The single degree of freedom can be viewed as a combination of three pure indipendent system.

• Mass component: The roof mass without stiffness or damping of the frame. The external force fi on the mass component is releated to the acceleration by fi=mu
• Stiffness component: The face without damping or mass. the external forces fs on the stiffness is related to displacement u by fs=ku, if the system is linear elastic. if not the stifness constant varies with the displacement.
• Damping component: The frame with damping property but no mass and stiffness. The external force on damping component is related to velocity by fd=cv.

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

ANS:

Natural frequency (f) and natural period (Tn) are two important parameters used to describe the dynamic behavior of a mechanical or structural system.

Natural frequency (f) is the frequency at which a system vibrates when it is disturbed from its equilibrium position and then released to oscillate freely. It is a characteristic of the system and depends on the mass, stiffness, and damping of the system. Natural frequency is measured in Hertz (Hz).

Natural period (Tn) is the time required for a system to complete one full cycle of vibration when it is disturbed from its equilibrium position and then released to oscillate freely. It is the reciprocal of natural frequency and is also a characteristic of the system. Natural period is measured in seconds (s).

The relationship between natural frequency and natural period is given by the equation:

f = 1/Tn

This means that if the natural frequency of a system is known, its natural period can be calculated by taking the reciprocal of the natural frequency, and vice versa.

In mechanical and structural engineering, the natural frequency and natural period are important parameters used in the design of structures and machines that are subject to dynamic loads such as earthquakes, wind, and vibrations. The design must ensure that the natural frequency and natural period of the system are outside the range of the excitation frequency to avoid resonance, which can cause catastrophic failure.

5Q:Explain in detail about Response spectrum & its Graph.

• Acentral concept in earthquake engineering, response spectrom is a practical means of characterizing groung motions and their effects on structures of different Tn.
• It provides a convenient means to summarize the peak response of all possible linear systems to a particular component of ground motion.
• It also provides approach to apply the knowledge of structural dynamics to the design of structures and development of lateral force requirements in building codes.
• A plot of the peak value of a response quality(u,v,and a) as a function of the natural vibration period Tn of the system, is called resonse spectrum for that quality.
• Each such plot is for SDF system having a fixed damping ratio and several such plots for different values of damping ration are included to cover the range of damping values encountered in actual structures.
• The response spectrum can be 3 types. The deformation response spectrum is a plot of peak u, again Tn for fixed damping ratio. A similar plot for peak u, is the relative velocity response spectrum, and for peak Ut, is the acceleration response spectrum.