Structural dynamics

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

Static Analysis

A static analysis can only be performed if the system being simulated does not depend on time, and if the loads being applied are constant.

Static analyses don’t consider inertia, but the elastic force should be accounted

In a static analysis, only the stiffness matrix of the FEA model is solved.

Static refers to the force that has constant size, position and direction on or within a structure

Dynamic Analysis

In a dynamic analysis, the system itself, the load application, or both might change with time.

In a dynamic analysis, the inertial loads developed by the system due to acceleration are taken into account.

In a dynamic analysis, in addition to the stiffness matrix, the mass matrix (and damping matrix, if not zero) is solved as well. That is one of the reasons why dynamic analyses require more computational time than static analyses, for the same structure.

For a dynamic analysis, the loads can be in either the time domain or in the frequency domain.

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

Elastic behavior

Elasticity is the property of a body to regain its original shape and size when deforming forces are removed. It exhibits an opposition to change.

Examples of elasticity are suspensions, steel, and rubber bands.

Below graph represents force displacement graph for elastic and inelastic system.

Inelastic behavior

The material response in relation to the stress-strain diagram that is non-linear and that retains a permanent strain or returns to an unstrained state on complete unloading.

Inelastic materials: brittle (material that cracks or fractures easily without much stretching, such as glass) and ductile (material that can be drawn, stretched, or compressed into a deformed shape without breaking, such as silly putty at room temperature).

Plastic behavior

The material deforms irreversibly 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 wrap is an example of plasticity

Non-linear Inelastic behavior

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 non-linear elastic behavior

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

Mass represents the inertia of a system, or its resistance to acceleration when a force is applied. It determines how much the system resists motion in response to external forces.

In the equation of motion, the mass is often written as m, and the term involving mass typically appears as mü(t), where:

• ü(t), is the acceleration (second derivative of displacement with respect to time).
• The force F applied to the mass leads to a proportional acceleration based on the mass.

Stiffness describes the resistance of a system to deformation when a force is applied. It defines how much force is needed to displace a system by a certain amount. In simpler terms, it determines the system's ability to return to its equilibrium position when displaced.

In the equation of motion, stiffness is typically represented by k, and the term involving stiffness often appears as ku(t), where:

• u(t) is the displacement of the system.
• The restoring force is proportional to the displacement, based on Hooke’s Law for linear systems.

Damping represents the resistance to motion due to energy dissipation, typically from friction or other resistive forces. it reduces the amplitude of oscillations over time, eventually leading to the cessation of motion in the absence of external forces.

In the equation of motion, damping is represented by c, and the term involving damping appears as cú(t) where:

• ú(t) is the velocity (first derivative of displacement with respect to time).
• The damping force is proportional to the velocity of the system.

For a single-degree-of-freedom (SDOF) system, the equation of motion often takes the form:

mü(t)+cú(t)+ku(t)=F(t)

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

Natural Frequency (f_n)

• The natural frequency of a system is the frequency at which the system tends to oscillate when disturbed from its equilibrium position, without any external force acting on it (after an initial displacement). It is a characteristic property of the system, dependent on its mass and stiffness.
• Unit: Hertz (Hz), which is cycles per second.

Natural Period (T_n​):

• The natural period is the time it takes for the system to complete one full oscillation (cycle) at its natural frequency. It is the inverse of the natural frequency.
• Unit: Seconds (s).

The relationship between natural period (T_n ​) and natural frequency (f_n​) is given by the following equation:

· T_n=1/f_n

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

The response spectrum is a graphical representation that describes the maximum response (such as displacement, velocity, or acceleration) of a set of single-degree-of-freedom (SDOF) oscillators, each with a different natural frequency, subjected to a particular seismic event (earthquake). It is used in earthquake engineering and structural analysis to predict the dynamic response of structures under transient dynamic loading, particularly during seismic events.

Purpose of Response Spectrum:

The response spectrum provides a convenient way to estimate the peak response of various systems without performing complex time-history analyses.

It helps engineers design structures to withstand seismic forces by showing how a structure might respond at different frequencies.

Types of Response Spectra:

Displacement Response Spectrum: Shows the maximum displacement of each oscillator.

Velocity Response Spectrum: Shows the maximum velocity.

Acceleration Response Spectrum: Shows the maximum acceleration.

Response Spectrum Graph:

The X-axis shows periods ranging from 0 (high-frequency, stiff systems) to several seconds (low-frequency, flexible systems).

The Y-axis shows the peak response (usually in acceleration).

The curve rises sharply at first, then levels off after reaching a peak around the period that corresponds to the system's most vulnerable natural frequency.