Structural dynamics

Structural dynamics

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

Illustrative Diagrams:

Static Analysis

Dynamic Analysis

2. Explain the following with relevant Force-Displacement Graphs (40 Marks)

Elastic Behavior (10 Marks)

• Definition: Returns to original shape upon removal of load.

• Graph: Straight line from origin (Hooke’s Law)

• Equation: $F = kx$F=kx

Inelastic Behavior (10 Marks)

• Definition: Partial recovery after removing load, some deformation remains.

• Graph: Curve that does not return fully to origin.

Plastic Behavior (10 Marks)

• Definition: Permanent deformation even after load is removed.

• Graph: Sharp yield point followed by flat/softened line.

Non-linear Inelastic Behavior (10 Marks)

• Definition: Neither linearly proportional nor fully recoverable. Complex behavior under large deformations.

• Graph: Non-linear, hysteresis loop under cyclic loading.

Here's a proper set of Force-Displacement Graphs for each of the behaviors. These graphs help visualize how different materials or systems behave when subjected to external force.

1. Elastic Behavior

• Key Point: Fully recoverable deformation.

• Graph: Linear relationship (Hooke’s Law)

$$\text{Force (F)} \uparrow \quad \text{Displacement (x)} \rightarrow$$Force (F)↑Displacement (x)→

|
|         /
|        /
|       /
|______/
Explanation: The structureS returns to its original position after the load is removed. The slope of the line = stiffness 
$k = F/x$k=F/x.
2. Inelastic Behavior


Key Point: Partial recovery; some deformation remains.


Graph: Curved path with residual displacement after unloading.
|
|        /
|       /
|      /
|_____/_____
      \     
       \___
Explanation: After unloading, the material does not return completely to the origin; some permanent strain remains.
3. Plastic Behavior


Key Point: Beyond yield point, permanent deformation occurs.

Graph: Linear till yield, then flat or less steep
|
|        /
|       /
|      /
|_____/---------
Explanation: Initially elastic, but once the yield point is reached, the material flows plastically and doesn’t return to original shape even after unloading.
4. Non-Linear Inelastic Behavior


Key Point: Complex, non-linear, often cyclic behavior; energy is dissipated.


Graph: Hysteresis loop, often used in seismic or cyclic loading
|
|     /\
|    /  \
|   /    \
|__/      \__
Explanation: In repeated loading and unloading (e.g., during earthquakes), the structure exhibits non-linear behavior and dissipates energy, shown by the area of the loop.
Here are the digitally represented Force-Displacement Graphs for:


Elastic Behavior


Inelastic Behavior


Plastic Behavior


Non-linear Inelastic Behavior (Hysteresis)

These visuals clearly illustrate the deformation characteristics under loading and unloading conditions.

3. Mass, Stiffness & Damping in Equation of Motion (30 Marks)

Equation of Motion:

$[M]\{\ddot{u}(t)\} + [C]\{\dot{u}(t)\} + [K]\{u(t)\} = \{F(t)\}$[M]{u¨(t)}+[C]{u˙(t)}+[K]{u(t)}={F(t)}

4. Relationship Between Natural Period (Tn) & Natural Frequency (f) + Definitions (25 Marks)

Definitions:

• Natural Period (Tn): Time taken by a structure to complete one full vibration cycle.
  $T_n = \frac{2\pi}{\omega_n}$Tn​=ωn​2π​

• Natural Frequency (f): Number of cycles a system completes in one second.
  $f = \frac{1}{T_n} \quad \text{(Hz)}$f=Tn​1​(Hz)

Relationship:

$$f = \frac{1}{T_n} \quad \text{or} \quad T_n = \frac{1}{f}$$f=Tn​1​orTn​=f1​

Graph: Shows frequency vs period (inverse relationship)

5. Response Spectrum & Its Graph (25 Marks)

Definition:

• A Response Spectrum shows maximum response (like displacement, velocity, acceleration) of a single degree-of-freedom (SDOF) system to a base excitation (typically an earthquake) across a range of natural periods or frequencies.

Types:

• Displacement Spectrum

• Velocity Spectrum

• Acceleration Spectrum

Applications:

• Used in seismic design to determine the peak response of structures without time history analysis.

Graph:

• X-Axis: Natural Period (Tn)

• Y-Axis: Spectral Acceleration $Sa$Sa, Velocity $Sv$Sv, or Displacement $Sd$Sd