Notes on Stiffness and Stiffness Matrices

Introduction

• Topic: Stiffness and Stiffness Matrices
• Overview of forces, reactions, and displacement in a body
• Detailed lectures will be posted in the future

Definition of Stiffness

• Stiffness: Force or moment required to produce a unit deflection or rotation.
• Example with a loaded beam:
  • Loaded with force P (kN).
  • Deflection denoted as Δ.
• Deflection formula:
  • [ \Delta = \frac{P \cdot L}{A \cdot E} ]

Stiffness Calculation

• Stiffness (K):
  • Rearranging the deflection equation:
  • If ( \Delta = 1 ):
    • [ K = \frac{A \cdot E}{L} ]
  • This represents the force required to produce unit deflection.

Rotation and Moments

• Anti-clockwise rotation at point B:
  • Resultant deflection seen in the beam.
  • Moment causing rotation is given by:
    • [ M = \frac{4EI \cdot \theta_B}{L} ]
  • Reaction at A:
    • [ M_A = \frac{2EI \cdot \theta_B}{L} ]
  • Vertical reactions:
    • [ R_A = R_B = \frac{6E I \cdot \theta_B}{L^2} ]

Understanding Reaction Directions

• Direction of reactions at A and B depends on the applied moment at each point.
• Moment at B (anti-clockwise) causes upward reaction at A.
• Moment at A (clockwise) causes downward reaction at B.

Stiffness with Clockwise Moment

• Clockwise moment applied at B:
  • Corresponds to a clockwise rotation and reactions.
  • Moments:
    • At B: [ M = \frac{4EI \cdot \theta_B}{L} ]
    • At A: [ M_A = \frac{2EI \cdot \theta_B}{L} ]
    • Vertical reactions: [ R_A = R_B = \frac{6EI \cdot \theta_B}{L^2} ]

Stiffness with Rotation at Point A

• Clockwise rotation at A:
  • Resulting moments and reactions calculated similarly.
  • Moment at A: [ M = \frac{4EI \cdot \theta_A}{L} ]
  • Reaction at B: [ M_B = \frac{2EI \cdot \theta_A}{L} ]
  • Vertical reactions: [ R_A = R_B = \frac{6EI \cdot \theta_A}{L^2} ]

Unit Deflections

• Unit deflection at A (upward):

  • Force causing deflection: [ F = \frac{12EI \cdot \Delta}{L^3} ]
  • Moments: [ M_A = \frac{6EI \cdot \Delta}{L^2} ]

• Unit deflection at A (downward):

  • Similar calculations apply:
  • Force: [ F = \frac{12EI \cdot \Delta}{L^3} ]
  • Moments: [ M = \frac{6EI \cdot \Delta}{L^2} ]

Actual Deflection

• Applying an actual force at A:
  • Force magnitude: [ F = \frac{A \cdot E \cdot \Delta}{L} ]
  • Reaction will equal and oppose this force.

Sign Conventions

• Positive Forces:
  • Positive directions along axes:
    • X: right
    • Y: up
    • Z: towards you
• Positive Moments:
  • Anti-clockwise moments considered positive.
  • Use right-hand thumb rule for determining moment direction.

Conclusion

• Importance of stiffness matrices in problem-solving and computer applications discussed.
• Future lectures will cover generating stiffness matrices for beam elements and plane frame elements.
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