Evaluation of Seismic Energy in Structures with Rigidend Offsets

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Evaluation of Seismic Energy in Structures with
Rigid-end Offsets

• By Kevin K. F. Wong

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Outline

• 1. Introduction
• 2. Force Analogy Method
• 3. Optimal Linear Control
• 4. Energy Balance
• 5. Rigid-end Offsets
• 6. Panel Zone Deformation
• 7. Numerical Simulation
• 8. Conclusion

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INTRODUCTION
Consider Earthquake from Energy Perspectives
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FORCE ANALOGY METHOD

• Uses a change in displacement instead of
  stiffness to characterize nonlinear
  force-displacement relationship
• Provides a simple way of separating strain energy
  and plastic energy

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FORCE ANALOGY METHOD

• Total Displacement Elastic Inelastic
• Total Moment Elastic Inelastic
• Uses 3 stiffness matrices to relate the force and
  displacement at any given time
• where n is the number of DOFs

(1) K Stiffness Matrix (n x n)
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FORCE ANALOGY METHOD
(2) K Relates Force at DOFs with
Rotation at Plastic Hinges (n x m)
A

• where m number of plastic hinges

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FORCE ANALOGY METHOD
(3) K Relates Moments and Rotation at
Plastic Hinges (m x m)
B

• Governing Equations

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OPTIMAL LINEAR CONTROL
Control
Earthquake

• Equation of Motion
• State Space Representation

Based on Force Analogy Method
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OPTIMAL LINEAR CONTROL

• Then
• Solving for the differential equation

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OPTIMAL LINEAR CONTROL

• Performing Numerical Integration
• Feedback Control
• where Q and R are weighting matrices.

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ENERGY BALANCE

• Equation of Motion
• Integrating over the path of response
• First Term

CE
DE
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ENERGY BALANCE

• Third Term
• Recall from Force Analogy Method

SE
KE
IE
PE
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ENERGY BALANCE

• Left Hand Side
• 1st Term Kinetic Energy
• 2nd Term Damping Energy
• 3rd Term Strain Energy
• 4th Term Plastic Energy
• 5th Term Control Energy
• Right Hand Side
• Input Energy

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ENERGY BALANCE

• Note the Plastic Energy Term
• Plastic Energy among each plastic hinge
• In Summary

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RIGID-END OFFSETS

• Structural Member with Rigid-ends and Two Plastic
  Hinge Locations (PHLs)
• Three-element Model with Added Degrees of Freedom
  Labeled

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RIGID-END OFFSETS

• Member Stiffness Matrices can be Partition into
  Submatrices Containing Original DOFs (ODOFs) and
  Additional DOFs (ADOFs)

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RIGID-END OFFSETS

• Static Condensation is Applied to Reduce DOFs 5,
  6, 7, and 8 so that the Member Once Again
  Contains n 4 and m 2
• In Summary,

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RIGID-END OFFSETS

• Modified Stiffness Matrices for Members

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PANEL ZONE DEFORMATION

• A scissors model
• Relative rotation between x4 and x5 simulates
  panel zone deformation.

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NUMERICAL SIMULATION
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NUMERICAL SIMULATION
Control Weighting Matrices
Earthquake
Spring Properties
kN-m/rad
kN-m
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NUMERICAL SIMULATION
No Rigid-End Offset
Rigid-End Offset
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NUMERICAL SIMULATION
No Panel Zone Deformation
Panel Zone Deformation
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NUMERICAL SIMULATION
No Control
Active Control
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CONCLUSION

• Structure with rigid-ends tends to increase the
  seismic demand in the beams and decrease the
  demand in the columns.
• Structure with nonlinear panel zone deformation
  tends to attract seismic demand from the beams
  and columns to the joints.
• Structure with active control absorbs seismic
  demand in the form of control energy and reduces
  the plastic energy dissipation in every part of
  the structure.