METHODS OF ANALYSIS

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METHODS OF ANALYSIS

• Stick Analysis (this class)
• Matrix Stiffness Analysis (504)
• Finite Element Analysis (506, 608)
• Nonlinear Pushover Analysis (504)
• Response Spectra Analysis (508)
• Nonlinear Time History Analysis (608)

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SHANGHAI WFC

• Les Robertson and Associates, New York
• Dynamic Response Spectrum Analyses
• Time History Analyses (6 records)
• Nonlinear Pushover Analyses

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LIMITATIONS OF STICK ANALYSIS

• Plane sections do not remain plane (outriggers
  not infinitely stiff)
• Loads and properties are distributed
  three-dimensionally
• Torsion may exist
• Nonlinearity may exist
• Dynamics may exist

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MATRIX STIFFNESS ANALYSIS

• Structures are collections of springs ku f
• u displacement vector
• f force vector
• k stiffness matrix

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12x12 STIFFNESS MATRIX
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SYSTEM STIFFNESS MATRIX

• KU F
• R rotation matrix from global coordinates to
  local coordinates
• M mapping matrix from system DOFs to member
  DOFs

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FINITE ELEMENT ANALYSIS

• Membrane element

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• Plate element
• Shell element membrane plate
• (20 DOFs)

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PUSHOVER ANALYSIS
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P-D EFFECT
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NONLINEAR ANALYSIS

• Material nonlinearity (yielding, fracture)
• Geometric nonlinearity (P-D effect, buckling)
• KUF becomes Z(U) F
• Z resistance vector, a nonlinear function of
  displacement vector U

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NEWTON-RAPHSON ITERATION
Linear Approximation

• Tangent Stiffness Matrix

Iteration
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RESPONSE SPECTRA ANALYSIS

• Mass (Inertia) Matrix M, Time-varying F(t)

Modal Eigenproblem
Eigenvector Orthogonality
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MODE SUPERPOSITION
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RESPONSE SPECTRA

• Seismic ground acceleration

Maximize yi over time for given record
Plot vs wi (or Ti)
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NONLINEAR TIME HISTORYANALYSIS

• Above equilibrium equation must be solved at each
  time step i
• Force F(t) given at each time step i
• Assume time steps equally spaced Dt

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EXPLICIT (MIDPOINT)
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IMPLICIT (TRAPEZOIDAL)
Explicit is unstable unless Dt is very
smallLinear Approximation
Tangent Stiffness and Damping Matrices
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ITERATIONj TIME STEPi

• Upon convergence

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