Dynamic Analysis of Structures by 2003 한국전산구조공학회 봄 학술발표회 2003년 4월 12일 Dynamic Analysis of Structures by Superposition of Modified Lanczos Vectors Byoung-Wan Kim1), Hyung-Jo Jung2), Woon-Hak Kim3) and In-Won Lee4) 1) Ph.D. Candidate, Dept. of Civil and Environmental Eng., KAIST 2) Assist. Professor, Dept. of Civil and Environmental Eng., Sejong Univ. 3) Professor, Dept. of Civil Engineering, Hankyong National Univ. 4) Professor, Dept. of Civil and Environmental Eng., KAIST

Contents Introduction Proposed method Numerical examples Conclusions

Introduction Background Dynamic analysis of structures - Direct integration method - Vector superposition method Vector superposition method - Eigenvector superposition method - Ritz vector superposition method - Lanczos vector superposition method Eigenvalue analysis No eigenvalue analysis The Lanczos vector superposition method is very efficient.

Literature review Objective Nour-Omid(1984) first proposed. Nour-Omid(1995): Unsymmetric nonclassically damped system Chen(1990): Symmetric nonclassically damped system Ibrahimbegovic(1990), Mehai(1995): Dam-foundation system Drawback: The method is costly when multi-input-loaded structures are analyzed. Objective Improvement of the Lanczos vector superposition method to overcome the shortcoming.

Proposed method Conventional method Dynamic equation of motion of structures mass matrix(n  n) stiffness matrix(n  n) displacements vector(n  1) force vector(n  1) Rayleigh damping coefficients

Lanczos algorithm ith Lanczos vector

Reduced tridiagonal equation of motion premultiply

Single input loads spatial load distribution vector(n  1) time variation function(scalar) Multi-input loads spatial load distribution matrix(n  k) time variation function vector(k  1) the number of input loads

Proposed method Modified Lanczos algorithm  main idea conventional:

Reduced tridiagonal equation of motion premultiply Single input load Multi-input loads

Summary Single input loads - conventional - proposed Multi-input loads

Numerical examples Structures Normalized RMS error Simple span beam(Pan and Li, 2002) Multi-span continuous bridge(Park et al., 2002) Normalized RMS error results by the direct integration method time duration

Simple span beam Geometry and material properties

Loading configurations - Single input load(concentrated sinusoidal force) - Multi-input load(moving load)

Error (a) Single input load (b) Multi-input load

Computing time (a) Single input load (b) Multi-input load

Multi-span continuous bridge Geometry and material properties Dongjin bridge(PSC box girder type)

Loading configurations - Single input load(El Centro earthquake) - Multi-input load(moving load)

Error (a) Single input load (b) Multi-input load

Computing time (a) Single input load (b) Multi-input load

Conclusions The Lanczos and Ritz vector superposition methods have almost the same accuracy. For the single input loading case, the Lanczos and Ritz vector superposition methods have better accuracy than the eigenvector superposition and mode acceleration methods. For the multi-input loading case, the eigenvector superposition and mode acceleration methods have better accuracy than the Lanczos and Ritz vector superposition methods.

The Lanczos and Ritz vector superposition methods have better computing efficiency than the eigenvector superposition and mode acceleration methods. For the single input loading case, proposed and conventional Lanczos vector superposition methods have almost the same computing efficiency. For the multi-input loading case, proposed method has better computing efficiency than the conventional Lanczos vector superposition method.