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This report explores stiffness and flexibility derivation for rods and beams using dual integral equations, emphasizing rigid body and spurious modes. Dual boundary integral equations are applied to yield accurate results matching finite element method outcomes. Various mathematical techniques such as Fredholm theorem and SVD updating facilitate the analysis.
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Derivation of stiffness and flexibility for rods and beams by using dual integral equations 海洋大學河海工程學系 報 告 者:謝正昌 指導教授:陳正宗 特聘教授 日期:2006/04/01 中工論文競賽(土木工程組)
Outlines • Introduction • Dual boundary integral formulation for rod and beam problems • Discussion of the rigid body mode and spurious mode • Conclusions
Outlines • Introduction • Dual boundary integral formulation for rod and beam problems • Discussion of the rigid body mode and spurious mode • Conclusions
Introduction For undergraduate students, it is well-known in mechanics of material. stiffness For graduate students, they revisited it in the finite element course. flexibility Dual boundary integral equations were employed to derive the stiffness and flexibility of the rod and beam .
Influence matrix nonsingular Influence matrix singular Degenerate scale problem Fredholm theorem and SVD updating technique Rigid body mode Spurious mode
Outlines • Introduction • Dual boundary integral formulation for rod and beam problems • Discussion of the rigid body mode and spurious mode • Conclusions
Rod and beam problems Rod Beam Governing equation: Governing equation: Fundamental solution Fundamental solution
Boundary integral equations Rod Beam
Degenerate kernels Rod Beam
Influence matrices By approaching to and into the boundary integral equations Rod Beam
Translation matrix Rod Beam
The stiffness matrix of rods Stiffness matrix for rod problems using dual BEM
Singular value decomposition The matrix can be expressed as The denoted by satisfies the four Penrose conditions. The pseudo-inverse is identified as
The flexibility matrix of rods Flexibility matrix for rod problems using dual BEM
Outlines • Introduction • Dual boundary integral formulation for rod and beam problems • Discussion of the rigid body mode and spurious mode • Conclusions
Fundamental solution Rod Beam formulation
Influence matrix Rod Beam , , When When and and
Mathematical SVD structures of the influence matrices According to Fredholm alternative theorem
Spurious modes and the rigid body modes for a rod and a beam in BEM Rod Beam
Conclusion • Dual boundary integral equations were employed to derive the stiffness and flexibility of the rod and beam which match well with those of FEM. • Both direct and indirect methods were used. • The displacement-slope and displacement-moment formulations in the direct method can construct the stiffness matrix. • The single-double layer approach and single-triple layer approach work for the constructing of stiffness matrix in the indirect method.
Conclusion • The rigid body mode and spurious mode are imbedded in the right and left unitary vectors of the influence matrices through SVD.
The end Thank you for your kind attention!