By Tai Hun Kwon

ISBN-10: 1584885297

ISBN-13: 9781584885290

**Read Online or Download Introduction to Finite Element Method PDF**

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**Extra info for Introduction to Finite Element Method**

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E In fact, this specific δy(x) limits the arbitrariness of δy in the variational principle so that the solution becomes just an approximation, not an exact one. © 2005 by T. H. Kwon 54 Assembly procedure is a matter of summation of scalar quantities yielding δΠ = ∑ δΠ e 2 e = δy i (K ij y j − Fi ) = 0 for any δy i which provide us with the following global matrix equation: [K ]{y} = {F } {F } being the work equivalent nodal force3. Consider specifically a linear element for simplicity. y e ( x) = N 1 ( x) y1e + N 2 ( x) y 2e N 1 (ξ) = 1 − ξ 1 N 2 (ξ) = ξ with ξ≡ ∴ 1 ξ xl called a normalized coordinate.

H. Kwon 62 yields δΠ p = 0 . 34) Ω ∂U o 1 and ε ij = (u i , j + u j ,i ) given. 33) is valid for any material, whereas Principle of Minimum Potential Energy is valid only for elastic materials. C. Principle of Complementary Virtual Work Consider a variation of statically admissible stress field σ ij and external forces f i , t i while keeping kinematically admissible displacement u i . δσ ij , j + δ (ρf i ) = 0 in Ω δσ ij n j = δt i on S (∂Ω) Define a complementary virtual work δW * as δW* dε Ω = ∫ u i (δσ ij )n j dS + ∫ u i δ(ρf i )dV ∂Ω dσ δWext δW * = ∫ u i δt i dS + ∫ u i δ(ρf i )dV ∂Ω σ Ω [ ε ] = ∫ u i , j δσ ij dV + ∫ δ σ ij , j + ρf i u i dV Ω Ω = ∫ ε ij δσ ij dV Ω ∴ δW * = ∫ u i δt i dS + ∫ u i δ(ρf i )dV = ∫ ε ij δσ ij dV ∂Ω © 2005 by T.

F. n For equilibrium Fi = f i A + f i B + f i C + f i D ⎛ NDOF ⎞ ⎛ NDOF ⎞ ⎛ NDOF ⎞ = ⎜ ∑ K inA x nA ⎟ + ⎜ ∑ K inB x nB ⎟ + L + ⎜ ∑ K inD x nD ⎟ ⎝ n ⎠ ⎝ n ⎠ ⎝ n ⎠ Using the nodal compatibility xiA = xiB = xiC = xiD = xi one can rearrange the equilibrium equation as ⎛ NDOF ⎞ ⎛ NDOF ⎞ ⎛ NDOF ⎞ Fi = ⎜ ∑ K inA x nA ⎟ + ⎜ ∑ K inB x nB ⎟ + L + ⎜ ∑ K inD x nD ⎟ ⎝ n ⎠ ⎝ n ⎠ ⎝ n ⎠ A B C D A B C = K ii + K ii + K ii + K ii xi + K ii′ + K ii′ + K ii′ + K iDi′ xi′ ( + (K + K + (K + 0 +( 0 + 0 +( 0 +K A ij B ij A ik B im + 0 +K C ik +K C il + 0 ) ( ) + 0 )x + (K + K + 0 + 0 )x + 0 )x + (K + 0 + K + 0 )x + K )x + ( 0 + 0 + K + K )x + K )x + ( 0 + K + 0 + K )x D il D im j A ij ′ k A ik ′ B ij ′ C il ′ l m j′ C ik ′ B im ′ k′ D il ′ D im ′ l′ m′ which becomes one of the equations in the global matrix equation [K ]{x} = {F } .

### Introduction to Finite Element Method by Tai Hun Kwon

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