By Tai Hun Kwon
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Extra resources for Introduction to Finite Element Method
Bandwidth with an appropriate nodal numbering. f.. f. before proceeding the assembly procedure. f. f. f. ”. f. f. to an auxiliary memory device such as a hard disk (or a magnetic tape) from the core memory for the global stiffness matrix. One keeps on adding element contributions one by one until the assembly is completed. After the end of assembly, one has whole information for the upper triangular matrix in the auxiliary memory device that will be recovered from it to the core memory to be used one by one for the backward substitution to obtain the solution for the unknown nodal values.
Wext = −δV Therefore, δ (U + V ) = 0 Then Defining the total potential energy Π p as Π p = U +V © 2005 by T. 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.
Other methods are based on energy concept . 22) (or kinetic potential) where qi : generalized coordinate T : kinetic energy V : potential energy One can easily derive Lagrange’s equation via calculus of variation from Hamilton’s principle that is represented by the following (case B in the summary). 23) t2 minimize I = ∫ (T − V )dt . t1 Which implies that “actual path followed by a dynamic process is such as to make the integral of (T-V) a minimum”. Hamilton’s principle for deformable body can be stated as t2 t2 δ ⎡ ∫ (T − Π )dt ⎤ = δ ⎡ ∫ Ldt ⎤ = 0 ⎢⎣ t1 ⎥⎦ ⎢⎣ t1 ⎥⎦ with L = T − Π, © 2005 by T.