By Singiresu S. Rao Ph.D. Case Western Reserve University Cleveland OH
With the revolution in on hand computing energy, the finite point strategy has turn into probably the most very important instruments for the trendy engineer. This booklet bargains a entire creation to the rules concerned
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Additional info for The Finite Element Method in Engineering, Fifth Edition
8 DERIVATION OF FINITE ELEMENT EQUATIONS USING A DIRECT APPROACH The element stiffness (or characteristic) matrices and load (characteristic) vectors and the finite element equations can be derived by using a direct approach. In this method, direct physical reasoning relevant to the problem (such as consideration of equilibrium of the system) is used to establish the element properties (characteristic matrices and vectors) in terms of pertinent variables. The direct approach is applicable only to problems involving simple types of elements; hence most practical (complex) problems cannot be solved using this approach.
1 ðeÞ ! 44) where the degree of freedom numbers corresponding to the rows and columns of the characteristic matrix ½K ðeÞ are also indicated. 16. Determine the displacements of nodes 2, 3, and 4 using the finite element method. State the assumptions made in your solution. 16 Springs in Combination. Node 4 P k5 U2 U3 U4 CHAPTER 1 Overview of Finite Element Method Solution Assumption: The vertical bars with nodes 2, 3, and 4 are rigid and massless and move horizontally. 1) U4 where the degrees of freedom corresponding to the different rows and columns are also indicated.
As stated in Eq. 2), the nodal temperatures can be determined by minimizing the functional I. The conditions for the minimum of I are given by ! 10) where I has been replaced by the sum of elemental contributions, IðeÞ. ðeÞ ! ! ∂IðeÞ ðeÞ ! 11) 2 2 ! ðeÞ where ½K = ∑e = 1 ½K ðeÞ is the assembled characteristic matrix, P = ∑e = 1 P is the assembled 8 9 ~ ~ < T1 = characteristic vector, and T is the assembled or overall nodal temperature vector = T2 . 11) : ; ~ T3 gives the governing matrix equations as !
The Finite Element Method in Engineering, Fifth Edition by Singiresu S. Rao Ph.D. Case Western Reserve University Cleveland OH