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This material is copyright - all rights reserved. Reproduced under licence from The Steel Construction Institute on 12/2/2007 To buy a hardcopy version of this document call 01344 872775 or go to http://shop.steelbiz.org/ Steel Designers' Manual - 6th Edition (2003) and ÈD nD [ D]= Í nD1 Í ÎÍ 0 D1 0 3 Et Dx = Dy = D1= 12 1 - 1 Dxy = ( - ) D1 2 1 n n Finally the strain energy can be written as 1 T Ub= Ú [ c] [ D][ c] dA 2 Denoting the generalized nodal displacement by the vector {d e— }, d e { }= [ C]{ A} Finite element method 323 (9.38) (9.39) where {A – } is a vector of polynomial constants. From Equations (9.36) and (9.39), - { c}= [ B]{ A}= [ B][ C] { d } (9.40) 1 e Using Equations (9.38) and (9.40), the following equation for Ub in terms of nodal point displacement parameters {d e— } is obtained: e UbC B D B A C (9.41) A Differentiation of U with respect to the nodal displacements yields the stiffness e matrix [ K ]: T 1 -1, T È T ˘ -1 e = { d } [ ] ÍÚ [ ] [ ][ ] d d 2 Î ˚ e -1, T È T ˘ -1 [ K ]= [ C] ÍÚ [ B] [ D][ B] dA˙[ C] ÎA ˚ Equation (9.42) can now be written as e , T [ K ]= [ C] [ Q][ C] where T [ ]= [ ] [ ][ ] Ú Q B D B dA A A 1 1 0 0 D xy 2 ( ) -1 -1 ˘ ˙ ˙ ˚˙ ˙[ ] { } (9.42) (9.43) (9.44) As mentioned before, the accuracy is dependent on choosing a large number of elements. Many refined elements giving greater accuracy are described in standard books on finite element methods, which also provide details of assembling the
This material is copyright - all rights reserved. Reproduced under licence from The Steel Construction Institute on 12/2/2007 To buy a hardcopy version of this document call 01344 872775 or go to http://shop.steelbiz.org/ Steel Designers' Manual - 6th Edition (2003) 324 Introduction to manual and computer analysis elements and analysing the structure. 5–7 These methods are used for solving a wide range of problems. References to Chapter 9 1. The Steel Construction Institute (SCI) (2001) Steelwork Design Guide to BS 5950: Part 1: 2000, Vol. 1: Section Properties Member Capacities (6th edn) SCI, Ascot, Berks. 2. Nethercot D.A., Salter P.R. & Malik A.S. (1989) Design of Members Subject to Combined Bending and Torsion. The Steel Construction Institute, Ascot, Berks. 3. Timoshenko S. (1976) Strength of Materials – Part 2, 3rd edn. Van Nostrand & Co., New York. 4. Argyris J.H. (1960) Energy Theorems and Structural Analysis. Butterworths, London. 5. Zienkiewicz O.C. & Cheung Y.K. (2000) The Finite Element Method in Structural and Continuum Mechanics, 5th edn. Butterworth-Heinemann, Oxford. 6. Coates R.C., Coutie M.G. & Kong F.K. (1988) Structural Analysis, 3rd edn. Chapman & Hall, London. 7. Nath B. (1974) Fundamentals of Finite Elements for Engineers. Athlone Press, London. Further reading for Chapter 9 Brown D.G. (1995) Modelling of Steel Structures for Computer Analysis. The Steel Construction Institute, Ascot, Berks.
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