Steel Designers Manual - TheBestFriend.org

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)
320 Introduction to manual and computer analysis
Fig. 9.31 Stress components on a plane element
components. For the state of plane stress, the stress components are related to the
strain components as given below:
(9.25)
where E and n are Young’s modulus and Poisson’s ratio respectively.
The matrix [D] is referred to as the elasticity or property matrix. The strain energy
of the element is given by Equation (9.22) as
(9.26)
Denoting the generalized nodal displacement by the vector {d e },
(9.27)
where {A} = [a1, a2, ...,an] T d
, which is a vector of polynomial constants, and [C] is a
transformation matrix.
The strains are obtained through making appropriate differentiations of the
displacement function with respect to the relevant co-ordinate variable x or y.
Thus,
e
È
s
1 n 0˘
Ï x ¸
e x
Ô Ô E
Í ˙ Ï ¸
Ìs
y ˝ = Ín
1 0˙
Ô Ô
e
2
Ì y ˝
Ô
t
Ô 1 - n Í 1 - n ˙
Ó xy ˛ Í0
0 Ô
g
Ô
xy
Î 2 ˙ Ó ˛
˚
\{ s}= [ D]{}
e
1 T
U = Ú Ú {} e [ D]{}
e d( vol)
2
{ }= [ C]{ A}
Ïe
Ô
{}= e Ìe
Ô
Óg
x
y
xy
Ï ∂u ∂x
¸
¸ Ô Ô
Ô Ô ∂v ∂y
Ô
˝ = Ì ˝
˛
Ô Ô
∂u
∂v
+ Ô
Ô∂
y ∂x
Ô
Ó ˛
(9.28)