67 Substituting the above expressions for
0" 1
and o'2 into Eq. '2.120 yields:
d2y I'h~ d2y rh2 Et ~x 2 ]o z l d A - E -~-fix2 Jo z 2d A - o or
E t $1 --
E S2 - 0
(2.123)
where: statical moments of the cross-sectional areas to the left and right of the axis n - n, respectively.
$1 and $2
In order to represent the pipe section as a rectangular cross-section, pipe wall thickness, t, is considered as height, h, and the unit length, 1, as base (Refer to Fig 2-12(c)). Using this notation, Eq. 2.12:3 reduces to:
(2.124)
E h~ - Et h~
As shown in Fig 2-12(c), h - hi + h2. Thus, the changes in cross-sectional areas (hi x 1) and (h2 x 1) from the neutral axis are given by:
h4- ,
hi = x / ~ 4- x/~t
('2.125)
and
(2.126) = 4-E +
The moment of inertia of the deformed sections can be given as"
11 = bh~ 3'
I2-
bh32 and I 3
bh3 12
Combining Eqs. 2.125 and 2.126 and substituting for the moment of inertia, one can define an additional parameter, Er (reduced modulus)"
EF
~-
4 E . Et (v/--E 4- v/Et) 2
(2.127)