Page 59 - Pipeline Rules of Thumb Handbook

46 Pipeline Rules of Thumb Handbook

height of bend = 2.513ft = “V”. Take 8° as trial , tive solution. Builders will take advantage of sag bends where
T = 5ft, Incr. = 0.5°. Note Calc. V is close to given V. possible, depending on site conditions, space available, pipe
This is a possible solution. Note, however, this is an alterna- size, etc.

Calculate maximum bend on cold pipe

2
How much of a bend can cold straight pipe take without d D y 2 ZS A
= (9)
over stressing? In other words, in laying straight pipe cross- dl 2 3 EI
country, what is the smallest radius of curvature possible
within the allowable stress? Radius of curvature is given by the equation:
The following is an analytical solution to these questions.
32
2
È Ê D ˘
dy ˆ
Í 1 - Ë dl ¯ ˙
Î ˚
R = (10)
2
d D y
dl 2
The deflection equation for a length of straight pipe used
as a cantilever beam with a concentrated end load is: Substituting in Equation 10:
Pl 3 32
Dy = (1) È 2 ˘
3 EI Ê 2 lZS A ˆ
Í 1 + EI ¯ ˙
Î Ë 3 ˚
The maximum stress in the beam is: R = (11)
2 ZS A
M 3 EI
s = (2)
Z 1 82 3EI
limR = = (12)
The maximum bending moment is: 1Æ 0 2ZS A 2ZS A
3EI
MPl= (3)
Notation
If the allowable stress is S A , rewriting Equation 2 gives:
Dy deflection, in.
M l length, in.
S A = (4) P load, lb
Z
E modulus of elasticity, lb/sqin.
Pl I cross-sectional moment of inertia, in. 4
S A = (5) S Stress, lb/sqin.
Z
S A allowable stress, lb/sq in. from Code for Pressure Piping
Pl = ZS A (6) M bending moment, inch-pounds
Z section modulus, in. 3
Rewriting Equation 1: R Radius of curvature, in.

() 2 ZS l 2 Example. What is the smallest radius of curvature that can
Pl l
A
Dy = = (7) be used without overstressing straight 24-in. Schedule 20
3 EI 3 EI
seamless steel pipe ASTM 53? The pipe is at 70°F, and is to
Taking the first and second derivatives of Equation 7: be used for oil outside of refinery limits.
D
dy 21ZS A 3EI
= (8) R =
d1 3EI 2ZS A