
172 Ballistics: Theory and Design of Guns and Ammunition
For an open-ended tube σ
1
= 0 and Equation 5.4 becomes
σσ σ
23
=− +
(5.5)
Recalling Lamé’s formulas for stress in a thick-walled tube
σ
θθ
==
+
−
()
σ
2
1
2
2
22
22
p
r
r
rr
rr
i
o
oi
i
,maximum at =
(5.6)
σ
rr i
o
oi
p
r
r
rr
rr
==
−
−
()
σ
3
1
2
2
22
22
,maximum at =
(5.7)
Let us put Lamé’s formulas into a more useful form by letting
=>
o
and
x
(5.8)
then
ζ
ζ
θθ
==
−
+
σ
2
2
2
1
1
p
i
x
2
x
2
(5.9)
ζ
ζζ
ζ
rr i
p==
−
−
σ
3
2
2
1
1
x
2
x
2
(5.10)
and
ζ
ζ
θθ
max max
at = ==
+
−
σ
2
2
2
1
r
(5.11)
rr
max max
3
(5.12)
Failure is considered to have occurred when the equivalent stress, σ