dy w Lx2
dx 2E*Ir3 2
w Lx3 x4
2E*zr3 6 12
With boundary conditions at x = 0, y = 0, and at x
and C2, are found to be Equations 2.17 and 2.18
C, =0
+ C )
(2.15)
(2.16)
+ C C 2)
L, y = 0, constants of integration, Ci
(2.17)
(2.18)
Substituting Equations 2.17 and 2.18 into Equation 2.16 yields the vertical deflection
solution shown as Equation 2.19.
w Lx3 X4
y2E 6 12
2E* r3 6 12
SL3
12
(2.19)
2.3.3 Wrinkle Length
Wrinkle length is the length of the tube at which wrinkling first occurs. This
length depends on the moment necessary to initiate wrinkling. Comer and Levy first
determine the moment necessary to initiate wrinkling by equating expressions for om
(Comer & Levy, 1963). Equation 2.20 describes a force balance between the internal
pressure acting over the area of the circular end-plate and the longitudinal stress
integrated over the circumference of the tube.
pr2 = 2 o-*r dO
(2.20)
Substituting Equation 2.5 into Equation 2.20 and integrating yields Equation 2.21, the
first expression for om.