As shown in Figure 2.3, Gm and ao are the maximum and minimum longitudinal stresses
in the circumferential stress distribution, respectively. Equation 2.4 can be rewritten as in
Equation 2.5, by diving through by the material thickness. Stress, C, is replaced with
stress resultant c*.
( (1+ cos0) o (1 cos))
a= + (2.5)
2 2
Equation 2.6 describes the bending moment, M(x), as a product of the tensile forces,
F(O), and the distance, h(O), from the neutral axis of bending. Force is then replaced with
the product of stress and area. The area is defined by integrating the fabric thickness, t,
over the tube circumference using tube radius, r, and angle, 0, from 0427n.
M(x) = F(O)h(O) = (O)h(O)rt dO (2.6)
8=0 o
Defining height, h(O) = r cosO, and stress resultant, u (0) = c(O)t, Equation 2.6 can be
rewritten using symmetry as Equation 2.7.
M(x)= -2J *r2 cos 0dO (2.7)
0
Substituting Equation 2.5 into Equation 2.7 and integrating yields Equation 2.8.
Sx 2M((x)28)
&,m O 0 (2.8)
According to Main (1994), the unwrinkled curvature, pc, is given by Equation 2.9, where
E* is the resultant elastic modulus.
(2.9)
Pc 2rE*