Substituting Equation 2.8 into Equation 2.9 and approximating curvature as d y/dx2,
where x and y are coordinates, yields Equation 2.10.
1 M(x) d2y
= (2.10)
pc itrE* dx2
This leads to the final relationship, Equation 2.11, which corresponds to the traditional
beam bending relation with I* representing the resultant area moment of inertia. Note the
units of resultant area moment of inertia are cubic which corresponds to using the stress
resultant, c*, and resultant elastic modulus, E*.
d y M(x) I*
d2y M (x) ,where I = r' by association (2.11)
dx2 E I
This differential equation describes the bending behavior of an inflated circular tube. It
will be integrated to determine deflection, y, for a specific set of boundary conditions.
This theoretical deflection solution is then used to verify simulation deflection results in
Chapter 4.
2.3 Case Specific Non-wrinkled Inflatable Theory
2.3.1 Bending Moment
Figure 2.4 shows the assumed drag force loading, w, and model boundary
conditions, with L representing the tube length. The specific bending moment, M(x), for
this situation is derived from free body diagrams of a cut section of the beam as shown in
Figure 2.5. Shear, V, and moment, M are shown as reaction forces. Variable x is used as
a local coordinate along the tube length. Equations 2.12 and 2.13 show the sum of
moments about point A assuming static equilibrium.