Elastic Bend Behaviour

This post is the second in the new ‘beyond the box’-category of application scope examples for Ple4Win. See the starting page for more information.

Elastic bend behaviour

Consider an elastic bend, fixed at one side (ident Start) and free at the other side (ident End), as shown below.

An elastic bend has been elastically bent as the name already says. This means that a constant bending moment

M = E · I · k / R is introduced in the entire bend, where:

E = Young’s modulus of the bend material,
I  = inertia moment of the bend cross-section,
k = stiffness reduction factor (<1),
R = bend radius.

This moment M is initially applied at all elements of the bend resulting in rotations of the element nodes made possible because of the free end. To be able to control the iteration process well, there is a maximum rotation allowed in each iteration (default 0.1 rad). Due to the node rotations the curvature of the elements κ and thus the radius R changes resulting in a reduction of the bending moments in the elements and in translations of the nodes as well. The result at the end of the iteration process is a straight pipeline section along the X-axis with zero bending moments. The displacements of the bend nodes after each iteration are shown in the animation below.

The following graph illustrates the decrease of the bending moments both at the fixed end (Mstart) and at the free end (Mend) to zero.