Each particular combination of material, thickness, and convolution geometry has a different axial spring rate (per convolution) associated with it. Bending and lateral convolution spring rates can be computed from the axial spring rate.
The behavior of a bellows under load is described by the following equations:
F = fex
Where:
F = axial force in each convolution (also the axial force throughout the entire bellows), lb
f = axial stiffness per convolution, lb/in
N * Kax
N = number of convolutions in the joint
Kax = total expansion joint axial stiffness, lb/in
ex = axial displacement per convolution, in
= X/N
X = total axial displacement of joint, in
Mr = f Der / 4
Where:
Mr = bending moment in each convolution (also the bending moment supported by the entire bellows), in-lb
D = is the effective diameter of the joint (equal to the inside diameter plus the height of one convolution), in
er = axial displacement per convolution resulting from a rotation of the convolution,in
= (r x D)/(2N)
r = bending rotation of single convolution, radians
V = f D ey / (21)
Where:
V = shear force in each convolution (also the shear force supported by the entire bellows), lb
ey = axial displacement per convolution resulting from a lateral deflection of the convolution.
= 3 Dy / ( Nl)
y = total lateral displacement of the joint, in
l = length of the bellows, in
These expressions can easily be converted into stiffness and flexibility coefficients
Axial Stiffness: Kax = F/x
Bending Flexibility: Mr/r = (1/8) (Kax) (D^2)
Lateral Stiffness: V/y = (3/2) (D^2) (Kax) / (l^2)
These stiffness values are provided in most manufacturer's catalogs. In the event that the manufacturer only gives axial stiffness, the other two can be calculated once the effective diameter and length are known. (Note that torsional stiffnesses are not usually provided, since unprotected expansion joints are not designed to carry torsional loads and may fail catastrophically if inadvertently exposed to even moderate torsional moments.)
Note however that the bending flexibility coefficient should not be used in any piping program. The bending stiffness that should be used is exactly four times the bending flexibility.
This is because the so-called bending flexibility is calculated by applying a moment (Mr) to the free end of an expansion joint and observing its end rotation (θ). A computer model, however, expects a bending stiffness to be the ratio of the applied moment to the angular rotation at the end of an expansion joint that is fixed against translation — i.e., a representation of guided cantilever. This angular stiffness for a guided cantilever expansion joint model is calculated as:
Some pipe stress programs only offer "point", or zero-length expansion joint models. (In CAESAR II the user can define "finite length" or "point" expansion joints.) There is a difference in terms of how the two models are entered. As seen above, for finite length expansion joints, the lateral and bending stiffnesses are related by the equation:
Because of this exact relation, and since the length is known, the user can only enter one of these two values. CAESAR II computes the other value using this equation. For a "point" expansion joint, the length is unknown, so all three stiffnesses must be defined for the model
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