Engineering task and calculation objective
The EJMR module calculates rectangular expansion joints to the EJMA Standards (8th Edition, 2003). Rectangular bellows are used wherever large, non-circular cross-sections have to absorb thermal expansion — typically in flue-gas and air ducts of power plants, at electrostatic precipitators, induced-draft fans and duct systems of exhaust trains.
Unlike a round bellows, a rectangular bellows carries its convolutions as long straight sides with corner regions; the long and the short side are therefore considered separately. The module converts axial, lateral and angular movements into equivalent axial movements per side, determines the meridional membrane and bending stresses and the circumferential bending stresses from pressure and deflection, forms the combined stress from these, and determines the allowable number of load cycles via the EJMA fatigue relation. In addition, the axial force, the deflection moments, the stiffness factor, the moment of inertia and the theoretical axial spring constant are reported.
Anyone who wants to calculate a rectangular expansion joint to EJMA thus obtains the complete stress and fatigue verification including the reaction forces and moments for the duct structural analysis.
Standard and calculation basis: EJMA Standards, 8th Edition 2003
Calculation workflow
- Enter geometry and loading: The duct dimensions (long and short side), convolution profile, wall thickness and number of convolutions describe the bellows; added to these are pressure, temperature and the movements to be absorbed: axial deflection, lateral deflections and rotations about both axes.
- Form equivalent axial movements: Lateral and angular movements are converted separately for the long and the short side into equivalent axial convolution movements and superimposed with the axial deflection to give the total movement of the expansion joint per side.
- Calculate pressure stresses: From the pressure, the meridional membrane and bending stresses of the straight sides and the circumferential pressure bending stress are obtained; the corner influence factors for the long and short side capture the supporting effect of the corners. The results are compared with the allowable stresses for operation and test.
- Form deflection stresses and combined stress: The circumferential bending stress due to deflection is calculated from the total movement and superimposed with the pressure stress components to give the combined stress; material manufacturing factors and the manufacturer's stress concentration factor enter here.
- Determine cycle life and reactions: From the combined stress, the allowable number of load cycles follows via the EJMA fatigue relation. In parallel, the axial force, the lateral and angular deflection moments and the theoretical axial spring constant are output for the design of the duct guides and anchor points.
Input quantities
| Quantity | Symbol | Unit |
|---|---|---|
| Design pressure | P | N/mm^2 |
| Design temperature | T | °C |
| Bellows material | Balges | - |
| Nom. design strength at 20°C | K20 | N/mm^2 |
| Nom. design strength at design temperature | K | N/mm^2 |
| Safety factor test | Ss | - |
| Safety factor operation | S | - |
| Reduction factor | c1 | mm |
| Corrosion allowance | c2 | mm |
| Modulus of elasticity at 20°C | Eb20 | N/mm^2 |
| Length of longest side | Ll | mm |
| Length of shortest side | Ls | mm |
| Bellows length | Lb | mm |
| Mean bellows diameter longest side | Lml | mm |
| Mean bellows diameter shortest side | Lms | mm |
| Inside radius of convolution | ri | mm |
| Convolution height | w | mm |
| Thickness of plate | tp | mm |
| Number of plies | np | - |
| Convolution pitch | q | mm |
| Number of convolutions | N | - |
| Outer distance of bellows | Lu | mm |
| Axial movement | X | mm |
| Lateral movement of longest side | yl | mm |
Frequently asked questions
Why are the long and the short side calculated separately?
In a rectangular bellows, each side acts like a straight convoluted strip supported at the corners. The bending load from pressure grows with the free side length, so the long side is usually governing; lateral and angular movements also act differently on the two pairs of sides depending on their direction. The EJMA method therefore keeps all stress and movement quantities separate for the long and short side and evaluates the more unfavourable case.
What role do the corner influence factors play?
The corners stiffen the adjacent side regions and change the moment distribution along the side length — to a different degree depending on the corner design (e.g. rounded or mitre-welded corners). The corner influence factors for the long and short side represent this supporting effect in the stress formulas; a wrong corner assumption directly shifts the calculated bending stresses.
Where do the material manufacturing factor and the stress concentration factor come from?
Both are manufacturer-specific quantities: the material manufacturing factor captures the influence of forming and fabrication on the material strength, the stress concentration factor the local stress peak in the convolution profile. They are established by the expansion joint manufacturer on the basis of its production and tests and must be confirmed by the manufacturer for reliable cycle numbers — standard assumptions provide guidance values only.
Does the calculation also apply to high pressures?
Rectangular expansion joints are inherently low-pressure components: the flat sides carry the pressure in bending rather than in membrane tension, so even low pressures produce large stresses. Typical applications are flue-gas and air ducts with pressures in the millibar range up to a few hundred millibar. For higher pressures, round expansion joints are the right choice.