Engineering task and calculation objective
The AP26 module calculates unreinforced expansion joints with U-shaped bellows convolutions to ASME BPVC Section VIII, Division 1, Mandatory Appendix 26. Metal bellows compensate axial, lateral, and angular displacements between equipment and piping — for example the differential expansion between tube bundle and shell of a fixed-tubesheet heat exchanger — and at the same time must safely contain the internal pressure.
The procedure corresponds methodically to the EJMA standards: from the convolution geometry (inside diameter, convolution height, convolution pitch, mean convolution radius, number of plies, and ply thickness — corrected for thinning during forming), the pressure- and deflection-induced stresses in the circumferential, meridional, and membrane directions are calculated via the coefficients of Figures 26-4 through 26-6. Verified are the pressure capacity of bellows, end tangents, and collar, the stability against column instability and in-plane instability, and the fatigue life for the specified number of fatigue cycles.
Anyone who wants to calculate an expansion joint to ASME — as a manufacturer for design or as an operator for reassessment — obtains with this module the complete verification scope of Appendix 26, including the allowable number of cycles from the fatigue correlation for the selected bellows material and material condition (annealed or as-formed).
Standard and calculation basis: ASME BPVC VIII-1 Appendix 26, 2019 Edition
Calculation workflow
- Define geometry and loads: Inputs are the design pressure and temperature, the inside diameter of the convolutions and end tangents, convolution height, convolution pitch, number of convolutions, number of plies and the nominal thickness of one ply, as well as the displacement to be absorbed and the required number of fatigue cycles. The ply thickness is corrected for thinning during forming.
- Provide the material properties: The moduli of elasticity of the bellows and collar materials at design and room temperature, Poisson's ratio, allowable stresses, yield strength, and the material condition (as-formed or annealed) enter the stress, stability, and fatigue verifications.
- Verify the pressure stresses: For internal pressure, the circumferential membrane stresses of bellows, end tangent (with the stiffening factor for the end region), and collar as well as the meridional membrane and bending stresses of the convolution are calculated via the coefficients Cp, Cf, and Cd and compared with the allowable values.
- Check stability: The allowable pressure against column instability of the bellows as a whole (depending on bellows length and spring rate) and against in-plane instability (depending on the yield strength) is determined; the design pressure must remain below both limits.
- Deflection stresses and spring rate: From the total equivalent axial displacement range per convolution follow the meridional membrane and bending stresses due to deflection as well as the axial spring rate of the bellows, which acts as a restoring force on the connected components.
- Perform the fatigue verification: From the pressure- and deflection-induced stress components, the total stress range is formed, and via the fatigue correlation of Appendix 26 — where applicable with a fatigue strength reduction factor — the allowable number of cycles is calculated. It must reach the specified number of fatigue cycles, otherwise the convolution geometry is adjusted.
Input quantities
| Quantity | Symbol | Unit |
|---|---|---|
| Cross-sectional metal of one convolution | A | mm² |
| Coefficients for U-shaped convolutions, given by Figs. 26-4, 26-5, and 26-6 | Cp | - |
| Coefficients for U-shaped convolutions, given by Figs. 26-4, 26-5, and 26-6 | Cf | - |
| Coefficients for U-shaped convolutions, given by Figs. 26-4, 26-5, and 26-6 | Cd | - |
| Coefficients given by equations, used to determine coefficients Cp | C1 | - |
| Coefficients given by equations, used to determine coefficients Cp | C2 | - |
| Inside diameter of bellows convolution and end tangents | Db | mm |
| Mean diameter of collar | Dc | mm |
| Mean diameter of bellows convolution | Dm | mm |
| Modulus of elasticity of belIows material at design temperature | Eb | N/mm² |
| Modulus of elasticity of collars material at design temperature | Ec | N/mm² |
| Modulus of elasticity of bellows material at room temperature | Eo | N/mm² |
| Resultant total internal pressure force acting on the bellows and reinforcement | H | N |
| Bellows axial stiffness | Kb | N/mm |
| Forming method factor | Kf | - |
| Factor considering the stiffening effect of the attachment weld and the end convolution on the pressure capacity of the end tangent | k | - |
| BelIows collar length | Lc | mm |
| Effective shell length | Ls | mm |
| End tangent length | Lt | mm |
| Number of convolutions | N | - |
| Allowable number of fatigue cycles | Nalw | - |
| Specified number of fatigue cycles | Nspe | - |
| Number of plies | np | - |
| Design pressure | P | MPa |
Frequently asked questions
Why are bellows built with multiple plies?
Several thin plies achieve the same pressure capacity as one thick wall, but keep the bellows much softer: the bending stiffness grows with the cube of the individual ply thickness, so the spring rate stays low and the deflection-induced stresses small. In addition, multi-ply bellows offer redundancy against through-cracking of one ply. The price is more complex fabrication and more difficult inspection between the plies.
What distinguishes column instability from in-plane instability?
Column instability is the lateral buckling of the entire bellows like a compression strut — it occurs in long bellows with many convolutions and depends on spring rate and bellows length. In-plane instability is the tilting or shifting of individual convolutions within their plane at high pressure and is governed by the material's yield strength. Both failure modes limit the allowable pressure independently of the pure stress verification.
What influence does the material condition (as-formed vs. annealed) have on the verification?
Work-hardened (as-formed) bellows have a higher yield strength and thus higher allowable pressures against in-plane instability; annealed bellows are more ductile and often advantageous for low temperatures or corrosive media. Appendix 26 explicitly distinguishes the conditions in the instability and fatigue assessment — specifying the bellows material condition is therefore not a formality, it changes the results.
Why must an expansion joint never transmit a torsional moment?
The thin-walled bellows has practically no torsional stiffness; a twisting moment about the axis produces shear stresses that are not included in any standard design and quickly leads to instability of the convolutions. Piping systems with expansion joints must exclude torsion by design (guides, hinged tie rods). Equally important: the pressure thrust force (pressure times effective cross-section) must be taken by anchors or tie rods, never by the bellows itself.