# Spring Rate Calculation for Expansion Bellow

Expansion bellows are applicable to fixed tube sheet heat exchanger which help in reducing the longitudinal stresses or tube to tube sheet joint loads by allowing the axial displacement between shell and tubes. RCB-8 flexible Shell Elements (FSE) of TEMA standards deals with the calculation of spring rate and stresses induced in the FSE by using Finite Element Analysis method.

Historically engineers used to calculate the stresses and spring rate using the plate and beam theory nut due to limitations in the plate and beam theory Finite Element Analysis method is the most widely used method for calculating the spring rate and stress induced in the FSE.

Now let us take practical example of FSE for calculation of the spring rate –

Fig – Expansion Bellow

**Spring rate calculation as per TEMA RCB-8.5 will be carried out as per following steps –**

The flexible shell element (FSE) will be modeled as per section RCB-8.2 & RCB-8.3 i.e. Axisymmetric model will be created for Finite Element Analysis. Meshing is done in such a way that there much be at least 8 elements across the length of the flexible shell element.

- An axial load F
_{axial}as described in RCB-8.42 shall be applied at the smaller end of end FSE. F_{axial }will be calculated as per (p/4) x G^{2}x 100 lbf/in^{2}. We need to model dummy flange at the smaller end by considering the shell up to the minimum length of 2.5SQRT(RT) of the FSE to reduce the stress concentration factor in the stresses induced. - Finite Element Analysis will be performed and displacement in the axial direction (d) shall be noted for the applied force (F
_{axial}). - The spring rate of the axisymmetric FSE, K
_{AS}will be computed as the force applied (F_{axial}) / displacement induced (d). - The Spring rate of the entire FSE is K
_{FSE}= K_{AS}/2 - When only one FSE is present then the spring rate is given by K FSE above. When multiple FSE’s are present, the spring rate is given by K
_{E}= 1/ ((1/K_{FSE1}) + (1/K_{FSE2}) + …. (1/_{KFSEn})) Where K_{E}is the equivalent spring rate of the entire system.

Most important note – Spring calculation to be done for the nominal and minimum condition of thickness for the corroded and un-corroded condition.

**EXPANSION BELLOW SPRING STIFFNESS CALCULATION (NOMINAL UNCORRODED)**

Finite Element analysis based on the application of a unit axial load as per above steps is carried out to evaluate spring rate of the bellow.

Un-corroded condition –

SHELL ID 840 mm

F_{axial} 382091.5365 N

Figure A.1: Axisymmetric Geometry of Bellow (Un-Corroded)

Figure A.2: Meshed Model

Figure A.3: Loading & Boundary condition plot for Bellow (Un-Corroded)

Figure A.4: Axial Deflection plot (Un-Corroded)

**Spring Rate Stiffness Calculation –**

SHELL ID 840 mm

F _{axial} 382091.5365 N

Deflection 1.0585 mm (refer figure A.4)

K_{FEA }= 360974.5266 N/mm

**K _{FSE }= ¼ * KFEA = 90243.63166/mm ( for Nominal Un-Corroded )**

Similarly, Calculate spring rate for all remaining cases i.e. nominal corroded case, minimum un-corroded and minimum corroded case.

Nominal Un-corroded = 90243.63166 N/mm.

Nominal Corroded = 66455.6724 N/mm.

Minimum Un-corroded = 67929.79954 N/mm.

Minimum Corroded = 44986.70738 N/mm.

# Summary

We have successfully executed this method to provide expansion bellow reports justifying designs reviewed by third-party inspectors (TPI) and professional engineers. FEA tool can be used to address ASME code rules as per ASME Section VIII Div 2 Part 5. Finite Element Analysis is an excellent technique for performing design by analysis (DBA) and in some cases the only option to validate a design.

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