Bellows Expansion Joint Analysis using FEA

FSE Drawing

This article explains the use of advanced FEA analysis for evaluating the structural behaviour of Economizer Heat Exchanger equipment and compares the designs obtained through different design codes.

This example is a heat exchanger with a bellow expansion joint and the reason of performing this study is because of a transition zone between two parts with different thickness, in order to assess the failure risk in that zone. On one hand, the heat exchanger design has been performed according to ASME VIII Div. 1 code using design by rules and shell thickness value obtained is 40 mm. On the other hand, the design of expansion joint has been performed according to Standards of the Expansion Joint Manufacturers Association (EJMA) and the minimum required thickness obtained is 30 mm. Is there a discrepancy? Keep reading if you are interested in finding what is going on. 

The SDEA advanced engineering team put their hands on the problem using design by analysis on the bellow design. 

Fixed Tubesheet Heat Exchanger

Flexible Shell Elements (FSE) are often used in fixed tubesheet heat exchanger in order to reduce shell and tube longitudinal stresses or tube-to-tubesheet joint loads. Ligth gauge bellows type expansion joints within the scope of the EJMA Standard are not included within the purview of this section.

The Tubular Exchanger Manufacturers Association (TEMA) Code, according to RCB-8 part, provided guidelines for determining stresses using a 2-dimensional Axisymmetric Finite Element Model (FEA) for the FSE or FSE combinations.

According to the problem described above, and considering the guidelines provided by TEMA for modelling FSE, the procedure used to obtain the answer to this problem is exposed in the following lines:

Boundary Conditions for FSE analysis RCB-8.42

At first, the CAD model (2D Axisymmetric geometry of FSE) is meshed considering eight node quadratic axisymmetric elements and the boundary conditions shown in figure RCB-8.42.  The axial translation of the FSE is restringed at the FSE axial plane, pressure of shell side is applied in the inner face of FSE and an axial displacement is applied for stress determination.

In order to evaluate the stress, it’s necessary to establish the minimum number of stress classification lines (SCL) and to compute linearized both membrane and membrane+bending stress intensities at each SCL in order to compare these stress values with the allowable stress limits defined in the design Code.

Stress Classification Lines for Evaluating Stresses RCB-8.62

The methodology for evaluating stresses is not specified in part RCB-8, so the code chosen for this purpose is ASME VIII Divisions 1 and 2.

In the context of bellows expansion joints design, Division 1 provides the allowable stress in bellows, according to part 6 of Mandatory Appendix 26 (Design of U-shaped unreinforced bellows):

  • S1 ≤ S
  • S2 ≤ S 
  • S3 + S4 ≤ KMS

where S1 is the circumferential membrane stress in bellows tangent; S2, circumferential membrane stress in bellows; S3 and S4 are meridional membrane and bending stresses, respectively, all of them obtained due to pressure. S is the allowable stress of bellows material at design temperature. Stresses in bellows due to deflection are only considered in fatigue analysis.

Stress in Bellows

Km is equal to 1.5·Ysm for as-formed bellows and 1.5 for annealed bellows, where Ysm is the yield strength multiplier which depends on the  material. The value of Km varies between 1.5 and 3.0Similarly, EJMA Standard collects the same stresses criteria in 4.13.1 (Design Equations for Unreinforced Bellows).
It is important to remark the fact that the combination of membrane plus bending stress on the convolution peak is compared against the allowable stress, scaled by a factor of 3 in this case. This is paramount and takes into account the hardening effect from the forming manufacture method, where the material goes well beyond yielding, resulting in hardening after the process.

Design Equations for Unreinforced Bellows [EJMA Standard]

On the other hand, according to Table 5.6 of ASME Section VIII Division 2, the next stresses classification should be used, depending on the location and the origin of the stresses:

  • Pm < S
  • PL < S 
  • PL + PB < 1.5S

where Pm is the maximum primary membrane stress; PL, local membrane stress and PB, primary bending stress.

So, in order to analyse the stresses in the different Stress Classification Lines of FSE, stresses classification provided by Division 2 and allowables stresses in bellows provided by Division 1 (and EJMA) are considered.

Stress Classification Lines for Stress Evaluation

Bellows SCLs Contours

Von Mises Stress Contours of FSE

2D-Axisymmetric FEA is performed for the FSE applying the boundary contidions described above and the stress values are evaluated in one of each SCL chosen for this model. According to the criteria exposed previously, table with SCL values is presented and the stress values obtained are below of the allowable ones:

SCL Results

Stress Classification Lines Results

This work was addressed in order to give an answer to the question if there is some discrepancy between design codes used for the design of the heat exchanger (ASME VII Div 1 Code used for calculating the minimum required thickness of the shell and EJMA Code used for mechanical design of bellow expansion joint) and to predict the structural behaviour in the transition zone between both. In this scope, a FEA method was used in order to analyse this problem. In view of the FEA analysis results , it can be reasuring to see the good behaviour of the system and the transition zone between the parts with different thickness: it is safe under the design conditions for different pressure and temperature parameters considered. It’s a good practice to combine Design by Rules with Design by Analysis, using FEA methods in order to validate the design by rules performed beforehand.

Pressure Vessels Design

Stationary Tubesheet Configurations

Pressure Vessels are used in the industry to store process fluids under certain temperature and pressure conditions. Commonly, pressure vessels have cylindrical or spherical shape, can be disposed in a horizontal or vertical way and are provided of heads with different shapes (Ellipsoidal, Hemispherical, Toroidal, etc.). These equipments are fitted with different elements such as nozzles (used to control the entry and/or exit of the fluids), cooling or heating jackets to maintain the temperature of the stored fluid, or other non-pressurized elements capable of supporting and transporting the equipment such as saddles, skirts, legs and lugs. In the SDEA_Engineering solutions team we accumulate ample experience in pressure vessels among other process engineering equipment design.

Design by Rules approach is a widely used method in several industrial sectors. This approach is based on a set of rules that specifies the design method, loads, allowable stress and materials requirements. The objective of design by rules is to determine the minimum thickness of the part considering design pressure, allowable materials and the specific formulas for the component geometry.

The most used and recognized code was proposed by the American Society of Mechanical Engineers (ASME), and is known as ASME Boiler and Pressure Vessel Code (BPVC). Other commonly used codes are EN-13445 (Europe), BS 5500 (UK), CODAP (France) and AD Merkblatt (Germany).

In particular, ASME Section VIII, Division 1 provides formulas for determining thickness and maximum allowable stress of basic components for Pressure Vessels. Equipment design according to this code does not require a detailed assessment of stresses.

The design method uses design pressure, allowable stress and specific formulas according to geometry of the component subject to design. ASME requires that pressure vessels are designed considering the most severe conditions subjected to the vessel, including the normal operation and other conditions such as start-up and shut-down. It’s necessary to have in mind several considerations for the adequate design. One of them is the design pressure, in order to determine it is necessary to know the Maximum Allowable Working Pressure (MAWP) of the component. That is, the maximum pressure permissible at the top of the vessel in its normal operating position at a specific temperature. Other consideration is the Design Temperature: the maximum and minimum design temperatures will determine the maximum (and minimum) allowable stress for the material to be used in the vessel. The minimum design temperature would be MDMT (Minimum Design Metal Temperature). Other important factor is the maximum allowable stress of the material which, as mentioned, depends on the temperature design and these values of stresses include certain safety margins. Also, it’s necessary to take into account an extra thickness in order to avoid the negative effects that may appear due to corrosion. All design considerations for the complete design of a pressure vessel are collected in paragraphs UG-16 to UG-35 of ASME VIII Division 1.

The designer must be familiar with the different failures that may occur and to take them into account in the design stage. These different failure modes are grouped in four categories:

  • Material: Can be due to improper material selection and / or material corrosion, among others.
  • Design: An incorrect input data or design procedure can lead to design failure
  • Fabrication: Due to gross fabrication errors or poor welding qualities
  • Service: Change in service condition or unexpected operations

The four previous categories describe “why” the vessel failure may occur. The failure types are: Elastic deformation, excessive plastic deformation, brittle fracture, stress rupture, plastic instability, high strain, stress corrosion and corrosion fatigue. More details about fatigue failure can be found here

In order to avoid equipment failure, the designer must take into account which loads will be involved in the process: steady loads applied continuously such as internal or external pressure, thermal loads and wind loads among others, are sometimes combined with Non-Steady loads, variable and short duration, like earthquakes, erection, transportation or start up – shut down. In particular, part UG-22 of the mentioned code collects all loadings to be considered in the design of a vessel.

On the other hand, several processes involve cyclic loads and thermal stresses, and other approaches more adequate for these cases are needed. Usually, instead of design by rules, design by analysis is used. This approach is collected by ASME in Section VIII Division 2. The combination of both approaches (Design by rules and Design by Analysis) results in a safer, more accurate and economically efficient design.

Thermal Stress Analysis Of Dissimilar Welding Joints using FEA

Weld Residual Stress Cut

Finite Element to foreseen residual stress from the welding process

Most welding process involve local heating, therefore temperature distribution is not uniform and structural thermal stresses beyond the material yield [strongly dependent on the temperature] show up, which can create issues on the final design related to fatigue or even geometry distortions.

Material Yield Temperature Graphics

This local temperature field is enough to create metallurgical and phase changes. As the pool solidifies and shrinks, it begins to exert stresses on the surrounding weld metal and heat affected zones. When it first solidifies and the weld metal is hot, yield is very low and exerts little stresses. As it cools to ambient temperature, this stresses in the weld area increase and eventually reach the yield point leading to a tensile stress state prone to giving problems during the equipment operational life.

As it cools to ambient temperature, this stresses in the weld area increase and eventually reach the yield point leading to a tensile stress state prone to giving problems during the equipment operational life.

Often pressure vessel suppliers are asked by their clients on how the residual stresses would influence the expected structural performance. Those residual stresses, that in tension may lead to high local stress in weld regions of low notch toughness and, as a result, may initiate brittle cracks that can be propagated by low overall stresses that are normally present. Also, notice that residual stresses can be reduced using thermal or mechanical relief methods, but not always post weld treatments are performed for different reasons.

Residual Stress and Temperature contours

SDEA engineering team used a Finite Element Analysis [FEA] approach to foresee the residual stresses from a dissimilar weld between the tubular plate and the shell [see figure 1]. SA-516 Gr 70 against SA-965 F304 will be welded with ER 309L Mo.

The welding procedure includes the geometry detail of each pass and a two-staged coupled simulation involving a thermal stage and a posterior mechanical step that use the temperature field as an input from the previous model. This can be done because changes in the mechanical state do not cause a change in the thermal state, but the opposite doesn’t apply; so that in this study, computation of the temperature history during welding and subsequent cooling is completed first and then this temperature field is applied to the mechanical model as a body force to perform the residual stress analysis.

Heat input during welding is modelled by a distributed heat flux applying on individual element based data provided by the client for each pass. The amount of heat input is obtained from the next expression, that depends on electrode speed, energy applied and a welding efficiency factor.

Principal stress and contours after weld is finished

Is important to highlight the fact that the reduction in yield for temperatures above 700-800 °C will govern the melting transition. Image above shows the expansion coefficient used for the both materials to be welded, notice the different thermal properties are the ones responsible for different heat diffusion and different thermal strain response.

As expected, the high temperature in each pass relaxes the stresses as yield drops with temperature. Weld metal melts and gets to flow state where stresses are relieved; this effect is clear and can be seen in the images around.

As expected tensile stresses appear on the weld surface. Next plot represents the Von Mises stress on the weld surface for the top and bottom side:

The model was also assessed for protection against ratcheting. Ratcheting is a behavior in which plastic deformation accumulates due to, in this case, thermal stress. To evaluate the model against ratcheting, as well as in the previous point from this document, an elastic-plastic analysis was used, according to the guidelines provided by “ASME BPVC.VIII.2-2017” Part 5, point 5.5.7.

For this analysis, same model from plastic collapse was used. In addition to its boundary conditions, three thermal cycles were calculated, as required in the normative, by setting the following temperature profiles on the interior faces:


  • Weld residual stresses were calculated using a non-linear FEA coupled thermo-mechanical analysis. As expected, tensile stresses appear at ambient temperature on the weld surface close to the material yield [in the range of 400 MPa].
  • Stresses on the model were calculated using a non-linear FEA coupled thermo-mechanical analysis under operating loads, achieving convergence according to “ASME BPVC.VIII.2-2017 Part5” and thus satisfying the elastic plastic protection criteria
  • Elastic-plastic ratcheting assessment was performed after three thermal cycles, according to “ASME BPVC.VIII.2-2017 Part5”. Two main dimensions on the shell were measured during the cycles, showing no permanent deformation due to the effect of the thermal loads. This condition satisfies the elastic plastic ratcheting assessment criteria as stipulated by the standard.

If you are interested in stress analysis please contact us.

Keep your curiosity in good shape.  

TLD Tuned Liquid Dampers

FEA for TLD Tuned Liquid Dampers

One of the greatest challenges when it comes to design steel stack structures such as chimneys, is to deal with oscillating loads as dynamic wind forces and seismic spectrum, as commented in this article. These loads threaten the design with coupling the excitation with its natural frequency, leading the structure to failure through resonance due to the high stresses produced by the oscillation amplitude.

Two main strategies can be followed to address this issue: either increasing the natural mode of the structure or damping its response to the excitation.

The objective of the first way (natural mode improvement) is to run away from the dangerous resonance frequencies. 

Figure 1 - Vortex induced oscillating loads on a cylindrical profile due to the wind action

This can be achieved by geometrical changes on the design such as plate thickness enhancement or a gain of chimney’s diameter, or by the addition of new structural elements as stiffeners or supports. Although this method is conceptually simple and does not require an extra designing effort other than adding some more material, is has other limitations. Usually, geometrical changes as wider diameter or the addition of new elements are restricted by other features of the whole project. 

Figure 2 - TLD pools coupled oscillation

Neighbour structures limiting the available volume or chimney’s exhaust properties are examples of these limitations. Other inconvenience for this solution, and the one who uses to have the last word, is the cost-efficiency. An increment on plate’s thickness involve a not negligible increment on the project’s expense moving it away further than the structure’s natural mode.

The other path to follow in order to guard the structure against oscillations is dramatically increasing the structure’s damping by the addition of a damping system. 

Even though there are different methods to achieve this, this article will be focused on the tuned liquid dampers (TLD).

The mission of the TLD is to dissipate the energy absorbed by the structure from oscillating loads, through a mass-spring-damper equivalent system that moves desynchronized with the chimney at its natural frequency. 

It is, in essence, a series of pools connected to the structure that moves against its pendulum movement. The phenomenon responsible of the energy mitigation effect inside the pools is called sloshing, and is defined as the movement of liquid inside another object. An example of sloshing familiar to almost everyone can be seen every time you walk with a soup plate. 

The human gait frequency (acting as the excitation) is close to the plate mode, so each step amplifies the soup’s wave motion, finally causing it to ruin your menu by splitting out of the rim. 

Figure 3 - Amplitude vs Frequency graph comparing a chimney with and without damper

On the other hand, if you are carrying a cup of coffee at a standard walking speed, despite it creates waves on the surface, the excitation frequency remains lower that the coffee’s natural mode, and as long as you don’t speed up, you breakfast will be safe.

Figure 4 - Plate & cup first oscillating mode shapes

The parameters that define geometrically a TLD, controlling its behaviour, are its interior [Rint] and exterior radius [Rext], the number of pools in which the TLD is divided, and the liquid height [H] that controls the liquid volume and thus the liquid mass. This liquid mass is divided in two distinguishable masses: the impulsive mass, the one that moves creating the sloshing phenomenon, and the convective mass, that has no relative movement with the TLD. Note that despite it has its influence on the sloshing frequency, the amount of convective mass also acts as dead load on the chimney, moving its natural frequency, so it has to be taken into consideration when tuning the assembly.

Figure 5 - TLD main dimensions and water mass distribution

The interior radius is used as the start point to calculate the previously mentioned variables [as it is more or less set by the chimneys geometry itself] along with the chimneys natural frequency and modal mass, acting as the focus result for the tune and as a reference for the total liquid mass respectively. Then, using the linear wave theory and the Graham & Rodríguez equations, the first group of parameters is obtained. Because of the influence of the TLD’s convective mass on the base chimney’s natural mode, an interpolation process has to be performed in order to obtain a fine tuning for the liquid damper.

Figure 6 - TLD sloshing mode shape at tuned frequency

To optimize this method’s accuracy and asses its results, SDEA’s advanced engineering team assembled a coupled FEA modal analysis plus harmonic response computational model. A fluid-structure interaction model along with a fluid free surface boundary condition is needed to capture the sloshing effect on the TLD’s pools. The next image shows the pools from a TLD where it can be appreciated that the liquid from all pools move in the same direction when they get coupled with the chimney’s response to external excitation.

Although this solution for the problem introduced in this article has a bigger technical load than enhancing the structure plate’s thickness, the cost-efficiency and adaptability of the damper installation compensate these cons by relieving the projects budget and providing certain design flexibility to overcome hypothetical installation issues.

FSI Steel Stack Structural FEA Analysis

Steel stack wind contours

Chimney miniature

When designing steel stack structures, several rules from standards as “ASME STS-1” or “ASCE 7” have to be taken into consideration to assure security during its useful life. This is particularly important for those located in areas subjected to heavy environmental loads, such as high activity seismic zone 4, due to the slender nature of this kind of structures thus low resonant primary modes prone to be easily excited by seismic or wind actions. In this article a brief outline about the designing process and considerations is provided based on the experience in FEA use gathered by the SDEA engineering team along different projects.

The main goal to achieve for the structure is to keep its design stresses under certain load combinations below a maximum allowable value. That is a common design strategy through many sectors; the difference here is the forces acting over the structure are somehow easy to misunderstand.

The first step is to determine the load combinations that the structure is going to be subjected to. Standards as “ASCE 7” or national regulations define the basic group of combinations to be considered during the assessment of the design, for example:

Stack structure load combinations

Load Combinations

The whole set of load combinations should be taken from the standards, from project’s characteristics or from customer’s requirements.

Once the load combinations are defined and the most relevant loads are identified, it’s time to calculate each of these values separately by using a finite element analysis (FEA) method over a detailed 3D model. Next, a summary of three of the most common loads shared for the most part of the projects is presented:

The first one is the dead load, and can be defined as the weight of all material incorporated into the structure, including its fixed permanent equipment such as ladders, platforms, etc. A constant when calculating stresses for different loads is that the worst situation is the one to be considered. Thus, in this case unlike the following ones, effects that reduce the total mass such as corrosion have not to be taken into consideration to not underestimate the stresses induced by the dead load expected in a real situation. A static structural FEA model can be used to precisely calculate the stresses induced by this load on every section of the stack.

The second common load is the wind load, or better said, are the wind loads. Wind load can be divided in two different loads: static and dynamic. The first one represents the action of the constant part of the wind pressure and is calculated using a static FEA model where a height dependant pressure load is applied to upwind and leeward surfaces. This pressure is obtained using standard’s equations, and usually depends on factors as the local wind speed, the geometry of the structure and the height of the point where the pressure is being applied.

The other wind load is the dynamic one, and it has to be considered due to the lightweight and flexible nature of steel stacks. It is defined as the load produced by the oscillations generated by the wind-structure interaction. “ASME STS-1” standard contains a set of rules to calculate the dynamic wind response. A more complex CFD fluid structure interaction model should be used in this case to assess the structure-vortex interaction in order to then evaluate the stresses produced. More details about the dynamic wind can be seen in this article.

Stack structure static wind profile

Static wind profile

Last but not least, we have the earthquake load, defined as the horizontal and vertical loads related to the response of the structure to seismic motions, and are calculated using the response spectrum method. In terms of the FEA model, a coupled modal analysis plus response spectrum model is required. The pseudo-spectral acceleration (Sa) used as input for the seismic response spectrum is calculated according to the following equation:

Spectral Acceleration Formula

Spectral acceleration

Where g is the gravity acceleration. R and U are factors that depend on the structure itself, being R the seismic force reduction factor that depends on the geometric characteristics of the structure and U the use factor, which defines the category of the building depending on its function and importance. The ground environment is defined by the factors Z and S. The seismic zone factor Z depends on the magnitude of the seismic forces presented on the region where the stack structure is going to be standing, and the ground factor S is a value used to define the geological conditions of the terrain.

Stack structure U.S. seismic zones map

U.S. Seismic Zones Map

The last factor C is the seismic amplification factor. It is the non-constant value that provides Sa with the frequency dependant behaviour. An example of a real pseudo-spectral acceleration is showed in the next graph:

Stack structure seismic pseudo spectral acceleration

Seismic pseudo spectral acceleration

Both the dynamic and earthquake loads have a paramount importance when designing steel stack structures due to its slender and flexible nature. Frequency excitations close to the structure’s natural mode of vibration can lead it to failure because of the appearance of resonance phenomenon. To mitigate this effect, cost-effective solutions as tuned liquid dampers (TLD, as explained in this article) are widely used.

Once every relevant load is calculated and combined according to appropriate normative, the maximum allowed values have to be obtained to assess the design, with the idea of improving it in case of not meeting the requirements, or optimizing it reduce the project expenses! These value’s calculation methodology is described in standards such as “ASME STS-1”, and are directly related to the structure’s parameters such as diameter, plate thickness, material properties, etc. Various values have to be obtained depending on the project conditions, being the most common maximum allowable values for longitudinal compression and the longitudinal compression and bending combination.