December 2025

Valves, Pumps, Turbomachinery and Compressors

The use of Belleville hold-down supports in piping to overcome vibration problems

IP: 10.1.103.123

Dynamic loading due to pulsation is a critical consideration in piping systems connected to reciprocating machinery, such as plunger pumps and reciprocating compressors. The Belleville hold-down support is an effective pulsation support that provides three-directional restraints, all with limited resistance but required stiffness. This article explores the concept of Belleville hold-down supports and explains the Belleville spring curve and stacking principle. Also, a case study is explained, demonstrating how the stiffness and preload of a Belleville hold-down support can be calculated. At the end of this article, the modeling approaches for Belleville hold-down supports in a proprietary pipe system safety software^a are also discussed to achieve more accurate results while performing the modal analysis to determine the piping system's natural frequencies.  

Working with a reciprocating compressor or pump also means dealing with potential vibration in the piping. With this scenario, the piping system must be designed somewhat differently than normal practices. One such practice is to have a secure clamping effort. The connection between the pipe and the support structure is very critical to the effectiveness of the support. A good connection starts with good clamps, which are the first link between the pipe and its supports.¹ 

Without a good connection, a purposely designed heavy support structure is just a waste of resources. FIG. 1 shows some of the clamps used in vibration piping. The clamp must be stiffened, as shown in FIGS. 1a and 1b. In FIG. 1a, the clamp is also aligned with the belting material to offer some damping effect in addition to securing a good connection. 

FIG. 1.Hold-down clamp for piping to combat potential vibration.¹

FIG. 1b shows two squeezing wedges to ensure a snug fit between the pipe and clamp. This is often used in large piping. FIG. 1c shows hold-down beams with controlled hold-down force provided by coil springs or Belleville springs (washers). The spring-loaded, hold-down provides three-directional restraints, all with limited resistance but high stiffness. The downward resistance is significant, but upward resistance is limited to the initial force of the spring. The lateral resistance is the friction from the weight of the piping and the initial spring force.¹ 

The importance of accurate modal analysis and pipe support stiffness. Modal analysis evaluates the natural frequencies and mode shapes of piping systems.² It is essential to avoid resonance conditions when connected to pulsation systems like reciprocating compressors and pumps. For example, if pulsation from a pump produces a dominant frequency near 45 Hz, the piping system must be analyzed for natural frequencies either below 36 Hz or above the 54 Hz range (± 20% separation margin as per API 674).³ To match this frequency requirement for pipe supports with actual site conditions in a piping stress analysis software like the aforementioned proprietary software^a, it is vital to input the actual pipe support stiffness values behavior into the software. The natural frequency of piping varies with the spacing between supports and with the stiffness of the supports themselves.  

The stiffness of a pipe support system depends on two things:² 

• The stiffness of the clamp/spring hold-down support itself 

• The stiffness of the structure underneath the pipe to which the pipe is clamped.  

An industry-wide problem exists due to incorrect assumptions about pipe support stiffness in software like the proprietary software^a that will give inaccurate results in terms of modal analysis. The current practice by many engineering firms is to assume that pipe supports are rigid (e.g., 1E12 lb./in.). The clamp and support systems have some flexibility (e.g., 1E5 lb/in.–1E6 lb./in.). This means the piping engineer’s assumptions can be > 1 MM times stiffer than the actual installation. A case study is provided in this article in which the Belleville hold-down support stiffness and preload were calculated, and how they were modeled in the software^a is also explained.²  

Belleville spring concept with stacking and load curve. The Belleville spring was patented and named after the inventor Julien Belleville in France in 1867, and is one of the frequently used springs that support very large loads with a small installation space. 

Belleville springs are shallow conical rings⁴ that are subjected to axial loads. Normally, the ring thickness is constant, and the applied load is evenly distributed over the upper inside and lower outside edges. Belleville springs are generally manufactured from spring steel. 

Belleville springs have the following characteristics:⁴ 

• They provide forces or stored energy that are easily controlled by tightening or loosening a bolt to vary the spring deflection. 

• They have a high load capacity with a small spring deflection. 

• They offer high space utilization when compared with other types of springs. 

• They possess a high fatigue life and low set loss/creep when properly sized. 

• They offer different combinations of springs that can be designed to achieve the desired load characteristics.  

• They offer a variety of special materials and surface coatings that can be used. 

• They are a cost-effective as a result of standardized sizes. 

The characteristic load curve is a representation of the force-deflection behavior of the spring. Depending upon the dimensional ratios, the characteristic load curve of a Belleville spring is more or less digressive up to the flat position (FIG. 2).

FIG. 2. Typical characteristic curve.⁴ 

Typically, Belleville springs are used as modular components. A group of individual Belleville springs stacked facing the same way is called a parallel spring stack (FIG. 3/Type-2). A group of individual Belleville springs that are stacked facing alternate ways is called a series spring stack (FIG. 3/Type-3). In a parallel spring stack, the deflection of the stack is equal to that of the individual spring. The load at a given deflection is proportional to the number of individual springs in the stack. In a series spring stack, the deflection of the stack is the sum of the deflections of the individual springs. The load of the stack is equal to the load on the individual spring. The combination of both series and parallel stacking is a means of multiplying both force and deflection. It is the user’s responsibility to minimize the number of springs in the stack by examining the various alternatives in FIG. 3.⁵ 

FIG. 3. Belleville spring stack (in series: parallel and combination).⁵ 

When calculating the spring deflection and load capacity of a stack composed of individual springs or spring packs, the factors in FIG. 4 must be considered:⁵ 

• FIG. 4a shows a stack of three individual springs in parallel: the force is multiplied by 3. 

• FIG. 4b shows a stack of four individual springs in series: the deflection is multiplied by 4.  

• FIG. 4c shows a series stacking of three parallel stacks—each parallel stack contains two individual Belleville springs, so the deflection is multiplied by 3 and the force is multiplied by 2. 

FIG. 4. Different combinations of Belleville springs.⁴ 

CASE STUDY

Two positive-placement plunger pumps (one in operation and the other used as a spare) (FIG. 5) were used in the boiler package in one project with a maximum suction pressure of 5 kg/cm2 and a discharge pressure of 345 kg/cm2 at 111°C (232°F) pumping temperature and running at a maximum speed of 400 revolutions per minute (rpm). 

FIG. 5. Plunger pump.  

The suction line of this pump was 2-in. piping coming from the deaerator vessel. Since the boiler vendor was unable to perform the pulsation study per API-674,3 an in-house pulsation study (not part of this article) was performed with the following results: 

• Identified critical frequency = 45 Hz at the 12th harmonic.  

• Force peak-to-peak (pk-pk) at this critical frequency= 150 kg. 

The next step was to conduct a modal analysis (with accurate stiffnesses) of the suction piping—using the proprietary software^a—to identify any resonant frequencies close to 45 Hz ± 20% (separation margin as per API-674).3 In this case, since the pulsation from a pump produces a dominant frequency of 45 Hz, the piping system must be analyzed to ensure that the piping natural frequencies should be either below 36 Hz or above the 54 Hz range (± 20% separation margin as per API 674).⁶ 

A Belleville spring hold-down support was used as per the client’s support standard (as shown in FIG. 6) in which the preload and stiffness of the Belleville spring hold-down support was not mentioned. 

FIG. 6. Belleville spring hold-down support arrangement. 

Therefore, to properly calculate the preload and the stiffness of the Belleville spring hold-down support, an outside vendor’s^b data in FIG. 7 and TABLE 1 was considered per the client’s reference.  

FIG. 7. High carbon steel disc spring washer parameter.⁴ 

The preload and stiffness of the Belleville spring hold-down support shown in FIG. 6 (Belleville in series, FIG. 3/Type-3 was used in this case) is calculated and provided in TABLE 2 for ready reference.  

SPRING HOLD-DOWN MODELING APPROACHES

 Approach A: Connecting node modeling. This method provides a more realistic and comprehensive representation of the Belleville hold-down support, particularly suitable for dynamic systems. In this case, a 2-in. [outer diameter (OD)] stainless-steel Schedule 40 pipe with a length of 5 m anchored at node 1482 is considered (FIG. 8).⁶ 

FIG. 8. Analysis software^a input graphics snapshot. 

The procedure for Belleville hold-down support modeling is: 

• The Belleville spring hold-down is applied at a specific node (e.g., Node 1480). 

• This 1480 node is linked to a C-node (e.g., Node 14770) with stiffness in all translational directions. +Y, Z and X stiffnesses are actual structural stiffness (values mentioned below are for reference only, FIGS. 9a, 9b and 9c) and must be taken from the civil/structural group, while the –Y stiffness is a spring hold-down support spring rate value, as calculated in TABLE 2.  

 

 

 

 

 

 

FIGS. 9a, 9b and 9c. Analysis softwarea input parameter snapshots. 

• Intermediate element creation: A short rigid element (1 mm in length) is created between nodes 14770 and 14800. The actual surface sliding conditions should be entered at node 14800. A force vector is introduced at node 14770, which will act as the spring clamp force with the preload value, as calculated in TABLE 2. 

• Detailed restraint at node 14800: This will normally be a Y support for a spring hold-down with a clamp spring force. An X and Z restraint will be used, if needed. A rotational restraint (RX, RY, RZ) is provided to avoid a large increase in the solution time. 

Approach B: Simplified restraint modeling.  This method is convenient for its relative simplicity and ease of implementation. While this method is simple and effective for static and basic dynamic conditions, it may not capture detailed interactions like sliding or rotational friction at the clamp-pipe interface. In this case, a 2-in. (pipe OD) stainless-steel Schedule 40 pipe with a length of 5 m is considered, which is anchored at node 1482 (FIG. 10).⁶ 

FIG. 10. Analysis software^a input graphics snapshot. 

The procedure for Belleville hold-down support modeling is: 

• The spring hold-down is applied at a specific node (e.g., node 1480). The +Y, X and Z supports can be entered with actual structural stiffness (values mentioned are for reference only, FIGS. 11a and 11b) and must be taken from the civil/structural group. The –Y stiffness is a spring hold-down support spring rate value, as calculated in TABLE 2. 

FIGS. 11a and 11b. Analysis software^a input parameter snapshots.  

• Intermediate element creation: A short rigid element (1 mm in length) is created between nodes 1477 and 1480 to allow for more precise simulation of the restraint interface. In this node, a force vector is introduced that will act as the spring clamp force with a preload value as calculated in TABLE 2. 

Takeaways. Both approaches have their place, depending on project needs, time constraints and system criticality. Approach A, with its enhanced realism and flexibility, is highly recommended for systems under dynamic loads or where accurate clamp response is essential. Approach B offers a quick and simplified method, suitable for less critical applications. Engineers are encouraged to evaluate both methods during analysis and calibration based on validation with field measurements or detailed structural input. 

NOTES 

a Hexagon’s CAESAR II version 13 software 

b Maryland Precision Spring    

LITERATURE CITED 

1 Peng, L. C. and T.-L. Peng, Pipe stress engineering, 1st Ed., ASME Press, New York, New York, January 2009. 

2 BETA Machinery Analysis, “Pipe support stiffness,” GMRC, 2014, online: https://vdn.woodplc.com/assets/pdfs/Technical_Articles/Pipe_Support_Stiffness_-_GMRC_Project.pdf 

3 American Petroleum Institute (API) Standard 674, “Positive displacement pumps—Reciporocating,” 3rd Ed., December 2010.  

4 Mubea, “Mubea disc springs,” online: x engl Umschlag 

5 Belleville Springs, “Stacking disc springs,” online: Stacking Disc Springs - Belleville Springs 

6 American Society of Mechanical Engineers (ASME) B31.3, “Process piping,” 2020, online: https://www.asme.org/wwwasmeorg/media/codes-standards/find-codes/b31-3_process-piping-2020-toc.pdf 

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