1. Importance of internal pressure load of flue expansion joint

In flue system design, the expansion joint is used to absorb the thermal displacement of the pipe and reduce the structural stress, but many engineers often only pay attention to its compensation ability, and neglect another key factor-internal pressure load. The internal pressure load of flue expansion joint refers to the axial force on the inner wall and the end of the expansion joint due to its corrugated structure characteristics under the action of working pressure. Unlike the straight pipe section, the internal pressure load of the expansion joint is not a simple "pressure multiplied by the cross-sectional area", but also includes the additional thrust caused by the bellows waveform. If this part of the load is under-calculated or improperly handled, it may lead to overload failure of the fixed bracket, excessive extension of the expansion joint and even rupture of the bellows. For large-diameter flues and high-pressure flue gas systems, such as booster fan outlets, the effect of internal pressure loads is particularly significant. This paper will systematically analyze this core design parameter from generation mechanism, calculation method to structural countermeasures.

2. Generation mechanism of internal pressure load

2.1 The concept of effective area

To answer the internal pressure load of flue expansion joint, we must first understand the concept of "effective area". For straight pipes, the axial thrust generated by internal pressure is:

F = P × A

Where A is the internal cross-sectional area of the pipe. However, for bellows expansion joints, due to the undulating structure of the corrugations, the internal pressure acts on the wall surfaces in different directions at the same time, and the net effect is equivalent to acting on an imaginary area called "effective area" (A_eff).

The effective area is usually between the inner cross-sectional area and the outer cross-sectional area of the pipe, and its exact value is determined by the waveform, wave height, wall thickness and material properties of the bellows.

2.2 Two components of internal pressure thrust

The internal pressure load of the flue expansion joint can be decomposed into two main components:

In the absence of external constraints, the internal pressure causes the expansion joint to "elongate" deformation, which may be superimposed on the thermal displacement, further increasing the pipe end displacement.

2.3 Difference between pressure thrust and blind plate force

In engineering practice, the internal pressure load of flue expansion joint is often referred to as "pressure thrust" or "blind plate force". The physical nature is the same as the pressure on the blind plate: if one end of the expansion joint is blocked, the thrust generated by the internal pressure is equal to the pressure multiplied by the effective area. Even if the ends are open (the flue is in normal flow state), the corrugated structure of the expansion joint will still transmit this thrust to the pipes and supports at both ends.

3. Calculation method of internal pressure load

3.1 Determination of effective area

The core of calculating the internal pressure load of flue expansion joint is to determine the effective area A_eff. Common methods are as follows:

Method 1: Empirical formula method (for standard waveforms)

For common U-shaped bellows, the effective area can be calculated approximately as follows:

A_eff = (π/4) × (D_b) ^2

Where D_b is the average diameter of the bellows (the average value of the crest diameter and the trough diameter).

Method 2: Stiffness method (inverse deduction by test)

Knowing the axial stiffness K of the expansion joint and the amount of elongation δ under internal pressure, then:

A_eff = (K × δ) /P

Method 3: Finite Element Analysis

For non-standard or large diameter expansion joints, the finite element method is recommended to accurately calculate the effective area and stress distribution.

3.2 Calculation of Internal Pressure Thrust

After determining the effective area, the internal pressure thrust F_p is:

F_p = P × A_eff

Among them:

• P: Operating pressure (Pa or MPa)
• A_eff: Effective area (m² or mm²)

Example of calculation:

Assuming that the working pressure of the flue expansion joint is 15kPa (0.015MPa) and the effective area is 1.2m², then:

F_p =0.015×10⁶ ×1.2=18,000 N ≈ 1.8 tonne force

This means that at normal operating pressure, the expansion joint will generate about 1.8 tons of axial thrust on the supports at both ends. For systems with large diameter flues (e.g. above DN4000) or higher pressures, internal pressure loads can reach tens of tons and must be borne by fixed supports.

3.3 Modification of Multi-wave Expansion Joints

For duplex or multi-wave expansion joints, the internal pressure load of the flue expansion joint needs to be modified. In multi-wave structure, each corrugated segment will generate internal pressure thrust, but the thrust of adjacent bands partly cancel each other. In the tie rod type expansion joint, the internal pressure thrust is mainly borne by the large tie rod and is not transmitted to the external support. For a duplex expansion joint without a tie rod, the total thrust is equal to the single wave thrust multiplied by the wave number.

4. Influence of internal compression load on structure

4.1 Requirements for fixed brackets

Internal pressure load is one of the main control loads in the design of fixed support. The fixing bracket must simultaneously withstand:

• Internal pressure thrust generated by expansion joint
• The frictional or elastic force caused by the thermal expansion of a pipe
• Dead weight of pipeline and medium weight
• Wind load, snow load (outdoor flue)

If the internal pressure load of flue expansion joint is underestimated, the fixed support may yield, instability or foundation failure, which may lead to the overall displacement of flue system exceeding the standard, and even cause the expansion joint to be overstretched and damaged.

4.2 Arrangement of guide brackets

The guide bracket cannot withstand the internal pressure thrust, but its arrangement spacing will affect the working state of the expansion joint. The guide brackets shall be arranged at both ends of the expansion joint, and the spacing shall generally not exceed 4 times the diameter of the flue. A reasonable guide bracket setting can prevent the expansion joint from generating column instability (lateral bending) when under pressure.

4.3 Influence on the body of the expansion joint

The internal pressure generates not only axial thrust, but also circumferential and meridional stresses at the crests and troughs of the bellows:

• Circumferential stress: internal pressure causes the crest to swell outward and the trough to contract inward
• Meridian stress: internal pressure attempts to elongate or compress the bellows

When the working pressure exceeds the design pressure of the bellows, it may cause plastic deformation, crest instability, or trough crushing.

V. Determination of pressure level of flue system

5.1 Common flue pressure ranges

Different types of flue systems, the working pressure is very different, and the internal pressure load of flue expansion joint is also different accordingly:

5.2 Difference between positive pressure and negative pressure conditions

Positive pressure condition: The internal pressure thrust force is pushed outward, trying to elongate the expansion joint, and the fixed bracket is subjected to tension force

Negative pressure condition: the internal pressure is negative, the pressure thrust is sucked inward, and the expansion joint is attempted to be compressed, and the fixed bracket is under pressure. At the same time, the vacuum stability of the expansion joint needs to be considered (to prevent trough suction and deflation)

5.3 Consideration of Transient Pressure

The transient conditions that may occur during operation of the flue system (such as fan start-stop, baffle door closure) will produce pressure fluctuations, and the peak pressure may reach 1.5-2 times the normal working pressure. These transient conditions must be considered when calculating the internal pressure load of the flue expansion joint, and the permissible pressure of the expansion joint should be higher than the maximum expected pressure.

VI. Measures to cope with internal pressure load in engineering

6.1 Set up a tie rod or hinge

For pipelines or equipment interfaces that cannot withstand the internal pressure thrust (e.g. flue gas heat exchanger, dust collector inlet), an expansion joint with a tie rod or hinge should be selected:

• Large tie rod expansion joint: the tie rod directly bears all internal pressure thrust and does not transmit it to the pipe support
• Hinge expansion joint: the hinge is subjected to the bending moment generated by internal pressure thrust and lateral displacement
• Universal Hinge Expansion Joints: For applications where angular displacement is required to be absorbed

After choosing this kind of expansion joint, the fixed bracket only needs to bear the friction force caused by the thermal displacement of the pipeline, and the load is significantly reduced.

6.2 Reinforcing fixed brackets

For situations where internal pressure must be withstood by the bracket, the fixed bracket should be specially designed:

• The support structure should have sufficient rigidity and strength, and it is recommended to use steel truss or reinforced concrete pier
• The tensile and shear resistance of the support foundation shall be checked
• The connection between the support and the flue shall be made by welding or high strength bolts, and shall not be fixed by friction alone

6.3 Structural strengthening of bellows

For high-pressure flues, the ability of the expansion joint to withstand internal pressure can be improved by:

• Increase the wall thickness of bellows
• Use multi-layer bellows (stress sharing)
• Provision of reinforcing ring (adding circumferential reinforcement at trough)
• Reduce the ratio of wave height to pitch

VII. Engineering case analysis of internal pressure load

Case background: The outlet flue of a desulfurization booster fan in a coal-fired power plant has a diameter of DN3200 and a design pressure of 12kPa. Single axial expansion joint without tie rod is selected. After 3 months of operation, it was found that the foundation of the fixed bracket was cracked, and the expansion joint appeared obviously elongated.

Cause Analysis:

• When calculating the internal pressure load of the flue expansion joint, the internal cross-sectional area of the pipe (0.785×3.2² =8.04m²) was incorrectly used
• The actual effective area is about 7.2m², and the internal pressure thrust F =12×7.2=86.4kN (about 8.8 tonnes force)
• The fixed bracket design only considers thermal expansion friction (about 2 tons force), which is underestimated by more than 4 times

Corrective measures:

• Reinforcing fixed brackets, adding diagonal braces and ground anchors
• Replace with expansion joints with large tie rods, so that the internal pressure thrust is borne by the tie rods
• Revised design specifications to explicitly require that all positive pressure flue expansion joints must be subjected to internal pressure load calculation

VIII. Summary

The internal pressure load of flue expansion joint is an important load that cannot be ignored in the design of flue system, and its influence is often equal to or even more critical to the thermal expansion displacement. The core points can be summarized as follows: "The effective area is fixed thrust, the fixed bracket should be held, and the high tie rod can solve the worry". Specifically, the calculation of internal pressure thrust must be based on the true effective area of the bellows rather than the simple cross-sectional area of the pipe. This value usually needs to consult the data provided by the expansion joint manufacturer or be calibrated by experiments. For flue systems with working pressure exceeding 5kPa, the internal pressure bearing capacity of the fixed bracket should be checked one by one. If necessary, an expansion joint with tie rod or hinge should be selected to self-balance the internal pressure thrust and avoid transmitting it to the external structure. At the same time, the positive pressure and negative pressure conditions should be treated differently, and the vacuum stability of the expansion joint should be checked during negative pressure. Through correct internal pressure load calculation and corresponding structural measures, serious accidents such as fixed bracket damage, excessive deformation of expansion joint and even bellows rupture caused by thrust overload can be avoided, and the long-term safe operation of flue system can be ensured. In the design document, it is recommended that the calculation process of the internal pressure load be included as a mandatory review item and labeled in the technical specification of the expansion joint.