In flue system design, the expansion joint not only needs to absorb thermal displacement, but also produces a significant "pressure thrust" due to internal medium pressure. If this thrust is neglected in the design stage, it may lead to the failure of the fixed bracket, flue deformation and even the instability of the expansion joint itself. Therefore, mastering the calculation formula of flue expansion joint thrust is the key link to ensure the safety of flue structure. This paper will systematically explain the thrust source, calculation formula and engineering application examples of metal expansion joint and non-metal expansion joint.
1. Why do you need to calculate the thrust of the expansion joint
The expansion joint is installed in the flue, and when there is pressure (positive or negative) acting inside, the pressure creates an axial force on the effective area of the bellows or skin. This force will be transmitted to the fixed brackets at both ends and will also act on the expansion joint body.
The core value of the thrust calculation formula of flue expansion joint lies in:
- Determining the structural dimensions and anchoring mode of the fixed bracket
- Check the pressure stability of the expansion joint itself
- Prevent flue interface cracking or expansion joint inversion due to thrust exceeding limit
The consequences of ignoring the thrust calculation are often disastrous: a power plant did not calculate the pressure thrust of the metal expansion joint, which led to the flue fixing bracket at the outlet of the induced draft fan being pushed 30mm away from the foundation, and the flue welds cracked in many places.
2. Thrust calculation of expansion joint of metal bellows
2.1 Sources of Pressure Thrust
Under the action of internal pressure, the effective area of the metal bellows expansion joint will produce an axial expansion force. The magnitude of this force is proportional to the pressure value and the effective area of the bellows.
The basic form of the calculation formula of the thrust of the flue expansion joint (metal bellows) is:
F_p = P × A_eff
Among them:
- F_p — — Pressure thrust, unit: N
- P — — Working pressure in flue (gauge pressure), unit: Pa (Note: the thrust direction is opposite under negative pressure)
- A_eff-Effective area of bellows in m²
2.2 Determination of effective area A_eff
The effective area of the bellows is not equal to the cross-sectional area of the flue because the corrugated structure of the bellows makes its pressure-bearing area between the inner diameter area and the outer diameter area. The following methods are commonly used in engineering to obtain:
Method 1: Check the product sample
The A_eff value is given directly in the technical parameter sheet provided by the manufacturer.
Method 2: Empirical Formula
For standard U-bellows:
A_eff ≈ (π/4) × (D_m) ²
Where D_m is the mean diameter of the bellows = (D_in + D_out) /2, D_in is the inner diameter, and D_out is the peak outer diameter.
Method 3: Inverse calculation by stiffness method
For installed expansion joints, it can be calculated back from the length change under pressure:
A_eff = K × Δ L/P
Where K is the axial stiffness of the bellows (N/mm) and Δ L is the elongation under pressure (mm).
2.3 Corrections in actual operating conditions
Metal bellows expansion joints are usually equipped with tie rods or hinges. The role of the tie rod is to withstand the pressure thrust, thus protecting the bellows. Therefore, the thrust calculation needs to distinguish between two cases:
| Structural form | Thrust bearer | The thrust to be withstood by the stent |
|---|---|---|
| No tie rod (free type) | Both end fixing bracket | F_p (all) |
| With tie rod (restraint type) | Tie rod + bracket | F_p =0 (balance in tie rod) |
Key conclusion: For metal expansion joints with tie rods, the pressure thrust is balanced by the tie rods themselves and is not transmitted to the external bracket. However, the design strength of the tie rod must be able to withstand F_p (usually taking 1.5 times the safety factor).
2.4 Calculation Examples
Known:
- Circular flue diameter DN1200mm, metal bellows expansion joint
- Inner diameter D_in =1200mm, crest outer diameter D_out =1320mm
- Operating pressure P = +5000Pa (5kPa positive pressure)
- Try to calculate the pressure and thrust and judge whether the tie rod needs to be installed
Calculation:
- Average diameter D_m = (1200+1320) /2=1260mm =1.26m
- Effective area A_eff = π/4× (1.26) ² =1.247 m²
- Pressure thrust F_p =5000×1.247=6235 N ≈ 636 kgf
Conclusion: If the free expansion joint is used, the fixed bracket at both ends needs to bear a thrust of about 636kgf, which must be included in the design of the bracket. If the type with tie rods is adopted, it can be easily withheld by 4 M16 tie rods (each with a bearing capacity of about 3000kgf).
Thrust calculation of non-metallic fabric expansion joint
A non-metallic expansion joint has a different source of thrust than a metal. Because the fabric skin is so soft that it can barely withstand pressure thrust, the thrust is all borne by the external metal frame and platen.
The calculation formula of flue expansion joint thrust (non-metal) is:
F_p = P × A_duct
That is, the effective area is directly taken as the internal cross-sectional area of the flue (instead of the average area of the bellows).
For rectangular flue:
A_duct = W × H
For circular flues:
A_duct = π/4× D²
3.1 Thrust transmission path of non-metallic expansion joint
The thrust of the non-metallic expansion joint is not borne by the skin, but is transmitted through the following path:
- The flue gas pressure acts on the end face of the flue
- The end plate transmits force to the flange connected to the expansion joint
- The flange is transmitted to the outer metal frame by a platen bolt
- The frame is then transmit to the flue fixing bracket by a pull rod or bracket
Therefore, for non-metallic expansion joints, installation must ensure that the platen bolts have sufficient strength and pre-tightening force to prevent internal pressure from blowing off the skin.
3.2 Calculation Examples
Known:
- Rectangular flue, width 1500mm, height 1200mm
- Operating pressure P = -8000Pa (8kPa negative pressure, i.e. suction)
- Trial calculation of the thrust to be withstood by the fixed bracket
Calculation:
- Flue cross-sectional area A_duct =1.5×1.2=1.8 m²
- Thrust F_p = P × A_duct = (-8000) ×1.8= -14400 N (negative sign indicates directional inward contraction)
- About 1469 kgf in absolute
Conclusion: The fixed bracket must withstand a tensile force of about 1470kgf (because the negative pressure is inward suction). This value is required for anchor checking during bracket design.
4. Elastic reaction force generated by temperature load
In addition to the pressure thrust, the expansion joint also produces an elastic reaction force when absorbing thermal displacement. This force also needs to be factored into the total load.
Calculation formula of elastic reaction force:
F_e = K × Δ L
Among them:
- K-axial stiffness of the expansion joint (N/mm), supplied by the manufacturer
- Δ L — — Thermal displacement absorbed after actual installation (mm)
For metal bellows expansion joints, the K value is usually 100~500 N/mm; For non-metallic expansion joints, the K value is small (typically
The total thrust (acting on the fixed bracket) is:
F_total = F_p + F_e (When the metal expansion joint has no tie rod)
F_total = F_e (when the metal expansion joint is equipped with a tie rod, the pressure thrust is balanced by the tie rod)
F_total = F_p (non-metallic expansion joint, elastic reaction force can be ignored)
V. Precautions in engineering application
5.1 Thrust direction under negative pressure
When the flue is under negative pressure (such as behind the induced draft fan), the direction of thrust is opposite to the positive pressure, which is the "suction" of inward contraction. At this time, the fixed bracket needs to be subjected to tension instead of pressure. Many engineers only focus on positive pressure thrust and ignore negative pressure suction, resulting in insufficient pull-out ability of the bracket and being pulled out of the foundation.
5.2 Effect of temperature change on thrust
For metal expansion joints, if cold pre-compression/pre-stretching is not performed at the design temperature during installation, the actual Δ L will deviate from the design value, resulting in more than expected elastic reaction force F_e. For example, if the thermal elongation is designed to be 40mm, if it is not pre-compressed during installation, the actual Δ L may reach 2 times the design value and F_e may double, which may lead to overload of the fixed bracket.
5.3 Introduction of safety factors
Regardless of pressure thrust or elastic reaction force, when finalizing the bracket load, the safety factor shall be multiplied by:
- Normal operating load: safety factor 1.5
- Extreme working conditions (e.g. start-stop, failure): Safety factor 1.2 (check according to material yield strength)
That is:
F_design ≥ F_total ×1.5
5.4 Coupling Effect of Multiple Expansion Joints
When multiple expansion joints are arranged in series on the same section of flue, the forces on the fixed bracket are not simply superimposed. A pipe flexibility analysis is required because the stiffness of the expansion joints interacts with each other and the displacement distribution may not be consistent with the initial design. At this point it is recommended to use CAESAR II or AutoPIPE software for simulation.
6. Quick table look-up for thrust calculation
To facilitate engineering site estimation, the following table gives the pressure thrust F_p (non-metallic expansion joint) at ±5kPa pressure for common flue sizes:
| Flue size (circular diameter mm) | Sectional area (m²) | F_p at 5kPa (kgf) |
|---|---|---|
| 500 | 0.196 | 100 |
| 800 | 0.503 | 257 |
| 1000 | 0.785 | 401 |
| 1200 | 1.131 | 578 |
| 1500 | 1.767 | 903 |
| 2000 | 3.142 | 1606 |
| Flue size (rectangular width × height mm) | Sectional area (m²) | F_p at 5kPa (kgf) |
|---|---|---|
| 1000×800 | 0.8 | 408 |
| 1500×1200 | 1.8 | 918 |
| 2000×1500 | 3.0 | 1530 |
| 2500×2000 | 5.0 | 2550 |
Instructions for use: The values in the table are approximate values (converted to 9.8 N/kgf). In practical application, for the expansion joint of metal bellows, the effective area A_eff should be calculated instead of the flue cross-sectional area A_duct, and the value will be slightly lower.
Common Mistakes and Avoidance
| Error Type | performance | consequence | Correct practice |
|---|---|---|---|
| Confusion effective area | Calculation of thrust of metal expansion joint by flue cross-sectional area | The thrust value is 20~30% larger, and the bracket is over-designed | Check the sample or calculate according to D_m |
| Negative pressure suction is ignored | Calculate only positive pressure, not negative pressure | Insufficient pullout resistance of stent, pulled out of foundation d> | Positive and negative pressures were checked separately |
| Forget the internal balancing effect of the tie rod | The expansion joint with tie rod is still loaded on the bracket according to F_p | Excessively strong bracket design, increased cost | Verify that the tie rod is subjected to F_p |
| Excluding elastic reaction forces | Compute pressure thrust only, ignore F_e | Small diameter high stiffness metal expansion joints may be overloaded | Check the stiffness K and calculate F_e |
| Insufficient safety factor | Access by 1.0 | Long-term stent fatigue | Take at least 1.5 |
VIII. Summary
The calculation formula for flue expansion joint thrust varies depending on the type of expansion joint:
- Metal bellows expansion joint: thrust F_p = P × A_eff (A_eff is the effective area of bellows). When there is no pull rod, F_p is carried by the supports at both ends; When there is a tie rod, it is balanced inside the tie rod, and the bracket only bears the elastic reaction force F_e = K × Δ L.
- Expansion joint of non-metallic fabric: Thrust force F_p = P × A_duct (A_duct is the internal cross-sectional area of flue), and elastic reaction force can be ignored. The thrust force is all transmitted from the external metal frame to the fixed bracket.
Correct calculation of expansion joint thrust is an indispensable step in flue structure design. In engineering practice, the process of "first distinguishing the types of expansion joints, then selecting the correct effective area, and finally counting the elastic reaction force and safety factor" should be strictly followed. For complex pipeline systems, it is recommended to use professional stress analysis software for overall calibration. Through scientific calculation and reasonable type selection, serious accidents such as fixed bracket damage, expansion joint inversion and flue cracking can be effectively avoided.