Calculation of Expansion Joint Support: Full Analysis of Internal Pressure Thrust and Stiffness Reaction Force

1. Importance of calculation of expansion joint stent

In pipeline systems, expansion joints are used to absorb thermal displacement and reduce stress, but they themselves cannot withstand internal pressure thrust and pipeline weight. If the bracket is set up incorrectly or calculated incorrectly, the expansion joint cannot compensate properly at least, and the pipeline system will be instable, the bracket will be damaged or even the bellows will be torn at worst. Therefore, the calculation of expansion joint support is a key step after pipeline design and expansion joint selection. Many engineers and technicians either ignore the internal pressure thrust or confuse the force difference between the fixed bracket and the guide bracket when calculating, resulting in serious safety hazards in the design documents. The core of stent calculation is to accurately determine the thrust generated by the expansion joint and rationally distribute it to the fixed stent, guide stent and intermediate stent. This paper will systematically expound the calculation methods of different types of brackets from force analysis, calculation formulas to engineering examples.

2. Types and functions of expansion joint brackets

2.1 Fixed bracket

To understand the calculation of expansion joint stent, first of all, the role of fixed stent should be clarified. A fixing bracket is a rigid structure that completely fixes the pipe and does not allow the pipe to displace or rotate in any direction. In pipes fitted with expansion joints, the fixed brackets bear the following loads:

• Internal pressure thrust generated by expansion joint (blind plate force)
• The frictional or elastic force caused by the thermal expansion of a pipe
• Elastic reaction force generated by bellows stiffness
• Dead weight of pipeline and medium weight
• Wind load, snow load (outdoor pipeline) and earthquake load

Fixed brackets are typically provided at both ends of the expansion joint, at the branch pipe, or where the pipe changes direction.

2.2 Guide Bracket

The guide bracket allows the pipe to move freely in the axial direction, but limits lateral and angular displacement. In the calculation of the expansion joint bracket, the calculation of the guide bracket is essentially different from that of the fixed bracket-the guide bracket does not bear internal pressure thrust, but only bears:

• Vertical load caused by pipeline dead weight
• Axial friction generated by thermal expansion of pipe
• Lateral constraints required to prevent instability

The spacing of the guide brackets is directly related to the stability of the pipeline under the action of compressive force, and excessive spacing will lead to buckling of the pipeline.

2.3 Intermediate bracket (ordinary bracket and hanger)

The intermediate support only bears the pipe's dead weight and vertical load, and does not limit the axial thermal displacement of the pipe. The calculation is relatively simple, and the conventional support and hanger design can be carried out according to the pipeline span and weight.

3. Force calculation of fixed bracket

3.1 Internal pressure thrust (blind plate force)

To answer the core question of the calculation of expansion joint bracket, the calculation of internal pressure thrust is the first task in the design of fixed bracket. Its calculation formula is:

F_p = P × A_eff

Among them:

• F_p: internal pressure thrust (N)
• P: working pressure (Pa)
• A_eff: Effective area of expansion joint (m²)

The effective area A_eff is not a simple internal cross-sectional area of the pipe, but the equivalent area of the bellows that generates thrust under pressure. For standard waveforms, the effective area is usually between the inner and outer cross-sectional areas, which can be approximated as follows:

A_eff = (π/4) × D_m²

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

Example of calculation:
Assuming that the inner diameter of the pipe is DN500, the average diameter of the bellows is D_m =550mm, and the working pressure is P =0.6MPa, then:

A_eff = π/4×0.55² =0.2376 m²
F_p =0.6×10⁶ ×0.2376=142,560 N ≈ 14.5 tonne force

This means that the internal pressure thrust generated by this expansion joint is up to 14.5 tons, which must be borne by the fixed brackets on both sides.

3.2 Elastic reaction force generated by bellows stiffness

When thermal displacement occurs in the pipe, the expansion joint bellows can be compressed or stretched, creating an elastic reaction force:

F_s = K × Δ L

Among them:

• K: axial stiffness of expansion joint (N/mm)
• Δ L: Actual displacement (mm)

3.3 Pipe Friction and Weight

Friction when the pipe moves on the guide bracket:

F_f = μ × W × L

Among them:

• μ: friction coefficient (0.3 for steel-to-steel sliding and 0.1 for rolling bracket)
• W: Weight of pipe and media per unit length (N/m)
• L: length of pipe from expansion joint to fixed bracket (m)

3.4 Total load of fixed bracket

The final result of the calculation of the expansion joint bracket-the total load of the fixed bracket is the vector sum of the respective partial loads. In the axial arrangement, the loads on both sides of the fixed bracket are in opposite directions, and the total load is the difference between the loads on both sides (take the greater value).

When there are pipes on both sides of the expansion joint, the total axial force on the fixed bracket is:

F_total = max (F_left, F_right) -min (F_left, F_right) The absolute value of, plus the algebraic sum of the co-directional loads on both sides.

4. Arrangement and spacing calculation of guide brackets

4.1 Functional positioning of guide bracket

In the calculation system of the expansion joint bracket, the guide bracket does not bear the internal pressure thrust, and its core functions are:

• Ensure that the expansion joint moves along the axis direction and does not bend laterally
• Prevent column instability of bellows under pressure
• Limit the lateral displacement of the pipe to the specified range

4.2 Determination of spacing between guide brackets

The maximum spacing of the guide brackets is related to the pipe diameter and the rigidity of the bellows. Experience Recommended by EJMA Standards:

The distance between the first guide bracket and the expansion joint port shall not exceed 4 times the nominal diameter of the expansion joint and not exceed 4m.

4.3 Forces on guide brackets

The guide bracket mainly bears the vertical load of the pipeline's own weight and the friction force during axial movement, but does not bear the internal pressure thrust. Its vertical load is calculated by weight distribution within the pipe span.

Note: There should be a gap (usually 2-5mm) between the guide surface of the guide bracket and the pipe to avoid sticking and causing the expansion joint to be unable to expand and contract freely.

5. Calculation characteristics of bracket with tie rod expansion joint

5.1 Function of the tie rod

For expansion joints with large tie rods (such as large tie rod transverse type, hinge type, etc.), the tie rod directly bears internal pressure thrust and is not transmitted to the fixed bracket. This is an important change in the calculation of the expansion joint bracket-the fixed bracket only needs to withstand the elastic reaction force and pipe friction generated by the rigidity of the bellows, and the load is significantly reduced.

5.2 Forces on the fixed brackets on both sides of the tie rod expansion joint

When the expansion joint is set with a tie rod:

F_fixed = F_s + F_f

Where F_p has been internally balanced by the tie rod and no longer acts on the pipe support.

This characteristic makes the tie rod expansion joint particularly suitable for retrofit projects where fixed brackets are difficult to set or carrying capacity is insufficient.

VI. Actual engineering calculation steps

6.1 Collection of underlying data

Before performing the expansion joint stent calculation, the following data need to be prepared:

• Pipe diameter, wall thickness, material
• Operating pressure, operating temperature, installation temperature
• Expansion joint model, effective area, axial stiffness
• Pipe layout drawing, pipe rack position
• Pipes and media weight, insulation weight

6.2 Calculate the total displacement of the expansion joint

Δ L_total = α × L × (T_work-T_inst)

Among them:

• α: Linear expansion coefficient (carbon steel about 12×10⁻⁶/℃)
• L: length of pipe between two fixed brackets (m)
• T_work, T_inst: Operating temperature, installation temperature (℃)

6.3 Calculate the total load of fixed bracket

According to the formula in Section 3, calculate the internal pressure thrust, elastic reaction force, and friction force item by item, and then sum them.

6.4 Check the strength of the stent

The calculated load is taken as the design input, and the strength of the steel structure, anchor bolts and foundation of the fixed bracket is checked. The safety factor is generally 1.5-2.0.

Common Calculation Errors and Correction

VIII. Calculation Examples

Working condition: DN600 steam pipeline, P =0.8MPa, T_work =250℃, installation temperature 20℃, distance between two fixed brackets 30m. An axial expansion joint with effective area of 0.32m² and axial stiffness K =2000N/mm was selected. Friction coefficient μ =0.3, unit weight of pipe and medium is 1200N/m.

Thermal displacement: Δ L =12×10⁻⁶ ×30× (250-20) =0.0828m =82.8mm

Internal pressure thrust : F_p =0.8 x 10⁶ x 0.32=256,000 N

Elastic reaction force : F_s =2000×82.8=165,600 N

frictional force : F_f =0.3×1200×30=10,800 N

total load of fixed bracket: F_total = F_p + F_s + F_f =256,000+165,600+10,800=432,400 N ≈ 44 tonnes force

The fixed bracket should be structurally designed according to the axial thrust of 44 tons, and the safety factor should be 1.5 times.

IX. SUMMARY

The calculation of expansion joint support is a fundamental and critical task in pipeline stress analysis. The core can be summarized as follows: "The internal pressure thrust depends on the effective area, the stiffness reaction force depends on the displacement, the fixed bracket carries the sum, and the tie rod structure can be self-balanced".

In the calculation process, the different functions of the fixed bracket and the guide bracket must be clearly distinguished: the fixed bracket bears the sum of internal pressure thrust, elastic reaction force and friction force, among which the internal pressure thrust often accounts for the largest proportion, which can not be ignored by calculating the effective area formula F_p = P × A_eff; The guide bracket only bears the vertical load and friction of self-weight, and does not bear the internal pressure thrust. The spacing of the guide bracket should be controlled within the specification range to prevent pipe buckling. For expansion joints with tie rods, the internal pressure thrust is self-balanced by the tie rods, and the fixed bracket load is significantly reduced.

In the specific operation, the data of pipeline parameters, working conditions and expansion joint characteristics should be collected first, and the thermal displacement, internal pressure thrust, elastic reaction force and friction force should be calculated in turn. Finally, the safety factor of 1.5-2.0 should be summed and considered. Common mistakes include replacing the effective area with the internal cross-sectional area of the pipe, ignoring the internal pressure thrust, and confusing the functional positioning of the fixed bracket with the guide bracket. Through the correct calculation and design of the expansion joint bracket, the expansion joint can play a safe and reliable compensation role in the pipeline system, and avoid pipeline deformation, expansion joint damage and even safety accidents caused by the failure of the bracket.