Fundamental theories of pipeline stress analysis

Thread Content

Basic theories of pipeline stress analysis: Pipeline stress analysis mainly involves three aspects: correctly establishing the model, accurately describing the boundary conditions, and properly analyzing the calculation results. The so-called model establishment involves discretizing the mechanical model of the pipeline system under analysis in a certain manner, thereby simplifying it into a mathematical model that meets the requirements of the program. The accuracy of this model is a prerequisite for conducting proper stress analysis. The fundamental issue in stress analysis is the problem of boundary conditions; in engineering terms, this manifests as the simulation of specific issues such as supports and pipe ends. Only by accurately representing these boundary conditions can correct calculation results be obtained. To be able to analyze results skillfully and accurately, it is first necessary to design supports and brackets properly; relevant theoretical knowledge such as engineering mechanics, fluid mechanics, chemical processing equipment, and machinery is also required. In addition, continuous experimentation over a period of time is needed to identify patterns and regularities. Chapter 1: Topics Related to Pipeline Stress Analysis · §1.1 Purpose of Pipeline Stress Analysis There are many issues related to pipeline stress analysis, and the main problems that CAESARII addresses include: 1. Keeping the stress levels in various parts of the pipeline within the limits permitted by regulations. 2. Ensure that the loads on the pipe connections connected to the equipment meet the stress conditions specified by the manufacturer or recognized standards (such as NEMASM23, API610, API617, etc.). 3. Maintain the local stresses at the containers connected to the pipes within the allowable stress range specified in ASME Section VIII. 4. Calculate the loads acting on each constraint. 5. Determine the displacement of the pipeline under various operating conditions. 6. Solve pipeline dynamics problems, such as mechanical vibration, water hammer, earthquakes, and relief valve discharge. 7. Assist piping designers in the optimal design of piping systems. §1.2 Classification of Stresses Acting on Pipes 1.2.1 Definition of Basic Stresses Axial stress: Axial stress is a normal stress parallel to the axis of the pipe, caused by forces acting axially on the pipe. It is given by SL = FAX/Am, where SL represents the axial stress in MPa, FAX is the internal force in the cross-section in N, and Am is the cross-sectional area of the pipe wall in mm2, calculated as π(do² – di²)/4. The axial stress resulting from the design pressure of the pipe is given by SL = Pdo/4t. Since both the axial force and the design pressure induce stresses that are distributed evenly across the cross-section, these stresses must remain within the allowable stress limit σ]t. Bending stress: The axial normal stress generated by a torque with the normal vector perpendicular to the pipeline axis. SL = Mbc/I, where: Mb = the bending moment acting on the pipe cross-section, in N.m; C = the distance from the neutral axis of the pipe cross-section to the given point, in mm; I = the moment of inertia of the pipe cross-section, in mm4 = π((do4 – dl4)/64). The bending stress is greatest when C reaches its maximum value. Smax = MbR0/I = Mb/Z. The bending stress is distributed linearly across the cross-section; the stress at the outermost part of the cross-section is the highest, while other areas remain in an elastic state. Therefore, the stress is kept within the range of 1.5σ. Circumferential stress: The normal stress caused by the internal pressure in the tangential direction of the pipe wall circumference. Radial stress of thin-walled tubes: SH = Pdo/2t. This stress is caused by the internal pressure acting in the radial direction of the tube; Sr = P((ri2 – ri2)/(ro2 – ri2)). Shearing stress: This stress is resulting from the shear forces acting on the cross-section of the tube. tmax = VQ/Am. Here, tmax represents the maximum shear stress in MPa; V is the shear force, and Q is the shear coefficient. The shear force resulting from torque is given by tmax = MTC/R, where MT is the torque acting on the cross-section in N.m, C is the distance from a point on the cross-section to the torsion center in mm, and R is the torsional section modulus in mm^4. Since 4I = π(d_o^4 – d_i^4)/32, the torsional stress is greatest when C is at its maximum value, that is, when C equals the outer radius. In this case, τmax = MTR_o/2I = MT/2Z. By summing up the various components of the shear stress, the maximum shear stress acting on the pipe’s cross-section is given by τmax = VQ/Am + MT/2Z. CAESAR II calculates stress values that include bending stress, axial stress, and torsional stress; these values are then compared with the specified stress limits and allowable stresses. Most American pipeline specification standards require the use of the following formulas for stress calculations: Axial stress: SL = Mb/Z + Fmax/Am + Pdo/4t; Shear stress: τ = MT/2Z; Circumferential stress: SH = Pdo/2t. 1.2.2 Stress classification Pipeline failure is primarily caused by fracture due to primary stresses and fatigue fracture due to secondary stresses. Primary stresses are the normal and shear stresses resulting from mechanical external loads, and they must satisfy the laws of equilibrium for external and internal forces and moments. Feature: Primary stress is non-self-limiting; it always increases as the applied load increases. When it exceeds the material’s yield limit or endurance strength, it will cause plastic failure or overall deformation of the pipe. Therefore, in the stress analysis of piping systems, the primary stress must first be brought within the allowable stress range. Secondary stress: The normal or shear stress resulting from the constraint of deformation; it is not directly in equilibrium with external forces. Features: ① Secondary stresses in pipes are usually caused by displacement loads (such as thermal expansion, additional displacements, installation errors, and vibration loads). ② These secondary stresses are self-limiting; when local yielding occurs and a small amount of plastic deformation takes place, the stresses can be reduced through deformation coordination. ③Secondary stress is periodic (excluding the secondary stress caused by installation). ④ The allowable limit for secondary stress is based on periodicity and fatigue fracture patterns; it does not depend on the stress level over a particular period, but rather on the range of alternating stresses and the number of cycles. Peak stress refers to high stresses caused by local stress concentration, local structural discontinuities, or local thermal stresses. §1.3 Criteria for pipeline stress analysis: Petrochemical pipelines generally follow the B31 or B31.1 standards. 1.3.1 The B31.1 standard for power pipelines: The primary stress corresponds to the stress under the sustained (SUS) condition as defined in CAESARII. SSuS = S1 = 0.75iMA/Z + Pdo/4t ≤ Sh, where SSUS… S1 = sustained stress in MPa; i = strength coefficient (a single coefficient for various types of moments), as specified in Appendix D of standard B31.1. MA = total moment resulting from sustained loads = Sh – allowable stress of the material at the design temperature. Secondary stress corresponds to the stress under the EXP condition in CAESAR II. SE = IMC/Z ≤ f(1.25Sc + 1.25Sh – S1) in MPa. Where: SE = range of secondary stress; i = strength coefficient (a single coefficient for various types of moments), as specified in Appendix D of standard B31.1. Mc = range of moments caused by secondary loads = Sc – allowable stress of the material at the ambient temperature. Accidental stress, corresponding to stresses generated by accidental loads such as wind loads: Soce = Where: Socc – Total bending moment caused by accidental loads, in N.m; K – Coefficient for accidental loads (the coefficient is 1.2 when the occurrence rate of accidental loads is less than 1% of the operating time, and it is 1.15 when the occurrence rate is 10% of the operating time). 1.3.2 B31.3: Standard for chemical plants and oil refining pipelines. Primary stress: B31.3 does not provide a specific equation for defining primary stress; it only requires engineers to calculate the axial stress resulting from gravity and pressure, ensuring that this stress does not exceed Sh. The general formula for this is: S1 = FAX/Am + 1/2/Z + Pdo/4t ≤ Sh, where: Fax – Axial force resulting from permanent loads; Mi – Bending moment in the plane resulting from permanent loads; Mo – Bending moment out of the plane resulting from permanent loads; ii, io – Stress amplification factors for bending in the plane and out of the plane, as specified in Appendix D of B31.3. Secondary stress: SE = Where: Mi – Bending moment in the plane caused by temperature (secondary) loads; Mo – Bending moment out of the plane caused by temperature (secondary) loads; MT – Torsional moment caused by temperature (secondary) loads; Sc – Basic allowable stress of the material at ambient temperature, as specified in Appendix A of B31.3. Accidental stress: B31.3 does not specify an equation for calculating accidental stress; under simple conditions, the total axial stress resulting from both permanent and accidental loads should not exceed 1.33 times Sh. 1.3.3 Differences between B31.1 and B31.3: ① B31.3 takes into account the effect of torque, whereas B31.1 does not. ② B31.1 lacks a clear definition for the calculation theories related to continuous and intermittent loading conditions, while B31.1 provides explicit regulations on this matter. ③In most standard annotations, B31.1 ignores torque under sustained load conditions, whereas B31.3 includes it. ④In the default description, B31.1 ignores all forces; under continuous load conditions, B31.3 takes into account the different allowable stress values specified in various standards for Fax ⑤. ⑥In each standard specification, the stress increase caused by accidental loads varies. 1.3.4 Other standards followed in the stress analysis of CAESRII pipelines include ASME Section III NC or ND standards for nuclear industry pipelines, B31.4 standards for oil and gas pipelines, B31.8 standards for gas transmission and distribution systems, Canada’s 2183/2184 standards for oil and gas pipelines, and the UK’s BS806 pipeline standard, among others. §1.4 Combinations of operating conditions for pipeline stress analysis: The loads acting on pipelines can be classified according to their nature as static loads, dynamic loads, and thermal loads. Static loads mainly include the weight of the pipeline itself (including valves, fittings, and insulation), the weight of the fluid inside the pipeline, the design pressure, as well as other continuous loads such as the elastic reaction forces from springs and bellows. Dynamic loads mainly include loads generated by pressure fluctuations or impacts, seismic loads, and the discharge pressure of safety valves. The load conditions in CAESERII are: W – gravitational load condition, D – additional displacement load condition, T – temperature load condition, P – pressure load condition, F – concentrated load condition, Wind – wind load condition. These conditions can be combined in any way as required by the analysis, or stress calculations can be performed separately. The results obtained from combined conditions are the linear sum of the results calculated for each individual condition; for example, the result for the (OPE)W + D + T + P + F condition is the sum of the results for the (SUS)W + P + F condition and the (EXP)Dσ = D2 – D1 condition. §1.5 Stress analysis and evaluation of pipes and pipe fittings: When loads act on the pipe fittings of pumps, compressors, steam turbines, and heat exchangers, excessive loads can cause significant deformation in the equipment’s pipes, thereby affecting their proper operation. Therefore, it is necessary to limit the forces acting on these pipe fittings. Typically, manufacturers specify the allowable loads that the pipe fittings can withstand; otherwise, general standards such as NEMASM-23 (for steam turbines), API610 (for centrifugal pumps), API617 (for centrifugal compressors), and API661 (for air coolers) can be referred to. The film stress and bending stress generated on the container by the pipes connected to it can be evaluated in accordance with Section 2 of Part 8 of the ASME Boiler and Pressure Vessel Code; accurate results can be obtained using finite element analysis. Part WRC107 of CAESARII allows for the determination of the allowable load values for container nozzles in a conservative manner, based on limits on the calculated stresses. 1.5.1 Analysis of loads on the nozzles of rotating equipment: The most reliable way to accurately assess the load-bearing capacity of equipment nozzles is through testing; the best alternative to testing is finite element analysis. CAESARII provides a ROT program that uses relevant standards to automatically evaluate nozzle loads; when assessing the forces acting on the nozzles of the equipment, the nozzle load is taken as the higher value between the cold and hot condition results obtained from the pipeline stress analysis. The equipment standards include: ① Steam turbines – NEMA Standard SM23; ② Centrifugal pumps – API Standard 610, editions 6 and 7; ③ Centrifugal compressors – API Standard 617; ④ Air coolers – API Standard 661; ⑤ Hermetically sealed feedwater heaters – HEI Standard. When using these programs, users need to enter the structural dimensions and operating loads of the relevant equipment. 1.5.2 Calculating container stresses based on nozzle loads. For the calculation of container stresses due to nozzle loads, since the early 1960s, WRC 107 from the Welding Research Council has been widely used by design engineers to assess local stresses at the interfaces between containers and attachments. CAESARII uses WRC107 to calculate the stresses induced in the container by nozzle loads; WRC107 is a parametric method based on finite element analysis results for stresses generated by external loads on the container. It includes equations and dimensionless curves (basic parameters such as the ratio of nozzle to container diameter, and the ratio of container diameter to thickness). The dimensionless curves are used to calculate the stresses at the connections where attachments are attached to the container, through root-finding techniques. WRC107 can be used to analyze the stresses at the attachments of cylindrical or spherical vessels. When using WRC107 to verify nozzle loads, these loads are taken as the force values at the constraints under the corresponding operating conditions as obtained from CAESARII stress analysis. 1.5.3 Limiting conditions for container stress The stress induced on the container wall by the nozzle load must satisfy the following condition: Pm