Basic Concepts of Stress Analysis - Flexibility Analysis


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Part 4 of our continuing series will take an historical perspective of how earlier analysis techniques were developed in the absence of today's computer technology. Learn how earlier techniques have evolved ultimately leading to today's finite element practices. The basic concepts of analysis will be covered, including failure theories, stress intensification factors and the overall purpose of stress analysis.

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  • The goal of this slide is to show how meticulously the system has been broken down from the total structure into small pieces and how those small pieces are in local coordinate systems when necessary. Secondarily we see how even in a simple piping structure how much effort is necessary to separate the individual elements and perform the calculations by hand.
  • Where W=load per unit length, l is the length, E is elasticity, I is the moment of inertia, Z is section modulus
  • Where L = the length in feet, S sub A is the allowable stress in pounds per square inch, E is the elasticity modulus in pounds per square inch and D sub O is the outside diameter in inches.
  • Conservative correction factorsLed designers to incorporate more flexibility than was truly required.Earlier computational methods were not as accurate as the above computer methods which produced more accurate results.This slide is a review of the results of earlier analysis techniques which utilized correction factors. Ultimately these correction factors provided too high bending stresses compared to the actual (computer generated) results. Consequently the use of these conservative correction factors led to the design of earlier systems that incorporated too much flexibility.
  • Excessive flexibility comes from excessive loops and offsets. Externally, the piping system with excessive loops becomes flimsy and is prone to vibration and they are also weak in resisting occasional loads such as seen in wind and earthquakes.Also excessive pipe bends generate excessive pressure drops and create unsteady flow. The increased pressure drops would require increased power. And the pressure drops can cause vaporization of fluids operating at or near saturated temperature. The near saturated condition occurs in condensate lines and bottom fluid lines at distillation towers. When vaporization occurs at portions of the piping away from the pipes, it would generate shaking forces due to slug formation and then when slug formation occurs, the entire piping system shakes violently making the system unsafe for operation.
  • Technological progress and the trend to larger scale, and more complex units have continued unabated while the increase in pressure, temperature, and structural complexities have resulted in the use of larger pipe sizes and heavier wall thickness pipes and an increase in the use of alloy piping systems as well. Consequently today’s piping design involves more than just flexibility analysis. The overall problem analysis is much more complex.
  • Basic Concepts of Stress Analysis - Flexibility Analysis

    1. 1. New Webinar SeriesSession #4: Basic Concepts of Stress Analysis – Flexibility Analysis by Piping Technology & Products, Inc.
    2. 2. Jerry Godina Hyder HusainIf you have any questions, comments or suggestions, please email us at To request a PDH certificate, email
    3. 3. PT&P Subsidiaries PIPING TECHNOLOGY & PRODUCTS, INC. Member of MSS, SPED, APFA,U.S. Bellows, Inc. Sweco Fab, Inc. Fronek Anchor Pipe Shields, Inc. Darling Ent., Inc.Member of EJMA ASME U-Stamp ISO 9001-2000 Certified R-Stamp ASME Nuclear Qualified
    4. 4. Topic Session / DateIntroduction I. Overview of Piping (completed)Preliminary I. Piping System II. The Total SystemPiping Design Components (completed) (completed)Basic Concepts I. Flexibility Analysis II. Design Basesof Stress August 1, 2012 August 8, 2012AnalysisInfluences on I. Rigid Supports II. Spring Supports III. RestraintsPipe Support August 15, 2012 August 22, 2012 August 29, 2012Design
    5. 5. • During and immediately following World War II, oil refining, petrochemical and natural gas transportation industries in the United States grew extensively.• This expansion led to the installation of numerous gas processing plants and, in particular, gas transmission compression facilities and pipelines.• Machinery and piping for these facilities were sometimes hastily designed (primarily via hand calculations) and installed.• Many equipment consisted primarily of reciprocating compressors and pumps.• Primary focus was the flexibility analysis• One of the earliest books on flexibility analysis entitled, “Design of Piping Systems” by M.W. Kellogg Co. was published in 1941.Piping Technology & Products, Inc. 5
    6. 6. Basic Analysis of an In-plane System • Single plane analysis • Break-down of the system • Accumulation of the individual resultsSource: M.W. Kellogg Co., “Design of Piping Systems.”Piping Technology & Products, Inc. 6
    7. 7. Source: M.W. Kellogg Co., “Design of Piping Systems.”Piping Technology & Products, Inc. 7
    8. 8. Source: M.W. Kellogg Co., “Design of Piping Systems.”Piping Technology & Products, Inc. 8
    9. 9. Piping systems should have sufficient flexibility so that thermal expansion or contraction ormovements of supports and terminal points will not cause: a) Failure of piping or support from over stress or fatigue b) Leakage at joints c) Detrimental stresses or distortion in piping or connected equipment (pumps, vessels, valves, for example)The purpose of piping flexibility analysis is to produce a piping layout that causes neitherexcessive stresses nor excessive end reaction.• Should not be stiff• Unnecessarily flexible (extra material)• More length with many bends increases pressure drop, hence operating cost.Piping Technology & Products, Inc. 9
    10. 10. Source: S. Kannappan, “Introduction to Pipe Stress Analysis.”Piping Technology & Products, Inc. 10
    11. 11. AN EXAMPLE FORCE DEVELOPED IN A STIFF JOINTConsider a 10 in sch. 40 carbon steel pipe A53 Grade B subjected to 200oF from an installationtemperature of 70oF.The metal area of a 10 in sch. 40 pipe is : 11.9 sq.inCoefficient of expansion is: 0.99 in/100 ft @ 200 deg. FE = 27.9 x 106 psiF = EαA = 27.9 x 106 x(0.99/100 x12) x 11.9 F= 273,908 lb. Piping Technology & Products, Inc. 11
    12. 12. Piping system: group of beamsconnected together to form theshape required for transportingfluid from one point to another:Because the actual piping in aplant is supported somewherebetween simple support andfixed support, the followingaverage formulas are generallyused for evaluating S and D. M wL2 S Z 10 Z and 3wL4 384 EIPiping Technology & Products, Inc. 12
    13. 13. Dictated by THREE main factors a) Bending stress b) Vertical deflection c) Natural frequencyThe formulation and equation depends on the end conditionThe end condition can be assumed as a mean between a uniformly loaded beam with simplysupported end and a uniformly loaded beam with fixed ends. With this assumption theequations are: 0.4 ZS h based on limitation of stress L w 1/ 4 EI L based on limitation of deflection 13.5wWhere L = allowable pipe span (ft) Z= section modulus of pipe (in3) Sh= allowable tensile stress of the pipe at the design temperature w= Total weight of the pipe per unit length (lb/ft) (metal weight + content weight + insulation weight) = Allowable deflection (inch) I= area moment of inertia of pipe (in4) E= modulus of elasticity of the pipe at the design temperature (psi)Piping Technology & Products, Inc. 13
    14. 14. One of the simplified methods used in piping design is the guided cantilever method.Deflections are assumed to occur in a single plane under guided approximation. Under this assumption the deflection is given by: 144L2SA =Source: S. Kannappan, “Introduction to Pipe Stress Analysis.” 3EDoPiping Technology & Products, Inc. 14
    15. 15. Limitations • The system has only two terminal points and composed of straight legs of a pipe with uniform size and square corner. • All legs are parallel to the coordinate axes. • Thermal expansion is absorbed only by the legs in a perpendicular direction. • The amount of thermal expansion that a given leg can absorb is inversely proportional to its stiffness. • In accommodating thermal expansion, the legs acts as guided cantilever; can bend in the plane but no rotation.Piping Technology & Products, Inc. 15
    16. 16. Some Analytical & Computer Simulated Results Bending Stress in Symmetrical Loop Bending Stress in L-shaped PipingPiping Technology & Products, Inc. 16
    17. 17. Some Analytical & Computer Simulated Results Bending Stress in Z-shaped PipingPiping Technology & Products, Inc. 17
    18. 18. • Increased material cost • Increased pressure drop • Phase Change • Increased vibration • Loss of pump efficiency Since the 1950’s (Advent of Flexibility Analysis), failure due to insufficient flexibility has become rare. The majority of failures occur now as a result of: • Vibration • Thermal bowing • Creep • Thermal Fatigue • Steam or water hammerPiping Technology & Products, Inc. Image Source: 18
    19. 19. • Piping design involves more than just flexibility analysis: the problem is much more complex…• In a nut shell, piping design problems can be shown in the following block diagram:Piping Technology & Products, Inc. 19
    20. 20. Design considerations: • Tank/Vessel shell displacement, rotation, and hydrostatic bulge • Valve and flange leakage • Pipe temperature gradients over the pipe cross-section • The pipe connections at rotating equipmentPiping Technology & Products, Inc. 20
    21. 21. Piping Technology & Products, Inc. 21
    22. 22. • Complex Projects • Extended Liabilities • Tight Cost Controls • Strict Quality StandardsTo this end, the following should be done precisely: • Optimal Design • Design Documents • Fabrication Details • Procedures • Specifications Developed to Communicate • Monitor • Fabrication and Erection of Piping SystemsPiping Technology & Products, Inc. 22
    23. 23. The technology behind piping system design is extensive and diverse.Requires the knowledge from a number of engineering disciplines • Statics • Dynamics (vibration) • Mechanics of Materials • Fluid Mechanics (two phase flow) • Heat Transfer • Physical Metallurgy • Linear algebraProvides necessary techniques for engineers to design piping systems withoutoverstressing.Piping Technology & Products, Inc. 23
    24. 24. • Objective of codes and standards is to achieve minimum requirements for safe construction;• To provide public protection by defining those material, design, fabrication, and inspection requirements whose omission may radically increase operating hazards.• Responsibility of the designer to recognize the applicable codes and standards based upon the function and layout of the piping system.Piping Technology & Products, Inc. 24
    25. 25. STRESS: Stress is the internal resistance per unit area to the deformation caused byapplied load.STRAIN: Strain is unit deformation under applied load.STRESS –STRAIN CURVE: It is a curve in which unit load or stress is plotted againstunit elongation, technically known as strain. • O– A represents the stress is directly proportional to strain, and point A is known proportional limit. • Point B represents elastic limit beyond which the material will not return to its original shape when unloaded but will retain a permanent deformation called permanent set. • Point C is called yield point and is the point at which there is an appreciable elongation or yielding of the material without any corresponding increases of load. • Point D is ultimate stress or ultimate strength of material. • Point E is the stress at failure known as rupture strength.Piping Technology & Products, Inc. 25
    26. 26. Piping stress analysis is a term applied to calculations, which address the static anddynamic loading resulting from the effects of: • Gravity • Temperature Changes • Internal Pressures • Changes in Fluid Flow Rate • Seismic Activity • External Loads (wind, snow, ice)Codes and standards establish the minimum requirements of stress analysis.Piping Technology & Products, Inc. 26
    27. 27. Purpose to ensure: • Safety of piping and piping components • Safety of connected equipment and supporting structure • Piping deflections are within the limitsPiping Technology & Products, Inc. 27
    28. 28. There are various failure modes, which could affect a piping system. The pipingengineers can provide protection against some of these failure modes byperforming stress analysis according to piping codes.The modes of failures are: • Failure by general yielding: excessive plastic deformation • Yielding at sub-elevated temperature: body undergoes plastic deformation under slip action of grains. • Yielding at elevated temperature: after slippage, material re-crystallizes and hence yielding continues without increasing load. This phenomenon is known as creep. • Brittle fracture: Complete breaking under tension with no apparent plastic deformation. • Fatigue: Due to cyclic loading initially a small crack develops that grows after each cycle and results in sudden failure.Piping Technology & Products, Inc. 28
    29. 29. Failure of a structural part occurs when a certain function of the stress or straincomponent reaches a critical value.There are six main theories of failure suggested as below: 1. Maximum principle stress 2. Maximum shearing stress 3. Maximum strain 4. Total strain energy 5. Strain energy of distortion 6. Octahedral shearing stressPiping Technology & Products, Inc. 29
    30. 30. The maximum principal stress theory forms thebasis for piping systems governed by ASME B31 andSubsections NC and ND (Classes 2 and 3) of SectionIII of the ASME Boiler and Pressure Vessel Codes.This theory states that yielding in a pipingcomponent occurs when the magnitude of any of thethree mutually perpendicular principal stressesexceeds the yield strength of the material.The maximum shear stress theory is more accurate than the maximum principal stress theoryfor predicting both yielding and fatigue failure in ductile metals. This maximum shear stresstheory forms the basis for piping of Subsection NB (Class 1) of ASME Section III.The maximum shear stress at a point τmax is defined as one-half of the algebraic differencebetween the largest and the smallest of the three principal stresses σ1, σ2, and σ3. If σ1 > σ2 > σ3(algebraically), then τmax = (σ1 - σ3)/2The maximum shear stress theory states that failure ofa piping component occurs when the maximum shearstress exceeds the shear stress at the yield point in atensile test. In the tensile test, at yield, σ1 = σy (yieldstress), σ2= σ3=0. So yielding in the component occurswhen tmax = (σ1 - σ3)/2 = σy /2Piping Technology & Products, Inc. 30
    31. 31. Source: Seely & Smith, “Advanced Mechanics of Material.”Piping Technology & Products, Inc. 31
    32. 32. • The stress-intensification factor (SIF) is defined as the ratio of the maximum stress intensity to the nominal stress, calculated by the ordinary formulas of mechanics.• It is used as a safety factor to account for the effect of localized stresses on piping under a repetitive loading. In piping design, this factor is applied to: 1. Welds 2. Fittings 3. branch connections 4. other piping component where stress concentrations and possible fatigue failure might occur. * Usually, experimental methods are used to determine these factors. • It is recognized that some of the SIFs for the same components are different for different codes. In some cases, different editions of the same code provide different SIFs for a given component. • Therefore, the stress analyst has to ensure that the appropriate SIFs from the applicable code (i.e., committed code) are used.Piping Technology & Products, Inc. 32
    33. 33. Source: S. Kannappan, “Introduction to Pipe Stress Analysis.”Piping Technology & Products, Inc. 33
    34. 34. Next Webinar Session:August 8, 2012Session 5: Basic Concepts of Stress Analysis – Design BasesIn Part 5 of our Webinar series, discover how vibration analysis tools will be used inconjunction with flexibility analysis tools introduced in the last session. Learn about thevarious design bases which form the foundation of all our analyses, including physicalattributes, loading conditions and joint design. We will also discuss how the model isdeveloped, and how vibration affects the piping system.If you have any questions, comments or suggestions, please emailus at enews@pipingtech.comTo request a PDH certificate, email