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    SA-E - Fundamentals of Pipe Stress Analysis with Introduction to CAESAR II

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    Piping system design constitutes a major part of the design and engineering effort in any facility. Stress analysis is a critical component of piping design through which important parameters such as piping safety, safety of related components and connected equipment and piping deflection can be addressed. The objective of pipe stress analysis is to prevent premature failure of piping and piping components and ensuring that piping stresses are kept within allowable limits.

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    Fundamentals of Pipe Stress Analysis with Introduction to CAESAR II - Introduction to Pipe Stress Analysis

    1. Introduction to Pipe Stress Analysis

    This chapter provides a brief introduction to pipe stress analysis and explains the need for stress analysis in piping systems.  The phenomenon of overstressing in piping systems and its consequences are touched upon in detail. An attempt is also made to obtain a clear understanding of the fundamental physical parameters used in stress analysis such as force, stress and strain and modulus of elasticity. Details of the various physical quantities and units used in pipe stress analysis are also discussed along with a description of the concepts of tensile testing and yield strength of materials. The various aspects related to the thermal expansion / contraction and flexibility of piping systems are also adequately covered during the course of the discussion.

                   Learning objectives

    • Need for stress analysis in piping systems.
    • Consequences of overstressing in piping systems.
    • Understanding of the various fundamental physical quantities used in pipe stress analysis: Force, Stress, Strain, Modulus of Elasticity and Linear Coefficient of Thermal Expansion.
    • Tensile testing of materials and Stress – Strain curve.
    • Yield Strength of materials.
    • Hooke’s Law.
    • Thermal effects and flexibility of piping systems.
    • Practical Exercises.

     

    1. Need for stress analysis in piping systems

    Piping systems need to be analyzed for stresses, to ensure that the components of the system are not overstressed.  Piping systems typically consist of straight pipes, fittings (elbow, tees, reducers), flanges, valves and accessories such as actuators. The stresses on equipment nozzles where the pipe connects to the equipment also need to be analyzed.  The pipe wall resists both the internal and external forces experienced by the piping system.  The force per unit metal area of the pipe wall is the resulting pipe stress. The objective of pipe stress analysis is to ensure that the stresses do not exceed allowable values specified by the design codes. Pipe stress analysis provides the necessary techniques and methods for designing piping systems without overstressing the piping components and the connected equipment.

     

    Piping stress analysis applies to calculations that address the static and dynamic loading arising on account of the effects of temperature changes, gravity, external and internal pressures and changes in fluid flow rate.

     

    Some of the important reasons why piping stress analysis is needed include:

     

    • Complying with legislation.
    • Ensuring that the piping is well supported and does not sag or deflect under its own weight.
    • Ensuring that the loads and moments imposed on the machinery as well as the vessels due to the thermal expansion of the attached piping are not excessive.
    • Ensuring that deflections are kept under control when thermal and other loads are applied.
    • Ensuring that stresses in the pipe work in both the cold and hot conditions are below permissible values.
    • Ensuring that the piping meets intended service and loading condition requirements while optimizing the layout and support design.
    • Ensuring the safety of piping and piping components.
    • Ensuring the safety of connected equipment and supporting structures.

     

     

     

    1. Consequences of overstressing in piping systems

    Overstressing can lead to premature failure of the piping system, causing leaks and safety hazards. Overstressing can lead to cracks, breakages and other secondary failures and failures such as bowing and opening of flanges. In some cases, the failure can be catastrophic, causing the collapse of the system and with the potential for loss of life and property.  Some situations may even require the entire plant to be shut down.  Thus, the objective of pipe stress analysis is to ensure the safe operation of piping systems within a plant, while simultaneously meeting the performance requirements of the plant.

     

     

    Overstressing in piping can result in the following:

     

    • Permanent deformation of the piping.

     

    • Cracking and breakage of piping.

     

    • Degradation of material with time.

     

    • Higher creep rate resulting in premature piping failure.

     

    • Excessive plastic deformation leading to failure.

     

    • Fatigue related failures due to cyclic loading.

     

    In general, overstressing can result from many different sources. Common examples include inadequate input such as insufficient pipe thickness, over-constraint, excessive thermal expansion or presence of other loads. The remedy for overstressing can be both, to add or in certain instances remove constraints such as releasing degrees of freedom of pipe supports or hangers. Although this process is often carried out on a trial and error basis, major piping layout related problems can usually be anticipated by experienced piping engineers during the design stage itself.

     

     

    1. Fundamental physical parameters used in stress analysis: Force, stress, strain, modulus of elasticity and linear coefficient of thermal expansion

    1.3.1          Force

     

    Force is a vector quantity that has both magnitude and direction. It can be defined as a push or pull on an object resulting from its interaction with another object. Force is no longer experienced when this interaction ceases. Piping systems experience both tensile and compressive forces.  Forces experienced by piping systems are also known as “piping loads”.  Commonly used units for force are: Newton (N), kilogram force (kgf) and pound force (lbf). The units of force are explained here. 

     

    Newton is the force required to accelerate a one kilogram-mass at 1 m/s2.  Thus,

     

     

     

     

     

     

     

    Kilogram force is the force required to accelerate a one kilogram-mass at 9.81 m/s2.  Thus,

     

     

    Pound force is the force required to accelerate a one pound-mass at 32.2 ft/s2.  Thus,

     

     

     

    The definition of pound force creates a need for using the conversion constant gc while performing calculations in the US Customary System (USCS).


     

     

    The conversion factors for force units are:

     

    1 kgf = 9.81 N

    1 kgf = 2.205 lbf

    1 lbf = 4.4462 N

     

    The concept of force can be better understood with the help of the following exercise.

     

    Sample Exercise

     

    Problem

     

    A 5 kg .object is moving horizontally at a speed of 10m/sec. Determine the Net force required to keep the object moving at this speed and in the same direction.

     

    Solution

     

    Zero N.  This is because, an object in motion will maintain its state of motion and the presence of an unbalanced force results in a change in its velocity.

     

    1.3.2          Engineering stress

     

    Engineering stress S is the force per unit area of the metal cross section.  A stress may be normal, shear or torsion, leading to corresponding deformations. While stress cannot be measured directly, deformations can be measured.

     

    Units for engineering stress:

     

    N/m2 (Pascal, Pa)

    lbf/in(psi)

    kgf/cm2

     

    Commonly used units for stress:

     

    Kilo pounds per square inch (ksi) = 10psi

    Megapascals (MPa) = 106 Pa

     

    Commonly used conversion factors for stress:

     

    1 lbf/in2 (psi) = 0.0703 kgf/cm2 = 6.896 kPa

    1 lbf/in2 (psi)  = 6.896 kPa

    1 MPa = 145 psi

    1 ksi = 6.88 MPa

     

    1.3.3          Deformation of materials and engineering strain

     

    When the elements of materials are subjected to tensile or compressive loads, they undergo small deformations. These deformations can be “elastic” or “plastic”.  Within the elastic limit, the deformation is “elastic”, i.e. the material springs back to its original shape when the load is removed. Thus, elastic deformation is temporary in nature and exists only when the load is present.  After the material begins to yield, the deformation is permanent and remains even after the load is removed.  This is called “plastic” deformation. Figure illustrates the manner in which deformation of a material occurs, when it is subjected to tensile and compressive loads.

     

                                 

                     Tension                                              Compression

     

    Figure 1.1

    Deformation of a material when subjected to tensile and compressive loads

     

     

    Engineering strain is the change in length divided by the original length, i.e.

                                

                                

     

    Where

    DL is the change in length

    Lo is the original length 

                    

    Units of strain: in/in or mm/mm.

     

    While an object in tension has resulting tensile strain, an object in compression has resulting compressive strain. The above equation for strain is only valid if the deformation of the object is uniform throughout its volume.

     

    1.3.4          Modulus of elasticity (E)

     

    Modulus of Elasticity E is a material property that is indicative of the strength of the material.  The modulus of elasticity values for steel and aluminum are given here.  The values indicate that steel is about three times stronger than aluminum.

     

    Esteel = 30 x 106 psi = 2.07 x 105 MPa

      

     Ealuminum = 10 x 106 psi = 0.70 x 105 MPa

     

    The modulus of elasticity of materials decreases with increase in temperature.  This is due to the thermal expansion of materials.  At higher temperatures, thermal expansion results in a lesser force being required to cause a given amount of strain, resulting in a lower modulus of elasticity.  The modulus of elasticity for different materials and at various temperatures is listed in Table 1.1.  Modulus of elasticity is also referred to as “Young’s Modulus”.

     

     

    Table 1.1

    Modulus of Elasticity of Different Materials at Various Temperatures

    (Modulus of Elasticity is given in 105 MPa. The values in parenthesis are in 106 psi)

     

     Material

     -130°C

    (-203°F)

    20°C

    (68°F)

    260°C

    (500°F)

    540°C

    (1004°F)

    810°C

    (1490°F)

     

    Carbon steels (<3% C)

     

    2.03

    (29.5)

     

     

    1.92

    (27.9)

     

     

    1.82

    (26.4)

     

     

    1.06

    (15.4)

     

     

    -

    -

     

     

    Low, Intermediate alloy steels

     

     

    1.96

    (28.5)

     

     

    1.88

    (27.4)

     

     

    1.79

    (26.0)

     

     

    1.57

    (22.8)

     

     

    -

    -

     

     

    Austenitic stainless steels

     

     

    2.06

    (29.9)

     

     

    1.95

    (28.3)

     

     

    1.80

    (26.1)

     

     

    1.56

    (22.7)

     

     

    1.23

    (17.9)

     

     

    Monel

    (67Ni, 30Cu)

     

     

    1.83

    (26.6)

     

     

    1.79

    (26.0)

     

     

    1.75

    (25.4)

     

     

    1.10

    (16.0)

     

     

    -

    -

     

     

    Cupro-Nickel

    (70Cu, 30Ni)

     

    -

    -

     

     

    1.49

    (21.6)

     

     

    1.40

    (20.3)

     

     

    -

    -

     

     

    -

    -

     

     

    Aluminum

    Alloys

     

     

    Copper

     

     

     

    Brass

    (66Cu, 34Zn)

     

     

    Bronze

    (88Cu, 6Sn, 4.5Zn, 1.5Pb)

     

    0.750 (10.9)

     

     

    1.15

    (16.7)

     

     

    1.01

    (14.7)

     

     

    0.945

    (13.8)

     

     

    0.695

    (10.1)

     

     

    1.10

    (16.0)

     

     

    0.963

    (14.0)

     

     

    0.894

    (13.0)

     

     

    0.530

    (7.7)

     

     

    1.01

    (14.7)

     

     

    0.874

    (12.7)

     

     

    0.805

    (11.7)

     

     

    -

    -

     

     

    -

    -

     

     

    -

    -

     

     

    -

    -

     

     

    -

    -

     

     

    -

    -

     

     

    -

    -

     

     

    -

    -

     

     

     

     

     

     

     

     

     

     

     

     

    1. Physical quantities and units used in pipe stress analysis

    The different physical quantities of force, stress, strain, modulus of elasticity and their respective units have already been discussed.   Let us go ahead and discuss some of the other physical quantities used in pipe stress analysis.

     

    1. Density (r)

    The density of a substance is its mass per unit volume.  It is represented by the symbol “r”. Density for a given substance can be calculated from the following equation,

     

         Density (r) = Mass of the substance (m) / Volume of the substance (V)

     

         Density has the units, lbm/ft3 or kg/m3.

     

    If equal masses of cotton and lead are taken (say 1 kg each), we will find that the volume of cotton is much larger than the volume of lead. This is because lead is heavier (denser) than cotton. The particles of lead are closely packed while those of cotton are more diffused.

    Density tends to change with change in temperature.

     

     

    1. Specific Gravity (SG)

    The specific gravity of a substance is the ratio of the density of a substance to the density of some standard substance. The standard substance is usually water (at 4°C) for liquids and solids, while for gases it is usually air. Specific gravity is also known as Relative Density.

     

                       Relative density for liquids and solids (s) = Density of substance        

                                                                       Density of water at 4°C

                                

                       Relative density for gases (s) =      Density of substance                 

                                                                      Density of air

     

                       Density of substance = Density of water at 4°C ´ Relative density of liquid or solid 

               

                        i.e.:           r (for liquids and solids) = 1000 ´ s   and

     

                  r (for gases) = 1.29 ´ s

     

    Specific gravity is a dimensionless number.

     

    1. Specific Weight (g)

    The specific weight of a substance is the weight per unit volume.  It has units of kN/m3 or kgf/m3 or lbf/ft3.   The specific weight of water at standard conditions is 9.81 kN/m3 or 1000 kgf/m3 or 62.4 lbf/ft3.   The specific weight of any substance is the product of the specific gravity of the substance and the specific weight of water at standard conditions.

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     

    Table 1.2

    Specific Gravity, Density and Specific Weights of Materials

     

    Material

    Specific Gravity

    Density kg/m3

    Density lbm/ft3

    Specific Weight kN/m3

    Specific Weight kgf/m3

    Specific Weight lbf/m3

    CS (<0.3% C)

    7.84

     

    7840

    489

    76.91

    7840

    489

    Intermediate Alloy Steels (5% Cr, Mo to 9% Cr, Mo)

     

    7.84

    7840

    489

    76.91

    7840

    489

    Austenetic Stainless Steel

     

    7.98

     

    7980

    498

    78.28

    7980

    498

    Brass (66% Cu, 34% Zn)

     

    8.75

    8750

    546

    85.84

    8750

    546

    Aluminum Alloys

    2.77

    2770

    173

    27.17

    2770

    173

     

     

    1.4.4          Poisson’s Ratio

     

    When a material is subjected to a tensile load, it elongates.  Since the volume of the material is constant, the elongation in the longitudinal direction results in compression in the lateral direction. Similarly, compression along the longitudinal direction is accompanied by elongation along the lateral direction. Poisson’s ratio is the ratio of lateral strain to the longitudinal strain and is mathematically represented as 

     

    n = - elateral / elongitudinal

     

    In the case of a perfectly incompressible material that is deformed elastically at small strains, the Poisson's ratio would be exactly 0.5. Most practical engineering materials have values between 0 and 0.5. While cork has a value close to 0, most steels have values around 0.3. Rubber has a value of almost 0.5. Some materials, mostly polymer foams, have a negative Poisson's ratio. A value of 0.3 is used for most materials. Typical Poisson’s Ratio values for some common materials are given in table 1.3.

     

     

     

     

     

     

     

     

     

     

     

     

     

     

    Table 1.3

    Typical Poisson’s Ratio Values for Different Materials

     

    Material

    Poisson’s  Ratio 

    Rubber

    Lead

    Phosphor Bronze

    Copper

    Magnesium

    Molybdenum

    Magnesium alloy

    Beryllium Copper

    Wrought Iron

    Nickel Silver

    Aluminum

    0.48 – 0.50

    0.431

    0.359

    0.355

    0.350

    0.307

    0.281

    0.285

    0.278

    0.322

    0.334

    Clay

    0.3 - 0.45

    Zinc

    Brass (70-30)

    Titanium

    Stainless steel 18-8

    0.331

    0.331

    0.320

    0.305

    Mild steel

    0.303

    High carbon steel

    Nickel steel

    0.295

    0.291

    Cast steel

    Glass Ceramic

    Glass

    Cast iron - grey

    Concrete

    Bronze

    Cork

    0.265

    0.290

    0.240

    0.211

    0.200

    0.140

    0.000

     

     

     

    1.4.5          Linear coefficient of thermal expansion (a)

     

    Thermal expansion and contraction of piping systems is an important aspect of pipe stress analysis.  The Linear Coefficient of Thermal Expansion a is useful in determining the thermal displacements of piping systems and connected equipment.

    Coefficient of Thermal Expansion is defined as the thermal strain per unit degree change in temperature.  Thermal strain is the change in length (DL) divided by the original length(Lo).

                                

     

    Table 1.4 gives the Thermal Expansion Coefficients for different materials.

     

     

     

     

     

    Table 1.4

    Thermal Expansion Coefficients for Selected Materials at 21°C (70°F)

     

     Material

    a  (mm/mm)/°C

    a        (in./in.)/°F

    Carbon and Low Alloy Steel Through 3 Cr-Mo

     

    10.93

    6.07

    Intermediate Alloy Steel

    5 Cr-Mo Through 9 Cr-Mo

     

    10.30

    5.73

    Austenitic Stainless Steel

    18 Cr-8 Ni

     

    16.40

    9.11

    Copper

     

    16.68

    9.27

    Aluminum

    22.85

    12.69

     

     

    Sample Exercise

     

    Problem

     

    A steel rod of 10 mm diameter is subjected to a tensile load of 5000 N. Calculate the following:

     

    1. Stress in the rod.

     

    1. If the original length of the rod is 3 m, calculate the increase in length of the rod due to the load.  The modulus of elasticity of steel is 2.03 x 105

     

    Solution

     

     

        

         A.

                    

                    

     

     

                     

     

     

    B.

                 

     

     

                     Increase in length,

     

     

                    

     

     

     

    The physical quantities used in pipe stress analysis and their units are listed in Table 1.5.

     

     

    Table 1.5

    Physical Quantities and Units Used in Pipe Stress Analysis

     

     Physical

    Quantity

    Symbol

    SI System

    USCS

    Length

    L

    Meter (m)

    Feet (ft)

    Diameter

    D

    Millimeter (mm)

    Inch (in)

    Thickness

    Dx

    Millimeter (mm)

    Inch (in)

    Mass

    M

    Kilogram (kg)

    Pound mass (lbm)

    Weight

    W

    Newtons (N)

    Pound force (lbf)

    Time

    t

    Seconds (s)

    Seconds (sec)

    Temperature

    T

    Degree Celcius (°C)

    Degree Farenheit (°F)

    Area

    A

    Square meter (m2)

    Square feet (ft2)

    Volume

    V

    Cubic meter (m3)

    Cubic feet (ft3)

    Density

    r

              kg / m3

              lbm / ft3

    Acceleration

    a

    Meters/sec2 (m/s2)

    Feet/sec2 (ft/sec2)

    Force

    F

    Newton (N)

    Pound force (lbf)

    Pressure

    P

    Pascal (Pa)

    Pounds/in2 (psi)

    Stress

    s

    Megapascal (Mpa)

    Pounds/in2 (psi)

    Strain

    e

    mm/mm

    in/in

    Work

    W

    Newton-meter (N.m)

    Foot pound force (ft-lbf)

    Energy

    E

    Joule (J)

    British thermal unit (Btu)

    Modulus of Elasticity

    E

    MPa

    Kilopounds / in2 (ksi)

    Moment

    M

    N.m

    ft-lbf

    Moment of Inertia

    I

    mm4

    in4

    Section Modulus

    Z

    mm3

    in3

     

     

     

    Unit Prefixes:

     

    Kilo (k) = 103                                                         Micro (m) = 10-6                       

    Mega (M) = 106                                                    Nano (n) = 10-9

    Giga (G) = 109                                                                          Milli (m) = 10-3             

     

    1. Tensile testing and stress – strain curves

    Tensile tests are conducted on material specimens to determine material properties such as modulus of elasticity and yield strength.  The yield strength of a material is frequently used in determining allowable stresses for piping systems. Tensile tests are conducted using procedures and guidelines established by the “American Society for Testing of Materials (ASTM)”.   Tensile tests are carried out using “Universal Testing Machine” or UTM.

     

    The result of tensile testing is a “Stress – Strain Curve”.   The stress – strain curve for a typical ductile material such as mild steel is shown in Figure 1.2.

     

    Figure 1.2

    Stress – Strain Curve for a Ductile Material (Source: “Introduction to Pipe Stress Analysis”, Sam Kannappan, John Wiley & Sons, 1986)

    The following points can be observed from the Stress – Strain Curve for the ductile material shown in the figure.

     

    • The stress – strain curve is linear until the yield point of the material.   Until the yield point of the material, the strain or deformation is elastic.  Hence, yield point is known as the “elastic limit” of the material.  Beyond the yield point, the deformation is plastic. The characteristics of elastic and plastic deformations have already been described in the preceding discussion.

     

    • The stress at the yield point is known as the “Yield Strength” of the material. 

     

    • The “Allowable Stress” for materials (to be discussed in detail later) at different temperatures is a fraction of the yield strength of the material.  Therefore, yield strength forms the basis for determining the allowable stresses as per the codes.

     

    • As the load or stress is further increased beyond the yield point, the stress-strain curve becomes non-linear. The stress continues to increase and reaches a maximum value.  The maximum stress in the stress-strain curve is known as the “Ultimate Tensile Strength (UTS)” of the material.   Most often, the UTS of a material is simply referred to as the “Tensile Strength” of the material.

     

    • Beyond the UTS, the stress decreases slightly until the point of failure, where the material fractures.  The stress at failure is known as the “Fracture Strength”.

     

    Figure 1.3 illustrates the stress-strain curve for a non-ductile material such as cast iron.  In the case of a ductile material, there is significant plastic deformation after yielding and before failure. In contrast, failure occurs without significant plastic deformation in the case of a non-ductile material. The area under the stress-strain curve is a measure of the energy required to cause failure.  It is clear that this area is much larger for ductile materials as compared to non-ductile materials. 

     

                           

    Figure 1.3

    Stress – Strain Curve for a Non-Ductile Material (Source: “Introduction to Pipe Stress Analysis”, Sam Kannappan, John Wiley & Sons, 1986)

    1.5.1          Yield strength based on 0.2% offset

    Sometimes, the results from tensile testing of materials do not exhibit a sharp, well-defined yield point. In such cases, the “0.2% Offset Method” is used in determining the yield point.  This is based on the observation that most materials can have a plastic strain of 0.2% without failing.  0.2% strain is equivalent to a strain of 0.002.  The technique involving the 0.2% offset method is illustrated in Figure 1.4. A strain value of 0.002 is used as the starting point and a line parallel to the linear portion of the stress – strain curve is drawn.  The intersection of this line with the stress – strain curve gives the yield point.

     

     

                    

    Figure 1.4

    Yield Strength Based on the 0.2% Offset Method

     

    The Yield Strength and Tensile Strength of selected piping materials are given in Table 1.6.

     

    Table 1.6

    Yield Strength and Tensile Strength of Selected Piping Materials

     

     Material

    Specification

    TS (ksi)

      TS (MPa)

      YS                           (Ksi)

      YS (MPa)

    Carbon Steel

    A106 Gr.B

     

    60

    414

    30

    207

    Carbon Steel

    API 5L Gr.B

     

    60

    414

    35

    241

    Carbon Steel

    API 5LX Gr.X52

     

    66

    455

    52

    359

    Low and Intermediate Alloy Steel

    A333 Gr.3

    65

    448

    35

    241

     

    Low and Intermediate Alloy Steel

     

    A334 Gr.8

     

    100

     

    689

     

    75

     

    517

     

    Low and Intermediate Alloy Steel

     

    A369 Gr.FP1

     

    55

     

    379

     

    30

     

    207

     

    Stainless Steel

     

     

    A312 Gr.TP304

     

    75

     

    517

     

    30

     

    207

    Stainless Steel

     

    A312 Gr.TP310

    75

    517

    30

    207

    Stainless Steel

    A312 Gr.TP316L

    70

    483

    25

    172

     

    1.5.2          Hooke’s Law

    This law states that

     

    “Within the elastic limit, the strain of a material is proportional to the applied stress. This can be represented as

                                                                          e a S

     

                          
    Using the reciprocal of Modulus of Elasticity as the constant of proportionality, Hooke’s Law can be mathematically written as

     

                   

    From Hooke’s Law, it can be concluded that for a given applied stress, the engineering strain will be lesser for a material having higher Modulus of Elasticity.

     

    Along with this, various pipe properties such as DN (or Nominal Diameter), wall thickness and pipe schedule play a very significant role in Stress Analysis.

     

    Sample Exercise

     

    Problem

     

    A steel rod of 25 mm diameter indicates a strain of 0.001 when subjected to a tensile load.   Find the applied load.   Esteel is 2.03 x 105 MPa.

     

    Solution

     

     

                        

     

              

    1.6 Thermal effects and flexibility of piping systems

    Piping systems should have the flexibility to expand or contract as required, due to differences between the operating and installation temperatures. This flexibility is achieved by providing loops in the pipe routing as shown in Figure 1.5 or by providing expansion bellows as shown in Figure 1.6.  The stiff piping system illustrated in Figure 1.7 lacks flexibility. This will result in overstressing of the system due to thermal expansion.

        

    Figure 1.5

    Providing Flexibility for Piping Systems by Using Expansion Loops (Source: “Introduction to Pipe Stress Analysis”, Sam Kannappan, John Wiley & Sons, 1986)

     

     

    Figure 1.6

    Providing Flexibility for Piping Systems by Using Expansion Bellows (Source: “Introduction to Pipe Stress Analysis”, Sam Kannappan, John Wiley & Sons, 1986)

     

     

    Figure 1.7

    Piping System that Lacks Flexibility (Stiff Piping) (Source: “Introduction to Pipe Stress Analysis”, Sam Kannappan, John Wiley & Sons, 1986)

    1.6.1          Calculating thermal growth

     

    Tables in piping codes provide thermal data in the form of thermal expansion/ contraction, in mm/m and in/100 ft, between 21°C (70°F) and indicated temperatures. This data is used for determining the displacement in piping systems on account of thermal expansion/contraction.  The thermal data for common piping materials is presented in Table 1.7.

     

    Table 1.7

    Total Thermal Expansion between 21°C (70°F) and Indicated Temperatures for Common Piping Materials

     

     Temperature

      °C            °F

    Carbon Steel mm/m      in./100ft

    Inter. Alloy Steel

    mm/m      in./100ft

    Austenitic SS

    mm/m      in./100ft

    -184        -300

       -1.90           -2.24

     -1.70         -2.10

      -3.00        -3.63

    -129        -200

       -1.40           -1.71

     -1.30         -1.62

      -2.30        -2.73

      93          200

        0.80            0.99

      0.80          0.94

       1.20         1.46

     204         400

        2.20            2.70

      2.10          2.50

       3.20         3.80

     316         600

        3.80            4.60

      3.50          4.24

       5.20         6.24

     427         800

        5.60            6.70

      5.10          6.10

       7.30         8.80

     538       1000

        7.40            8.89

      6.70          8.06

       9.60         11.48

     649       1200

        9.20           11.10

      8.30          10.00

      11.80        14.20

     760       1400

       11.10          13.34

     10.00         12.05

      14.10        16.92

     

     

     

     

    Fundamentals of Pipe Stress Analysis with Introduction to CAESAR II - Introduction to Pipe Stress Analysis

    1. Introduction to Pipe Stress Analysis

    This chapter provides a brief introduction to pipe stress analysis and explains the need for stress analysis in piping systems.  The phenomenon of overstressing in piping systems and its consequences are touched upon in detail. An attempt is also made to obtain a clear understanding of the fundamental physical parameters used in stress analysis such as force, stress and strain and modulus of elasticity. Details of the various physical quantities and units used in pipe stress analysis are also discussed along with a description of the concepts of tensile testing and yield strength of materials. The various aspects related to the thermal expansion / contraction and flexibility of piping systems are also adequately covered during the course of the discussion.

                   Learning objectives

    • Need for stress analysis in piping systems.
    • Consequences of overstressing in piping systems.
    • Understanding of the various fundamental physical quantities used in pipe stress analysis: Force, Stress, Strain, Modulus of Elasticity and Linear Coefficient of Thermal Expansion.
    • Tensile testing of materials and Stress – Strain curve.
    • Yield Strength of materials.
    • Hooke’s Law.
    • Thermal effects and flexibility of piping systems.
    • Practical Exercises.

     

    1. Need for stress analysis in piping systems

    Piping systems need to be analyzed for stresses, to ensure that the components of the system are not overstressed.  Piping systems typically consist of straight pipes, fittings (elbow, tees, reducers), flanges, valves and accessories such as actuators. The stresses on equipment nozzles where the pipe connects to the equipment also need to be analyzed.  The pipe wall resists both the internal and external forces experienced by the piping system.  The force per unit metal area of the pipe wall is the resulting pipe stress. The objective of pipe stress analysis is to ensure that the stresses do not exceed allowable values specified by the design codes. Pipe stress analysis provides the necessary techniques and methods for designing piping systems without overstressing the piping components and the connected equipment.

     

    Piping stress analysis applies to calculations that address the static and dynamic loading arising on account of the effects of temperature changes, gravity, external and internal pressures and changes in fluid flow rate.

     

    Some of the important reasons why piping stress analysis is needed include:

     

    • Complying with legislation.
    • Ensuring that the piping is well supported and does not sag or deflect under its own weight.
    • Ensuring that the loads and moments imposed on the machinery as well as the vessels due to the thermal expansion of the attached piping are not excessive.
    • Ensuring that deflections are kept under control when thermal and other loads are applied.
    • Ensuring that stresses in the pipe work in both the cold and hot conditions are below permissible values.
    • Ensuring that the piping meets intended service and loading condition requirements while optimizing the layout and support design.
    • Ensuring the safety of piping and piping components.
    • Ensuring the safety of connected equipment and supporting structures.

     

     

     

    1. Consequences of overstressing in piping systems

    Overstressing can lead to premature failure of the piping system, causing leaks and safety hazards. Overstressing can lead to cracks, breakages and other secondary failures and failures such as bowing and opening of flanges. In some cases, the failure can be catastrophic, causing the collapse of the system and with the potential for loss of life and property.  Some situations may even require the entire plant to be shut down.  Thus, the objective of pipe stress analysis is to ensure the safe operation of piping systems within a plant, while simultaneously meeting the performance requirements of the plant.

     

     

    Overstressing in piping can result in the following:

     

    • Permanent deformation of the piping.

     

    • Cracking and breakage of piping.

     

    • Degradation of material with time.

     

    • Higher creep rate resulting in premature piping failure.

     

    • Excessive plastic deformation leading to failure.

     

    • Fatigue related failures due to cyclic loading.

     

    In general, overstressing can result from many different sources. Common examples include inadequate input such as insufficient pipe thickness, over-constraint, excessive thermal expansion or presence of other loads. The remedy for overstressing can be both, to add or in certain instances remove constraints such as releasing degrees of freedom of pipe supports or hangers. Although this process is often carried out on a trial and error basis, major piping layout related problems can usually be anticipated by experienced piping engineers during the design stage itself.

     

     

    1. Fundamental physical parameters used in stress analysis: Force, stress, strain, modulus of elasticity and linear coefficient of thermal expansion

    1.3.1          Force

     

    Force is a vector quantity that has both magnitude and direction. It can be defined as a push or pull on an object resulting from its interaction with another object. Force is no longer experienced when this interaction ceases. Piping systems experience both tensile and compressive forces.  Forces experienced by piping systems are also known as “piping loads”.  Commonly used units for force are: Newton (N), kilogram force (kgf) and pound force (lbf). The units of force are explained here. 

     

    Newton is the force required to accelerate a one kilogram-mass at 1 m/s2.  Thus,

     

     

     

     

     

     

     

    Kilogram force is the force required to accelerate a one kilogram-mass at 9.81 m/s2.  Thus,

     

     

    Pound force is the force required to accelerate a one pound-mass at 32.2 ft/s2.  Thus,

     

     

     

    The definition of pound force creates a need for using the conversion constant gc while performing calculations in the US Customary System (USCS).


     

     

    The conversion factors for force units are:

     

    1 kgf = 9.81 N

    1 kgf = 2.205 lbf

    1 lbf = 4.4462 N

     

    The concept of force can be better understood with the help of the following exercise.

     

    Sample Exercise

     

    Problem

     

    A 5 kg .object is moving horizontally at a speed of 10m/sec. Determine the Net force required to keep the object moving at this speed and in the same direction.

     

    Solution

     

    Zero N.  This is because, an object in motion will maintain its state of motion and the presence of an unbalanced force results in a change in its velocity.

     

    1.3.2          Engineering stress

     

    Engineering stress S is the force per unit area of the metal cross section.  A stress may be normal, shear or torsion, leading to corresponding deformations. While stress cannot be measured directly, deformations can be measured.

     

    Units for engineering stress:

     

    N/m2 (Pascal, Pa)

    lbf/in(psi)

    kgf/cm2

     

    Commonly used units for stress:

     

    Kilo pounds per square inch (ksi) = 10psi

    Megapascals (MPa) = 106 Pa

     

    Commonly used conversion factors for stress:

     

    1 lbf/in2 (psi) = 0.0703 kgf/cm2 = 6.896 kPa

    1 lbf/in2 (psi)  = 6.896 kPa

    1 MPa = 145 psi

    1 ksi = 6.88 MPa

     

    1.3.3          Deformation of materials and engineering strain

     

    When the elements of materials are subjected to tensile or compressive loads, they undergo small deformations. These deformations can be “elastic” or “plastic”.  Within the elastic limit, the deformation is “elastic”, i.e. the material springs back to its original shape when the load is removed. Thus, elastic deformation is temporary in nature and exists only when the load is present.  After the material begins to yield, the deformation is permanent and remains even after the load is removed.  This is called “plastic” deformation. Figure illustrates the manner in which deformation of a material occurs, when it is subjected to tensile and compressive loads.

     

                                 

                     Tension                                              Compression

     

    Figure 1.1

    Deformation of a material when subjected to tensile and compressive loads

     

     

    Engineering strain is the change in length divided by the original length, i.e.

                                

                                

     

    Where

    DL is the change in length

    Lo is the original length 

                    

    Units of strain: in/in or mm/mm.

     

    While an object in tension has resulting tensile strain, an object in compression has resulting compressive strain. The above equation for strain is only valid if the deformation of the object is uniform throughout its volume.

     

    1.3.4          Modulus of elasticity (E)

     

    Modulus of Elasticity E is a material property that is indicative of the strength of the material.  The modulus of elasticity values for steel and aluminum are given here.  The values indicate that steel is about three times stronger than aluminum.

     

    Esteel = 30 x 106 psi = 2.07 x 105 MPa

      

     Ealuminum = 10 x 106 psi = 0.70 x 105 MPa

     

    The modulus of elasticity of materials decreases with increase in temperature.  This is due to the thermal expansion of materials.  At higher temperatures, thermal expansion results in a lesser force being required to cause a given amount of strain, resulting in a lower modulus of elasticity.  The modulus of elasticity for different materials and at various temperatures is listed in Table 1.1.  Modulus of elasticity is also referred to as “Young’s Modulus”.

     

     

    Table 1.1

    Modulus of Elasticity of Different Materials at Various Temperatures

    (Modulus of Elasticity is given in 105 MPa. The values in parenthesis are in 106 psi)

     

     Material

     -130°C

    (-203°F)

    20°C

    (68°F)

    260°C

    (500°F)

    540°C

    (1004°F)

    810°C

    (1490°F)

     

    Carbon steels (<3% C)

     

    2.03

    (29.5)

     

     

    1.92

    (27.9)

     

     

    1.82

    (26.4)

     

     

    1.06

    (15.4)

     

     

    -

    -

     

     

    Low, Intermediate alloy steels

     

     

    1.96

    (28.5)

     

     

    1.88

    (27.4)

     

     

    1.79

    (26.0)

     

     

    1.57

    (22.8)

     

     

    -

    -

     

     

    Austenitic stainless steels

     

     

    2.06

    (29.9)

     

     

    1.95

    (28.3)

     

     

    1.80

    (26.1)

     

     

    1.56

    (22.7)

     

     

    1.23

    (17.9)

     

     

    Monel

    (67Ni, 30Cu)

     

     

    1.83

    (26.6)

     

     

    1.79

    (26.0)

     

     

    1.75

    (25.4)

     

     

    1.10

    (16.0)

     

     

    -

    -

     

     

    Cupro-Nickel

    (70Cu, 30Ni)

     

    -

    -

     

     

    1.49

    (21.6)

     

     

    1.40

    (20.3)

     

     

    -

    -

     

     

    -

    -

     

     

    Aluminum

    Alloys

     

     

    Copper

     

     

     

    Brass

    (66Cu, 34Zn)

     

     

    Bronze

    (88Cu, 6Sn, 4.5Zn, 1.5Pb)

     

    0.750 (10.9)

     

     

    1.15

    (16.7)

     

     

    1.01

    (14.7)

     

     

    0.945

    (13.8)

     

     

    0.695

    (10.1)

     

     

    1.10

    (16.0)

     

     

    0.963

    (14.0)

     

     

    0.894

    (13.0)

     

     

    0.530

    (7.7)

     

     

    1.01

    (14.7)

     

     

    0.874

    (12.7)

     

     

    0.805

    (11.7)

     

     

    -

    -

     

     

    -

    -

     

     

    -

    -

     

     

    -

    -

     

     

    -

    -

     

     

    -

    -

     

     

    -

    -

     

     

    -

    -

     

     

     

     

     

     

     

     

     

     

     

     

    1. Physical quantities and units used in pipe stress analysis

    The different physical quantities of force, stress, strain, modulus of elasticity and their respective units have already been discussed.   Let us go ahead and discuss some of the other physical quantities used in pipe stress analysis.

     

    1. Density (r)

    The density of a substance is its mass per unit volume.  It is represented by the symbol “r”. Density for a given substance can be calculated from the following equation,

     

         Density (r) = Mass of the substance (m) / Volume of the substance (V)

     

         Density has the units, lbm/ft3 or kg/m3.

     

    If equal masses of cotton and lead are taken (say 1 kg each), we will find that the volume of cotton is much larger than the volume of lead. This is because lead is heavier (denser) than cotton. The particles of lead are closely packed while those of cotton are more diffused.

    Density tends to change with change in temperature.

     

     

    1. Specific Gravity (SG)

    The specific gravity of a substance is the ratio of the density of a substance to the density of some standard substance. The standard substance is usually water (at 4°C) for liquids and solids, while for gases it is usually air. Specific gravity is also known as Relative Density.

     

                       Relative density for liquids and solids (s) = Density of substance        

                                                                       Density of water at 4°C

                                

                       Relative density for gases (s) =      Density of substance                 

                                                                      Density of air

     

                       Density of substance = Density of water at 4°C ´ Relative density of liquid or solid 

               

                        i.e.:           r (for liquids and solids) = 1000 ´ s   and

     

                  r (for gases) = 1.29 ´ s

     

    Specific gravity is a dimensionless number.

     

    1. Specific Weight (g)

    The specific weight of a substance is the weight per unit volume.  It has units of kN/m3 or kgf/m3 or lbf/ft3&a

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