Pipe Stress Analysis

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Pipe Stress Analysis

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Table of Contents 1 Introduction to Pipe Stress Analysis 1

1.1 Need for stress analysis in piping systems 1 1.2 Consequences of overstressing in piping systems 2 1.3 Fundamental physical parameters used in stress analysis 3 1.4 Physical quantities and units used in pipe stress analysis 6 1.5 Tensile testing and stress-strain curves 12 1.6 Thermal effects and flexibility of piping systems 15 1.7 Summary 17

2 Piping Materials 19 2.1 Introduction 19 2.2 Material properties 20 2.3 Material classification system and specifications 24 2.4 Piping specifications 29 2.5 Material selection 31 2.6 Quality control, testing and material certification 32 2.7 Summary 32

3 Codes Governing Piping Design and Pipe Stress Analysis 33 3.1 Philosophy, objectives and intent of piping codes 33 3.2 Role and scope of piping codes 34 3.3 Information available from piping codes 35 3.4 ASME B31.3 Process Piping 36 3.5 ASME B31.4 Pipeline Transportation Systems for Liquid

Hydrocarbons and Other Liquids 37 3.6 ASME B31.8 Gas Transmission and Distribution Piping Systems 38 3.7 Other ASME B31 Codes 39 3.8 AS 4041 – 2006: Pressure piping 39 3.9 Codes governing long distance pipelines 41 3.10 Summary 42

4 Principal Stresses and Failure Theories 45 4.1 Longitudinal, Circumferential and Radial stresses in pipe walls 45 4.2 Principal axes and Principal stresses 48 4.3 Failure theories 49 4.4 Maximum Principal stress failure theory 49 4.5 Maximum Shear stress failure theory 49 4.6 Shear strain energy theory 50 4.7 Maximum strain energy theory 51 4.8 Summary 51

5 Design Conditions and Allowable Stresses 53 5.1 Design pressure and design temperature 53 5.2 Basis for code allowable stresses 54 5.3 Allowable stress data 56 5.4 Primary and secondary stresses 57 5.5 Summary 60

6 Design of Pipe Wall Thickness for Internal Pressure 63 6.1 Wall thickness design equations 63 6.2 Maximum allowable working pressure 67 6.3 Pressure-temperature class ratings for flanges 69 6.4 Determination of pressure class ratings for flanges 69 6.5 Summary 71

7 Loads on Piping Systems and Code Criteria for Design 73 7.1 Primary and secondary loads 73 7.2 Self-limiting and non self-limiting characteristics of loads 74 7.3 Sustained and Occasional loads 74 7.4 Static and dynamic loads 74 7.5 Load cases used in stress analysis calculations 75 7.6 Bending stresses in pipes 75 7.7 Total longitudinal stress in pipes 76 7.8 Torsional stress in pipes 78 7.9 Code criteria for design 78 7.10 Summary 80

8 Thermal Stresses in Piping Systems 81 8.1 Stresses due to thermal expansion and contraction 81 8.2 Thermal expansion and contraction of materials 82 8.3 Thermal fatigue and cyclic stress reduction factor 84 8.4 Design criteria for expansion stress 85 8.5 Code allowable stress range for thermal expansion 86 8.6 Flexibility and stress intensification factors (SIF) 86 8.7 Calculation of expansion stress range 88 8.8 Summary of code requirements for safe design of piping systems 91 8.9 Summary 91

9 Pipe Stress Analysis Software–Introduction to CAESAR II 93 9.1 Introduction to pipe stress analysis software 93 9.2 Overview of CAESAR II stress analysis software 94 9.3 Piping input spreadsheet and creation of the stress model 95 9.4 Static analysis 97 9.5 Dynamic analysis 98 9.6 Building the load cases and running the analysis 99 9.7 Output and Results 100 9.8 Summary 103

10 CAESAR II Practical Exercises 105

Appendix A: Exercises 109 Appendix B: Answers to Exercises 115 Appendix C: Practical Problems 117 Appendix D: Answers to Practical Problems 119 Appendix E: Pipe Data and Fitting Dimensions 125

Preface

Piping and piping components are subjected to different types of stresses just like other mechanical components. Overstressing can result in premature failure of piping and piping components and it is therefore important to ensure that piping stresses are kept within allowable limits. This is precisely why piping systems are subjected to stress analysis. Piping stress analysis involves those calculations that address static and dynamic loading in piping that result from various factors such as internal and external pressures, changes in temperature and fluid flow rate and changes due to gravity and seismic activity, to name a few. There are various codes and standards that establish the minimum requirements for carrying out stress analysis in piping. Some of the important parameters that can be addressed through stress analysis include piping safety, safety of related components and connected equipment, piping deflection etc. Material selection is quite critical to ensuring the efficiency and long service life of piping systems. Selection of appropriate piping material ensures the safety and integrity of piping systems. In this context, it is essential that a clear understanding of various material properties such as hardness, strength and toughness be obtained. Aiding in the process of material selection are the various material classification systems and standard piping specifications and classes.

Piping codes define requirements regarding design, fabrication, materials, tests and inspection of pipes and piping systems, while piping standards define application design and construction rules and requirements for piping components such as flanges, elbows, tees, valves etc. Piping design and stress analysis are governed by important piping codes such as ASME B31.3, ASME B31.4, ASME B 31.8 and AS 4041-2006. The intent and scope of these codes must be properly understood, since they provide guidelines for safe design and construction of piping, fabrication, material use and testing and inspection criteria.

Another very important aspect of piping stress analysis is to understand the fundamentals related to pipe stresses and more specifically the principal stresses that include longitudinal, tangential and radial stresses. In order to predict the load levels that a structure can actually withstand, the stress values must be used in conjunction with failure criteria that are further defined by failure theories of the likes of maximum principal stress and maximum shear stress theories.

After the system functions, service conditions, materials and codes have been finalized, the design conditions of pressure and temperature must be established. These refer to the conditions that the system will work under, during its design life. This data is further used to calculate parameters such as piping stresses and piping wall thickness. Allowable stresses can be determined from data available in codes. It is obvious that the calculated stresses for a system at design conditions should be below code allowable limits.

Another important aspect of piping design is the calculation of pipe wall thickness. This assumes great significance considering that the pipe wall must have sufficient thickness to overcome the internal pressure that is generated. Maximum allowable working pressure is another critical parameter related to piping design. Various code equations are employed in order to determine this pressure for a given pipe wall thickness. Also, piping flanges in a piping system are subjected to various forces during operation and must therefore possess the required diameter and thickness to withstand the same. Piping flanges are categorized under

different classes, based on their ability to withstand a given pressure at set temperature conditions. Piping systems are subjected to different load types. These various load cases are taken into account, when calculating the stresses in a piping system. The nomenclature and terminology used in describing the different load cases are explained during the course of the discussion and this is the same as the one used in the CAESAR II stress analysis program. Piping systems experience thermal stresses on account of thermal expansion and contraction. These are caused due to restraints present in the system. They are cyclical in nature and may lead to fatigue and ultimate failure of the piping component. Calculations involving thermal stresses must take into account the stress reduction and stress intensification factors and these are adequately explained in the manual. In view of the wide variety of loads that piping systems are subjected to and also taking into account their various sizes and complexities, a quick and accurate analysis is very much essential. This is where the role of stress analysis software programs assumes importance. There are various softwares that are used for carrying out stress analysis in piping systems, but CAESAR II is considered the industry standard against which all others are measured and compared. CAESAR II incorporates a wide range of capabilities and tools and these are extensively touched upon, during the course of the discussion. To enable a better understanding of how the software works, the various tools and procedures involved in the creation of the stress model are outlined along with a detailed discussion on the procedures adopted to run the analysis and ways of interpreting the output and results.

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.

1.1 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.

2 Pipe Stress Analysis 2

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.

1.2 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.

Introduction 3

1.3 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).

2

32.2 lbm-ftlbf-seccg =

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.

m 2

m1 N = 1 kg 1 s

×

f m 2

m1 kg 1 kg 9.81s

= ×

f m 2

ft1 lb = 1 lb 32.2 sec

×

4 Pipe Stress Analysis 4

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/in2 (psi) kgf/cm2

Commonly used units for stress: Kilo pounds per square inch (ksi) = 103 psi 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.

Introduction 5

Tension Compression

Figure 1.1 Deformation of a material when subjected to tensile and compressive loads

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

o

LL

ε Δ=

Where ΔL 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

Load, P

P

Area Ao

Lo

ΔL/2

ΔL/2

Load, P

P

Area Ao

Lo

ΔL/2

ΔL/2

6 Pipe Stress Analysis 6

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)

- - - - - - - -

- - - - - - - -

1.4 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.

Introduction 7

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

Density (ρ) = 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.

1.4.2 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.: ρ (for liquids and solids) = 1000 × s and

ρ (for gases) = 1.29 × s

Specific gravity is a dimensionless number.

1.4.3 Specific Weight (γ) 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.

8 Pipe Stress Analysis 8

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 ν = - εlateral / εlongitudinal 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.

Introduction 9

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 (α) Thermal expansion and contraction of piping systems is an important aspect of pipe stress analysis. The Linear Coefficient of Thermal Expansion α 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 (ΔL) divided by the original length(Lo).

in./in. mm/mm Units: or

F Co

LLT

α

Δ⎛ ⎞= ⎜ ⎟Δ ° °⎝ ⎠

Table 1.4 gives the Thermal Expansion Coefficients for different materials.

10 Pipe Stress Analysis 10

Table 1.4 Thermal Expansion Coefficients for Selected Materials at 21°C (70°F)

Material α

(mm/mm)/°C α

(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: A. Stress in the rod. B. 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 Mpa.

Solution A.

( )22

-5 20.01mCross section area of the rod = = =7.85 x 10 m

4 4d ππ

-5 2 6

ForceStress = Metal Area

5000 N 1 MPa

7.85 x 10 m 10 Pa = 63.69 MPa

⎛ ⎞= ⎜ ⎟⎝ ⎠

B.

-45

S 65.69 MPaStrain, = 3.14 x 10 /E 2.03 x 10 MPa

mm mmε = =

Introduction 11

Increase in length,

( )-4( )( ) 3.14 x 10 3000 mm 0.942 mmommL Lmm

ε ⎛ ⎞Δ = = =⎜ ⎟⎝ ⎠

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 Δx 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 ρ 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 ε 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 (μ) = 10-6 Mega (M) = 106 Nano (n) = 10-9 Giga (G) = 109 Milli (m) = 10-3

12 Pipe Stress Analysis 12

1.5 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

Introduction 13

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.

14 Pipe Stress Analysis 14

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

Introduction 15

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

ε α 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 5S = E = (0.001)(2.03 x 10 MPa) = 203 MPaε

( )( ) ( )( )( )23CSLoad = S A 203 x 10 kPa / 4 0.025 m 99.65 kN

π= =

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)

S = or S = EE

∈ ∈

16 Pipe Stress Analysis 16

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

Introduction 17

Sample Exercise Problem The vertical leg of a piping system carrying high temperature gas is 8 m long. Calculate the thermal growth of this pipe if the design temperature of the system is 800°C. The pipe material is Austenitic SS with a total expansion of 15 mm / m at 800°C.

Solution

Thermal expansion of the vertical leg

= (Length in m)

15 mm= (8 m)m

= 120 mm or 4.72 in.

LmΔ⎛ ⎞

⎜ ⎟⎝ ⎠⎛ ⎞⎜ ⎟⎝ ⎠

1.7 Summary Stress analysis of piping systems is required to ensure that the system is not overstressed. If the system experiences excessive stresses, it can fail, causing leaks and other safety problems. A good understanding of physical quantities such as force, stress, strain and their units is essential for performing the stress analysis. Tensile testing of material specimens is conducted to determine important material properties such as yield strength and modulus of elasticity. Most often, the allowable stresses specified by the codes are based on the yield strength of the material. Piping systems also need to be configured to handle thermal expansion or contraction. It is important to understand that lack of flexibility in piping systems can result in overstressing.

18 Pipe Stress Analysis 18