Code ASME B31.1 for Piping Stress Analysis

Piping stress analysis can use code ASME B31.1 as reference calculation and design. Code ASME B31.1 for power piping analysis is belonging to the American society of mechanical engineers. In code ASME B31.1 there are empirical formulas that apply to the sustain load, expansion load, and the combination of sustain and expansion load (during operations), and also occasional load are described as the following below:

1. Sustain load

The stresses (S) that occur due to sustain the load such as pressure, weight and other mechanical loads can be expressed by the equation as follows:

(P D_o / 4 t_n) + 1000(0.75 i M_a / Z) ≤ 1.0 S_h

2. Range Thermal Expansion Load

The stresses that occur due to thermal expansion can be expressed as the equation below:

S_E = 1000(i Mc / Z) ≤ S_A + f(S_h – S_L)

3. Combination Load of Sustain load and Thermal Expansion Load

The stresses due to combination of sustained load and thermal expansion load (S_ls + S_E), can be calculated with the equation:

S_ls + S_E = (P D_o / 4 t_n) + 1000(0.75 i M_a / Z) + 1000(i Mc / Z) ≤ (S_h + S_a)

4. Occasional Load

The stresses that occur due to pressure, weight, and other sustain load can be expressed as the equation below:

(P D_o / 4 t_n) + 1000(0.75 i M_a / Z) + 1000(i Mc / Z) ≤ K S_h

Where:

P          = internal design pressure (psi)

Do       = outside diameter (in)

Ma       = Moment due to sustain load (in-lbs)

Mb       = Moment due to occasional load (in-lbs)

Mc       = Range of the moment due to thermal expansion (in-lbs)

Z          = section modulus of the pipe (in³)

tn         = nominal wall thickness of pipe (in)

i           = stress intensification factor

K equal to 1.15 for the occasional load which work less than 1% of the operating period and is equal to 1.20 for the occasional load which work less than 10% of the operating period.

Octahedral Shear Theory

Octahedral shear theory is referred to as Von Mises theory too. Based on octahedral shear theory, failure takes place if the octahedral shear stress within a system is higher when compared with the octahedral shear stress at yield in a specimen put through to uniaxial tension experiment. This is equivalent in text to the assertion of the previous 2 theories besides that octahedral shear stress is applied as consideration with regard to comparability as towards maximum theory stress used in the Rankine principle or maximum shear stress used in Tresca principle.

This octahedral shear stress is explained with regard to the 3 stresses theory as following below:

τ_oct = 1/3 [(S₁-S₂)² + (S₂-S₃)² + (S₃-S₁)²]^0.5

Where:

τ_oct        = Octahedral shear stress

Becuase of the stresses theory explained pertaining to a specimen under uniaxial load previously, that octahedral shear stress on yield in the specimen could be established to become as equation below:

τ_oct = √2 . Sy / 3

Furthermore, the Von Mises theory expresses that failure takes place in a piping system when octahedral shear stress while in the piping system surpasses (√2 . Sy / 3) or more.

Regarding stress analysis associated computations, almost all of the current time piping rules use a modified version of Tresca principle

Maximum Shear Principle

Maximum shear principle is also known as Tresca principle. Based on this principle, failure happens if the maximum shear stress within a system τ_max is higher when compared with the maximum shear stress at yield in a specimen put through to uniaxial tension experiment. Be aware that it is equivalent in text, to the record of the previous principle besides that maximum shear stress is utilized as considerations for comparability as towards maximum stress principle used within the Rankine principle.

Within uniaxial test, maximum shear stress at yield situation of maximum shear experiment provided previous can be

τ_max = 0.5 [(S_L - S_C )² + 4 τ²]^0.5

       = S_L/2 = Sy/2

The Tresca principle therefore only states that failure happens if the maximum shear stress within a system is much more as compared to half the yield stress in the material (Sy). The maximum shear stress within the system must be determined as previous.

It must also be exciting to verify the implication of this principle upon the situation while a pipe is put through to internal pressure. When the circumferential stress caused by internal pressure (S_C) is two times the axial stress (S_L) and the shear stress is not caused specifically (τ = 0) the uppermost level of shear stress in the pipe according to the previous provided method could be following below:

τ_max = 0.5 [(S_L - S_H)² + 4 τ²]^0.5

       = 0.25 S_H

This condition must be fewer as compared with 0. 5Sy based on Tresca principle to get secure design. This results in a various criteria that circumferential tension in a pipe must be fewer than two times the yield stress. The Tresca principle and the design consideration applied in the membrane principle for pipe will be incompatible.

Maximum Stress Principle

The maximum stress principle is usually known as Rankine principle. In accordance with this principle, failure happens if the highest basic principle stress within a system (S₁) is actually higher when compared with the highest tensile stress principle at yield within a specimen put through to uniaxial tension experiment.

Uniaxial tension experiment can be essentially the most popular experiment performed regarding to every material of construction. The tensile stress within a consistent cross-section specimen will be described as yield stress (Sy) at yield with regard to every material as well as usually available. In uniaxial experiment, the utilized load provides increase solely to axial stress (S_L) and also circumferential stress (S_C), radial stress (S_R) and also shear stresses tend to be omitted.

Furthermore, the following equation below can be taken from uniaxial experiment:

S₁ = S_Y                        S₂ = 0              S₃ = 0

S_L = S_Y            S_H = 0              S_R = 0

The highest tensile stress principle at yield is so equivalent to the typically described yield stress. The Rankine principle hence only states that failure happens if the maximum stress principle in a system (S₁) is much more when compared with the yield stress on the material (Sy). The maximum stress principle in the system must be determined as before. This will be exciting to examine the implication of that principle upon the event if a pipe is put through to internal pressure.

According to the membrane principle pertaining to design pressure of pipe, provided that the circumferential stress is a smaller amount compared to the yield stress from the material construction, the design will be secure. That can be recognized that circumferential stress (S_C) caused by external pressure is two times the axial stress (S_L).

The stresses in the pipe depending on the previously assigned formulation could be as following below:

S₁ = (S_L + S_C)/2 + [{(S_L - S_C)/2}² + τ²]^0.5

     = S_L

S₁ = (S_L + S_C)/2 - [{(S_L - S_C)/2}² + τ²]^0.5

     = S_H

The maximum stress principle within the following scenario is S₂ (=S_C). The Rankine principle and the design considerations used in the membrane principle will be consequently appropriate. This principle is commonly applied for calculation of strain thickness for piping layout and pressure vessels employs Rankine principle as considerations to get failure.

Maximum Shear Stress Theory

Maximum shear stress theory states that failure will occur if maximum shear stress equal to or greater than maximum shear stress when perform failure test to uniaxial tension. The formula can be shown as follows:

| σ₁ -  σ₂ |  ≥  | σ_f |

| σ₂ -  σ₃ |  ≥  | σ_f |

| σ₃ -  σ₁ |  ≥  | σ_f |

Where:

σ_f                     = uniaxial tensile strength

σ₁, σ₂, σ₃          = main stresses

Graphical representation of maximum shear stress theory is shown is Figure 1 below:

Figure 1: Graphic of Maximum Shear Stress Theory

Maximum Normal Stress Theory

Maximum normal stress theory states that failure will occur if maximum main stress equal to or greater than maximum normal stress on failure in testing of uniaxial tension. The equation mathematically can be shown as follows:

σ₁ ≥ σ_f             σ₁ ≥ σ_c

σ₂ ≥ σ_f             σ₂ ≥ σ_c

σ₃ ≥ σ_f             σ₃ ≥ σ_c

Where:

σ₁, σ₂, σ₃          = main stresses

σ_f                     = uniaxial tensile strength

σ_c                     = uniaxial compressive strength

According to maximum normal stress equation above, failure will occur when uniaxial tensile or compressive strength is greater than main stress. Graphical representation of maximum normal stress theory is shown is Figure 1 below:

Figure 1: Graphic of Maximum Normal Stress

Stress Equation

At the stress component, the equations of normal stress (σ) and shear stress (τ) as following below:

Equation 1       → σ = (σ_x + σ_y)/2  +  ((σ_x - σ_y) cos 2θ)/2 + (τ_xy sin 2θ)

Equation 2       → τ = - ((σ_x - σ_y) sin 2θ)/2 + (τ_xy cos 2θ)

The equation 1 is derived to θ and make the result equal to zero will be obtained the following stress equation:

Equation 3       → σ₁, σ₂ = (σ_x + σ_y)/2  ±  √(((σ_x - σ_y)/2)² + τ_xy²)

Equation 4       → τ₁, τ₂ = ±  √(((σ_x - σ_y)/2)² + τ_xy²)

σ₁ shows maximum normal stress and σ₂ shows minimum normal stress. Meanwhile τ₁ shows maximum shear stress and τ₂ shows minimum shear stress. From stress equation 3 and stress equation 4, Mohr Circle can be made as shown in Figure 1 below. 

Figure 1: Mohr Circle

Stress Components

In three dimensions, the elements of primary stress can be described in Figure 1 where the three normal stresses are shown in σ_x, σ_y, and σ_z and six shear stresses are shown in τ_xy, τ_yx, τ_yz, τ_zy, τ_zx, and τ_xz in positive direction. The condition of biaxial stress or plane stress is shown in Figure 2 which all of them are shown in positive direction. The stress is depicted in xy plane.

Figure 1: Primary Stress Elements

Figure 2: (a) Biaxial Stress (b) Stress on Inclined Cut

If figure 2 (a) is cut by inclined plane with angle θ to x axis as shown in Figure 2 (b), so by adding all of forces which caused by all of stresses is equal with zero and the equation can be written as follow:

σ = (σ_x + σ_y)/2  +  ((σ_x - σ_y) cos 2θ)/2 + (τ_xy sin 2θ)

τ = - ((σ_x - σ_y) sin 2θ)/2 + (τ_xy cos 2θ)

Shear Stress due to Torque Moment

Shear stress on a straight beam due to torque moment can be shown in the figure 1. This stress will has maximum value on the surface which is located at longest distance from twist axis.

Figure 1: Shear Stress due to Torque Moment

Shear stress due to torque moment (σ) can be calculated using the following equation:

σ = T c / I_p

Where:

T          = torque moment

c          = distance element from center point  

I_p         = section polar of moment inertia

σ          = shear stress due to torque moment 

Normal Stress due to Bending Moment

Normal stress on a straight beam that gets bending moment is shown in the figure below. In contrast to shear stress due to shear force, the magnitude of normal stress due to bending moment will has maximum value in the longest distance from neutral axis.

Figure 1: Normal Stress due to Bending Moment

Normal stress due to bending moment can be calculated using the following equation:

σ = M / I

Where:

I           = section of moment inertia

M         = normal force

σ          = normal stress due to bending moment 

Shear Stress due to Shear Force

In contrast to normal stress due to axial force, the magnitude of shear stress due to shear force is not same for every part of beam. The amount depends on distance from neutral axis. Highest shear stress occurs in area close to neutral axis, on contrary the lowest stress occurred in area located on surface of beam. The figure below shows the shear force on straight beam.

Figure 1: Shear Stress due to Shear Force

Shear stress due to shear forces can be calculated using the following equation:

Where:

I           = section of moment inertia

t           = width of cross section

y          = distance element to neutral axis

τ          = shear stress

Normal Stress due to Axial Force

Normal stress which acting on straight beam due to axial force will has maximum value on the plane perpendicular to axis because the plane has smaller area than other planes. The following below is the equation of normal stress due to axial force:

σ = P / A

Where:

A         = plane area

P          = normal force

σ          = normal stress

The following below is figure that show normal stress due axial force.

Figure 1: Normal Stress due to Axial Force

Stress and Strain Analysis

Stress is the result of forces acting in specified element parallel or perpendicular to the sliced plane of element. The force perpendicular to the plane is called normal force and generates normal stress. Meanwhile, the force parallel to the plane is called shear force and produce shear stress. Both of them are shown in the following equation:

σ = P / A   and   τ = V / A

Where:

σ          = normal stress

τ           = shear stress

P          = normal force

V         = shear force

A         = plane area

The external force which is given on the element will cause internal reaction force that called as deformation. Deformation can be a change in geometry and dimension. Deformation that occurs on a unit length is called strain and can be shown in the following equation:

ε = (l – l_o) / l_o

Where:

ε          = strain

l_o          = length before deformation

l           = length after deformation

Tensile test on specimen material can be carried out to determine the strength of material against forces. This specimen is slowly pulled until failure (fracture). Tensile force and length specimen is always measured continuously during tensile test.

Figure 1: Specimen of Tensile Test

Based on the result of test, can observed and get conclusion of properties of material which show relationship between stress and strain. This linear relationship is expressed in Hooke Law as shown in the following equation:

E = σ / ε

Analysis and diagram of stress and strain in mild steel and brittle steel can be seen in stress analysis in piping system.

Stress Analysis in Piping System

Stress in piping system must be analyzed to know the strength of overall pipe and its components. Stress analysis is required to determine material strength against all of load which occurs in piping system. Stress will be most important parameter to know the strength of material. These parameters can be known by testing material strength and show it into graphic diagram stress and strain. Generally there are two types of stress-strain diagram; stress-strain diagram for mild steel and brittle steel. Both figures can be seen below.

Figure 1: Stress-Strain Diagram

On both figures above, there is straight line profile at the beginning of loading which indicate the existence of elastic deformation phenomenon in steel material. The end point of this straight line is called the yield point. Phenomenon of plastic deformation is initiated by curvature of the curve which can be known as ludders band. The highest point on the stress-strain diagram shows the ultimate strength of material. The ultimate point is a stress value of steel material at begin experience phenomenon of necking. The end boundary point in diagram show the failure point of material and show stress value when material begins to crack.

There are differences of curve characteristic in mild steel and brittle steel.  It is influenced by properties of both materials. Mild steel tend to has relatively long strain in the elastic region and plastic region, while brittle steel tend to has relatively short strain. In the piping system, it is used as the basic consideration to select pipe material with optimum performance. 

Allowable Stress Based on ASME B31.3

Allowable stress of material which is used is an important parameter in the stress analysis of piping system. Working stress in the piping system should not exceed allowable stress of that material based on code and standard. In the piping system, stress ratio is generally used. Stress ratio is comparison between actual working pressure in the piping system and allowable stress based on code and standard.

Allowable stress has different value in each code. For code ASME 31.3, the pipe material can be expressed in safe condition when the stresses which are experienced by the material meet the following criteria:

1.      Sustain Load

The resultant of longitudinal stress (S_L) in each component of piping system due to sustain load such as internal load and weight of pipe load, should not exceed the value of multiplication between Sh and W. Sh is allowable stress of pipe material at maximum temperature of operating condition. W is reduction factor of welding joint in the manufacturing process of pipe. W has value equal with 1.0 if longitudinal connection type is used in the manufacturing process of pipe.

Displacement stress range (S_E) in piping system should not exceed allowable displacement stress range (S_A) which can be calculated as follow:

S_A = f(1.25 Sc + 0.25 Sh)

Where:

F              = stress range factor

Sc             = allowable stress of pipe on minimum working temperature.

Sh                        = allowable stress of pipe on maximum working temperature

2.      Occasional Load

The resultant of longitudinal stress due to sustained load and all of loads which are caused by occasional load such as wind load and seismic load, should not exceed the value of 1.33 is multiplied with allowable stress of pipe on maximum working temperature (Sh). For casting material, the Sh value must be multiplied by factor quality casting (Ec). Loading due to wind and seismic are not generally occur simultaneously, so in the pipe stress analysis, the calculation is performed only to one of that loads in accordance with environmental condition.  

3.      Thermal Expansion Load

The stress that occurs due to thermal expansion load is stress range from the resultant of bending stress and torsional stress due to thermal expansion. Range expansion stress should not exceed allowable stress S_A according to the following equation:

S_E = √(Sb² + St²) ≤ f(1.25 Sc + 0.25Sh)

The value of Sb and St are bending stress and torsional stress respectively.

Stress in Piping System

The theory of stress on the piping system is development of the theory of stress that has developed in mechanics. Definitions that are used as force, moment, stress, strain, and others are same with the definitions used in mechanics. Stress that occurs in piping system can be caused by internal pressure, external pressure, weight of pipe itself, and thermal expansion load in the piping system because of temperature differences. At both pipe stress analysis and mechanics, some terms of stress had been known based on direction as follow:

1. Longitudinal stress, the stress along the axis parallel to the direction of the pipe.
2. Circumferential stress or hoop stress, the stress with circular direction and direct with circular wall pipe.
3. Radial stress,   the stress with a direction parallel to the straight line that runs to the axis of pipe and out through the wall pipe.

Stresses that occur in piping system will be totaled and analyzed to take the largest and dominant value of stress to be used as input for stress analysis on piping system. Based on stress equation, from the three type stress above, radial stress has relatively small value, so sometimes that value can be ignored. Longitudinal stress has the largest and dominant value of stress than others so it can be included as an input for the stress analysis. Stress analysis is performed at all points along the pipeline to generate distribution of stress value along the pipeline.