Bolt Torque Formula

24
GUIDELINES TO DERIVATION OF BOLT TORQUES This document has been prepared by Peter Kempster as an appendix to AS 2885.1, and/or to be included in a future handbook document to AS 2885. It is an adjunct to Issue Paper 1.9 It contains the basis for determining the appropriate bolt torque to achieve flange sealing.

Transcript of Bolt Torque Formula

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GUIDELINES TO DERIVATION OF BOLT TORQUES

This document has been prepared by Peter Kempster as an appendix to AS 2885.1, and/or to be included in a future handbook document to AS 2885. It is an adjunct to Issue Paper 1.9

It contains the basis for determining the appropriate bolt torque to achieve flange sealing.

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APPENDIX XYZ TO AS 2885.1

GUIDELINES FOR THE TENSIONING OF BOLTS IN THE FLANGED JOINTS OF PIPING SYSTEMS

1. INTRODUCTION This Appendix has been written to provide a guideline basis for the derivation of the value of torque necessary to provide adequate tension in the bolts of a flanged joint for an effective gasket seal after the nuts have been tightened up by a torque wrench. It also provides information relating to the consideration of applied loads during operation as this aspect of bolt tension is related in some instances to the remaining allowable stress after pre-tensioning the bolt prior to being put into service. Current Standards limit the design strength of bolts to a relatively low value of stress, typically 24% SMYS for ASTM A 193-B7 steel bolts. The construction industry has found that when the bolts of some flanged joints are tensioned to the full permitted stress levels the gaskets do not provide a tight seal during service. These guidelines provide a basis for calculating the value of torque to be applied to the nuts based on the gasket/bolt manufacturer’s recommendation of permitted bolt stresses for effective gasket sealing. The basis used in the calculation of the worked example is bolt stress. As an alternative to bolt stress, gasket compression load can be used, by relating the load and effective stress area of the gasket back to the total bolt load. In general the bolt stresses suggested by the manufacturer will be higher than the values permitted by current standards. These guidelines recognise therefore that additional precautions should be taken to calculate the sealing and operating bolt stresses to ensure that bolt yielding does not occur. In this respect it is considered necessary that the design of the piping take into account fully all of the applied loads that may exist during the operating life of the pipeline system and in particular the stress levels during installation. Under some conditions it may not be possible to achieve the manufacturer’s recommended residual bolt loads due to high installation stress levels. A worked example is provided in section 15 of this Appendix to demonstrate the methodology of these guidelines. 2. NOTATION Throughout this Appendix the following notation has been adopted:

Symbol Description Units µ Coefficient of friction λ Lead angle of the helix degree α Angle between flank of thread and degree

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plane perpendicular to helix π Constant ƒb Recommended bolt stress psi ƒy Yield stress MPa A Nominal bolt area mm2 Ab Stress area of bolt mm2 Ag Internal area at gasket force mm2 Ap Internal area at gasket force mm2 Ar Root area of bolt in2 c Radius to outermost fiber mm C Celcius degree d Nominal bolt diameter inch db Minor diameter inch dc Mean radius of nut face inch dp Pitch diameter of bolt inch F Factor F Applied force lb Fd Design factor fs Factor of safety G Reaction load diameter mm or inch h Projected thread height J Polar moment of inertia of cross

section mm4

k Constant kb Stiffness of bolt material N/m Kf Stress intensification factor kj Stiffness of joint material N/m ksi Stress kips/inch2 L Lead of screw inch M Bending moment in.lb N Number of bolts in a joint NPS Nominal pipe size inch P Load capacity of bolt N p Pitch of thread p Static internal fluid pressure MPag pd Dynamic internal fluid pressure

increment MPag

psi Pressure or stress lb/in2 Pav Average load N Pd Dynamic load N Pext External load N Pi Initial load N Pm Manufacturer’s recommended

load N

Pp Load from test pressure N Ps Force in bolt from fluid pressure N

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in joint Q Axial load in bolt N or lb S1 Stress in bolt from tensile load MPa S2 Stress in bolt from applied torque MPa Sa, Sall Allowable stress MPa Sb Stress in bolt ksi Sc Stress in joint from applied loads MPa Sd Stress in joint from surge MPa SE Stress due to expansion MPa Se Endurance limit MPa Sg Compressive stress in bolt to

compress gasket MPa

Sp Static stress in bolt from pressure in joint

MPa

Sr Alternating stress MPa Ss Shear stress MPa St Total stress MPa Su Ultimate tensile strength MPa Sy Yield stress MPa SMYS Specified minimum yield stress MPa T Torque N.m or lb.ft Tt Torque on threads N.m or lb.ft TPI Threads per inch

3. THE EFFECT OF THE GASKET ON THE LOAD CARRIED The load on the bolt depends on the initial tension Pi and the external load Pext. The load on the bolt also depends on the relative elastic yielding (springiness) of the bolt and the connected members as follows:

• If the connected members are very yielding compared with the bolt the resultant load on the bolt Pav will closely approximate the sum of the initial tension Pi and the external load Pext.

• If the bolt is very yielding compared with the connected members the resultant

load will be either the initial tension or the external load whichever is the greater.

To estimate the resultant load on the bolt the following formula can be used:

Pav = Pi + (kb / 2(kb + kj)) Pext. For flanged joints with a flexible gasket the value in brackets approaches unity, for a solid gasket such as a metallic ring jointed gasket the bracketed value is small and the resultant load is due mainly to the initial tension Pi (or to Pext if it is greater than Pi).

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4. STRENGTH CAPACITY OF A BOLT It is relatively easy to calculate the static tensile strength of a bolt. The load may be assumed to be uniformly distributed across the root section of the bolt, and stress concentration can be neglected. The stress area of the bolt can be obtained from the dimensions of the standard to which the bolt is manufactured and used with the yield strength ƒy of the bolt material to determine the load carrying capacity P of the bolt as follows: P = Ar ƒy. 5. INITIAL LOAD The initial load is highly indeterminate but can be estimated from the following formula which is attributed to J. H. Barr: Pi = k d, where k = a constant and d = the nominal diameter of the bolt in inches. The constant k for a “steamtight joint” is 16000. The judgement of the person applying the force with a wrench cannot be predicted accurately. Theoretically it is possible to relate the tightening load to the dimensions of the bolt screw thread and the applied torque. In practice there is considerable error in the calculation of the torque required, because of the wide variation of the effect of surface finish and lubrication of the sliding components on the torque required to overcome frictional resistance. 6. RELATIONSHIP BETWEEN APPLIED TORQUE AND TENSION The torque required to turn the nut can be related to the axial load in the bolt by the following formula: T = Q dp F / 2, where Q = the axial load, dp = pitch diameter of the screw and T = the applied torque. The factor F is a function of the lead angle of the helix λ, the angle between the flank of the thread and a plane perpendicular to the helix of the thread α, the coefficient of friction µ and dc the mean radius of the nut face as follows:

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F = ([(cos α tan λ + µ) / (cos α - µ tan λ)] + µ dc / dp) where L = the lead of the screw thread and tan λ = L / π dp. Alternatively the torque can be calculated using the simplified screw jack formula of A P Farr, rewritten using the notation of these guidelines, as follows: T = Q / 12 [L / (2 π) + µ dp / (2 cosα) + µ dc / 2]. Coefficients of friction vary between 0.06 and 0.40. These are practically independent of load and vary only slightly with different combinations of materials and rubbing speed. 7. IMPOSED LOADS ON A BOLT Loads may be separated into two categories, loads imposed during installation and externally applied loads after installation. The following is a list of the loads imposed on the bolts of a flanged joint during installation:

• Load on a bolt imposed by the connected piping from misalignment (note this load should be either eliminated or minimised by careful construction)

• Load on a bolt to compress the jointing gasket (bolt pretension)

The following is a list of the loads imposed on the bolts from operating conditions:

• Static load from internal pressure • Dynamic load from internal pressure • Loads applied externally from connected piping.

8. COMBINED STRESSES The stresses in a bolt will also fall into installation and operating stress categories. 8.1 Stresses During Installation During installation the minor diameter cross-section of the portion of the screw thread of the bolt between the nut and the flange will be subjected to a biaxial stress condition. This stress condition is comprised of a tensile stress due to the axial force and a shear stress only due to the applied bolting torque. The stress S1 in the bolt from the tensile load is as follows:

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S1 = Q / Ar. The stress S2 in the bolt from the applied torque is as follows: S2 = Tt c / J. Where Tt = torque on the threads, c = distance of neutral axis to the extreme fibre and J = Polar moment of inertia. The torque on the threads Tt is: Tt = Q dp [(cos α tan λ + µ) / (cos α - µ tan λ)] / 2. The maximum shear stress level in the bolt on the minor diameter Ss can be calculated from the following formula: Ss, max = ((S1 / 2)2 + (S2)2)0.5 According to the maximum shear theory the bolt will yield when the maximum shear stress Ss is equal to the shear yield strength of the material which is equal to half the yield stress in simple tension Sy / 2. The maximum tensile (principal) stress during torquing is: St = (S1 / 2) + Ss. The bolts should have sufficient strength to withstand the required applied torque during installation. After torquing has been completed the shear stress from the torque will cease to exist. 8.2 Stresses During Operation The design of the bolts should also have adequate strength to withstand the applied loads during operation. The stress in a bolt Sg to keep the gasket in compression can be calculated from the manufacturers minimum recommended bolt load as follows:

Sg = Pm / (Ar N) The static operational stress Sp in the bolt from internal fluid pressure p is given as follows: Sp = Ps / (Ar N),

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where Ps = p Ag (Ag is the internal area using the diameter at the location of the gasket force). The dynamic stress in the bolt from fluid pressure Sd will be a percentage of the static stress, as determined by analysis: Sd = Pd / (Ar N), Where Pd = p Ap (Ap is the internal area at the gasket force location). The loads from connected piping will be determined from analysis and the stress in the bolt Sc will be determined from these loads. The total stress St in the bolt from the operational loads will vary depending upon the type of gasket being used in the bolted joint. For flexible gaskets the total stress will be the sum of the individual stresses as follows: St = Sg + Sp + Sd + Sc. For rigid gaskets the total stress will be either:

St = Sg or St = Sp + Sd + Sc,

which ever is the greater. The required load capability of the bolt can then be back calculated from the greater of Ss and St above. 8.3 Stresses During Hydrostatic Pressure Test The design of the bolts should also have adequate strength to withstand the applied loads during the hydrostatic pressure test. The hydrostatic test pressure produces the following flange load: Pp = p π G2 / 4, and the stress in a single bolt is: Sp = Pp / (Ab / N).

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For flexible gaskets the total stress will be the sum of the individual stresses as follows: St = Sg + Sp. For rigid gaskets the total stress will be either:

St = Sg or St = Sp,

whichever is the greater. 9. FATIGUE FROM OPERATING LOADS It can be shown from the Soderberg triangle that the following is true:

fs = Sy / (Sav + (Sy / Se) Kf Sr), where Sr = (Smax – Smin) / 2 Sav = (Smax + Smin) / 2 Stress range = Smax – Smin. The equation above for fs effectively states that the total stress is the sum of the weighted stress reversed component and the steady stress component. The equation above can be used to calculate the total stress range due to the cyclic load. The total stress is given by: St = Sav + (Sy / Se) Kf Sr. Values of the endurance limit Se lie within the range 0.45 to 0.6 Su, with an upper limit of about 100 ksi, a value 0.5 Su is commonly used in design. 10. THE EFFECTS OF PIPING LOADS ON FLANGED JOINTS For routine design on the effects of loading on flanged joints other than internal pressure, i.e. loads from the connected piping, the method of M W Kellogg Company is provided. M W Kellogg found that, with a properly pretightened flange, the bolt load changes very little when a moment is applied to it.

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Further, M W Kellogg have found from experience that it is satisfactory to first calculate the maximum load per inch of gasket circumference due to the applied longitudinal bending moment and force. Then the internal pressure equivalent to this loading is then determined. The formula proposed by M W Kellogg is as follows: Pe = (16 M / (π G3) + 4 F / (π G2). The equivalent force in each bolt F = Pe Ag / N, and the stress in the bolt can be calculated as for other pressure load calculations i.e:

Sc = F / Ar. Simplistically if the mean moment resistance of the bolts about a line tangential to the reaction load diameter G is taken and if the load in the bolts is taken to be equal, then it can be shown that the equivalent pressure from an applied moment is: Pem = 8 M / (π G3). It is believed therefore that the M W Kellogg formula above may be conservative for the application of applied moments to the flanged joint. Regarding torsion, if the frictional resistance of the gasket is ignored and all of the bolts are put in shear it can be shown that the shear stress in the bolts is: Ss = 8 T / (π dp

3 N). Stresses can then be combined in accordance with the theory in section 8.1. 11. COEFFICIENT OF FRICTION There is a wide variance of the values of coefficient of friction for the calculation of applied torque. These variations are caused by a number of factors such as the condition of the threads the condition of the flange to the nut bearing surface and the type of lubricant used. The following table provides some indicative values for various conditions:

Coefficient of Friction for screw threads Average coefficient of

friction µ

Condition Starting Running

High grade materials and workmanship and best running conditions

0.14

0.10

Average quality of materials and workmanship and average running conditions

0.18

0.13

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Poor workmanship or very slow and infrequent motion with indifferent lubrication or newly machined surfaces

0.21

0.15

It is not possible to accurately determine a value of the coefficient of friction existing at site, some conservatism is therefore recommended in the selection of the value used in the calculations unless the conditions have been well established. It does not necessarily follow that the coefficient of friction of the lubricant is the same as the coefficient of friction of the moving components of the joint. 12. COMPONENTS OF THE FLANGE ASSEMBLY All of the components of the flange assembly should be designed to carry the required load capacity of the bolts. The other components of the assembly to be considered in the design of the flanged joint are as follows:

• The nut threads • The bolt threads • The gaskets • The flanges

If the flange is purchased as an assembly in accordance with a recommended standard at the appropriate design pressure and temperature then it may be assumed that the strength of the flange components will match the strength of the bolts. Whilst these guidelines provide a basis to review the strength of the bolts of the flanged joints, they do not provide any basis for reviewing the strength of the flange, the nuts or the gaskets. 13. DERATING OF ALLOWABLE STRESS AT ELEVATED TEMPERATURE The upper limit of temperature for the standard is 200oC fluid temperature. These guidelines only apply to steel bolts to ASTM standards up to 200oC. For bolt temperatures up to 120oC no de-rating of allowable bolt stress level is required. For temperatures between 120oC and 200oC the permitted allowable bolt stress level shall be de-rated in accordance with an approved standard.

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14. ALLOWABLE STRESS LIMITS Evaluation of loading of the bolts of flanged joints is treated as being similar to the evaluation of the pipe itself. On this basis the allowable stress limits in the bolts for steel materials where referenced and where otherwise provided by these guidelines are as follows:

Load Case Load Type Stress Type Stress Limit Reference Installation Torque + Axial Shear 45% Yield

(90% Shear Stress)

These guidelines

Installation Torque + Axial Tension 90% Yield These guidelines

Installation Residual (Pretension)

Axial 2/3rd Yield These guidelines

Hydrostatic Pressure Test

Sustained Axial <100% Yield AS 1978 Cl 4.3.3

Operation Sustained Axial 72% Yield AS 2885.1 Cl 4.3.6.5

Operation Cyclic stress range

Axial 72% Yield AS 2885.1 Cl 4.3.6.5

Operation Occasional Axial 110% Yield Fd

AS 2885.1 Cl 4.3.6.6

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15. WORKED EXAMPLE It is required to install an ASME B16.5 flange assembly using a 24 inch NPS Class 150 flange with raised face flanges. The bolt material is ASTM A 193-B7 material requiring 20 number 1¼ inch bolts. The yield strength of the bolts is 105 ksi, and the ultimate tensile strength of the bolts is 125 ksi. The manufacturer has advised that for a compressed fibre gasket the recommended bolt tension be such as to produce a bolt stress of 45 ksi in the minor area in order to provide the necessary pretension for an efficient seal for installation. The reaction load diameter G of the gasket has been taken to be 666.76 mm. The screw threads of the bolt have been stated to be 8UN with an external diameter of 1¼ inches, a pitch diameter of 1.1688 inches, a minor diameter of 1.0966 inches, a stress area of 0.9985 in2 and a dimension h = 0.866025 p. The vee formation of the screw thread is 60o and the relationship between the lead angle to the pitch diameter dp and the lead L is tan λ = L / π dp. The bolts are single screw thread (L = 1 / p) with 8 TPI. The width across the flats of the nut face is 1.875 inches. It is assumed that the coefficient of friction is 0.15 for the threads and 0.15 for the nut face. It is also assumed that the resultant load on the bolt is the sum of the gasket compression load and the external load. The maximum internal operating pressure is 1.5 MPag, the allowance for liquid surge is 10% of the operating pressure. The flange is subject to a sustained bending moment of 25,000 N.m and a thermal bending moment of 120,000 N.m from the connected piping. The thermal moment is cyclic in nature. The hydrostatic test pressure is 1.5 times the maximum internal operating pressure. 15.1 The Estimated Load for Tightness The estimated load Pi for a tight seal is: Pi = k d = 16,000 1.25 = 20,000 lb Pi = 88,960 N The corresponding stress in the bolt is: Sb = 20,000 / 0.9985 = 20.03 ksi The manufacturer recommends a pretension of 45 ksi or 2.25 times this value. 15.2 The Applied Load Q The applied load Q in the minor area is:

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Q = ƒb Ab = 45,000 π 1.09662 / 4 = 42,500 lb

Q = 189,000 N 15.3 The Applied Torque T The constants tan λ and cos α are:

tan λ = L / π dp = (1 / 8) / (π 1.1688) = 0.034 cos α = cos (60/2) = 0.866. Taking the mean radius of the nut face equal to the mean of the bolt diameter and width across the flats of the nut then: dc = (1.875 + 1.25) / 2 = 1.5625 inches. The applied torque T is: T = Q dp ([(cos α tan λ + µ) / (cos α - µ tan λ)] + µc dc / dp) / 2

T = 189,000 (1.1688 25.4) ([(0.866 0.034 + 0.15) / (0.866 – 0.15 0.034)] + .15 1.5625 / 1.1688) / 2 / 1000

T = 1147.34 N.m

T = 846.28 lb.ft. Alternatively, as the tangential force acts at the pitch radius, using the Farr formula: T = Q / 12 [L / (2 π) + µ dp / (2 cosα) + µc dc / 2] T = 42,500 / 12 [1 / 8 / (2 π) + .15 1.1688 / (2 cos 30) + .15 1.5625 / 2] T = 843.99 lb.ft. 15.4 Combined Stress Level During Installation During tightening the maximum combined shear stress level can be obtained as follows: S1 = 45 ksi = 310.35 MPa (45 ksi from the manufacturer) J = π db

4 / 32 = π 1.09664 / 32 = 0.142 in4 = 59,092 mm4

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Tt = 189,000 (1.1688 25.4) [(0.866 0.034 + 0.15) / (0.866 – 0.15 0.034)] / 2 / 1000 Tt = 584.77 N.m S2 = T c / J = 584.77 1000 (1.0966 25.4 / 2) / 59,092 = 137.82 MPa Ss = ((S1 / 2)2 + (S2)2)0.5 = ((310.35 / 2)2 + (137.82)2)0.5 Ss = 207.54 MPa Sy / 2 = 105 / 2 ksi = 52.5 ksi

Sy / 2 = 362.07 MPa Ss = 57.32 % yield in shear during tightening, and reduces to 310.35 MPa or 42.86 % yield in tension after tightening. The maximum combined tensile stress St is: St = (310.35 / 2) + 207.54 = 362.72 MPa St = 50.09% yield in tension during torquing. 15.5 Stress Level During Hydrostatic Pressure Test As the gasket is a flexible gasket the stress in the bolts Sg from the preload will add to the hydrostatic pressure test load. It is assumed that the piping is well supported during testing and that there are no additional imposed piping loads. Pp = 1.5 1.5 π 666.762 / 4 = 785,618 N Sp = Pp / (Ab / N) = 785,618 / (0.9985 25.42) / 20 = 60.98 MPa St = 310.35 + 60.98 MPa St = 371.33 MPa

or 51.28 % yield in tension. 15.6 Sustained Stress Level During Operation As the gasket is a flexible gasket the stress in the bolts Sg from the preload will add to the operating loads. During operation, the operating stresses are Sp + Sd + Sc. Note that the dynamic load from surge has been conservatively included in this sustained load case.

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Sg = 310.35 MPa Pp = 1.5 π 666.762 / 4 = 523,745 N Sp = Pp / Ab = 523,745 / (0.9985 25.42) / 20 = 40.65 MPa Sd = 0.1 40.65 = 4.07 MPa The pressure equivalent to the bending moment is: Pe = 16 M / (π G3) = 16 25,000 1000 / π 666.763 Pe = 0.4295 MPa. The applied load from the bending moment in each bolt is: F = 0.4295 π 666.762 / (4 20) F = 7,498 N. The stress in each bolt from the moment is: Sc = 7,498 / (0.9985 25.42) = 11.64 MPa. The total stress is: St = 310.35 + 40.65 + 4.07 + 11.64 MPa St = 366.71 MPa. or 50.64 % yield in tension Sa = 0.66 105 = 69.3 ksi = 477.93 MPa

15.7 Fatigue Stress Level During Operation As the gasket is a flexible gasket the stress in the bolts Sg from the preload will add to the operating loads. During operation the operating stresses are Sp + Sd + Sc, with Sc comprising the static component and the cyclic component of stress. The stress intensification factor of the vee thread is taken to be 2.5. The pressure equivalent to the bending moment is: Pe = 16 M / (π G3) = 16 120,000 1000 / π 666.763

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Pe = 2.0618 MPa. The applied load from the bending moment in each bolt is: F = 2.0618 π 666.762 / (4 20) F = 35,995 N. The stress in each bolt from the moment is: Sc = 35,995 / (0.9985 25.42) = 55.88 MPa. The stress due to the cyclic load Sr is equal to Sc. The steady stress is the same as that in 15.6 above. The maximum stress is: Smax = 366.71 + 55.88 = 422.59 MPa The minimum stress is: Smin = 366.71 – 55.88 = 310.83 MPa The average stress is: Sav = (422.59 + 310.83) / 2 = 366.71 MPa The total stress is: St = Sav + (Sy / Se) Kf Sr St = 366.71 + (724.14 / 0.5 862.07) 2.5 55.88

St = 601.41 MPa or 83.05 % yield in tension. The stress range from the alternating stress = stress range SE = Smax – Smin or = 2 Sr SE = 2 55.88 SE = 111.76 MPa. The allowable stress range is:

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Sall = 0.72 724.14 = 521.38 MPa. The following table summarises these results:

Table 15.0 Summary of Stress Levels

Case Type Total

value of stress

MPa

Total %SMYS

Total stress

excluding Residual

MPa

%SMYS

Installation Torque 207.54 - - - Installation Torque 362.72 - - - Installation Residual -

Pretension 310.35 42.86 310.35 42.86

Hydro Sustained 371.33 51.28 60.98 8.42 Operation Sustained 366.71 50.64 56.36 7.78 Operation Cyclic 601.41 83.05 291.06 40.19

It can be seen from the table above in this example that the bolt pretension comprises the majority of the total stress in the flanged joint apart from the cyclic loading. For stress compliance the following table summarises the calculated values:

Table 15.1 Summary of Stress Compliance

Case Type Value of

stress

MPa

Stress Limit

MPa

Allowable Stress

MPa

% Allowable

Installation Torque 207.54 90% Shear 362.07 57.32 Installation Torque 362.72 90% Yield 651.72 55.66 Installation Residual 310.35 2/3rd Yield 482.76 64.29

Hydro Sustained 371.33 100% Yield

724.14 51.28

Operation Sustained 366.71 72% Yield 521.38 70.34 Operation Cyclic 111.76 72% Yield 521.38 21.44

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APPENDIX XYZ2 TO AS 2885.1

GUIDELINES FOR THE TENSIONING OF BOLTS IN THE FLANGED JOINTS OF PIPING SYSTEMS

1. INTRODUCTION This Appendix has been written to provide a guideline basis for the calculation of the value of torque necessary to provide adequate tension in the bolts of a flanged joint for an effective gasket seal after the nuts have been tightened up by a torque wrench. Current Standards limit the design strength of bolts to a relatively low value of stress, typically 24% SMYS for ASTM A 193-B7 steel bolts for example. The construction industry has found that when the bolts of some flanged joints are tensioned to the full permitted stress levels the gaskets do not provide a tight seal during service. These guidelines provide a basis to calculate the value of torque to be applied to the nuts based on the gasket/bolt manufacturer’s recommendation of permitted bolt stresses for effective gasket sealing. The basis used in the calculation of the worked example is bolt stress. As an alternative to bolt stress, gasket compression load can be used, by relating the load and effective stress area of the gasket back to the total bolt load. In general the bolt stresses suggested by the manufacturer will be higher than the values permitted by current standards. Under some conditions it may not be possible to achieve the manufacturer’s recommended residual bolt loads due to high installation combined shear stress levels. The most important aspect of obtaining realistic torque values is to establish an accurate value of coefficient of friction used in the calculation. A worked example is provided in section 6 of these guidelines. 2. RELATIONSHIP BETWEEN APPLIED TORQUE AND TENSION The torque required to turn the nut can be related to the axial load in the bolt by the following formula: T = Q dp ([(cos α tan λ + µ) / (cos α - µ tan λ)] + µc dc / dp) / 2, where Q = the axial load in N, dp = pitch diameter of the screw in mm and T = the applied torque in N.m. The factor in brackets is a function of the lead angle of the helix λ, the angle between the flank of the thread and a plane perpendicular to the helix of the thread α, the coefficient of friction µ for the thread, µc for the nut face and dc the mean radius of the nut face.

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L = the lead of the screw thread and tan λ = L / π dp. Alternatively the torque can be calculated using the simplified screw jack formula of A P Farr (imperial units), as follows: T = Q / 12 [L / (2 π) + µ dp / (2 cosα) + µc dc / 2]. Coefficients of friction vary between 0.06 and 0.40. These are practically independent of load and vary only slightly with different combinations of materials and rubbing speed. 3. COMBINED STRESSES During installation the minor diameter cross-section of the portion of the screw thread of the bolt between the nut and the flange will be subjected to a biaxial stress condition. This stress condition is comprised of a tensile stress due to the axial force and a shear stress only due to the applied bolting torque. The stress S1 in the bolt from the tensile load is as follows: S1 = Q / Ar. The stress S2 in the bolt from the applied torque is as follows: S2 = T c / J. The maximum shear stress level in the bolt on the minor diameter Ss can be calculated from the following formula: Ss, max = ((S1 / 2)2 + (S2)2)0.5 According to the maximum shear theory the bolt will yield when the maximum shear stress Ss is equal to the shear yield strength of the material which is equal to half the yield stress in simple tension Sy / 2. The bolts should have sufficient strength to withstand the required applied torque during installation. After torquing has been completed the shear stress from the torque will cease to exist. 4. COEFFICIENT OF FRICTION There is a wide variance of the values of coefficient of friction for the calculation of applied torque. These variations are caused by a number of factors such as the condition of the threads the condition of the flange to the nut bearing surface and the type of

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lubricant used. The following table provides some indicative values for various conditions:

Coefficient of Friction for screw threads Average coefficient of

friction µ

Condition Starting Running

High grade materials and workmanship and best running conditions

0.14

0.10

Average quality of materials and workmanship and average running conditions

0.18

0.13

Poor workmanship or very slow and infrequent motion with indifferent lubrication or newly machined surfaces

0.21

0.15

It is not possible to accurately determine a value of the coefficient of friction existing at site, some conservatism is therefore recommended in the selection of the value used in the calculations unless the conditions have been well established. It does not necessarily follow that the coefficient of friction of the lubricant is the same as the coefficient of friction of the moving components of the joint. 5. ALLOWABLE STRESS LIMITS Evaluation of loading of the bolts of flanged joints is treated as being similar to the evaluation of the pipe itself. On this basis the allowable stress limits in the bolts for steel materials provided by these guidelines are as follows:

Load Case Load Type Stress Type Stress Limit Installation Torque + Axial Shear 45% Yield

(90% Shear Stress)

Installation Residual (Pretension)

Axial 60% Yield

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6. WORKED EXAMPLE It is required to install an ASME B16.5 flange assembly using a 24 inch NPS Class 150 flange with raised face flanges. The bolt material is ASTM A 193-B7 material requiring 20 number 1¼ inch bolts. The yield strength of the bolts is 105 ksi, and the ultimate tensile strength of the bolts is 125 ksi. The manufacturer has advised that for a compressed fibre gasket the recommended bolt tension be such as to produce a bolt stress of 45 ksi in the minor area in order to provide the necessary pretension for an efficient seal for installation. The screw threads of the bolt have been stated to be 8UN with an external diameter of 1¼ inches, a pitch diameter of 1.1688 inches, a minor diameter of 1.0966 inches, a stress area of 0.9985 in2 and a dimension h = 0.866025 p. The vee formation of the screw thread is 60o and the relationship between the lead angle to the pitch diameter dp and the lead L is tan λ = L / π dp. The bolts are single screw thread (L = 1 / p) with 8 TPI. The width across the flats of the nut face is 1.875 inches. It is assumed that the coefficient of friction is 0.15 for the threads and 0.15 for the nut face. 6.1 The Applied Load Q The applied load Q in the minor area is: Q = ƒb Ab = 45,000 π 1.09662 / 4 = 42,500 lb

Q = 189,000 N 6.2 The Applied Torque T The constants tan λ and cos α are:

tan λ = L / π dp = (1 / 8) / (π 1.1688) = 0.034 cos α = cos (60/2) = 0.866. Taking the mean radius of the nut face equal to the mean of the bolt diameter and width across the flats of the nut then: dc = (1.875 + 1.25) / 2 = 1.5625 inches. The applied torque T is: T = Q dp ([(cos α tan λ + µ) / (cos α - µ tan λ)] + µc dc / dp) / 2

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T = 189,000 (1.1688 25.4) ([(0.866 0.034 + 0.15) / (0.866 – 0.15 0.034)] + .15 1.5625 / 1.1688) / 2 / 1000

T = 1147.34 N.m

T = 846.28 lb.ft. Alternatively, as the tangential force acts at the pitch radius, using the Farr formula: T = Q / 12 [L / (2 π) + µ dp / (2 cosα) + µc dc / 2] T = 42,500 / 12 [1 / 8 / (2 π) + .15 1.1688 / (2 cos 30) + .15 1.5625 / 2] T = 843.99 lb.ft. 6.3 Combined Stress Level During Installation During tightening the maximum combined stress level can be obtained as follows: S1 = 45 ksi = 310.35 MPa (45 ksi from the manufacturer) J = π db

4 / 32 = π 1.09664 / 32 = 0.142 in4 = 59,092 mm4 S2 = T c / J = 1147.34 1000 (1.0966 25.4 / 2) / 59,092 = 270.41 MPa Ss = ((S1 / 2)2 + (S2)2)0.5 = ((310.35 / 2)2 + (270.41)2)0.5 Ss = 311.77 MPa Sy / 2 = 105 / 2 ksi = 52.5 ksi

Sy / 2 = 362.07 MPa Ss = 86.11 % yield in shear during tightening, and reduces to 310.35 MPa or 42.86 % yield in tension after tightening. Alternatively using the Farr formula: Ss = 311.14 MPa = 85.93 % yield

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For stress compliance the following table summarises the calculated values:

Table 6.0 Summary of Stress Compliance

Case Type Value of

stress

MPa

Stress Limit

MPa

Allowable Stress

MPa

% Allowable

Installation Torque 311.77 90% Shear 362.07 86.11 Installation Residual 310.35 60% Yield 434.48 71.43