Bolt Selection Approach LReis Final

Technical University of Lisbon

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BOLT SIZING FROM LOADING REQUIREMENTS

A SIMPLIFIED METHOD

Authors: Lus Reis, Albertino S. Pereira; Henrique Carinhas, Lamy Figueiras Corresponding author: Professor Lus Reis Instituto Superior Tcnico, DEM-SPM Av. Rovisco Pais, 1 , 1049-001- Lisboa, Portugal e-mail: [email protected] telef.: +351 21 8417481

July, 2007

Ps

Ps


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CONTENTS NOMENCLATURE........................................................................................................................................3 1. INTRODUCTION.................................................................................................................................4 2. SAFETY AGAINST SHEAR FORCE...................................................................................................5 3. SAFETY AGAINST JOINT SEALING..................................................................................................6 4. COMBINED SAFETY - SEPARATION AND SHEAR FORCE ............................................................7 5. BOLT SAFETY....................................................................................................................................7 6. GRAPHICAL INTERPRETATION .......................................................................................................9 7. BOLT SIZING PROCEDURE ............................................................................................................10

Step 1 Calculate the required preload .................................................................................................10 Step 2 Calculate the minimum tensile stress area...............................................................................11 Step 3 Select from tables the minimum bolt size.................................................................................11 Step 4 Calculate back the effective final safety ...................................................................................12

8. EXAMPLES FOR METRIC BOLTS...................................................................................................14 EXAMPLE 1: SHEAR FORCE SLIDING PROBLEM..........................................................................14 EXAMPLE 2: JOINT SEALING PROBLEM ...........................................................................................15 EXAMPLE 3: BOLT SAFETY PROBLEM..............................................................................................16

REFERENCES ...........................................................................................................................................18


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NOMENCLATURE

Fb Force - Resultant tensile effort on bolt, Fm Force Resultant compression effort on joint members. Fi Preload Initial/assembly preload given to joint/bolt assembly, C Stiffness constant of the joint, Kb Estimated effective stiffness of the bolt or cap screw in the clamped zone Km Estimated effective stiffness of the members in the clamped zone Ps External shear load on the joint in the most demanding bolt, P External tensile load on the joint, in the most demanding bolt, F0 Minimum required sealing compression force, per bolt. Divided by the sealing area will give

the sealing required pressure (when applicable) Safety factors :

nslide Safety factor against shear load to prevent the relative sliding of the joint members,

nsep - Safety factor against tensile load to prevent joint members separation nseal - Safety factor against tensile load to prevent joint leakage due to joint members

compression alleviation , i.e., to ensure that there will always be sufficient sealing pressure on gasketed joints,

nb - Bolt safety factor to ensure that the maximum load on bolt is always less than bolts proof load,

ncomb Combined safety factor to prevent both sliding and separation of the joint members,

nd Design safety factor for a specific design, fm Friction coefficient between the joint members, At Bolt tensile stress area, As Bolt shear area or minor diameter area, SP Bolt proof strength , FP - Bolt proof load, defined as : FP = SP At

Multiplication factor. Applied to the proof load gives the recommended preload for the specific bolt: = 0,75 for a reusable fasteners condition, = 0,9 for a permanent connection.


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1. INTRODUCTION The purpose of the present document is to describe a simple and integrated method for bolt sizing, based on the safety considerations for the bolt itself and for the joint. The method considers that a joint can be submitted to a loading, which, in a general case, is defined by two different kind of loads: an external tensile load P, acting normally to the joint and a external shear load Ps, acting parallel to the joint. The method is based on the assumption that both P and Ps loads on the bolt, are known. These forces are calculated from direct and torsion shear, and from direct and bending tensile loads on the joint, Based on the Shigleys Mechanical Engineering Design, 8th edition, 8-7 item: Tension Joints The External Load, the force in the bolt, Fb, and the force in the joint members, Fm, can be calculated by the following Eq.s, respectively:

CPFF ib += (1) ( )PCFF im = 1 (2)

Note that Fm in Eq. (2) is considered positive and represents the compression load on the joint members, and Fb is positive and represents the bolt load. Both depend on the preload Fi. C is the stiffness constant of the joint [C=Kb/(kb+km)]. One important condition assumed in this approach is that, in principle, the bolts shall not undergo shear load. The compression force on the joint should be sufficient to cancel, via friction, the shear force applied to the joint. And so, the bolt should be capable of withstanding the shear load in case of a joint compression failure. Therefore, the safety of the bolt against direct shear should be also investigated and assured. Both shear and tensile loads must be considered to establish the required safety against joint failure. The consequent relevant safety issues that need to be evaluated are the following:

I. Safety against shear force to prevent the relative sliding of the joint members,


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II. Safety against joint members separation1, to prevent the joint from opening with the loading, III. Safety against leakage due to joint members compression alleviation, i.e., to ensure that there will

always be sufficient sealing pressure on gasketed joints, IV. Bolt safety to ensure that the maximum load on the bolt is always less than its proof load, V. And finally, the bolt shear in the emergency case, where the bolt had not been tightened.

The approach consists in selecting the lowest calculated preload Fi that satisfies the requirements from I to IV above. The fifth requirement will be used to check the selected bolt size for residual shear strength. All safety factors, i.e., safety against shear force, safety against joint members separation, safety against joint leakage, and bolt safety, must be equal to or greater than the one specified by the designer to the whole design. These factors are defined in such a way that they only affect the external tensile load and the external shear load on the joint.

2. SAFETY AGAINST SHEAR FORCE To prevent the relative sliding of the joint members, the friction force capacity has to be greater than the actuating external shear external force. This is equated by the following expression:

smm PFf (3) Considering Eq. (2) and a safety factor, nslide , it becomes:

( )( ) sslideim PnPCFf = 1 (4) and the required preload:

( )PCfP

nFm

sslide

slidei += 1 (5)

For a known preload the safety factor can be calculated by:

( )m

s

islide

fP

PCFn = 1 (6)

and the respective safety assessment done. 1 This is a particular case from the next one. Case III Joint sealing also requires that no joint members separation occurs but with an extra requirement of some sealing force (permanent joint members in compression).


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3. SAFETY AGAINST JOINT SEALING To ensure that there will always be sufficient sealing pressure on gasketed joints, the compression force on the joint member shall be greater than the minimum required sealing force F0. This means that:

0FFm (7)

Considering Eq. (2) and a safety factor, nseal , it will result :

( ) 01 FPnCF sealseali = (8) or ( ) PnCFF sealseali += 10 (9)


( )PCFF

n iseal =

10 (10)

and the respective safety assessment done. If there is no sealing requirement, then the concern will be to prevent joint separation. Joint separation is the particular case where the sealing force, F0, is nil. Consequently, the requirement to prevent separation can be equated as:

0mF (11)

Considering Eq. (2) and a safety factor, nsep , it will result :

( ) 01 = PnCF sepsepi (12) or ( ) PnCF sepsepi = 1 (13)


( )PCFn isep = 1 (14)

and the respective safety assessment done.


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4. COMBINED SAFETY - SEPARATION AND SHEAR FORCE If both Eq. (5) and Eq. (13) are considered, and the same safety factor applied to both the external loads P and Ps, the equations for required preload and actual safety factor for a known preload, become:

( )

+= PCfP

nFm

scomb

combi 1 (15)

( )PCfP

Fn

m

s

icomb

+=

1 (16)

where ncomb stands for combined factor of safety.

Using Eq.s (15) and (16) the two effects (separation of joint and safety against shear force) will be simultaneously covered. Comparing Eq. (5) where P has been considered to be fixed with Eq. (15), and considering nslide


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CPFFn iPb

= (20) The maximum efficiency of the bolted joint calls for a as highest as possible preload. A recommended preload is taken for a specific bolt, and the here above safety factors calculated, its adequacy assessed and reiteration done whether necessary.

This recommended preload is related to the proof load by the multiplier factor as follows: Pi FF = (21)

or i

PFF = (22)

where the factor value considers two different conditions: a) a reusable fastener connection: =0.75 b) a permanent connection: =0.90

Figure 1 represents in a graphical way the influences of the factor in the proof load.

Figure 1 Bolt preload representation

Substituting Fp in Eq. (20) it will result:

i

biFPnCF =+ (23)

And the required preload, from the bolt strength safety point of view, is given by:

FP

Fi=0,75FP

0,7 0,8 0,9

Fi=0,9FP

nbCP=0,25FP

0,5 0

nbCP=0,1FP

0,6 0,1


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11 = PCnF bbi (24)

On the other hand, for a known preload, the safety factor can be calculated back by:

CP

Fn

i

b

=

11 (25)

6. GRAPHICAL INTERPRETATION The bolt and members forces versus the external separation load P are represented in Figure 2.

Figure 2 Influence of the external load P on the bolt and members forces

Fi

FP

F0

P P

Fm; Fb

Fb

FmFm =F

i -(1-C)P

Fb=Fi+CP 45

0

CP

(1-C)P

nbCP

n sea

l( 1-C

)P

F i-F

0

( )CFP i= 1

*2

P

P*2P*1

( )CFFP i

=1

0*1

(1-)FP

Joint Separation

Fi

FP

F0

P P

Fm; Fb

Fb

FmFm =F

i -(1-C)P

Fb=Fi+CP 45

0

CP

(1-C)P

nbCP

n sea

l( 1-C

)P

F i-F

0

( )CFP i= 1

*2

P

P*2P*1

( )CFFP i

=1

0*1

(1-)FP

Joint SeparationJoint Separation Load


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7. BOLT SIZING PROCEDURE Suggestion of a step by step approach that will lead to the right bolt size. It is assumed that the following parameters are known, before the bolt selection process starts:

Loads: Ps; P; Required minimum sealing compression, F0 Proof strength of the bolt material: SP; Required minimum safety factors against different effects: nslide; nseal; nsep; ncomb; nb. (Same value for all

of the safety factors are arbitrarily assumed);

Friction coefficient between the joint members: fm; One or another of the following two joint conditions are considered: reusable fasteners = 0,75, or

permanent connection = 0,90; As the bolt size is not known, the C value (stiffness constant of the joint) is also unknown. A good starting value for steel bolts and members can be C=0,25. As soon as we have achieved the first guess for the bolt size, the due C is calculated using the approaches of sections 8.5 and 8.6 of the book, and the bolt selection process reiterated as necessary.

Step 1 Calculate the required preload Use the previous developed expressions to calculate the minimum required preload, for the different effects as follows:

Preload required to prevent the relative sliding of the joint members: ( )PC

fPnFm

sslide

slidei += 1 (5)

Preload required to ensure that there will always be sufficient sealing pressure on gasketed joints: ( ) PnCFF sealseali += 10 (9)

Preload required to prevent joint separation:


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( ) PnCF sepsepi = 1 (13) Preload required for Combined safety - Separation and Sliding

( )

+= PCfP

nFm

scomb

combi 1 (15)

Preload required to ensure that the total force in the most demanding bolt does not exceed the proof load:

11 = PCnF bbi (24)

Should one wish to use Eq. (15) (combined safety) then Eq. (5) (sliding) and Eq. (13) (separation) need not to be used.

Select the maximum value of the required preload as: Fi max = Max{ slideiF ; sealiF ;

sepiF ;

combiF ;

biF }

Step 2 Calculate the minimum tensile stress area The selected bolt needs to accommodate the required preload, Fi max :

Pti SAF =max (26) So, the minimum tensile stress area will be ( = 0,75, or = 0,90):

P

it SFA

max (27)

Step 3 Select from tables the minimum bolt size Given the minimum required tensile stress area, At, it will be necessary to read from the Table 8.1, for metric threads, or from the Table 8.2, for UNC and UNF threads, the minimum bolt diameter compatible with the required tensile area. Now the bolt diameter and the corresponding tensile stress area - At , and shear resistant area (minor diameter area) - As, have been got. The C constant needs now to be evaluated, and the steps 1 to 3 need to be repeated until there is no change in the bolt diameter. Finally, it is needed to ensure that, in case of bolt unfastening, the shear external force Ps, will not cause shear failure in the bolt. To guarantee being on the safe side, the following check needs to be done, considering the Tresca criterion:


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d

P

s

s

nS

AP

2< (28)

Step 4 Calculate back the effective final safety After having set the bolt diameter, then the final/effective safety against the different effects, can be calculated using the previously developed theory:

The sliding safety factor: ( )m

s

islide

fP

PCFn = 1 (6)

The sealing safety factor (if applicable): ( )PCFF

n iseal =

10 (10)

The joint separation safety factor: ( )PCFn isep = 1 (14)

Combined safety factor: ( )PC

fP

Fn

m

s

icomb

+=

1 (16)

Bolt safety factor: CPFFn iPb

= (20)

or CP

Fn

i

b

=

11 (25)

When following the above procedure, increasing the bolt size, or the material strength class, or the initial number of bolts, or even improving the bolt arrangement, may turn out to be necessary in order to get a satisfactory bolted joint design. In most cases, the effective final safety against sliding is larger than calculated by Eq. (6). In this expression, the most demanding pair P and Ps is used. If the other bolts in the same joint are all similar and assembled with the same preload value, but less loaded, then their safety against sliding will be higher than the safety for the most demanding bolt. This leads to an overall sliding safety margin for the joint that is higher than the one calculated via expression (6).


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Summary of Bolt Sizing Procedure

Specify: Forces: Ps, P, F0; Minimum safety factors: nslide; nseal, nsep, ncomb, nb; Friction coefficient: fm; =0,75 or 0,9; Proof Strength: Sp. Assume C=0,25 (to be confirmed later)

Step # 1 - Calculate minimum required preload: Fislide using eq. (5); Fiseal using eq. (9); Fisep using eq. (13); Ficomb using eq. (15); Fib using eq. (24); Select the Maximum of the above values of preload: Fimax

Step # 2 - Calculate minimum tensile stress area using eq. (27)

Step # 3 Select from tables the minimum bolt size Standard bolt size diameter, At - tensile stress area, As Shear area. Calculate C.

o If it is different from previously assumed value, go to step #1

Calculate Shear stress for un-tighten bolt. o If it is bigger then Sp/(2nd), increase bolt size,

calculate C and go to Step # 1.

Itera

te

Step # 4 Calculate effective final safety With the final values of parameters, calculate the final safety

factors: nslide (6), nseal (10), nsep (14), ncomb (16), nb (20). Check that all safety factors are bigger than the minimum

required for the design.


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8. EXAMPLES FOR METRIC BOLTS Knowing that the friction coefficient between the joint members is fm=0,25, the stiffness constant of the joint is C=0,3, the joint uses reusable bolts, the bolt proof strength is SP= 380 MPa (CR 5.8 Table 8.11) and the minimum specified safety factor is nd=2, choose the minimum bolt metric size, coarse threads, adequate for the following examples:

EXAMPLE 1: SHEAR FORCE SLIDING PROBLEM The joint is loaded with the following external loading: P=10 kN; Ps=50 kN. The sealing requirement is to have a minimum compression force of F0 =5,0 kN. This force should always be kept, even when the external loads are acting. Step # 1: Minimum required preload using (5), (9), (13), (15) and (24) is, respectively:

slideiF =407,0 kN;

sealiF =19,0 kN;

sepiF =14,0 kN;

combiF =414,0 kN;

biF = 18,0 kN

The maximum value is: Fi required =414,0 kN, from combined safety. Step # 2: Minimum tensile stress area, using Eq. (27) is: At 1452,8 mm2 Step # 3: Minimum bolt size taken from Table 8.1, for coarse threads, is bolt M48. This bolt has a tensile stress area of: At=1470 mm2, giving FP=559,7 kN, and Fi=419,8 kN. The shear stress caused by Ps, in the minor diameter area of As=1380 mm2, is 36,2 MPa, and the proof shear stress is SP/2=190 MPa. Consequently, there is no risk of shear failure if the bolt is not tightened. Step # 4: The safety against the different effects is as follows: nslide=2,06; nseal=59,26; nsep=59,97; ncomb=2,03; nb=46,65.


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EXAMPLE 2: JOINT SEALING PROBLEM

The joint is loaded with the following external loading: P= 20,0 kN; Ps= 5,0 kN. The sealing requirement is to have a minimum compression force of F0 =50,0 kN. This force should always be kept, even when the external loads are acting. Step # 1: Minimum required preload using (5), (9), (13), (15) and (24) is, respectively:

slideiF =54,0 kN;

sealiF = 78,0 kN;

sepiF = 28,0 kN;

combiF = 68,0 kN;

biF = 36,0 kN

The maximum value is : Fi required = 78,0 kN from sealing requirement. Step # 2: Minimum tensile stress area, using Eq. (27) is: At273,7 mm2 Step # 3: MInimum bolt size from Table 8.1, for coarse threads, is bolt M24 would be adequate. This bolt has a tensile stress area of: At=353 mm2, giving FP=134,1 kN, and Fi=100,6 kN. The shear stress caused by Ps, in the minor diameter area of : As=324 mm2, is 15,4 MPa, and the proof shear stress is SP/2=190 MPa. Consequently, there is no risk of shear failure if the bolt is not tightened. Step # 4: The safety against the different effects is as follows: nslide=4,33; nseal=3,61; nsep=7,19; ncomb=2,96; nb=5,59. The results of this example are graphically shown in Figure 3. The tensile load which would cause the joint to reach its limit sealing force is 72,3 kN. The separation force occurs for F=143,7 kN.


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Example 2

0

20

40

60

80

100

120

140

160

180

0 20 40 60 80 100 120 140 160 180 200

P(kN)

Fb;F

m (k

N)

Figure 3 Influence of the external load P on the bolt and members forces for Joint Sealing Problem

EXAMPLE 3: BOLT SAFETY PROBLEM

The joint is loaded with the following external loading: P=75,0 kN; Ps= 3,0 kN. The sealing requirement is to have a minimum compression force of F0 = 5,0kN. This force should always be kept, even when the external loads are acting. Step # 1: Minimum required preload using (5), (9), (13), (15) and (24) is, respectively:

slideiF =76,5 kN;

sealiF = 110,0 kN;

sepiF =105,0 kN;

combiF = 129,0 kN;

biF = 135,0 kN

The maximum value is : Fi required = 135,0 kN from the bolt safety requirement.

P*1=72,3 kN P*2=143,7 kN


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Step # 2: Minimum tensile stress area, using Eq. (27) is : At473,7 mm2 Step # 3: Minimum bolt size got from Table 8.1, for coarse threads is bolt M30. This bolt has a tensile stress area of: At=561 mm2, giving FP=213,2 kN, and Fi=159,9 kN. The shear stress caused by Ps, in the minor diameter area of : As=519 mm2, is 5,8MPa, and the proof shear stress is SP/2=190MPa. Consequently, there is no risk of shear failure if the bolt is not tightened. Step # 4: The safety against the different effects is as follows: nslide=8,95; nseal=2,95; nsep=3,05; ncomb=2,48; nb=2,37. The results of this example are graphically shown in Figure 4.

Example 3

020406080

100120140160180200220240260

0 20 40 60 80 100 120 140 160 180 200 220 240 260P(kN)

Fb;F

m (k

N)

Figure 4 Influence of the external load P on the bolt and members forces for Bolt Safety Problem

P*1=221,3 kN P*2=228,4 kN


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REFERENCES [1] Shigleys Mechanical Engineering Design, 8th edition. [2] Robert C. Juvinall/ Kurt M. Marsheh Fundamentals of Machine Component Design, updated third edition.

Bolt Selection Approach LReis Final

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