Guideline For Bolted Joint Design And Analysis: Version 1

While this joint includes washers on both ends, many bolted joints do not use washers and themethodologies presented in this document apply to bolted joints with or without washers. Aclearance between the bolt and the clamped materials can be accounted for, however, themethodologies presented here assume a single clearance that applies to all the layers. Figure 2identifies important geometric parameters for a thread joint.Figure 2.Threaded Joint Geometry3 GENERAL GUIDELINESThe guidelines NASA [11] used for bolted joints on the space shuttle are generally applicableand are adopted here. The general guidelines are11

A preloaded joint must meet, as a minimum, the following three basic requirements1. The bolt must have adequate strength.2. The joint must demonstrate a separation factor of safety at limit load. This generallymeans the joint must not separate at the maximum load to be applied to the joint.3. The bolt must have adequate fracture and fatigue life.Bolt strength is checked at maximum external load and maximum preload, and joint separationis checked at maximum external load and minimum preload. To do this, a conservative estimateof the maximum and minimum preloads must be made, so that no factors of safety are requiredfor these preloads. Safety factors need only be applied to external loads.4 BOLT PRELOADA critical component of designing bolted joints is not only determining the number of bolts, thesize of them, and the placement of them but also determining the appropriate preload for the boltand the torque that must be applied to achieve the desired preload. There is no one right choicefor the preload or torque. Many factors need to be considered when making this determination.A basic guideline given in the Machinery’s Handbook [12] is to use 75% of the proof strength(or 75% of 85% of the material yield strength if the proof strength is not known) for removablefasteners and 90% of the proof strength for permanent fasteners. Things to consider include thetension in the bolt and therefore the clamping force, fatigue concerns (higher preload is generallypreferable), how much torque can easily be applied without risking damaging another part if thetool slips while applying the load, etc.The Machinery’s Handbook [12] and the NASA guide [11] give estimates for the accuracy ofbolt preload based on application method. The NASA guide states these uncertainties should beused for all small fasteners (defined as those less than ¾”). The results are summarized in Table3.Table 3: Accuracy of Bolt Preload Based on Application MethodMethodAccuracyTorque Wrench on Unlubricated Bolts [11] 35%Torque Wrench on Cad-Plated Bolts [11] 30%Torque Wrench on Lubricated Bolts [11] 25%Preload Indicating Washer [11] 10%Strain Gages[12] 1%Computer Controlled Wrench (Below Yield) [12] 15%Computer Controlled Wrench (Yield Sensing) [12] 8%Bolt Elongation [11] 5%Ultrasonic Sensing [11] 5%12

A general relationship between applied torque, T, and the preload in the bolt, Fp, can be writtenin terms of the bolt diameter, d, and the “Nut Factor”, K, asT K * d b * FP(1)Table 4 gives ranges for nut factors for a variety of materials and lubricants. The data is takenfrom the Standard Handbook of Machine Design [15]. Their data is based on multiple sources.As can be seen by examining the data, there can be large ranges of potential nut factors and assuch, it is recommended in the Standard Handbook of Machine Design [15] to only use nutfactors when approximate preload is sufficient for the design. For cases where strain gages cannot be used, bolt extension can not be measured, load sensing washers can not be used, etc., thereis no choice but use a nut factor. In these cases, any analysis should be done using a range of nutfactors to bound the results. A low nut factor gives a higher preload and clamping force but putsthe bolt closer to yield while a high nut factor gives a lower preload and clamping force but thecapacity of the joint to resist external tensile loads has been reduced.Table 4. Nut Factors for Various Lubricants.LubricantNut FactorMeanRangeCadmium Plating0.194-0.246 0.153-0.328Zinc Plate0.3320.262-0.398Black Oxide0.163-0.194 0.109-0.279Baked on PTFE0.092-0.112 0.064-0.142Molydisulfide Paste0.1550.14-0.17Machine Oil0.210.20-0.225Carnaba Wax (5% Emulsion) 0.1480.12-0.16560 Spindle Oil0.220.21-0.23As Received Steel Fasteners 0.200.158-0.267Molydisulfide Grease0.1370.10-0.16Phosphate and Oil0.190.15-0.23Plated Fasteners0.15Grease, Oil, or Wax0.12Additional information on nut factors can be found in Bickford [4] and the Machinery’sHandbook [12]. A summary of analytic approaches to compute a nut factor are given inAppendix A. At this point, the recommended method is to use a pre-computed nut factor fromTable 4 until the analytic methods are better understood, compared to the known methods, andconfidence is gained in the accuracy of the method. The analytic methods seem to produceartificially large nut factors (which produce very small preloads for a given torque). This issomething that will be looked at in follow-on work to the initial release of this report.13

5 ANALYTIC MODELING APPROACHESAll of the analytic approaches presented in this section implicitly assume an axisymmetic stressfield. Any geometric or material effects that significantly violate this assumption make theapproaches in this section invalid. This can include bolts very close together, bolts near aphysical boundary (see section 5.4), non axisymmetric geometries, etc. If the bolted joint ofinterest does not meet these assumptions (and the additional assumptions of the approachesbelow) then it is recommended that a finite element analysis be used for the joint.The general approach is to idealize a bolted joint into a pair of springs in parallel. One springrepresents the bolt and other represents the clamped material. If an estimate can be obtained forthe stiffness of the bolt (which is trivial) and the clamped material (which is difficult), thenexternally applied axial loads can be partitioned appropriately between the two and factors ofsafety can be computed to determine if the joint design is sufficient.It is generally assumed that the clamped material can be viewed as a set of springs in series andan overall stiffness for the clamped material, km, can be computed as11 11 L k m k1 k 2ki(2)where ki is the stiffness of the ith layer. The bolt stiffness, kb, can be estimated in terms of thecross sectional area of the bolt, Ab, Young’s modulus for the bolt, Eb, and the length of the bolt,Lb, askb Ab EbLb(3)The total stiffness of the joint, kj, can be computed (by assuming two springs in parallel) ask j kb k m(4)The remainder of this chapter is devoted to various methods of estimating the stiffness of theclamped material and comparing the various methods. It will be recommended that the FEAempirical models be used when they are applicable and to use Shigley’s frustum approach for allother cases.5.1 Cylindrical Stress Field Method (Q Factor)In this method it is assumed the true ‘barrel shaped’ stress field can be approximated as acylinder of diameter dc (see Figure 3, dc equals Qd). This was the original assumption made byShigley in his first edition mechanical engineering design book [8] and is what is chosen byBickford [4].14

A factor, Q, is defined as the ratio between the actual bolt diameter and the idealized cylindricalstress fieldQ dCd(5)Figure 3.Q Factor Stress Distribution for 2 GeometriesBy considering the layer as a one dimensional spring, the stiffness of the ith layer can becomputed aski Ai EiLi(6)The area of the ith layer can be computed, assuming the inner diameter is qidb (where qi 1 andis used to allow for clearance between the clamped material and the bolt) and the outer diameteris Qdb, as(π (Qd b ) (qi d b )Ai 422) π d (Q2b24 qi2)(7)The addition of qi is a logical extension to account for clearance holes that were included in thework of Pulling, et. al. [13] and is adopted here. The axial stiffness of the clamped material canbe written as2k axial 4 iπ dbLi(Ei Q 2 qi(8)2)15

Pulling, et. al. [13], went on to define a bending stiffness for the clamped material using the samemethodology. They assumed that the same material is loading in bending as was loaded axially.The approach is based on beam theory and as such they are assuming the ends (i.e., the edge ofthe assumed loaded material) are free (i.e., there is no rotation constraint posed by the materialbeyond that considered loaded). With these assumptions, the bending stiffness for each layer canbe computed to bekbendingi Ei I iLi(9). The moment of inertia, I, for the ith layer can be computed as(π (Qd b ) (qi d b )Ii 6444)(10)Once again assuming each layer is represented by a spring in series, the bending stiffness of theclamped material can be computed ask bending π db64 i(4(11)LiEi Q 4 qi4)For the case of a bolted flange of a pipe with the bending applied to the neutral axis of the pipe,the actual load on the bolt will be more like an axial load and less like a bending load. There isan additional concern with this method because it is probable that the actual load on the bolt dueto bending will be higher than what this theory predicts (i.e., this does not produce conservativeresults). This is a major concern and great care must be taken when considering bending loadson bolted joints with this method.The original guideline put out by Pulling, et. al. [13] used a value of 3 for Q. This was also thedefault value included in the spread sheet (boltfailurecalculationsheet.xls) that accompanied thereport. This is the value Shigley used in the 1st edition of Mechanical Engineering Design. Theaccuracy of this method is highly dependent on the choice of Q. As can be seen, Q is squared (orraised to the 4th power for bending), and therefore any errors in Q are magnified. As will beshown by comparing the different methods in a later section, the value of Q is variable anddepends on the geometry of the joint.Bickford [4] noted that spheres, cylinders and frustums could all be used. He also chose to use acylinder. He derived the same expressions for axial loading that were shown above (except hedid not include qi to account for clearance) and provided the following guidance for Q (actuallyhe provided guidance for the area of the cylinder which implies Q). His equations are modifiedhere to account for qi so that it can be compared to the work of Pulling [13]. For the case wherethe bolt head diameter (or washer diameter) is greater than the joint “diameter” of the materialbeing clamped, the entire area is used so16

A ()(ππ2222D J (qd b ) (Qd b ) (qd b )44)when d h D J(12)where DJ is the diameter of the joint. This impliesQ DJdwhen d h D J(13)For the case where the joint “diameter” is greater than the diameter of the bolt head (or washer)but less than three times the diameter, the area that should be used isA () d l l 2 ππ D22 when d h D J 3 d hd h (qd b ) J 1 h 48 dh100 5(14)The first term accounts for all the area under the bolt (or washer). The second term accounts foradditional material based on the thickness, l, of the joint. This implies a Q factor of DJ d h l l 2 12 when d h D J 3d h 1 Q d h d dh 10 200 (15)For the case where the joint “diameter” is greater than three times the diameter the of the bolt (orwasher), the area that should be used is2 π l 2A d h (qd b ) when D J 3 d h and l 8 d h4 10 (16)Again it can be seen that the equation above accounts for the materials under the bolt plusadditional material that is dependent on the thickness of the joint. This implies a Q factor ofQ l 1 d h when D J 3 d h and l 8 d hdb 10 (17)A plot of Q for various thicknesses and Dj/dh ratios is shown in Figure 4. The data was generatedassuming a 5/8” diameter bolt, d, with a bolt head diameter of 15/16” (1.5 time the boltdiameter), dh. From this data we can see there is a large variation in Q depending on thethickness of the joint relative to the bolt diameter and the joint diameter (i.e., how much materialis being clamped) relative to the bolt diameter.17

Q Factor (Bickford Method)2.60002.4000Bickford Dj/Db 12.2000Bickford Dj/Db 1.4QBickford Dj/Db 1.8Bickford Dj/Db 2.22.0000Bickford Dj/Db 2.6Bickford Dj/Db 31.8000Bickford Dj/Db Figure 4.Q factors for Various Geometries Using the Bickford Method.5.2 Shigley’s Frustum ApproachShigley [16] used a similar methodology but made a different assumption about the shape of thestress field to better correlate with experimental data. In this method, the stiffness in a layer isobtained by assuming the stress field looks like a frustum of a hollow cone (See Figure 5).By assuming a 1D (i.e., axial) compression (see Shigley [16] for the complete derivation), thestiffness of a layer can be computed aski π E d b tan (α )(18) (2l tan (α ) d h d b )(d h d b ) ln (2l tan (α ) d h d b )(d h d b ) 18

dhdhαldbA Bolt Through a PlateThe Assumed Stress FieldFigure 5.Shigley’s Stress Frustum.Various angles, α, have been used. 45 degrees is often used but this often over estimates theclamping stiffness. Shigley states that typically the angle to use should be between 25 and 33degrees and in general recommends 30 degrees (this is assuming a washer is used). There aretwo obvious examples when this falls apart. The first is for the case when there is not enoughmaterial for the frustum to exist (e.g., a bolt hole very near an edge of a plate). The second caseis for very thick clamping areas. For this case, the shape of the actual stress distribution looksmore like a barrel and the shape assumed by Shigley is inappropriate.There are a number of subtleties that must be noted based on the assumptions in this method.First, there must be ‘symmetric’ frustums across the entire joint regardless of the number ofmaterials (otherwise static equilibrium would not be met). The value of D used for a given layermust take into account the frustum of the previous layer and not just the bolt or washer diameter.The actual value of dh that really should be used is the start of the stress frustum and not thediameter of the bolt head and/or washer. Due to flexibility in the bolt or

connections. An overview of the current methods used to analyze bolted joint connections is given. Several methods for the design and analysis of bolted joint connections are presented. Guidance is provided for general bolted joint design, computation of preload uncertainty and preload loss, and the calculation of the bolted joint factor of safety.