Bolted Joints

ME 423: Machine Design Instructor: Ramesh Singh 1 Outline

• Introduction • Geometry and classification of fasteners • Types of fasteners

ME 423: Machine Design Instructor: Ramesh Singh 2 Bolted Joints

Bolts find application due to ease in following • Field assembly • Disassembly • Maintenance • Adjustment • Screw Jacks for lifting and power transmission • Screws are also used for linear actuation

Fun Fact: Boeing’s 747 require as many as 2.5 million fasteners

ME 423: Machine Design Instructor: Ramesh Singh 3 Bolted Joints

• Bolted joints are connectors which affect: – Stiffness (as they are not rigid) – Vibration, Damping and Stability – Load bearing capacity • Bolted Joints are needed for disassembly – Maximum benefits are obtained when they are preloaded – Threads can plastically deform – Reusing is no recommended • Note that the assemblies using bolts can resist tensile, shear loads and moments but bolts are only designed to take tension

ME 423: Machine Design Instructor: Ramesh Singh 4 Anatomy of a bolted joint l Grip Geometry of Bolt

Ad At Tensile stress area tw HBolt

Ad Major diameter area l t1 l d L t lt 2 lt Threaded length in grip H Nut tw LT ld Unthreaded length in grip A d t Major diameter (unthreaded)

©Martin Culpepper, All rights reserved 5 ME 423: Machine Design Instructor: Ramesh Singh 5 ManufactureBolted joint components: of Threads Bolt Rolled Threads NEVER in shear or bending Stress concentrations at the Head root of the teeth Fatigue crack propagation! Exception: Shoulder bolts

Keep threads clean & Shank lubed to minimize losses ~50% power to bolt head Cut Threads friction

~40% power to thread friction

~10% power to deforming Steel is most the bolt and flange common

©Martin Culpepper, All rights reserved 7 ME 423: Machine Design Instructor: Ramesh Singh 6 Functions ofBolted Washer joint components: Washers Purpose of Washers: • Spacer Spacer • Distribute load in clamped member Distribute load in clamped member • Reduce head-member wear Reduce head-member wear

• Lower coefficient of Lower coefficient of friction/losses friction/losses • Lock bolt into the joint Lock bolt into the joint (lock washer) (lock washer) Increase preload resolution (wave washer) • Increase preload resolution (wave washer)

ME 423: Machine Design Instructor: Ramesh Singh 7

• Pitch–distance between adjacent threads. Reciprocal of threads per inch Screws, Fasteners, and the Design of Nonpermanent Joints 411 • Major diameter–largestFigure diameter 8–1 of Major diameter Pitch diameter thread Terminology of screw threads. Minor diameter Sharp vee threads shown for Pitch p • Minor diameter–smallestclarity; diameter the crests and roots of are thread actually ﬂattened or rounded 45° chamfer • Pitch diameter–theoreticalduring the diameter forming operation. between major and minor diameters, where tooth and gap are same width Root • The lead, l, is the distance the nut moves Crest Thread angle 2α parallel to the screw axis when the nut is given one turn Figure 8–2 H 8 p 8 • Multiple threads are alsoBasic possibleproﬁle for metric M Internal threads and MJ threads. d ϭ major diameter p p 3H dr ϭ minor diameter 8 5H 2 2 d ϭ pitch diameter p ME 423: Machine DesignH 8 p ϭ pitch Instructor: Ramesh Singh p H 60° √3 4 4 8 H ϭ 2 p 60 H ° d 4 30° dp p

External threads dr

threads. The MJ profile has a rounded fillet at the root of the external thread and a larger minor diameter of both the internal and external threads. This profile is espe- cially useful where high fatigue strength is required. Tables 8–1 and 8–2 will be useful in specifying and designing threaded parts. Note that the thread size is specified by giving the pitch p for metric sizes and by giving the number of threads per inch N for the Unified sizes. The screw sizes in 1 Table 8–2 with diameter under 4 in are numbered or gauge sizes. The second column in Table 8–2 shows that a No. 8 screw has a nominal major diameter of 0.1640 in. A great many tensile tests of threaded rods have shown that an unthreaded rod having a diameter equal to the mean of the pitch diameter and minor diameter will have the same tensile strength as the threaded rod. The area of this unthreaded rod is called the tensile-stress area At of the threaded rod; values of At are listed in both tables. Two major Unified thread series are in common use: UN and UNR. The difference between these is simply that a root radius must be used in the UNR series. Because of reduced thread stress-concentration factors, UNR series threads have improved fatigue strengths. Unified threads are specified by stating the nominal major 5 diameter, the number of threads per inch, and the thread series, for example, 8 in-18 UNRF or 0.625 in-18 UNRF. Metric threads are specified by writing the diameter and pitch in millimeters, in that order. Thus, M12 1.75 is a thread having a nominal major diameter of 12 mm and a pitch of 1.75 mm.× Note that the letter M, which precedes the diameter, is the clue to the metric designation. Standards

• The American National (Unified) thread standard defines basic thread geometry for uniformity and interchangeability • American National (Unified) thread – UN normal thread – UNR greater root radius for fatigue applications • Metric thread – M series (normal thread) – MJ series (greater root radius)

ME 423: Machine Design Instructor: Ramesh Singh 9 Standards

• Coarse series UNC – General assembly – Frequent disassembly – Not good for vibrations – The “normal” thread to specify • Fine series UNF – Good for vibrations – Good for adjustments – Automotive and aircraft • Extra Fine series UNEF – Good for shock and large vibrations – High grade alloy – Instrumentation – Aircraft

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Screws, Fasteners, and the Design of Nonpermanent Joints 411

Figure 8–1 Major diameter Pitch diameter Terminology of screw threads. Minor diameter Sharp vee threads shown for Pitch p clarity; the crests and roots are actually ﬂattened or rounded Thread Profile45° chamfer during the forming operation. • Thread profileRoot M and MJ Crest Thread angle 2α

Figure 8–2 H 8 p 8 Basic proﬁle for metric M Internal threads and MJ threads. d ϭ major diameter p p 3H dr ϭ minor diameter 8 5H 2 2 d ϭ pitch diameter p H 8 p ϭ pitch p H 60° √3 4 4 H ϭ 2 p 60 H ° d 4 30° dp p

External threads dr

threads. The MJ profile has a rounded fillet at the root of the external thread and a larger minor diameter of both the internal and external threads. This profile is espe- cially useful where high fatigue strength is required. Tables 8–1 and 8–2 will be useful in specifying and designing threaded parts. Note that the thread size isME specified 423: Machine by giving Design the pitch p for metric sizes and by giving the number of threadsInstructor: per inch Ramesh N for Singh the Unified sizes. The screw sizes in 1 Table 8–2 with diameter under 4 in are numbered or gauge sizes. The second column 11 in Table 8–2 shows that a No. 8 screw has a nominal major diameter of 0.1640 in. A great many tensile tests of threaded rods have shown that an unthreaded rod having a diameter equal to the mean of the pitch diameter and minor diameter will have the same tensile strength as the threaded rod. The area of this unthreaded rod is called the tensile-stress area At of the threaded rod; values of At are listed in both tables. Two major Unified thread series are in common use: UN and UNR. The difference between these is simply that a root radius must be used in the UNR series. Because of reduced thread stress-concentration factors, UNR series threads have improved fatigue strengths. Unified threads are specified by stating the nominal major 5 diameter, the number of threads per inch, and the thread series, for example, 8 in-18 UNRF or 0.625 in-18 UNRF. Metric threads are specified by writing the diameter and pitch in millimeters, in that order. Thus, M12 1.75 is a thread having a nominal major diameter of 12 mm and a pitch of 1.75 mm.× Note that the letter M, which precedes the diameter, is the clue to the metric designation. Thread Nomenclature Bolt Specification

Threads per inch Material grade Thread series

¼-20 x ¾ in UNC-2 Grade 5 Hex head bolt

Nominal diameter length Class fit Head type

Metric Pitch

M12 x 1.75 ISO 4.8 Hex head bolt Nominal diameter Material class

Shigleys Mechanical Engineering Design

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412 Mechanical Engineering Design

Table 8–1 Nominal Coarse-Pitch Series Fine-Pitch Series Diameters and Areas of Major Tensile- Minor- Tensile- Minor- Diameter Pitch Stress Diameter Pitch Stress Diameter Coarse-Pitch and Fine- dpArea At Area Ar p Area At Area Ar Pitch Metric Threads.* mm mm mm2 mm2 mm mm2 mm2 1.6 0.35 1.27 1.07 2 0.40 2.07 1.79 2.5 0.45 3.39 2.98 3 0.5 5.03 4.47 3.5 0.6 6.78 6.00 4 0.7 8.78 7.75 5 0.8 14.2 12.7 6 1 20.1 17.9 8 1.25 36.6 32.8 1 39.2 36.0 10 1.5 58.0 52.3 1.25 61.2 56.3 12 1.75 84.3 76.3 1.25 92.1 86.0 14 2 115 104 1.5 125 116 16 2 157 144 1.5 167 157 20 2.5 245 225 1.5 272 259 24 3 353 324 2 384 365 30 3.5 561 519 2 621 596 36 4 817 759 2 915 884 42 4.5 1120 1050 2 1260 1230 48 5 1470 1380 2 1670 1630 56 5.5 2030 1910 2 2300 2250 64 6 2680 2520 2 3030 2980 72 6 3460 3280 2 3860 3800 80 6 4340 4140 1.5 4850 4800 90 6 5590 5360 2 6100 6020 100 6 6990 6740 2 7560 7470 110 2 9180 9080

*The equations and data used to develop this table have been obtained from ANSI B1.1-1974 and B18.3.1-1978. The minor diameter was found from the equation d d 1.226 869p, and the pitch diameter from d d r = − p = − 0.649 519p. The mean of the pitch diameter and the minor diameter was used to compute the tensile-stress area. ME 423: Machine Design Square and Acme threads,Instructor: whose profiles Ramesh are shownSingh in Fig. 8–3a and b, respectively, are used on screws when power is to be transmitted. Table 8–3 lists the pre- 13 ferred pitches for inch-series Acme threads. However, other pitches can be and often are used, since the need for a standard for such threads is not great. Modifications are frequently made to both Acme and square threads. For instance, the square thread is sometimes modified by cutting the space between the teeth so as to have an included thread angle of 10 to 15◦. This is not difficult, since these threads are usually cut with a single-point tool anyhow; the modification retains most of the high efficiency inherent in square threads and makes the cutting simpler. Acme threads Tensile Stress Calculations

• The tensile stress area, At, is the area of an unthreaded rod with the same tensile strength as a threaded rod • It is the effective area of a threaded rod to be used for stress calculations • The diameter of this unthreaded rod is the average of the pitch diameter and the minor diameter of the threaded rod

ME 423: Machine Design Instructor: Ramesh Singh 14 Bolt HeadsHead Type of Bolts • Hexagon head bolt Hexagon head bolt – Usually uses◦ nutUsually uses nut – Heavy duty◦ Heavy duty • Socket head Socketcap screw head cap screw Fig. 8–9 – Usually more◦ Usually precision more precision applications applications ◦ Access from the top – Access from the top Machine screws • Machine screws◦ Usually smaller sizes – Usually smaller◦ Slotted sizes or Philips head – Slotted or Philipscommon head common ◦ Threaded all the way – Threaded all the way Fig. 8–10 Shigleys Mechanical Engineering Design

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424 Mechanical Engineering Design Machine Screws Figure 8–11 ° Types of heads used on A D A D machine screws. 80 to 82 H H L L

(a) Round head (b) Flat head °

A D A D 80 to 82 H H L L

5° ±3°

A D A D R

H L L

(e) Truss head (f) Binding head

D D W W

H L H L

(g) Hex head (trimmed) (h) Hex head (upset)

Figure 8–12 W H H ME 423: Machine Design H Approx.1 in H Approx.1 in Hexagonal nuts: (a) end view, Instructor:64 Ramesh Singh 64 general; (b) washer-faced 16 regular nut; (c) regular nut chamfered on both sides; (d)jam nut with washer face; (e) jam nut chamfered on both sides. 30Њ 30Њ 30Њ 30Њ

(a)(b)(c)(d)(e)

8–4 Joints—Fastener Stiffness When a connection is desired that can be disassembled without destructive methods and that is strong enough to resist external tensile loads, moment loads, and shear loads, or a combination of these, then the simple bolted joint using hardened-steel washers is a good solution. Such a joint can also be dangerous unless it is properly designed and assembled by a trained mechanic. Hex Head Bolt • Hexagon-head bolts are one of the most common for engineering applications • Standard dimensions are included in Table A–29 (Shigley) bud29281_ch08_409-474.qxd 12/16/2009 7:11 pm Page 423 pinnacle 203:MHDQ196:bud29281:0073529281:bud29281_pagefiles: • W is usually about 1.5 times nominal diameter • Bolt length L is measured from below the head

Screws, Fasteners, and the Design of Nonpermanent Joints 423

H W Figure 8–9 1 Approx.64 in Hexagon-head bolt; note the washer face, the ﬁllet under the head, the start of threads, and the chamfer on both ends. Bolt lengths are always measured R from below the head. 30°

Figure 8–10 ME 423: Machine Design Instructor: Ramesh Singh Typical cap-screw heads: 17 (a)ﬁllister head; (b) ﬂat head; (c) hexagonal socket head. Cap screws are also manufactured A A A with hexagonal heads similar to the one shown in Fig. 8–9, as 80 to 82° well as a variety of other head H styles. This illustration uses one of the conventional H H D methods of representing D D threads.

L L L l l l

(a) (b) (c)

has elongated almost to the elastic limit. If the nut does not loosen, this bolt tension remains as the preload or clamping force. When tightening, the mechanic should, if possible, hold the bolt head stationary and twist the nut; in this way the bolt shank will not feel the thread-friction torque. The head of a hexagon-head cap screw is slightly thinner than that of a hexagon- head bolt. Dimensions of hexagon-head cap screws are listed in Table A–30. Hexagon-head cap screws are used in the same applications as bolts and also in applications in which one of the clamped members is threaded. Three other common cap- screw head styles are shown in Fig. 8–10. A variety of machine-screw head styles are shown in Fig. 8–11. Inch-series 3 machine screws are generally available in sizes from No. 0 to about 8 in. Several styles of hexagonal nuts are illustrated in Fig. 8–12; their dimensions are given in Table A–31. The material of the nut must be selected carefully to match that of the bolt. During tightening, the ﬁrst thread of the nut tends to take the entire load; but yielding occurs, with some strengthening due to the cold work that takes place, and the load is eventually divided over about three nut threads. For this reason you should never reuse nuts; in fact, it can be dangerous to do so. bud29281_appa_1003-1058.qxd 12/28/09 2:43 PM Page 1053

Useful Tables 1053

Table A–29 Dimensions of Square and Hexagonal Bolts

W R

Head Type Nominal Square Regular Hexagonal Heavy Hexagonal Structural Hexagonal Size, in WHWHRmin WHRmin WH Rmin bud29281_appa_1003-1058.qxdbud29281_appa_1003-1058.qxd 12/28/09 12/28/09 2:43 2:43 PM PMPage Page 1053 1053 1 3 11 7 11 4 8 64 16 64 0.01 5 1 13 1 7 16 2 64 2 32 0.01 3 9 1 9 1 8 16 4 16 4 0.01 7 5 19 5 19 16 8 64 8 64 0.01 1053 1 3 21 3 11 7 11 Useful7 Useful Tables Tables5 1053 2 4 64 4 32 0.01 8 32 0.01 8 16 0.009 5 15 27 15 27 1 27 1 25 TableTable A–298 A–29 16 64 16 64 0.02 116 64 0.02 1 16 64 0.021 3 1 1 1 1 1 1 0.02 1 1 1 0.02 1 1 15 0.021 DimensionsDimensions4 of Square of Square8 and andHexagonal 2HexagonalMetric Bolts8 Bolts 2 Bolt4 Sizes2 4 32 1 21 1 43 5 43 5 39 112H H 32 1 2 64 0.03 1 8 64 0.03 1 8 64 0.062 1 11 3 11 3 13 3 13 11 1 8 1 16 4 1 16 4 0.03 1 16 4 0.03 1 16 16 0.062 1 W 7 27 7 27 27 25 1W4 1 8 32 1 8 32 0.03 2 32 0.03 2 32 0.062 R R 3 1 29 1 29 3 29 3 27 1 8 2 16 32 2 16 32 0.03 2 16 32 0.03 2 16 32 0.062 1 1 1 Head3 Type 3 15 1 2 2 4 124 1 0.03Head 2Type8 1 0.03 2 8 16 0.062 Square Regular Hexagonal Heavy Hexagonal Structural Hexagonal NominalNominal Square Regular Hexagonal Heavy Hexagonal Structural Hexagonal Nominal Size,Size, in in WHWHRWHWHRmin WHRWHRmin WHWH R Rmin Size, mm min min min 1 3 11 7 11 1 3 11 7 11 0.010.01 4 M54 8 8 64 3.5864 16 16 864 3.5864 0.2 5 1 13 1 7 5 1 13 1 7 0.010.01 16 M616 2 2 64 64 2 2 1032 4.3832 0.3 3 3 9 9 1 1 9 9 1 1 0.01 8 M88 16 16 4 4 16 16 134 5.684 0.01 0.4 7 7 5 5 19 19 5 5 19 19 0.01 16 M1016 8 8 64 64 8 8 1664 6.8564 0.01 0.4 1 1 3 3 21 21 3 3 11 11 0.01 7 7 11 11 0.01 7 7 5 5 0.009 2 M122 4 4 64 64 4 4 1832 7.9532 0.01 0.68 218 32 7.9532 0.01 0.6 8 8 16 16 0.009 5 15 27 15 27 1 27 1 25 5 M1415 27 15 2127 9.250.02 0.61 12427 9.250.02 0.6 1 1 25 0.021 8 8 16 16 64 64 16 16 64 64 0.02 116 16 64 64 0.02 1 16 16 64 64 0.021 3 M163 1 1 1 1 1 1 1 1 241 10.751 0.02 0.61 1 271 1 10.751 0.02 0.61 1 271 15 10.7515 0.021 0.6 4 4 1 8 8 2 2 1 8 8 2 2 0.02 1 4 4 2 2 0.02 1 4 4 32 32 0.021 11M201 1 21 21 1 1 1 3043 13.4043 0.03 0.85 1 345 43 13.4043 0.03 0.85 1 345 39 13.4039 0.062 0.8 112 2 32 32 1 2 2 64 64 0.03 1 8 8 64 64 0.03 1 8 8 64 64 0.062 1 1M241 11 1 11 3 3 11 1 11 363 15.903 0.03 0.813 1 4113 3 15.903 0.03 0.813 1 4113 11 15.9011 0.062 1.0 1 8 8 1 16 16 4 4 1 16 16 4 4 0.03 1 16 16 4 4 0.03 1 16 16 16 16 0.062 1 1M301 7 1 7 27 27 7 1 7 4627 19.7527 0.03 1.0 2 5027 19.7527 0.03 1.0 2 5025 19.7525 0.062 1.2 1 4 4 1 8 8 32 32 1 8 8 32 32 0.03 2 32 32 0.03 2 32 32 0.062 M36 55 23.55 1.0 60 23.55 1.0 60 23.55 1.5 3 1 3 1 2 1 29 29 1 2 1 29 29 0.033 2 3 29 29 0.033 2 3 27 27 0.062 1 8 8 2 16 16 32 32 2 16 16 32 32 0.03 2 16 16 32 32 0.03 2 16 16 32 32 0.062 1 1 1 1 2 1 121 1 1 0.033 2 3 1 0.033 2 3 15 15 0.062 1 2 2 2 4 4 124 4 1 0.03 2 8 8 1 0.03 2 8 8 16 16 0.062

NominalNominal Size,Size, mm mm M5M5 8 8 3.58 3.58 8 8 3.58 3.58 0.2 0.2 M6M6 10 10 4.38 4.38 0.3 0.3 M8M8 13 13 5.68 5.68 0.4 0.4 ME 423: Machine Design M10M10 16 16 6.85 6.85 0.4 0.4 Instructor: Ramesh Singh M12 18 7.95 0.6 21 7.95 0.6 M12 18 7.95 0.6 21 7.95 0.6 18 M14M14 21 21 9.25 9.25 0.6 0.6 24 24 9.25 9.25 0.6 0.6 M16M16 24 24 10.75 10.75 0.6 0.6 27 27 10.75 10.75 0.6 0.6 27 27 10.75 10.75 0.6 0.6 M20M20 30 30 13.40 13.40 0.8 0.8 34 34 13.40 13.40 0.8 0.8 34 34 13.40 13.40 0.8 0.8 M24M24 36 36 15.90 15.90 0.8 0.8 41 41 15.90 15.90 0.8 0.8 41 41 15.90 15.90 1.0 1.0 M30M30 46 46 19.75 19.75 1.0 1.0 50 50 19.75 19.75 1.0 1.0 50 50 19.75 19.75 1.2 1.2 M36M36 55 55 23.55 23.55 1.0 1.0 60 60 23.55 23.55 1.0 1.0 60 60 23.55 23.55 1.5 1.5 LengthThreaded of Lengths Threads

English

Metric

Shigleys Mechanical Engineering Design

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Metric Thread SpecificationsScrews, Fasteners, and the Designfor of Nonpermanent Steel Joints 435 Bolts Table 8–11 Metric Mechanical-Property Classes for Steel Bolts, Screws, and Studs*

Minimum Minimum Minimum Size Proof Tensile Yield Property Range, Strength,† Strength,† Strength,† Class Inclusive MPa MPa MPa Material Head Marking 4.6 M5–M36 225 400 240 Low or medium carbon 4.6

4.8 M1.6–M16 310 420 340 Low or medium carbon 4.8

5.8 M5–M24 380 520 420 Low or medium carbon 5.8

8.8 M16–M36 600 830 660 Medium carbon, Q&T 8.8

9.8 M1.6–M16 650 900 720 Medium carbon, Q&T 9.8

10.9 M5–M36 830 1040 940 Low-carbon martensite, Q&T 10.9

12.9 M1.6–M36 970 1220 1100 Alloy, Q&T 12.9

*The thread length for bolts and cap screws is 2d 6 L 125 + ≤ L 2d 12 125 < L 200 ME 423: Machine Design T =  + ≤  2d 25 L > 200 + Instructor: Ramesh Singh where Lis the bolt length. The thread length for structural bolts is slightly shorter than given above. 20 †Minimum strengths are strengths exceeded by 99 percent of fasteners.

8–7 Tension Joints—The External Load Let us now consider what happens when an external tensile load P, as in Fig. 8–13, is applied to a bolted connection. It is to be assumed, of course, that the clamping force, which we will call the preload Fi , has been correctly applied by tightening the nut before P is applied. The nomenclature used is: F preload i = Ptotal Total external tensile load applied to the joint = API- Process Piping is in inches

ME 423: Machine Design Instructor: Ramesh Singh 21 API- Process Piping

ME 423: Machine Design Instructor: Ramesh Singh 22 Nuts

• See Appendix A–31 for typicalNuts specifications

• First three See threads Appendix Aof–31 nut for typicalcarry specifications majority of load • Localized First plastic three strain threads inof nutthe carry first majority thread of isload likely, so nuts Localized plastic strain in the first thread is likely, so nuts should should notnot bebe re re-used-used in critical in critical applications. applications

End view Washer-faced, Chamfered both Washer-faced, Chamfered regular sides, regular jam nut both sides, jam nut Fig. 8–12 Shigleys Mechanical Engineering Design

ME 423: Machine Design Instructor: Ramesh Singh 23 TensionTension LoadedLoaded Bolted Fasteners Joint

• GripGrip length l includesincludes everythingeverything beingbeing compressedcompressed byby bolt preload,preload, includingincluding washerswashers • WasherWasher underunder head head prevents prevents burrsburrs atat thethe hole hole from from gouginggouging into the filletfillet underunder thethe boltbolt head head

Fig. 8–13

ME 423: Machine Design Instructor: Ramesh Singh 24

Shigleys Mechanical Engineering Design Cylinder head to body

• Hex-head cap screw in tapped hole used to fasten cylinder head to Pressure Vessel Head

cylinder body Hex-head cap screw in tapped hole used to fasten • Note O-ring seal, not affectingcylinder head tothe cylinder stiffness of the membersbody within the Note O-ring seal, not grip affecting the stiffness of the members within the grip • Only part of the threaded Only part length of the threaded of length of the bolt contributes the bolt contributes toto the the effective effective grip l grip l Fig. 8–14

Shigleys Mechanical Engineering Design

ME 423: Machine Design Instructor: Ramesh Singh 25 Stiffness Modeling Bolted Joint Stiffnesses • During bolt preload Bolted Joint Stiffnesses During bolt preload – bolt is stretched During bolt preload ◦ bolt is stretched◦ bolt is stretched – members in grip are compressed ◦ members◦ members in grip arein grip are • Undercompressed an externalcompressed load P When external load P is When external load P is – Bolt stretchesapplied further applied – Members ◦inBolt grip stretches uncompress further a bit ◦ Bolt stretches◦ Members further in grip Fig. 8–13 • Joint ◦canMembers be modeleduncompress in grip someas a soft bolt springFig. 8in–13 parallel with a uncompressstiff memberJoint can some be modeled spring as a soft bolt spring in parallel Joint can be modeled as a with a stiff member spring soft bolt spring in parallel with a stiff member spring

Shigleys Mechanical Engineering Design ME 423: Machine Design Instructor: Ramesh Singh 26 Shigleys Mechanical Engineering Design Bolt Stiffness Modeling

• Axially loaded rod, 1 1 1 ()(* bud29281_ch08_409-474.qxd 12/16/2009 7:11 pm Page 426 =pinnacle +203:MHDQ196:bud29281:0073529281:bud29281_pagefiles:"# = partly threaded and "# "% "' ()+(* partly unthreaded ,*- ,)- • Each section can be "' = "% = .* .) 426consideredMechanical Engineering as Designa spring Table 8–7 ,),*- • These are two springs "# = Suggested Procedure for Finding Fastener Stiffness ,).*+,*.)

in series h t t t lt 1 2 ld H t

d d

lt L T

l l L T L ld L (a) ME 423: Machine Design (b) Instructor: Ramesh Singh Given fastener diameter d and pitch p in mm or number of threads per inch 27 Washer thickness: t from Table A–32 or A–33 Nut thickness [Fig. (a) only]: H from Table A–31 Grip length: For Fig. (a): l ϭ thickness of all material squeezed between face of bolt and face of nut

h t2/2, t2 < d For Fig. (b): l + = h d/2, t2 d ! + ≥ Fastener length (round up using Table A–17*): For Fig. (a): L > l H + For Fig. (b): L > h 1.5d + Threaded length LT:Inch series: 1 2d 4 in, L 6 in LT + ≤ = " 2d 1 in, L > 6 in + 2 Metric series: 2d 6 mm, L 125 mm, d 48 mm + ≤ ≤ LT 2d 12 mm, 125 < 200 mm =  L  + ≤  2d 25 mm, L > 200 mm + Length of unthreaded portion in grip: l L L d = − T Length of threaded portion in grip: l l l t = − d Area of unthreaded portion: A πd2/4 d = Area of threaded portion: At from Table 8–1 or 8–2 A A E Fastener stiffness: k d t b = A l A l d t + t d *Bolts and cap screws may not be available in all the preferred lengths listed in Table A–17. Large fasteners may not be available in fractional inches or in millimeter lengths ending in a nonzero digit. Check with your bolt supplier for availability. Bolt Stiffness

$%$&' !" = $%(&)$&(%

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426 Mechanical Engineering Design

Table 8–7 Suggested Procedure for Finding Fastener Stiffness

h t t t lt 1 2 ld H t

d d

lt L T

l l L T L ld L (a) (b) Given fastener diameter d and pitch p in mm or number of threads per inch Washer thickness:Effectivet from Table Grip A–32 orLength A–33 Nut thickness [Fig. (a) only]: H from Table A–31 •bud29281_ch08_409-474.qxdFor a screw effective 12/16/2009 grip7:11 pm is Page given 426 pinnacle by 203:MHDQ196:bud29281:0073529281:bud29281_pagefiles: Grip length: For Fig. (a): l ϭ thickness of all material squeezed between face of bolt and face of nut

Procedureh t2/2, to t2Find< d Bolt Stiffness 426 Mechanical ForEngineering Fig. Design(b): l + = h d/2, t2 d ! + ≥ Table 8–7 Fastener length (round up using Table A–17*): Suggested Procedure for Finding Fastener Stiffness For Fig. (a): L > l H + h For Fig. (b): L > h 1.5d t t t lt 1 2 ld +H t Threaded length LT:Inch series: Nut 2d 1 in, L 6 in + 4 ≤ d LT = 1 d " 2d 2 in, L > 6 in Tapped hole + Metric series: lt L T 2d 6 mm, L 125 mm, d 48 mm + ≤ l ≤ l L L T T  2d 12 mm, 125l < L 200 mm L = + d ≤  L  2d 25 mm, L > 200 mm (a) + (b) Given fastener diameter and pitch in mm or number of threads per inch Length of unthreaded portiondME  in 423: grip: Machinep l DesignL L d = − T Length ofWasher threaded thickness: portiont fromInstructor: in grip:Table A–32 Ramesh orl A–33Singhl l t = − d 29 Nut thickness [Fig. (a) only]: H from Table A–31 2 Area of unthreaded portion: Ad πd /4 Grip length: = Area of threadedFor Fig. portion: (a): l ϭ thickness of all materialAt from squeezed Table 8–1 or 8–2 between face of bolt and face of nut Shigleys Mechanical Engineering Design Ad At E Fastener stiffness: h t2/2, t2k< d For Fig. (b): l + b = h d/2, t2 d= Adlt At ld ! + ≥ + Fastener length (round up using Table A–17*): *Bolts and cap screws may not be available in all the preferred lengths listed in Table A–17. Large fasteners may not be available in fractional For Fig. (a): L > l H inches or in millimeter lengths ending in a nonzero digit. Check+ with your bolt supplier for availability. For Fig. (b): L > h 1.5d + Threaded length LT:Inch series: 1 2d 4 in, L 6 in LT + ≤ = " 2d 1 in, L > 6 in + 2 Metric series: 2d 6 mm, L 125 mm, d 48 mm + ≤ ≤ LT 2d 12 mm, 125 < 200 mm =  L  + ≤  2d 25 mm, L > 200 mm + Length of unthreaded portion in grip: l L L d = − T Length of threaded portion in grip: l l l t = − d Area of unthreaded portion: A πd2/4 d = Area of threaded portion: At from Table 8–1 or 8–2 A A E Fastener stiffness: k d t b = A l A l d t + t d *Bolts and cap screws may not be available in all the preferred lengths listed in Table A–17. Large fasteners may not be available in fractional inches or in millimeter lengths ending in a nonzero digit. Check with your bolt supplier for availability. Length Calculations Procedure to Find Bolt Stiffness

ME 423: Machine Design Instructor: Ramesh Singh 30 bud29281_ch08_409-474.qxd 12/16/2009 7:11 pmMember Page 428 pinnacle Stiffness 203:MHDQ196:bud29281:0073529281:bud29281_pagefiles: • Model compressed members as if they are frusta spreading from the bolt head and nut to the midpoint of the grip • Each frustum has a half-apex angle of a • Find stiffness for frustum in compression 428 Mechanical Engineering Design

D Figure 8–15 x

Compression of a member with ␣ the equivalent elastic properties y d y represented by a frustum of a w t x l t hollow cone. Here, l represents 2 the grip length. d dx d

x (a) (b)

The pressure, however, falls off farther away from the bolt. Thus Ito suggests the use of Rotscher’s pressure-coneME 423: method Machine forDesign stiffness calculations with a variable cone Instructor: Ramesh Singh angle. This method is quite complicated, and so here we choose to31 use a simpler approach using a fixed cone angle. Figure 8–15 illustrates the general cone geometry using a half-apex angle α. An 3 angle α 45◦ has been used, but Little reports that this overestimates the clamping stiffness.= When loading is restricted to a washer-face annulus (hardened steel, cast iron, or aluminum), the proper apex angle is smaller. Osgood4 reports a range of 25◦ α 33◦ for most combinations. In this book we shall use α 30◦ except in cases≤ in ≤which the material is insufficient to allow the frusta to exist.= Referring now to Fig. 8–15b, the contraction of an element of the cone of thickness dx subjected to a compressive force P is, from Eq. (4–3), Pdx dδ (a) = EA The area of the element is D 2 d 2 A π r 2 r 2 π x tan α = o − i = + 2 − 2 #$ % $ % & ! " (b) D d D d π x tan α + x tan α − = + 2 + 2 $ %$ % Substituting this in Eq. (a) and integrating gives a total contraction of P t dx δ (c) = π E [x tan α (D d)/2][x tan α (D d)/2] '0 + + + − Using a table of integrals, we find the result to be P (2t tan α D d)(D d) δ ln + − + (d) = π Ed tan α (2t tan α D d)(D d) + + − Thus the spring rate or stiffness of this frustum is P π Ed tan α k = δ = (2t tan α D d)(D d) (8–19) ln + − + (2t tan α D d)(D d) + + −

3R. E. Little, “Bolted Joints: How Much Give?” Machine Design, Nov. 9, 1967. 4C. C. Osgood, “Saving Weight on Bolted Joints,” Machine Design, Oct. 25, 1979. ME 423: Machine Design Instructor: Ramesh Singh 32 Integration

t P In[4]:= d = ‚x 0 p E m x + p m x + q 1 q q q q Out[4]= P‡If œ Reals 1 + Re £ 0 π 0 && Re ≥ 0 && „ p m HtH L H LLm t m t m t p p p p Re ≥ 0 && π 0 œ Reals 1 + Re £ 0 , B m t m»»t B m tF »» m tB F Log p - Log q - Log p + m t + Log q + m t 1 B F »» »», IntegrateB F , x, 0, t , m p - q m x + p m x + q q q q q Assumptions@ D @ ÆD ! @ œ RealsD 1@ + Re D £ 0 π 0 && Re ≥ 0 && m t m t mBt m t 8 < H L H L H L p p p p Re ≥ 0 && π 0 œ Reals 1 + Re £ 0 m t m t »» m t B F »» m t B F

B F »» »» B F FF

ME 423: Machine Design Instructor: Ramesh Singh 33 StiffnessMember of the Stiffness Member With typical value of a = 30º,

Use Eq. (8–20) to find stiffness for each frustum Combine all frusta as springs in series

ME 423: Machine Design Fig. 8–Instructor:15b Ramesh Singh Shigleys Mechanical Engineering Design 34 StiffnessMember Stiffness of multiple for Common layers Material inof Grip same If the grip consists of any number of members all of the same material, two identical frustamaterial can be added in series. The entire joint can be handled with one equation,

dw is the washer face diameter Using standard washer face diameter of 1.5d, and with a = 30º,

Shigleys Mechanical Engineering Design ME 423: Machine Design Instructor: Ramesh Singh 35 Example Problem

As shown in Fig. 8–17a, two plates are clamped by washer-faced 1/2 in-20 UNF × 1 1/2 in SAE grade 5 bolts each with a standard 1/2 N steel plain washer.

(a) Determine the member spring rate km if the top plate is steel and the bottom plate is gray cast iron. (b) Using the method of conical frusta, determine the member spring rate km if both plates are steel.

(d) Determine the bolt spring rate kb

ME 423: Machine Design Instructor: Ramesh Singh 36 bud29281_ch08_409-474.qxd 12/22/09 5:45 PM Page 431 epg Disk1:Desktop Folder:TEMPWORK:Don't-Delete Jobs:MHDQ196/Budynas:

FigureScrews, Fasteners, and the Design of Nonpermanent Joints 431 Figure 8–17

Dimensions in inches.

1.437

0.75 0.095

1 2 0.6725

0.0775

3 4 0.6725

1.527

(a) (b)

The distance between the joint line and the dotted frusta line is 0.6725 0.5 0.095 0.0775 in. Thus, the top frusta consist of the steel washer, steel plate,− and− = 0.0775 in of the cast iron. ME Since 423: theMachine washer Design and top plate are both steel with E 30(106) psi, they can be Instructor:considered Ramesh a singleSingh frustum of 0.595 in thick. The outer diameter= of the frustum of the steel member at the joint interface is 0.75 2(0.595) 37 tan 30° 1.437 in. The outer diameter at the midpoint of the entire joint+ is 0.75 2(0.6725)= tan 30° 1.527 in. Using Eq. (8–20), the spring rate of the steel is + =

6 0.5774π(30)(10 )0.5 6 k1 30.80(10 ) lbf/in = [1.155(0.595) 0.75 0.5](0.75 0.5) = ln + − + [1.155(0.595) 0.75 0.5](0.75 0.5) ! + + − " For the upper cast-iron frustum

6 0.5774π(14.5)(10 )0.5 6 k2 285.5(10 ) lbf/in = [1.155(0.0775) 1.437 0.5](1.437 0.5) = ln + − + [1.155(0.0775) 1.437 0.5](1.437 0.5) ! + + − " For the lower cast-iron frustum

6 0.5774π(14.5)(10 )0.5 6 k3 14.15(10 ) lbf/in = [1.155(0.6725) 0.75 0.5](0.75 0.5) = ln + − + [1.155(0.6725) 0.75 0.5](0.75 0.5) ! + + − " The three frusta are in series, so from Eq. (8–18)

1 1 1 1 6 6 6 km = 30.80(10 ) + 285.5(10 ) + 14.15(10 ) bud29281_ch08_409-474.qxd 12/16/2009 7:11 pm Page 430 pinnacle 203:MHDQ196:bud29281:0073529281:bud29281_pagefiles:

430 Mechanical Engineering Design

Figure 8–16 3.4 3.2 The dimensionless plot of 3.0 stiffness versus aspect ratio of the members of a bolted joint, 2.8 showing the relative accuracy 2.6 Ed of methods of Rotscher, ⁄ 2.4 m Mischke, and Motosh, k 2.2 compared to a ﬁnite-element 2.0 analysis (FEA) conducted by Wileman, Choudury, and 1.8 Green. 1.6 1.4

Dimensionless stiffness, Dimensionless stiffness, 1.2 1.0 0.8 0.6 0.4 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 Aspect ratio, d ⁄ l FEA Rotscher Mischke 45° Mischke 30° Motosh

Table 8–8 Material Poisson Elastic Modulus Stiffness Parameters Used Ratio GPa Mpsi AB of Various Member Steelbud29281_ch08_409-474.qxd 0.291 207 12/22/09 30.0 5:45 PM Page 0.787 431 15 epg Disk1:Desktop 0.628 73 Folder:TEMPWORK:Don't-Delete Jobs:MHDQ196/Budynas: Materials† Aluminum 0.334 71 10.3 0.796 70 0.638 16 † Copper 0.326 119 17.3Solution 0.795 68 0.635 53 Source: J. Wileman, M. Choudury, and I. Green, Gray cast iron 0.211 100 14.5 0.778 71 0.616 16 “Computation of Member Stiffness in Bolted General expression 0.789 52 0.629 14 Connections,” Trans. ASME, J. Mech. Design, vol. 113, There are three sections for stiffnessScrews, : Fasteners,first and the the Design washerof Nonpermanent Joints and431 the December 1991, pp. 432–437. bud29281_ch08_409-474.qxd 12/16/2009 7:11 pm Page top429 frustumpinnacle of203:MHDQ196:bud29281:0073529281:bud29281_pagefiles: steel, Figurea portion 8–17 of cast iron and the bottom frustum Dimensions in inches. As shown in Fig. 8–17 , two plates are clamped by washer-faced 1 in-20 UNF 1 1 in EXAMPLE 8–2of cast irona 2 2 1 × 1.437 SAE grade 5 bolts each with a standard 2 N steel plain washer. 0.75 (a) Determine the member spring rate km if the0.095 top plate is steel and the bottom plate is gray cast iron. (b) Using the method of conical frusta, determine the member spring rate km if both 1 plates are steel. 2 0.6725 (c) Using Eq. (8–23), determine the member spring rate km if both plates are steel. Compare the results with part (b). 0.0775 (d)Determine the bolt spring rate k . 3 b 4 0.6725

Solution From Table A–32, the thickness of a standard 1 N plain washer is 0.095 in. Screws, Fasteners,2 and the Design of Nonpermanent Joints 429 (a) As shown in Fig. 8–17b, the frusta extend halfway into the joint the distance 1.527 1 (0.5 0.75 0.095) 0.6725 in (a) (b) With α 30 , this becomes 2 + + = = ◦ The distance between the joint line and the dotted frusta line is 0.6725 0.5 0.5774π0.095Ed 0.0775 in. Thus, the top frusta consist of the steel washer, steel plate,− and− 0.0775= in of the cast iron. Since the washer and top plate are both steel with k E 30(106) psi, they can be considered a single frustum of 0.595 in thick. The outer = (1.155t D diameter=d)( Dof the frustumd) of the steel member at the joint interface(8–20) is 0.75 2(0.595) ln + −tan 30° 1.437+ in. The outer diameter at the midpoint of the entire joint+ is 0.75 (1.155t D 2(0.6725)d)(=D tan 30° d1.527) in. Using Eq. (8–20), the spring rate of the steel is + + + − = 6 0.5774π(30)(10 )0.5 6 k1 30.80(10 ) lbf/in =ME [1423:.155 (Machine0.595) 0. 75Design0.5](0.75 0.5) = Equation (8–20), or (8–19), must be solved separatelyln for+ each− frustum+ in the Instructor:[1.155(0 Ramesh.595) 0.75Singh0.5](0.75 0.5) ! " joint. Then individual stiffnesses are assembled to obtain km using+ + Eq. −(8–18). 38 If the members of the joint have the sameFor Young’s the upper cast-iron modulus frustum E with symmetrical 6 0.5774π(14.5)(10 )0.5 6 k2 285.5(10 ) lbf/in frusta back to back, then they act as two identical =springs[1.155(0 .0775in ) series.1.437 0. 5]From(1.437 0 .Eq.5) = (8–18) ln + − + [1.155(0.0775) 1.437 0.5](1.437 0.5) we learn that k k/2. Using the grip as l 2t and !dw as the+ diameter+ of− the" washer m = = face, from Eq. (8–19) we ﬁnd the spring rate ofFor thethe lower members cast-iron frustum to be

6 0.5774π(14.5)(10 )0.5 6 k3 14.15(10 ) lbf/in = [1.155(0.6725) 0.75 0.5](0.75 0.5) = π tan α ln + − + Ed [1.155(0.6725) 0.75 0.5](0.75 0.5) km ! + + − " = (l tan α dw d)(dw d) (8–21) 2 ln + −The three frusta are+ in series, so from Eq. (8–18) (l tan α dw d)(dw 1 d) 1 1 1 6 6 6 + + −km = 30.80(10 ) + 285.5(10 ) + 14.15(10 ) The diameter of the washer face is about 50 percent greater than the fastener diameter for standard hexagon-head bolts and cap screws. Thus we can simplify Eq. (8–21) by letting dw 1.5d. If we also use α 30 , then Eq. (8–21) can be written as = = ◦ 0.5774π Ed km = 0.5774l 0.5d (8–22) 2 ln 5 + 0.5774l 2.5d ! + " It is easy to program the numbered equations in this section, and you should do so. The time spent in programming will save many hours of formula plugging. To see how good Eq. (8–21) is, solve it for km/Ed:

km π tan α Ed = (l tan α dw d)(dw d) 2 ln + − + (l tan α dw d)(dw d) # + + − $

Earlier in the section use of α 30◦ was recommended for hardened steel, cast iron, or aluminum members. Wileman,= Choudury, and Green5 conducted a ﬁnite element study of this problem. The results, which are depicted in Fig. 8–16, agree with the α 30◦ recommendation, coinciding exactly at the aspect ratio d/l 0.4. Addition- ally,= they offered an exponential curve-ﬁt of the form =

km A exp(Bd/l) (8–23) Ed = with constants A and B deﬁned in Table 8–8. Equation (8–23) offers a simple calcu- lation for member stiffness km. However, it is very important to note that the entire joint must be made up of the same material.Fordeparturefromtheseconditions, Eq. (8–20) remains the basis for approaching the problem.

5J. Wileman, M. Choudury, and I. Green, “Computation of Member Stiffness in Bolted Connections,” Trans. ASME, J. Mech. Design, vol. 113, December 1991, pp. 432–437. bud29281_ch08_409-474.qxd 12/22/09 5:45 PM Page 431 epg Disk1:Desktop Folder:TEMPWORK:Don't-Delete Jobs:MHDQ196/Budynas:

Screws, Fasteners, and the Design of Nonpermanent Joints 431

Figure 8–17

Dimensions in inches.

1.437

0.75 0.095

bud29281_ch08_409-474.qxd 12/22/09 5:45 PM Page 431 epg Disk1:Desktop Folder:TEMPWORK:Don't-Delete Jobs:MHDQ196/Budynas: 1 2 0.6725

0.0775

3 4 0.6725 Screws, Fasteners, and the Design of Nonpermanent Joints 431

Figure 8–17

Dimensions in inches. 1.527

1.437 (a) (b) 0.75 0.095

1 2 0.6725

The distance between the joint line and the dotted frusta 0.0775 line is 0.6725 0.5

3 − − 4 0.095 0.0775 in. Thus, 0.6725 the top frusta consist of the steel washer, steel plate, and 0.0775= in of the cast iron. Since the washer and top plate are both steel with 6 E 30(10 ) psi, they can be considered a1.527 single frustum of 0.595 in thick. The outer diameter= of the frustum of the steel member at the joint interface is 0.75 2(0.595) Individual(a) Stiffness(b) tan 30° 1.437 in. The outer diameter at the midpoint of the entire joint+ is 0.75 2(0.6725)= tan 30° 1.527 in. Using Eq. (8–20), the spring rate of the steel is + = The distance between the joint line and the dotted frusta line is 0.6725 0.5 0.095 0.0775 in. Thus, the top frusta consist of 6 the steel washer, steel plate,− and− = 0.5774π(30)(10 )0.5 6 0.0775k1 in of the cast iron. Since the washer and top plate are both steel with30.80(10 ) lbf/in E 30(10= 6) psi,[1 they.155 can( be0. 595considered) 0 a. 75single frustum0.5]( 0of. 750.595 in0 .thick.5) The= outer diameter= ofln the frustum of the steel member+ at− the joint interface+ is 0.75 2(0.595) [1.155(0.595) 0.75 0.5](0.75 0.5) + tan 30° 1.437! in. The outer diameter+ at the +midpoint of the entire− joint" is 0.75 2(0.6725)= tan 30° 1.527 in. Using Eq. (8–20), the spring rate of the steel is + = For the upper cast-iron frustum 6 0.5774π(30)(10 )0.5 6 k1 30.80(10 ) lbf/in = [1.155(0.595) 0.75 0.5](0.75 0.5) = ln + − + [1.155(0.5950).57740.75π(014.5](.05.75)(1006.5))0.5 ! + + − " 6 k2 285.5(10 ) lbf/in = [1.155(0.0775) 1.437 0.5](1.437 0.5) = For the upperln cast-iron frustum + − + [1.155(0.0775) 1.437 0.5](1.437 0.5) ! + 6 + − " 0.5774π(14.5)(10 )0.5 6 k2 285.5(10 ) lbf/in = [1.155(0.0775) 1.437 0.5](1.437 0.5) = ln + − + For the lower [1cast-iron.155(0.0775 )frustum1.437 0.5](1.437 0.5) ! + + − "

For the lower cast-iron frustum 6 0.5774π(14.5)(10 )0.5 6 k3 14.15(10 ) lbf/in 6 = [1.1550.(57740.6725π(14.)5)(100).075.5 0.5](0.75 0.65) = k3 14.15(10 ) lbf/in =ln [1.155(0.6725) 0.75+0.5](0.75− 0.5) = + ln [1.155(0.6725+ ) − 0.75 +0.5](0.75 0.5) ![1.155(0.6725) 0.75+0.5](0.75+ 0.5) − " ! + + − " The Thethree three frusta frusta are are in series,in series, so from so Eq. from (8–18) Eq. (8–18)

1 1 1 1 6 6 6 km 1= 30.80(10 ) +1 285.5(10 ) + 14.151 (10 ) 1 6 6 6 km = 30.80(10 ) + 285.5(10 ) + 14.15(10 ) ME 423: Machine Design Instructor: Ramesh Singh 39 bud29281_ch08_409-474.qxd 12/16/2009 7:11 pm Page 429 pinnacle 203:MHDQ196:bud29281:0073529281:bud29281_pagefiles:

Screws, Fasteners, and the Design of Nonpermanent Joints 429

With α 30 , this becomes = ◦ 0.5774π Ed k = (1.155t D d)(D d) (8–20) ln + − + (1.155t D d)(D d) + + − Equation (8–20), or (8–19), must be solved separately for each frustum in the joint. Then individual stiffnesses are assembled to obtain km using Eq. (8–18). If the members of the joint have the same Young’s modulus E with symmetrical frusta back to back, then they act as two identical springs in series. From Eq. (8–18) we learn that km k/2. Using the grip as l 2t and dw as the diameter of the washer face, from Eq. (8–19)= we ﬁnd the spring rate= of the members to be

π Ed tan α km bud29281_ch08_409-474.qxd 12/16/2009 7:11 pm Page= 432(l tan pinnacleα dw 203:MHDQ196:bud29281:0073529281:bud29281_pagefiles:d)(dw d) (8–21) 2 ln + − + (l tan α dw d)(dw d) + + − The diameter of the washer face is aboutAssuming 50 percent greater all than steel the fastener diameter for standard hexagon-head bolts and cap screws. Thus we can simplify Eq. (8–21) by letting dw 1.5d. If we also use α 30 , then Eq. (8–21) can be written as = = ◦ 432 Mechanical Engineering Design 0.5774π Ed km = 0.5774l 0.5d (8–22) 2 ln 5 + Answer This results in k 9.378 (1006.)5774 lbf/in.l 2.5d m = ! + " (b) If the entire joint is steel, Eq. (8–22) with l 2(0.6725) 1.345 in gives It is easy to program the numbered equations in this section,= and you should= do so. 6 The time spent in programming will0.5774 saveπ many(30.0 )(hours10 )of0. 5formula plugging. 6 Answer km 14.64(10 ) lbf/in. To see how good Eq.= (8–21) is,0 .solve5774 (it1 .for345 k)m/Ed0.:5(0.5) = 2 ln 5 + 0.5774(1.345) 2.5(0.5) km ! " π tan α + #$ (c) From TableEd 8–8,= A (l0tan.787α 15d,w B d)(0d.628w d 73) . Equation (8–23) gives 2 ln = + −= + (l tan α dw d)(dw d) 6 # + + − $ 6 Answer km 30(10 )(0.5)(0.787 15) exp[0.628 73ME 423:(0 .Machine5)/1 Design.345] 14.92(10 ) lbf/in = Instructor: Ramesh Singh = Earlier in the section use of α 30◦ was recommended for hardened steel, cast iron, 40 = 5 or aluminumFor this members. case, the Wileman, difference Choudury, between and the Green resultsconducted for Eqs. a finite (8–22) element and (8–23) is less study ofthan this 2 problem.percent. The results, which are depicted in Fig. 8–16, agree with the α 30 / 0.4 (◦drecommendation,) Following the coinciding procedure exactly of Table at the 8–7, aspect the ratio threaded d l length. Addition- of a 0.5-in bolt is ally,= they offered an exponential curve-fit of the form = L 2(0.5) 0.25 1.25 in. The length of the unthreaded portion is l 1.5 T = + = d = − 1.25 0.25 in. The lengthkm of the unthreaded portion in grip is lt 1.345 0.25 = A exp(Bd/l) 2 (8–23)= −2 = 1.095 in. The major Ed diameter= area is Ad (π/4)(0.5 ) 0.196 3 in . From Table 8–2, the tensile-stress area is A 0.159= 9 in2. From Eq.= (8–17) t = with constants A and B defined in Table 8–8. Equation (8–23)6 offers a simple calcu- lation for member stiffness k . However,0.196 3 ( it0 . is159 very 9) 30 important(10 ) to note that the6 entire Answer kb m 3.69(10 ) lbf/in joint must be made up of = the0 .same196 3 material(1.095).Fordeparturefromtheseconditions,0.159 9(0.25) = + Eq. (8–20) remains the basis for approaching the problem.

5J. Wileman, M. Choudury, and I. Green, “Computation of Member Stiffness in Bolted Connections,” Trans. 8–6ASME, J.Bolt Mech. Design, Strengthvol. 113, December 1991, pp. 432–437. In the specification standards for bolts, the strength is specified by stating SAE or ASTM minimum quantities, the minimum proof strength, or minimum proof load, and the minimum tensile strength. The proof load is the maximum load (force) that a bolt can withstand without acquiring a permanent set. The proof strength is the quotient of the proof load and the tensile-stress area. The proof strength thus corresponds roughly to the proportional limit and corresponds to 0.0001-in permanent set in the fastener (first measurable deviation from elastic behavior). Tables 8–9, 8–10, and 8–11 provide minimum strength specifications for steel bolts. The values of the mean proof strength, the mean tensile strength, and the corresponding standard deviations are not part of the specification codes, so it is the designer’s responsibility to obtain these values, perhaps by laboratory testing, if designing to a reliability specification. The SAE specifications are found in Table 8–9. The bolt grades are numbered according to the tensile strengths, with decimals used for variations at the same strength level. Bolts and screws are available in all grades listed. Studs are available in grades 1, 2, 4, 5, 8, and 8.1. Grade 8.1 is not listed. ASTM specifications are listed in Table 8–10. ASTM threads are shorter because ASTM deals mostly with structures; structural connections are generally loaded in shear, and the decreased thread length provides more shank area. Specifications for metric fasteners are given in Table 8–11. It is worth noting that all specification-grade bolts made in this country bear a man- ufacturer’s mark or logo, in addition to the grade marking, on the bolt head. Such marks confirm that the bolt meets or exceeds specifications. If such marks are missing, the bolt may be imported; for imported bolts there is no obligation to meet specifications. Bolts in fatigue axial loading fail at the fillet under the head, at the thread runout, and at the first thread engaged in the nut. If the bolt has a standard shoulder under the head, it has a value of K f from 2.1 to 2.3, and this shoulder fillet is protected bud29281_ch08_409-474.qxd 12/16/2009 7:11 pm Page 427 pinnacle 203:MHDQ196:bud29281:0073529281:bud29281_pagefiles:

Screws, Fasteners, and the Design of Nonpermanent Joints 427 threaded portion. Thus the stiffness constant of the bolt is equivalent to the stiffnesses of two springs in series. Using the results of Prob. 4–1, we ﬁnd

1 1 1 k1k2 or k (8–15) k = k + k = k k 1 2 1 + 2 bud29281_ch08_409-474.qxdfor 12/16/2009 two springs 7:11 in pm series. Page 432 From pinnacle Eq. (4–4),203:MHDQ196:bud29281:0073529281:bud29281_pagefiles: the spring rates of the threaded and unthreaded portions of the bolt in the clamped zone are, respectively,

At E Ad E kt kd (8–16) = lt = ld 432 Mechanical whereEngineering DesignA tensile-stress area (Tables 8–1, 8–2) t = Answer lThist resultslength in ofk threaded9.378 (10 portion6) lbf/in. of grip = m = (b) If the entire joint is steel, Eq. (8–22) with l 2(0.6725) 1.345 in gives Ad major-diameter area of fastener = = = 0.5774π(30.0)(106)0.5 Answer ld lengthk of unthreaded portion in grip 14.64(106) lbf/in. = m = 0.5774(1.345) 0.5(0.5) = 2 ln 5 + 0.5774Bolt(1.345) Stiffness2.5(0.5) Substituting these stiffnesses! "in Eq. (8–15)+ gives #$ (c) From Table 8–8, A 0.787 15, B 0.628 73. Equation (8–23) gives = = 6 Ad At E 6 Answer km 30(10 )(0.5)(0.787 15) exp[0kb .628 73(0.5)/1.345] 14.92(10 ) lbf/in (8–17) = = Adlt Atld = For this case, the difference between the results+ for Eqs. (8–22) and (8–23) is less where kb thanis the 2 percent. estimated effective stiffness of the bolt or cap screw in the clamped zone. For (d short) Following fasteners, the procedure the one of in Table Fig. 8–7, 8–14, the threaded for example, length of the a 0.5-in unthreaded bolt is area is L 2(0.5) 0.25 1.25 in. The length of the unthreaded portion is l 1.5 small and soT =the first+ of the= expressions in Eq. (8–16) can be used to findd = kb.− For long 1.25 0.25 in. The length of the unthreaded portion in grip is lt 1.345 0.25 fasteners, the= threaded area is relatively small, and so2 the = second− 2 expression= in 1.095 in. The major diameter area is Ad (π/4)(0.5 ) 0.196 3 in . From Eq. (8–16)Table can 8– be2, theused. tensile-stress Table 8–7 area is is useful.A 0.159= 9 in2. From Eq.= (8–17) t = 0.196 3(0.159 9)30(106) Answer k 3.69(106) lbf/in b = 0.196 3(1.095) 0.159 9(0.25) = 8–5 Joints—Member Stiffness+ In the previous section, we determined the stiffness of the fastener in the clamped zone. In this section, we wish to study the stiffnesses of the members in the clamped zone. Both8–6 of theseBolt stiffnesses Strength must be known in order to learn what happens when the assembled connection is subjected to an externalME 423: Machine tensile Design loading. In the specification standards for bolts,Instructor: the strength Ramesh Singh is specified by stating SAE or ThereASTM may minimum be more quantities, than two the membersminimum proof included strength, inor theminimum grip proof of the load, fastener.and 41 All together thesethe minimum act like tensile compressive strength. The springs proof load inis series, the maximum and hence load (force) the thattotal a boltspring rate of the memberscan withstand is without acquiring a permanent set. The proof strength is the quotient of the proof load and the tensile-stress area. The proof strength thus corresponds roughly to the proportional1 limit1 and1 corresponds1 to 0.0001-in1 permanent set in the fastener (first measurable deviation from elastic behavior). Tables 8–9, 8–10, and 8–11 (8–18) provide minimum strengthkm = specificationsk1 + k2 +for ksteel3 +···+ bolts. Thek ivalues of the mean proof If one of strength,the members the mean is tensile a soft strength, gasket, and its the stiffnesscorresponding relative standard to deviationsthe other are members not is part of the specification codes, so it is the designer’s responsibility to obtain these usually sovalues, small perhaps that forby laboratoryall practical testing, purposes if designing the to othersa reliability can specification.be neglected and only the gasket stiffnessThe SAE used. specifications are found in Table 8–9. The bolt grades are numbered If thereaccording is no to gasket, the tensile the strengths, stiffness with of decimalsthe members used for is variations rather difficult at the same to obtain, strength level. Bolts and screws are available in all grades listed. Studs are available except by inexperimentation, grades 1, 2, 4, 5, 8, becauseand 8.1. Grade the compression8.1 is not listed. region spreads out between the bolt head andASTM the nut specifications and hence are listed the area in Table is 8–10. not uniform. ASTM threads There are shorter are, however, because some cases in whichASTM this deals area mostly can with be structures; determined. structural connections are generally loaded in Ito2 hasshear, used and ultrasonic the decreased techniques thread length to determine provides more the shank pressure area. distribution at the mem- Specifications for metric fasteners are given in Table 8–11. ber interface.It The is worth results noting show that all that specification-grade the pressure bolts stays made high in this out country to about bear 1.5a man- bolt radii. ufacturer’s mark or logo, in addition to the grade marking, on the bolt head. Such marks confirm that the bolt meets or exceeds specifications. If such marks are missing, the bolt may be imported; for imported bolts there is no obligation to meet specifications. 2 Y. Ito, J. Toyoda,Bolts and S.in Nagata,fatigue axial“Interface loading Pressure fail at theDistribution fillet under in athe Bolt-Flange head, at the Assembly,” thread runout, ASME paper no. 77-WA/DE-11,and at 1977. the first thread engaged in the nut. If the bolt has a standard shoulder under the head, it has a value of K f from 2.1 to 2.3, and this shoulder fillet is protected