Budynas-Nisbett:Shigley's Ill.Design of Mechanical 8.Screws,Fasteners,and T©The McGraw-Hil 413 Mechanical Engineering Elements the Design of Companies,2008 Design,Eighth Edition Nonpermanent Joints 410 Mechanical Engineering Design Figure 8-11 Types of heads used on machine screws. H H (a)Round head (b)Flat head H (c)Fillister head (d)Oval head (e)Truss head (f)Binding head H (g)Hex head(trimmed) (h)Hex head (upset) Figure 8-12 一H ←日+ Approx.in Approx.in →H Hexagonal nuts:(a)end view, general;(b)washer-foced regular nut;(d)regular nut chamfered on both sides; (d)jam nut with washer face; (e)jam nut chamfered on both sides. (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
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 8. Screws, Fasteners, and the Design of Nonpermanent Joints © The McGraw−Hill 413 Companies, 2008 410 Mechanical Engineering Design A H D L (a) Round head A D L H A (b) Flat head A H D L (c) Fillister head D L H (d) Oval head A H D L (e) Truss head W H D L (g) Hex head (trimmed) 80 to 82° 80 to 82° R A D L (f) Binding head 5° ±3° W H D L (h) Hex head (upset) Figure 8–11 Types of heads used on machine screws. 30 30 Approx. in W H (a) (b) (c) (d) (e) 30 H H 30 H Approx. in 1 64 1 64 Figure 8–12 Hexagonal nuts: (a) end view, general; (b) washer-faced regular nut; (c) regular nut chamfered on both sides; (d) jam nut with washer face; (e) jam nut chamfered on both sides. 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
414 Budynas-Nisbett:Shigley's Ill.Design of Mechanical 8.Screws,Fasteners,and T©The McGraw-Hill Mechanical Engineering Elements the Design of Companies,2008 Design,Eighth Edition Nonpermanent Joints Screws,Fasteners,and the Design of Nonpermanent Joints 411 Figure 8-13 A bolted connection loaded in tension by the forces P.Note the use of two washers.Note how the threads extend into the body of the connection. This is usual and is desired.I is the grip of the connection. Figure 8-14 Section of cylindrical pressure vessel.Hexagonhead cap screws are used to fasten the cylinder head to the body. Note the use of an O-ring seal I'is the effective grip of the connection (see Table 8-7). A section through a tension-loaded bolted joint is illustrated in Fig.8-13.Notice the clearance space provided by the bolt holes.Notice,too,how the bolt threads extend into the body of the connection. As noted previously,the purpose of the bolt is to clamp the two,or more,parts together.Twisting the nut stretches the bolt to produce the clamping force.This clamping force is called the pretension or bolt preload.It exists in the connection after the nut has been properly tightened no matter whether the external tensile load P is exerted or not. Of course,since the members are being clamped together,the clamping force that produces tension in the bolt induces compression in the members. Figure 8-14 shows another tension-loaded connection.This joint uses cap screws threaded into one of the members.An alternative approach to this problem (of not using a nut)would be to use studs.A stud is a rod threaded on both ends.The stud is screwed into the lower member first:then the top member is positioned and fastened down with hardened washers and nuts.The studs are regarded as permanent,and so the joint can be disassembled merely by removing the nut and washer.Thus the threaded part of the lower member is not damaged by reusing the threads. The spring rate is a limit as expressed in Eq.(4-1).For an elastic member such as a bolt,as we learned in Eq.(4-2),it is the ratio between the force applied to the member and the deflection produced by that force.We can use Eq.(4-4)and the results of Prob.4-1 to find the stiffness constant of a fastener in any bolted connection. The grip of a connection is the total thickness of the clamped material.In Fig.8-13 the grip is the sum of the thicknesses of both members and both washers In Fig.8-14 the effective grip is given in Table 8-7. The stiffness of the portion of a bolt or screw within the clamped zone will gen- erally consist of two parts,that of the unthreaded shank portion and that of the
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 8. Screws, Fasteners, and the Design of Nonpermanent Joints 414 © The McGraw−Hill Companies, 2008 Screws, Fasteners, and the Design of Nonpermanent Joints 411 P P P P l Figure 8–13 A bolted connection loaded in tension by the forces P. Note the use of two washers. Note how the threads extend into the body of the connection. This is usual and is desired. l is the grip of the connection. l' Figure 8–14 Section of cylindrical pressure vessel. Hexagon-head cap screws are used to fasten the cylinder head to the body. Note the use of an O-ring seal. l is the effective grip of the connection (see Table 8–7). A section through a tension-loaded bolted joint is illustrated in Fig. 8–13. Notice the clearance space provided by the bolt holes. Notice, too, how the bolt threads extend into the body of the connection. As noted previously, the purpose of the bolt is to clamp the two, or more, parts together. Twisting the nut stretches the bolt to produce the clamping force. This clamping force is called the pretension or bolt preload. It exists in the connection after the nut has been properly tightened no matter whether the external tensile load P is exerted or not. Of course, since the members are being clamped together, the clamping force that produces tension in the bolt induces compression in the members. Figure 8–14 shows another tension-loaded connection. This joint uses cap screws threaded into one of the members. An alternative approach to this problem (of not using a nut) would be to use studs. A stud is a rod threaded on both ends. The stud is screwed into the lower member first; then the top member is positioned and fastened down with hardened washers and nuts. The studs are regarded as permanent, and so the joint can be disassembled merely by removing the nut and washer. Thus the threaded part of the lower member is not damaged by reusing the threads. The spring rate is a limit as expressed in Eq. (4–1). For an elastic member such as a bolt, as we learned in Eq. (4–2), it is the ratio between the force applied to the member and the deflection produced by that force. We can use Eq. (4–4) and the results of Prob. 4–1 to find the stiffness constant of a fastener in any bolted connection. The grip l of a connection is the total thickness of the clamped material. In Fig. 8–13 the grip is the sum of the thicknesses of both members and both washers. In Fig. 8–14 the effective grip is given in Table 8–7. The stiffness of the portion of a bolt or screw within the clamped zone will generally consist of two parts, that of the unthreaded shank portion and that of the
Budynas-Nisbett:Shigley's Ill.Design of Mechanical 8.Screws,Fasteners,and ©The McGraw-Hil 415 Mechanical Engineering Elements the Design of Companies,2008 Design,Eighth Edition Nonpermanent Joints 412 Mechanical Engineering Design Table 8-7 Suggested Procedure for Finding Fastener Stiffness ←h +H+ (a) Given fastener diameter d and pitch p or number of threads Effective grip Grip is thickness I h+2/2,2<d Washer thickness from t={h+d2,n≥d Table A-32 or A-33 Threaded length Lr Inch series: |2d+}in, L≤6in Lr= 2d+in, L>≥6in Metric series: [2d+6mm,l≤125，d≤48mm 4= 2d+12mm,125<l≤200mm 12d+25mm,l>200mm Fastener length:L>+H Fastener length:L>h+1.5d Round up using Table A-17 Length of useful unthreaded Length of useful unthreaded portion:a=L-L灯 portion:l =L-L Length of threaded portion: Length of useful threaded 4=1-4 portion:I,=I'-l Area of unthreaded portion: A=πd14 Area of threaded portion: A.Table 8-1 or 8-2 Fastener stiffness: k6= AdA,E Adl:+Aild Boltsndrewsmayno be ve inthe preferred lengths listed in Table A-17.e fasteers may o bevbe ineslmentsnnin anon digit.Check with your bolt supplier for availability
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 8. Screws, Fasteners, and the Design of Nonpermanent Joints © The McGraw−Hill 415 Companies, 2008 412 Mechanical Engineering Design Given fastener diameter d and pitch p or number of threads Effective grip Grip is thickness l l = h + t2/2, t2 < d h + d/2, t2 ≥ d Washer thickness from Table A–32 or A–33 Threaded length LT Inch series: LT = 2d + 1 4 in, L ≤ 6 in 2d + 1 2 in, L > 6 in Metric series: LT = ⎧ ⎪⎨ ⎪⎩ 2d + 6 mm, 2d + 12 mm, 2d + 25 mm, L ≤ 125, d ≤ 48 mm 125 < L ≤ 200 mm L > 200 mm Fastener length: L > l H Fastener length: L > h 1.5d Round up using Table A–17∗ Length of useful unthreaded Length of useful unthreaded portion: l d L LT portion: l d L LT Length of threaded portion: Length of useful threaded l t l l d portion: l t l’ l d Area of unthreaded portion: Ad π d24 Area of threaded portion: At , Table 8–1 or 8–2 Fastener stiffness: kb = AdAtE Adl t + At l 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. l' L l d LT h t t 2 t 1 l d d l LT L t l t H l t (a) (b) d Table 8–7 Suggested Procedure for Finding Fastener Stiffness
416 Budynas-Nisbett:Shigley's Ill.Design of Mechanical 8.Screws,Fasteners,and T©The McGraw-Hil Mechanical Engineering Elements the Design of Companies,2008 Design,Eighth Edition Nonpermanent Joints Screws,Fasteners,and the Design of Nonpermanent Joints 413 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 111 or kik2 k=ki+kz (8-15) for two springs in series.From Eq.(4-4),the spring rates of the threaded and unthreaded portions of the bolt in the clamped zone are,respectively, k=AE 女= (8-161 where A,tensile-stress area (Tables 8-1,8-2) l,=length of threaded portion of grip Ad=major-diameter area of fastener l length of unthreaded portion in grip Substituting these stiffnesses in Eq.(8-15)gives AdAE k=Aa山+Ald (8-17刀 where k is the estimated effective stiffness of the bolt or cap screw in the clamped zone.For short fasteners,the one in Fig.8-14,for example,the unthreaded area is small and so the first of the expressions in Eq.(8-16)can be used to find k.For long fasteners,the threaded area is relatively small,and so the second expression in Eq. (8-16)can be used.Table 8-7 is useful. 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. Both of these stiffnesses must be known in order to learn what happens when the assembled connection is subjected to an external tensile loading. There may be more than two members included in the grip of the fastener.All together these act like compressive springs in series,and hence the total spring rate of the members is 1111 1 =+后+后++后 (8-18) If one of the members is a soft gasket,its stiffness relative to the other members is usually so small that for all practical purposes the others can be neglected and only the gasket stiffness used. If there is no gasket,the stiffness of the members is rather difficult to obtain, except by experimentation,because the compression spreads out between the bolt head and the nut and hence the area is not uniform.There are,however,some cases in which this area can be determined. Ito2 has used ultrasonic techniques to determine the pressure distribution at the mem- ber interface.The results show that the pressure stays high out to about 1.5 bolt radii. Y.Ito.J.Toyoda.and S.Nagata,"Interface Pressure Distribution in a Bolt-Flange Assembly,"ASME paper no.77-WADE-11,1977
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 8. Screws, Fasteners, and the Design of Nonpermanent Joints 416 © The McGraw−Hill Companies, 2008 Screws, Fasteners, and the Design of Nonpermanent Joints 413 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 k = 1 k1 + 1 k2 or k = k1k2 k1 + k2 (8–15) for two springs in series. From Eq. (4–4), the spring rates of the threaded and unthreaded portions of the bolt in the clamped zone are, respectively, kt = At E lt kd = Ad E ld (8–16) where At = tensile-stress area (Tables 8–1, 8–2) lt = length of threaded portion of grip Ad = major-diameter area of fastener ld = length of unthreaded portion in grip Substituting these stiffnesses in Eq. (8–15) gives kb = Ad At E Adlt + Atld (8–17) where kb is the estimated effective stiffness of the bolt or cap screw in the clamped zone. For short fasteners, the one in Fig. 8–14, for example, the unthreaded area is small and so the first of the expressions in Eq. (8–16) can be used to find kb. For long fasteners, the threaded area is relatively small, and so the second expression in Eq. (8–16) can be used. Table 8–7 is useful. 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. Both of these stiffnesses must be known in order to learn what happens when the assembled connection is subjected to an external tensile loading. There may be more than two members included in the grip of the fastener. All together these act like compressive springs in series, and hence the total spring rate of the members is 1 km = 1 k1 + 1 k2 + 1 k3 +···+ 1 ki (8–18) If one of the members is a soft gasket, its stiffness relative to the other members is usually so small that for all practical purposes the others can be neglected and only the gasket stiffness used. If there is no gasket, the stiffness of the members is rather difficult to obtain, except by experimentation, because the compression spreads out between the bolt head and the nut and hence the area is not uniform. There are, however, some cases in which this area can be determined. Ito2 has used ultrasonic techniques to determine the pressure distribution at the member interface. The results show that the pressure stays high out to about 1.5 bolt radii. 2 Y. Ito, J. Toyoda, and S. Nagata, “Interface Pressure Distribution in a Bolt-Flange Assembly,” ASME paper no. 77-WA/DE-11, 1977
Budynas-Nisbett:Shigley's Ill.Design of Mechanical 8.Screws,Fasteners,and T©The McGraw-Hill 417 Mechanical Engineering Elements the Design of Companies,2008 Design,Eighth Edition Nonpermanent Joints 414 Mechanical Engineering Design Figure 8-15 Compression of a member with the equivalent elastic properties represented by a frustum of a hollow cone. Here,represents the grip length. (a) (b) The pressure,however,falls off farther away from the bolt.Thus Ito suggests the use of Rotscher's pressure-cone method for stiffness calculations with a variable cone angle.This method is quite complicated,and so here we choose to use a simpler approach using a fixed cone angle. Figure 8-15 illustrates the general cone geometry using a half-apex angle a.An angle a=45 has been used,but Little3 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.Osgood reports a range of 25°≤a≤33°for most combinations..n this book we shall use a=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 thick- ness dx subjected to a compressive force P is,from Eg.(4-3), Pdx d8= (a) EA The area of the element is A=仔-分=[(ma+)-()] =(m+） 2 (6 Substituting this in Eq.(a)and integrating gives a total contraction of 8= P dx E Jo [x tana+(D+d)/2][x tana+(D-d)/2] (c) Using a table of integrals,we find the result to be P (2t tana+D-d)(D+d) 8= -In (d) πEd tano (2ttana+D+d)(D-d) Thus the spring rate or stiffness of this frustum is P πEd tan o k= 万=(2rtan&+D-d(D+D (8-191 (2t tana+D+d)(D-d) 3R.E.Little,"Bolted Joints:How Much Give?"Machine Design,Nov.9,1967. "C.C.Osgood."Saving Weight on Bolted Joints,"Machine Design.Oct.25.1979
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 8. Screws, Fasteners, and the Design of Nonpermanent Joints © The McGraw−Hill 417 Companies, 2008 414 Mechanical Engineering Design The pressure, however, falls off farther away from the bolt. Thus Ito suggests the use of Rotscher’s pressure-cone method for stiffness calculations with a variable cone angle. This method is quite complicated, and so here we choose to use a simpler approach using a fixed cone angle. Figure 8–15 illustrates the general cone geometry using a half-apex angle α. An angle α = 45◦ has been used, but Little3 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), dδ = P dx E A (a) The area of the element is A = π r 2 o − r 2 i = π x tan α + D 2 2 − d 2 2 = π x tan α + D + d 2 x tan α + D − d 2 (b) Substituting this in Eq. (a) and integrating gives a total contraction of δ = P π E t 0 dx [x tan α + (D + d)/2][x tan α + (D − d)/2] (c) Using a table of integrals, we find the result to be δ = P π Ed tan α ln (2t tan α + D − d)(D + d) (2t tan α + D + d)(D − d) (d) Thus the spring rate or stiffness of this frustum is k = P δ = π Ed tan α ln (2t tan α + D − d)(D + d) (2t tan α + D + d)(D − d) (8–19) (a) (b) t y t D x y l 2 d dw d x dx x Figure 8–15 Compression of a member with the equivalent elastic properties represented by a frustum of a hollow cone. Here, l represents the grip length. 3 R. E. Little, “Bolted Joints: How Much Give?” Machine Design, Nov. 9, 1967. 4 C. C. Osgood, “Saving Weight on Bolted Joints,” Machine Design, Oct. 25, 1979.