Bolt Pressure Distribution

ME 307 Machine Design I

CH-8 LEC 36 Slide 1Dr. A. Aziz Bazoune Chapter 8: Screws, Fasteners and the Design of Nonpermanent Joints


8-1 8-1 Thread Standards and DefinitionsThread Standards and Definitions8-28-2 The Mechanics of Power ScrewsThe Mechanics of Power Screws8-38-3 Strength Constraints Strength Constraints 8-48-4 Joints-Fasteners StiffnessJoints-Fasteners Stiffness8-58-5 Joints-Member Stiffness Joints-Member Stiffness 8-68-6 Bolt Strength Bolt Strength 8-78-7 Tension Joints-The External LoadTension Joints-The External Load8-88-8 Relating Bolt Torque to Bolt TensionRelating Bolt Torque to Bolt Tension8-98-9 Statically Loaded Tension Joint with PreloadStatically Loaded Tension Joint with Preload8-108-10 Gasketed JointsGasketed Joints8-118-11 Fatigue Loading of Tension JointsFatigue Loading of Tension Joints8-128-12 Shear JointsShear Joints8-138-13 SetscrewsSetscrews8-148-14 Keys and PinsKeys and Pins8-158-15 Stochastic ConsiderationsStochastic Considerations



8-38-3 Strength Constraints Strength Constraints 8-48-4 Joints-Fasteners StiffnessJoints-Fasteners Stiffness8-58-5 Joints-Member Stiffness Joints-Member Stiffness 8-68-6 Bolt Strength Bolt Strength


ME 307 Machine Design I 8-48-4 Joints: Fastener Joints: Fastener

Stiffness 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.


Twisting the nut stretches the bolt to produce the clamping force. This clamping force is called the pretention or bolt preload.

This force exists in the connection after the nut has been properly tightened.

Figure 8-13A 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. LG is the grip of the connection.


Figure 8-14Section of cylindrical pressure vessel. Hexagon-head caps crews are used to fasten the cylinder head to the body. Note the use of an O-ring seal. LG’ is the effective length of the connection (See Table 8-7) .

Joints: Fastener Joints: Fastener Stiffness Stiffness

Figure 8-14 shows another tension-loaded connection. This joint uses cap screws threaded into one of the members.


ME 307 Machine Design I Joints: Fastener Joints: Fastener

Stiffness Stiffness


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

Studs are regarded as permanent, and so the joint can be disassembled merely by removing the nut and washer.

ME 307 Machine Design I Joints: Fastener Joints: Fastener

Stiffness Stiffness


Spring Rate : The ratio between the force applied to the member and the deflection produced by that force.

The grip LG of a connection is the total thickness of the clamped material.

Total distance between the underside of the nut to the bearing face of the bolt head; includes washer, gasket thickness etc.

The grip LG here is the sum of the thicknesses of both members and both washers.



Table 8-7Suggested Procedure for Finding fastener Stiffness

To find different parameters use table 8-7


In joint under tension the members are under compression and the bolt under tension: kb = equivalent spring constant of bolt composed

of threaded kt and unthreaded kd parts acting as springs in series.


1 1 1

,

b d t

d tb

d t

d td t

d t

d tb

d t t d

k k k

k kk

k k

A E AEk k

l l

A AEk

A l Al

At: tensile stress area (Tables 8-1, 8-2), lt: length of threaded

portion of the grip; Ad: major diameter area of fastener; ld: length of unthreaded portion in grip. kb: is the estimated effective stiffness of the bolt or cap screw in the clamped zone.

For short bolts kb= kt

From Chapter 5 (Springs in series)



8-58-5 Joints- Member Joints- Member Stiffness Stiffness


There may be more than two members included in the grip of the fastener.

All together these act like compressive springs in series.

Equivalent spring constant km

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 gasket stiffness used.

With no gasket, the stiffness of the members is difficult to obtain, except by experimentation.

Compression spreads out between the bolt head and the nut and area is not uniform.

1 2 3

1 1 1 1 1....

m ik k k k k


8-58-5 Joints- Member Joints- Member Stiffness Stiffness



8-58-5 Joints Member Joints Member Stiffness Stiffness

Joint pressure distribution theoretical models

Ito used ultrasonic techniques to determine pressure distribution at the member interface. Results show that pressure stays high out to about 1.5 bolt radii.

Ito suggested the use of Rotscher’s pressure cone method for stiffness calculations with a variable cone angle. This method is quite complicated.




Figure 8-15


We choose a simpler approach using a fixed cone angle.

The contraction of an element of the cone of thickness dx is subjected to a compressive force P is, from Eq. (5-3),

The area of the element is

Pdxd

EA


Figure 8-15b general cone geometry using a half-apex angle .


2 2

2 20 tan

2 2

tan tan2 2

i

D dA r r x

D d D dx x


Substituting this into the previous equation and integrating the resulting equation from 0 to t gives.

For Members made of Aluminum, hardened steel and cast iron 25o<<33o

With =30o, this becomes



(2 tan )( )ln

tan (2 tan )( )tan

(2 tan )( )ln

(2 tan )( )

P t D d D dEd t D d D d

P Edk

t D d D dt D d D d

(8-19)

0.574(1.55 )( )

ln(1.55 )( )

Edk

t D d D dt D d D d

(8-20)


Substituting this into the previous equation and integrating the resulting equation from 0 to t gives.

With =30o, this becomes



(2 tan )( )ln

tan (2 tan )( )tan

(2 tan )( )ln

(2 tan )( )

P t D d D dEd t D d D d

P Edk

t D d D dt D d D d

(8-19)

0.574(1.55 )( )

ln(1.55 )( )

Edk

t D d D dt D d D d

(8-20)



If members of the joint have the same E with symmetrical frusta (l=2t), then they act as two identical springs in series km = k/2. For = 30° and D = dw = 1.5 d, this can be written as,

Finite element analysis agree with = 30o recommendation

coinciding exactly at the aspect ratio d/l = 0.4. Additionally, FEM offered an exponential curve-fit of the

form

where A and B are given in Table 8-8.

0.5774

0.5774 0.52ln 5

0.5774 2.5

m

Edk

l dl d

(8-22)

( / )Bd lmkAe

Ed (8-23)





Figure 8-16The dimensionless plot of stiffness versus aspect ratio of the members of a bolted joint, showing the relative accuracy of method of Rotsher, Michke and Motosh, compared to Finite Element Analysis (FEA) conducted by Wileman, Choudury, and Green.





Bolt strength is specified by: 1. minimum proof strength Sp 2. or minimum proof load Fp, 3. and minimum tensile strength, Sut

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 is about 90% of the

0.2% offset yield strength.CH-8 LEC 36 Slide 20Dr. A. Aziz Bazoune Chapter 8: Screws, Fasteners and the Design of

Nonpermanent Joints

8-68-6 Bolt StrengthBolt Strength


The SAE specifications are given in

Table 8-9 bolt grades are numbered

according to minimum tensile strength.

The ASTM Specs for steel bolts

(structural) are in Table 8-10.

Metric Specs are in Table 8-11.

If Sp not available use:

Sp =0.85 Sy

Fp = At Sp










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Bolt Pressure Distribution