Machinery's Handbook, 31st Edition
1664 TORQUE AND TENSION IN FASTENERS tension and the corresponding tightening torque at an arbitrary point in the 50 to 80 per- cent range of the bolt yield point or proof stress (for steel bolts, use the minimum value of the yield point or proof stress multiplied by the stress area of the bolt). Repeat this test several times and average the results. The tightening torque may be considered as the sum of the torque on the threads plus the torque on the bolt head- or nut-to-joint bearing surface. The torque coefficient can be found from K = T f ÷ F f × d , where F f is the measured axial tension, and T f is the measured tightening torque. To measure the coefficient of friction between threads or bearing surfaces, obtain the total tightening torque and that portion of the torque due to the thread or bearing surface friction. If only tightening torque and the torque on the bearing surfaces can be measured, then the difference between these two measurements can be taken as the thread- tightening torque. Likewise, if only the tightening torque and threaded-portion torque are known, the torque due to bearing can be taken as the difference between the known torques. The coefficients of friction between threads and bearing surfaces, respectively, can be obtained from the following: (18) (19) As before, T s is the torque attributable to the threaded portion of the screw, T w is the torque due to bearing, D w is the equivalent diameter of friction torque on bearing surfaces according to Equation (14) , and F f is the measured axial tension. Torque-Tension Relationships.— Torque is usually applied to develop an axial load in a bolt. To achieve the desired axial load in a bolt, the torque must overcome friction in the threads and friction under the nut or bolt head. In Fig. 5 , the axial load P B is a component of the normal force developed between threads. The normal-force component perpen - dicular to the thread helix is P N β and the other component of this force is the torque load P B tan β that is applied in tightening the fastener. Assuming the turning force is applied at the pitch diameter of the thread, the torque T 1 needed to develop the axial load is T 1 = P B × tan β × d 2 /2. Substituting tan β = l ÷ π d 2 into the previous expression gives T 1 = P B × l ÷ 2 π . cos cos tan ′ d F f 2 T 2 s s ′ µ α α β = − D F T 2 w f w w µ =
P B cos α =
P B
P N α
P N β
Bolt axis
1 P N α
P B
P B cos β =
α
P N β
P B tan β
l
β
d 2
π d 2
2
Fig. 5. Free Body Diagram of Thread Helix Forces Fig. 6. Thread Friction Force In Fig. 6, the normal-force component perpendicular to the thread flanks is P N α . With a coefficient of friction μ 1 between the threads, the friction load is equal to μ 1 P N α , or μ 1 P B ÷ cos α . Assuming the force is applied at the pitch diameter of the thread, the torque T 2 to overcome thread friction is given by: (20) T 2 1 µ =
cos d P 2 B α
2
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