TORQUE AND TENSION IN FASTENERS Machinery's Handbook, 31st Edition
1661 (10)
. . 09743 06495
2
d n d n # #
− −
UNJ
Unified
a
k
=
#
torque
torque
where d is the basic thread major diameter, and n is the number of threads per inch. The tensile stress area for metric threads is based on a diameter equivalent to the mean of the pitch diameter and a diameter obtained by subtracting 1 ∕ 6 the height of the fundamental thread triangle from the external-thread minor diameter. The Japanese Industrial Standard JIS B 1082 (see also ISO 898/1) defines the stress area of metric screw threads as follows: (11) In Equation (11) , A s is the stress area of the metric screw thread in mm 2 ; d 2 is the pitch diameter of the external thread in mm, given by d 2 = d − 0.649515 × P ; and d 3 is defined by d 3 = d 1 − H /6. Here, d is the nominal bolt diameter; P is the thread pitch; d 1 = d − 1.082532 × P is the minor diameter of the external thread in mm; and H = 0.866025 × P is the height of the fundamental thread triangle. Substituting the formulas for d 2 and d 3 into Equation (11) results in A s = 0.7854( d − 0.9382 P ) 2 . The stress area, A s , of Unified threads in mm 2 is given in JIS B 1082 as: (12) Relation between Torque and Clamping Force.— The Japanese Industrial Standard JIS B 1803 defines fastener tightening torque T f as the sum of the bearing surface torque T w and the shank (threaded) portion torque T s . The relationship between the applied tighten- ing torque and bolt preload F ft is as follows: T f = T s + T w = K × F f × d . In the preceding, d is the nominal diameter of the screw thread, and K is the torque coefficient defined as follows: (13) where P is the screw thread pitch; μ s is the coefficient of friction between threads; d 2 is the pitch diameter of the thread; μ w is the coefficient of friction between bearing surfaces; D w is the equivalent diameter of the friction torque bearing surfaces; and α ′ is the flank angle at the ridge perpendicular section of the thread ridge, defined by tan α ′ = tan α cos β , where α is the thread half angle (30 ° , for example), and β is the thread helix, or lead, angle. β can be found from tan β = l ÷ 2 π r , where l is the thread lead, and r is the thread radius (i.e., one-half the nominal diameter d ). When the bearing surface contact area is circular, D w can be obtained as follows: (14) where D o and D i are the outside and inside diameters, respectively, of the bearing surface contact area. The torques attributable to the threaded portion of a fastener, T s , and bearing surfaces of a joint, T w , are as follows: D D D D D 3 2 w o i o i 2 2 3 3 # = − − A d d + a k 2 3 4 2 s 2 = π . d n 0 7854 0 9743 25 4 . # − a . A s 2 = k sec 2 ′ = π +µ α + µ a K d P d 1 s D w w 2 k
F P
F D 2 f w w = µ
f
sec s 2 ′ = π + µ α a k d
T
(15)
(16)
T
2
s
w
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