Machinery's Handbook, 31st Edition
358 DISC SPRING FORCES AND STRESSES calculate stresses. The minus sign “ − ” indicates that compressive stresses are acting in a direction opposite to the tensile stresses.
4 E · t · s · K 4
3 r -
·
(14)
: Point 0
0 v =
(1 – µ 2 ) · K
1 · D
2 a
8 s 2 4 E · K 4 · s · K 4 · K 2 · h – (1 – µ 2 ) · K 1 · D `
+ K 3 · t
B
j
(15)
: Point 1
1 v =
2 a
` s 2 4 E · K 4 · s · K 3 · t – K 2 · K 4 · h – (1 – µ 2 ) · K 1 · D 8 2 a ` s 2 4 E · K 4 · s · K 4 · (2 K 3 – K 2 ) · h – (1 – µ 2 ) · K 1 · D 8 2 a · δ
B
j
(16)
: Point 2
2 v =
+ K 3 · t
B
j
(17)
: Point 3
3 v =
K 2 and K 3 are disc spring dimensional constants, defined as follows:
ln π δ δ δ − − a k · ln 6 1 1
K 3 3 =
1 − ^ h ·
ln π δ δ ·
(18)
(19)
K
=
2
where δ = D / d is the outside-to-inside diameter ratio. In static application, if disc springs are fully flattened (100 percent deflection), compres- sive stress at point 0 should not exceed the tensile strength of disc spring materials. For most spring steels, the permissible value is σ 0 1600 N/mm 2 or 232,000 psi. In dynamic applications, certain limitations on tensile stress values are recommended to obtain controlled fatigue life of disc springs utilized in various stacking. Maximum tensile stresses at points 2 and 3 depend on the Group number of the disc springs. Stresses σ 2 and σ 3 should not exceed the following values: Group 1 Group 2 Group 3 Maximum allowable tensile stresses at points 2 and 3 1300 N/mm 2 (188,000 psi) 1250 N/mm 2 (181,000 psi) 1200 N/mm 2 (174,000 psi) Fatigue Life of Disc Springs.— Fatigue life is measured in terms of the maximum number of cycles that dynamically loaded disc springs can sustain prior to failure. Dynamically loaded disc springs are divided into two groups: disc springs with unlimited fatigue life, which exceeds 2 3 10 6 cycles without failure, and disc springs with limited fatigue life between 10 4 cycles and less than 2 3 10 6 cycles. Typically, fatigue life is estimated from three diagrams, each representing one of the three Groups of disc springs (Fig. 10, Fig. 11, and Fig. 12). Fatigue life is found at the inter- section of the vertical line representing minimum tensile stress σ min with the horizontal line, which represents maximum tensile stress σ max . The point of intersection of these two lines defines fatigue life expressed in number of cycles N that can be sustained prior to failure. Example: For Group 2 springs in Fig. 11, the intersection point of the σ min = 500 N/mm 2 line with the σ max = 1200 N/mm 2 line, is located on the N = 10 5 cycles line. The estimated fatigue life is 10 5 cycles.
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