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Engineering Data Engineer Data Specifications Dimensions Knowledge Theory

Hardness

Brinell Hardness Test

In this test a hardened steel ball is pressed into the surface of the test material using a prescribed ball. The ball and load have to be selected to suit the material being tested.

The Brinell hardness test consists of indenting the test material with a 10 mm diameter hardened steel or carbide ball subjected to a load of 3000 kgf (29 430 N). For softer materials the load can be reduced to 1500 kgf (14 715 N) or 500 kgf (4 905 N ) to avoid excessive indentation. The full load is normally applied for 10 to 15 seconds for harder ferrous metals and for 30 seconds or more for other metals softer metals. The diameter of the indentation left in the test material is measured with a microscope.

The Brinell hardness number is calculated by dividing the load applied by the surface area of the indentation.

D = Diameter of Ball
D₁ = Diameter of indentation
F = applied force

Vickers Hardness Test

In this test a Diamond Indenter is pressed into the surface of the material being tested. Standard loads used include 5, 10, 20, 30, 50, and 100 kgf. (49,05, 98,1 196,2 490,5 and 981 N).
The load is stated in specifying the hardness number i.e HD(10) = 100. the indenter is a square based pyramid (136^o included angle )to suit the material being tested

The Hardness Number HD = 1.844 x Load/ Average diagonal length of indentation

Rockwell Hardness Test

In this test a Hard Steel Ball or a Diamond Cone Indenter is pressed into the surface of the material being tested/ The result of the test is read directly from machine.

The indenter is forced into the test material under a preliminary minor load (98N)and after equilibrium are achieved an indicating device, which follows the movements of the indenter, is set to the datum position. An additional major load is then applied with resulting increase in penetration. The conditions are then allowed to stabilise and then the additional major load is removed,leaving the minor load in place. The resulting permanent penetration esulting from the application and removal of the additional major load is used to calculate the Rockwell hardness number.

HR = E - e

F1 = additional major
e = permanent increase in depth of penetration due to major load , measured in units of 0.001 mm
E = a constant of 100 units for diamond and ball indenters
HR = Rockwell hardness number
D = diameter of steel ball


Scale	Indenter	Major Load F1 N	E	Applications
A	120^oDiamond cone	490.5	100	Sheet steel ; shallow case hardened
B	1/16" steel ball	882.9	130	Copper, Aluminium alloys, Low Carbon Steel
C	120^oDiamond cone	1373.4	100	Most Widely Used -Hardened Steels, Cast irons etc
D	120^oDiamond cone	882.9	100	Thin but hard steels, Ductile Iron (Pearlitic
E	1/8" steel ball	882.9	130	Cast Iron, Aluminium, Bearings alloys
F	1/16" steel ball	490.5	130	Annealed copper alloys , Soft thin metals
G	1/16" steel ball	1373.4	130	Phosphor bronze, beryllium copper, malleable irons, Lead etc
H	1/8" steel ball	490.5	130	Soft Metals Plastics etc
K	1/8" steel ball	1373.4	130	Soft bearing metals, Plastics, soft materials.
L	1/4" steel ball	490.5	130	Soft bearing metals, Plastics, soft materials.
M	1/4" steel ball	882.9	130	Soft bearing metals, Plastics, soft materials.
P	1/4" steel ball	1373.4	130	Soft bearing metals, Plastics, soft materials.
R	1/2" steel ball	490.5	130	Soft bearing metals, Plastics, soft materials.
S	1/2" steel ball	882.9	130	Soft bearing metals, Plastics, soft materials.
V	1/2" steel ball	1373.4	130	Soft bearing metals, Plastics, soft materials.

Superficial Rockwell Hardness Test

A more surface sensitive measurement of hardness than for the regular Rockwell scales. This technique is useful for samples with hardness gradients at the surface, to test small areas, and for thin samples. Superficial Rockwell hardness scales are N and T for metals and W, X, and Y for non-metallic materials and coatings. The Superficial Rockwell hardness test method consists of indenting the test material with a diamond cone (N scale) or hardened steel ball indenter. The indenter is forced into the test material under a preliminary minor (29.43N) and the conditions are allowed to settle. Then the indicating device that follows the movements of the indenter is set to a datum position. An additional major load, is applied with resulting increase in penetration . The conditions are allowed to settle again and then the additional major load is removed. The permanent increase in depth of penetration, resulting from the application and removal of the additional major load is used to calculate the Rockwell Superficial hardness number.

HR = E - e

e = permanent increase in depth of penetration due to major load F1, measured in units of 0.001 mm
E = a constant of 100 units for diamond and ball indenters
HR = Rockwell hardness number
D = diameter of steel ball


Scale	Indenter	Major Load ( N )	E	Application
15N	120^o Diamond cone	117.72	100	Similar to C scale, but for thin materials
30N	120^o Diamond cone	264.87	100	Same as 15N
45N	120^o Diamond cone	412	100	Same as 15N
15T	1/16" steel ball	117.72	100	Similar to B scale, but for thin materials
30T	1/16" steel ball	264.87	100	Same as 15T
45T	1/16" steel ball	412	100	Same as 15T
15W	1/8" steel ball	117.72	100	For very soft materials
30W	1/8" steel ball	264.87	100	For very soft materials
45W	1/8" steel ball	412	100	For very soft materials
15X	1/4" steel ball	117.72	100	For very soft materials
30X	1/4" steel ball	264.87	100	For very soft materials
45X	1/4" steel ball	412	100	For very soft materials
15Y	1/2" steel ball	117.72	100	For very soft materials
30Y	1/2" steel ball	264.87	100	For very soft materials
45Y	1/2" steel ball	412	100	For very soft materials

The Shore (Scleroscope ) Hardness Test

The Scleroscope test consists of dropping a diamond tipped hammer, which falls inside a glass tube under the force of its own weight from a fixed height, onto the test specimen. The height of the rebound travel of the hammer is measured on a graduated scale. The harder the material, the higher the rebound. The scale of the rebound is arbitrarily chosen and consists on Shore units, divided into 100 parts, which represent the average rebound from pure hardened high-carbon steel. The scale is continued higher than 100 to to allow for metals having greater hardness.
The shore scleroscope test does not normally mark the material under test. The Shore Scleroscope measures hardness in relation to the elasticity of the material.
Advantages of this method are portability and non-marking of the test surface.

Knoop

The Knoop indenter has a polished rhombohedral shape with an included longitudinal angle of 172� 30� and an included transverse angle of 130� 0�. The narrowness of the indenter makes it ideal for testing specimens with steep hardness gradients and coatings. Knoop is a better choice for hardness testing of hard brittle materials.

Jominy Hardenability

The Jominy test involves heating a test specimen of steel 25mm diameter and 100mm long to an austenitising temperature and quenching from one end with a controlled and standardized jet of water. After quenching, the hardness is measured at intervals taken form the quenched end. The hardness gradient along the test surface provides an indication of the material�s hardenability.

Moh's Hardness Scale

The Moh's hardness scale consists of 10 minerals arranged in order from 1 to 10. Diamond is rated as the hardest and is indexed as 10; talc as the softest with index number 1. Each mineral in the scale will scratch all those below it as follows:

Diamond	10
Corundum	9
Topaz	8
Quartz	7
Orthoclase (Feldspar)	6
Aptite	5
Fluorite	4
Calcite	3
Gypsum	2
Talc	1

Shore (Durometer) test

The hardness testing of plastics is most commonly measured by the Shore (Durometer) test.  This test measures the resistance of the plastic toward indentation.   The scale provide an empirical hardness value that doesn't relate directly to the mechanical properties of the material being tested.   Shore Hardness, using either the Shore A or Shore D scale, is the preferred method for rubbers/elastomers and is also commonly used for 'softer' plastics such as polyolefins, fluoropolymers, and vinyls.   The Shore A scale is used for 'softer' rubbers while the Shore D scale is used for 'harder' rubbers.
The Shore hardness is measured with an apparatus known as a Durometer and consequently is also known as 'Durometer hardness'. The hardness value is determined by the penetration of the Durometer indenter foot into the sample. If the indenter completely penetrates the sample, a reading of 0 is obtained, and if no penetration occurs, a reading of 100 results. Because of the resilience of rubbers and plastics, the hardness reading my change over time - so the indentation time is sometimes reported along with the hardness number.   The test method is identified in standard ISO 868.
The results obtained from this test are a useful measure of relative resistance to indentation of various grades of polymers.   However, the Shore Durometer hardness test does not serve well as a predictor of other properties such as strength or resistance to scratches, abrasion, or wear, and should not be used alone for product design specifications.

Harder Scales

VPN	ROCKWELL SCALES					ROCKWELL Superficial			BRINELL		SCLERO- SCOPE	U.T.S.
DPH HV/10	A	B	C	D	G	15N	30N	45N	BHN 500kg	BHN 3000kg		MPa
1865	92		80	87		97	92	87
1787	92		79	86		96	92	87
1710	91		78	85		96	91	86
1633	91		77	84		96	91	85
1556	90		76	83		96	90	84
1478	90		75	83		95	89	83
1400	89		74	82		95	89	82
1323	89		73	81		95	88	81
1245	88		72	80		95	87	80
1160	87		71	80		94	87	79
1076	87		70	79		94	86	78			101
1004	86		69	78		94	85	77			99
940	86		68	77		93	84	75			97
900	85		67	76		93	84	74			95
865	85		66	75		93	83	73			92
832	84		65	75		92	82	72		739	91
800	84		64	74		92	81	71		722	88
772	83		63	73		91	80	70		705	87
746	83		62	72		91	79	69		688	85
720	82		61	72		91	79	68		670	83
697	81		60	71		90	78	67		654	81	2206
674	81		59	70		90	77	66		634	80	2137
653	80		58	69		89	76	64		615	78	2069
633	80		57	69		89	75	63		595	76	2000
613	79		56	68		88	74	62		577	75	1944
595	79	120	55	67		88	73	61		560	74	1889
577	78	120	54	66		87	72	60		543	72	1834
560	78	119	53	65		87	71	59		523	71	1772
544	77	119	52	65		86	70	57		512	69	1689
528	77	118	51	64		86	69	56		496	68	1648
513	76	117	50	63		86	69	55		481	67	1607
498	75	117	49	62		85	68	54		469	66	1565
484	75	116	48	61		85	67	53		455	64	1524
471	74	116	47	61		84	66	51		443	63	1496
458	74	115	46	60		84	65	50		432	62	1462
446	73	115	45	59		83	64	49		421	60	1420
434	73	114	44	59		83	63	48		409	58	1379
423	72	113	43	58		82	62	47		400	57	1351
412	72	113	42	57		82	61	46		390	56	1317
402	71	112	41	56		81	60	44		381	55	1289
392	71	112	40	55		80	60	43		371	54	1255
382	70	111	39	55		80	59	42		362	52	1220
372	70	110	38	54		79	58	41		353	51	1193
363	69	110	37	53		79	57	40		344	50	1165
354	69	109	36	52		78	56	38		336	49	1138
345	68	109	35	52		78	55	37		327	48	1103
336	68	108	34	51		77	54	36		319	47	1076
327	67	108	33	50		77	53	35		311	46	1048
318	67	107	32	49		76	52	34		301	44	1014
310	66	106	31	48	91	76	51	33		294	43	993
302	66	105	30	48	91	75	50	31		286	42	965
294	65	104	29	47	89	75	50	30		279	41	945
286	65	104	28	46	88	74	49	29		271	41	917
279	64	103	27	45	87	73	48	28		264	40	889
272	64	103	26	45	86	73	47	27		258	39	869
266	63	102	25	44	85	72	46	26		253	38	855
260	63	101	24	43	84	72	45	24		247	37	834
254	62	100	23	42	83	71	44	23	201	240	36	814
248	62	99	22	42	81	71	43	22	195	234	35	793
243	61	98	21	41	79	70	42	21	189	228	35	772
238	61	97	20	40	78	69	42	20	184	222	34	752

VPN

ROCKWELL SCALES

ROCKWELL Superficial

BRINELL

SCLERO-
SCOPE

U.T.S.

DPH
HV/10

15N

30N

45N

BHN
500kg

BHN
3000kg

MPa

Softer Scales

VPN	ROCKWELL SCALES								ROCKWELL Superficial			BRINELL		SCLERO- SCOPE	U.T.S.
DPH HV/10	A	B	C	E	F	G	H	K	15T	30T	45T	BHN 500kg	BHN 3000kg		MPa
234	60	97	19			77			92	80	69	181	218	34	738
230	59	96	18			76			92	80	68	179	214	33	731
226	59	96	17			75			92	80	68	177	210	33	717
222	58	95	16			74			92	79	67	175	208	32	703
217	58	95	15			73			92	79	67	171	205	31	690
213	58	94	14			73			91	79	66	169	203	31	683
208	57	93	13			71			91	78	66	167	200	30	676
204	57	92	12			70		100	91	78	65	163	195	30	662
200	56	92	11			69		100	91	77	64	162	193	29	655
196	56	91	10			68		100	90	77	64	160	190	28	641
192	56	90	9			66		99	90	76	63	157	185	27	627
188	55	89	8			64		98	90	76	62	154	180	26	607
184	54	88	7			63		97	90	75	61	151	176	26	593
180	54	87	6			61		97	89	75	60	148	172	26	579
176	53	86	5			59		96	89	74	59	145	169	25	572
172	53	85	4			58		95	89	74	58	142	165	25	558
168	52	84	3			56		94	88	73	57	140	162	25	545
164	51	83	2			54		93	88	72	56	137	159	24	538
160	51	82	1			53		92	88	72	55	135	156	24	524
156	50	81	0			51		91	87	71	54	133	153	24	517
152	50	80				49		91	87	70	53	130	150		503
148	49	79				48		90	87	70	52	128	147
144	49	78				46		89	86	69	51	126	144
141	48	77				44		88	86	68	50	124	141
139	47	76				43		87	86	68	49	122	139
137	47	75			100	41		86	85	67	49	120	137
135	46	74			99	39		85	85	66	48	118	135
132	46	73			99	38		85	85	66	47	116	132
130	45	72			98	36		84	84	65	46	114	130
127	45	71		100	98	35		83	84	64	45	112	127
125	44	70		100	97	33		82	84	64	44	110	125
123	44	69		99	96	31		81	83	63	43	109	123
120	43	68		98	96	30		80	83	62	42	107	121
118	43	67		98	95	28		79	83	62	41	106	119
116	42	66		97	95	27		78	82	61	40	104	117
115	42	65		96	94	25		78	82	60	39	102	116
114	42	64		96	94	24		77	82	60	38	101	114
113	41	63		95	93	22		76	81	59	37	99	112
112	41	62		95	92	21		75	81	58	36	98	110
111	40	61		94	92	19		74	81	57	35	96	108
110	40	60		93	91	18		73	81	57	34	95	107
108	39	59		93	91	16		72	80	56	32	94	106
107	39	58		92	90	15		71	80	55	31	92	104
106	38	57		91	90	13		71	80	55	30	91	102
105	38	56		91	89	12		70	79	54	29	90	101
104	38	55		90	88	10		69	79	53	28	89	99
103	37	54		90	88	9		68	79	53	27	87
102	37	53		89	87	7		67	78	52	26	86
101	36	52		88	87	6		66	78	51	25	85
100	36	51		88	86	4		65	78	51	24	84
100	35	50		87	86	3		65	77	50	23	83
99	35	49		87	85			64	77	49	22	82
98	35	48		86	85			63	77	49	21	81
97	34	47		85	84			62	76	48	20	80
96	34	46		85	83			61	76	47	19	79
95	33	45		84	83			60	76	46	18	79
95	33	44		84	82			59	75	46	17	78
94	32	43		83	82			58	75	45	16	77
93	32	42		82	81			58	75	44	15	76
92	31	41		82	81			57	74	44	14	75
91	31	40		81	80			56	74	43	13	74
90	31	39		80	79			55	74	42	11	74
90	30	38		80	79			54	73	42	10	73
89	30	37		79	78			53	73	41	9	72
88	29	36		79	78		100	52	73	40	8	71
88	29	35		78	77		100	52	72	40	7	71
87	28	34		77	77		99	51	72	39	6	70
87	28	33		77	76		99	50	72	38	5	69
86	28	32		76	75		99	49	71	38	4	68
86	27	31		76	75		98	48	71	37	3	68
85	27	30		75	74		98	47	71	36	2	67
85	26	29		74	74		98	46	70	36	1	66
84	26	28		74	73		97	45	70	35		66
84	25	27		73	73		97	45	70	34		65
83	25	26		73	72		97	44	69	33		65
83	24	25		72	71		96	42	69	33		64
82	24	24		71	71		96	42	69	32		64
82	24	23		71	70		96	41	68	31		63
81	23	22		70	70		95	40	68	31		63
81	23	21		70	69		95	39	68	30		62
80	22	20		69	69		95	38	68	29		62
80	22	19		68	68		94	38	67	29		61
79	21	18		68	67		94	37	67	28		61
79	21	17		67	67		93	36	67	27		60
78	21	16		67	66		93	35	66	26		60
78	20	15		66	66		93	34	66	26		59
77		14		65	65		92	33	66	25		59
77		13		65	65		92	32	65	24		58
76		12		64	64		92	32	65	24		58
76		11		64	64		91	31	65	23		57
75		10		63	63		91	30	64	22		57
75		9		62	62		91	29	64	22		56
74		8		62	62		90	28	64	21		56
74		7		61	61		90	27	63	20		56
73		6		61	61		90	26	63	20		55
73		5		60	60		89	26	63	19		55
72		4		59	60		89	25	62	18		55
72		3		59	59		88	24	62	17		54
71		2		58	58		88	23	62	17		54
71		1		58	58		88	22	61	16		53
70		0		57	57		87	21	61	15		53
VPN	ROCKWELL SCALES								ROCKWELL Superficial			BRINELL		SCLERO- SCOPE	U.T.S.
DPH HV/10	A	B	C	E	F	G	H	K	15T	30T	45T	BHN 500kg	BHN 3000kg		MPa

Figure comparing hardness scales

Table of Components showing relevant surface Hardness values

Important Notes: Values below are typical of high specification components. Lower values are likely in more general applications.

Component	Hardness	Scale
Hardened Track for Roller	670-840	HV
Hardened Machine slideway	70-75	HS
Surface Hardened Gear (High spc'n)	58-63	RC
Modern Rails (Used for Normal traffic)	300-340	HV
Modern Rails (High Duty)	340 -420	HV
Shear Blades	81-85	RC
Carbide Dies (Drawing , Forming etc)	80-92	RA
Mild steel components	120	HV
Wrought iron	102	HV
Axe	600	HV
Fork/Rake	700	HV
Kitchen knife	680	HV
Stainless table knife	550	HV
Hammer Face	660	HV
Chisel	600	HV
File	60-61	RC
Drill Bit HSS	63-65	RC
Drill Bit Cobalt	66-67	RC

Shaft / Hole Tolerances

Example limits and fits using hole basis

Description Hole Shaft

Loose Running H11 c11

Free Running H9 d9

Loose Running H11 c11

Easy Running - Good quality easy to do- H8 f8

Sliding H7 g6

Close Clearance - Spigots and locations H8 f7

Location/Clearance H7 h6

Location- slight interference H7 k6

Location/Transition H7 n6

Location/Interference- Press fit which can be separated H7 p6

Medium Drive H7 s6

Force H7 u6

Machining Process associated with ISO IT Tolerance Grade.

.

IT Grade 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Lapping

Honing

Superfinishing

Cylinderical grinding

Diamond turning

Plan grinding

Broaching

Reaming

Boring, Turning

Sawing

Milling

Planing, Shaping

Extruding

Cold Rolling, Drawing

Drilling

Die Casting

Forging

Sand Casting

Hot rolling, Flame cutting

ISO Shaft Limit Nearest Zero (Fundamental Deviation)
Deviations in μmetres = (m^-6)

Fundamental Deviatons a to j

Over	Up -to (Incl.)	Fundamental Deviation (es )												(ei )
Over	Up -to (Incl.)	a	b	c	cd	d	e	ef	f	fg	g	h	js	j5	j6	j7
	3	-270	-140	-60	-34	-20	-14	-10	-6	-4	-2	0	ITn/2	-2	-2	-4
3	6	-270	-140	-70	-46	-30	-20	-14	-10	-6	-4	0	ITn/2	-2	-2	-4
6	10	-280	-150	-80	-56	-40	-25	-18	-13	-8	-5	0	ITn/2	-2	-2	-5
10	14	-290	-150	-95		-50	-32		-16		-6	0	ITn/2	-3	-3	-6
14	18	-290	-150	-95		-50	-32		-16		-6	0	ITn/2	-3	-3	-6
18	24	-300	-160	-110		-65	-40		-20		-7	0	ITn/2	-3	-3	-8
24	30	-300	-160	-110		-65	-40		-20		-7	0	ITn/2	-3	-3	-8
30	40	-310	-170	-120		-80	-50		-25		-9	0	ITn/2	-4	-4	-10
40	50	-320	-180	-130		-80	-50		-25		-9	0	ITn/2	-4	-4	-10
50	65	-340	-190	-140		-100	-60		-30		-10	0	ITn/2	-5	-5	-12
65	80	-360	-200	-150		-100	-60		-30		-10	0	ITn/2	-7	-7	-12
80	100	-380	-220	-170		-120	-72		-36		-12	0	ITn/2	-9	-9	-15
100	120	-410	-240	-180		-120	-72		-36		-12	0	ITn/2	-9	-9	-15
120	140	-460	-260	-200		-145	-85		-43		-14	0	ITn/2	-11	-11	-18
140	160	-520	-280	-210		-145	-85		-43		-14	0	ITn/2	-11	-11	-18
160	180	-580	-310	-230		-145	-85		-43		-14	0	ITn/2	-11	-11	-18
180	200	-660	-340	-240		-170	-100		-50		-15	0	ITn/2	-13	-13	-21
200	225	-740	-380	-260		-170	-100		-50		-15	0	ITn/2	-13	-13	-21
225	250	-820	-420	-280		-170	-100		-50		-15	0	ITn/2	-13	-13	-21
250	280	-920	-480	-300		-190	-110		-56		-17	0	ITn/2	-16	-16	-26
280	315	-1050	-540	-330		-190	-110		-56		-17	0	ITn/2	-16	-16	-26
315	355	-1200	-600	-360		-210	-125		-62		-18	0	ITn/2	-18	-18	-28
355	400	-1350	-680	-400		-210	-125		-62		-18	0	ITn/2	-18	-18	-28
400	450	-1500	-760	-440		-230	-135		-68		-20	0	ITn/2	-20	-20	-32
450	500	-1650	-840	-480		-230	-135		-68		-20	0	ITn/2	-20	-20	-32
Over	Up -to (Incl.)	a	b	c	cd	d	e	ef	f	fg	g	h	js	j5	j6	j7
Over	Up -to (Incl.)	Fundamental Deviation (es )												(ei )

Fundamental Deviatons k to zc

Over	Up to (Incl.)	Fundamental Deviation ( ei )
Over	Up to (Incl.)	k4-k7 (inc)	other k	m	n	p	r	s	t	u	v	x	y	z	za	zb	zc
	3	0	0	2	4	6	10	14		18		20		26	32	40	60
3	6	1	0	4	8	12	15	19		23		28		35	42	50	80
6	10	1	0	6	10	15	19	23		28		34		42	52	67	97
10	14	1	0	7	12	18	23	28		33		40		50	64	90	130
14	18	1	0	7	12	18	23	28		33	39	45		60	77	108	150
18	24	2	0	8	15	22	28	35		41	47	54	63	73	98	136	188
24	30	2	0	8	15	22	28	35	41	48	55	64	75	88	118	160	218
30	40	2	0	9	17	26	34	43	48	60	68	80	94	112	148	200	274
40	50	2	0	9	17	26	34	43	54	70	81	97	114	136	180	242	325
50	65	2	0	11	20	32	41	53	66	87	102	122	144	172	226	300	405
65	80	2	0	11	20	32	43	59	75	102	120	146	174	210	274	360	480
80	100	3	0	13	23	37	51	71	91	124	146	178	214	258	335	445	585
100	120	3	0	13	23	37	54	79	104	144	172	210	254	310	400	525	690
120	140	3	0	15	27	43	63	92	122	170	202	248	300	365	470	620	800
140	160	3	0	15	27	43	65	100	134	190	228	280	340	415	535	700	900
160	180	3	0	15	27	43	68	108	146	210	252	310	380	465	600	780	1000
180	200	4	0	17	31	50	77	122	166	236	284	350	425	520	670	880	1150
200	225	4	0	17	31	50	80	130	180	258	310	385	470	575	740	960	1250
225	250	4	0	17	31	50	84	140	196	284	340	425	520	640	820	1050	1350
250	280	4	0	20	34	56	94	158	218	315	385	475	580	710	920	1200	1550
280	315	4	0	20	34	56	98	170	240	350	425	525	650	790	1000	1300	1700
315	355	4	0	21	37	62	108	190	268	390	475	590	730	900	1150	1500	1900
355	400	4	0	21	37	62	114	208	294	435	530	660	820	1000	1300	1650	2100
400	450	5	0	23	40	68	126	232	330	490	595	740	920	1100	1450	1850	2400
450	500	5	0	23	40	68	132	252	360	540	660	820	1000	1250	1600	2100	2600
Over	Up to (Incl.)	k4-k7 (inc)	other k	m	n	p	r	s	t	u	v	x	y	z	za	zb	zc
Over	Up to (Incl.)	Fundamental Deviation ( ei )

ISO Hole Nearest Dim to Zero (Fundamental Deviation)
Deviations in μmetres = (m^-6)

Over	Up to (Incl.)	Fundamental Deviation (El )											Fundamental Deviation (Es )
Over	Up to (Incl.)	A	B	C	CD	D	E	EF	F	FG	G	H	JS	J6	J7	J8	K7	K8	>K8
	3	270	140	60	34	20	14	10	6	4	2	0	IT/2	2	4	6	0+	0	0
3	6	270	140	70	46	30	20	14	10	6	4	0	IT/2	5	6	10	3	5
6	10	280	150	80	56	40	25	18	13	8	5	0	IT/2	5	8	12	5	6
10	14	290	150	95		50	32		16		6	0	IT/2	6	10	15	6	8
14	18	290	150	95		50	32		16		6	0	IT/2	6	10	15	6	8
18	24	300	160	110		65	40		20		7	0	IT/2	8	12	20	6	10
24	30	300	160	110		65	40		20		7	0	IT/2	8	12	20	6	10
30	40	310	170	120		80	50		25		9	0	IT/2	10	14	24	7	12
40	50	320	180	130		80	50		25		9	0	IT/2	10	14	24	7	12
50	65	340	190	140		100	60		30		10	0	IT/2	13	18	28	9	14
65	80	360	200	150		100	60		30		10	0	IT/2	13	18	28	9	14
80	100	380	220	170		120	72		36		12	0	IT/2	16	22	34	10	16
100	120	410	240	180		120	72		36		12	0	IT/2	16	22	34	10	16
120	140	460	260	200		145	85		43		14	0	IT/2	18	26	41	12	20
140	160	520	280	210		145	85		43		14	0	IT/2	18	26	41	12	20
160	180	580	310	230		145	85		43		14	0	IT/2	18	26	41	12	20
180	200	660	340	240		170	100		50		15	0	IT/2	22	30	47	13	22
200	225	740	380	260		170	100		50		15	0	IT/2	22	30	47	13	22
225	250	820	420	280		170	100		50		15	0	IT/2	22	30	47	13	22
250	280	920	480	300		190	110		56		17	0	IT/2	25	36	55	16	25
280	315	1050	540	330		190	110		56		17	0	IT/2	25	36	55	16	25
315	355	1200	600	360		210	125		62		18	0	IT/2	29	39	60	17	28
355	400	1350	680	400		210	125		62		18	0	IT/2	29	39	60	17	28
400	450	1500	760	440		230	135		68		20	0	IT/2	33	43	66	18	29
450	500	1650	840	480		230	135		68		20	0	IT/2	33	43	66	18	29
Over	Up to (Incl.)	A	B	C	CD	D	E	EF	F	FG	G	H	JS	J6	J7	J8	K7	K8	>K8
Over	Up to (Incl.)	Fundamental Deviation (El )											Fundamental Deviation (Es )

Important Note: For Fundamental deviations P-ZC ITn's > 7 only applies . For ITs 6 & 7 refer to table below..

Over	Up to (Incl.)	Fundamental Deviation (Es )
Over	Up to (Incl.)	M7	M8	>M8	N7	N8	>N8	P	R	S	T	U	V	X	Y	Z	ZA	ZB	ZC
	3	-2	-2	-2	-5	-4	-4	-6	-10	-14		-18		-20		-26	-32	-40	-60
3	6	0	2	-4	-4	-2	0	-12	-15	-19		-23		-28		-35	-42	-50	-80
6	10	0	1	-6	-4	-3	0	-15	-19	-23		-28		-34		-42	-52	-67	-97
10	14	0	2	-7	-5	-3	0	-18	-23	-28		-33		-40		-50	-64	-90	-130
14	18	1	5	-7	-5	-3	0	-18	-23	-28		-33	-39	-45		-60	-77	-108	-150
18	24	0	4	-8	-7	-3	0	-22	-28	-35		-41	-47	-54	-63	-73	-98	-136	-188
24	30	1	6	-8	-7	-3	0	-22	-28	-35	-41	-48	-55	-64	-75	-88	-118	-160	-218
30	40	0	5	-9	-8	-3	0	-26	-34	-43	-48	-60	-68	-80	-94	-112	-148	-200	-274
40	50	2	7	-9	-8	-3	0	-26	-34	-43	-54	-70	-81	-97	-114	-136	-180	-242	-325
50	65	0	5	-11	-9	-4	0	-32	-41	-53	-66	-87	-102	-122	-144	-172	-226	-300	-405
65	80	2	8	-11	-9	-4	0	-32	-43	-59	-75	-102	-120	-146	-174	-210	-274	-360	-490
80	100	0	6	-13	-10	-4	0	-37	-51	-71	-91	-124	-146	-178	-214	-258	-335	-445	-585
100	120	2	10	-13	-10	-4	0	-37	-54	-79	-104	-144	-172	-210	-254	-310	-400	-525	-690
120	140	0	8	-15	-12	-4	0	-43	-63	-92	-122	-170	-202	-248	-300	-365	-470	-620	-800
140	160	0	8	-15	-12	-4	0	-43	-65	-100	-134	-190	-228	-280	-340	-415	-535	-700	-900
160	180	2	11	-15	-12	-4	0	-43	-68	-108	-146	-210	-252	-310	-380	-465	-600	-780	-1000
180	200	0	9	-17	-14	-5	0	-50	-77	-122	-166	-236	-284	-340	-425	-520	-670	-880	-1150
200	225	0	9	-17	-14	-5	0	-50	-80	-130	-180	-258	-310	-385	-470	-575	-740	-960	-1250
225	250	3	12	-17	-14	-5	0	-50	-84	-140	-196	-284	-340	-425	-520	-640	-820	-1050	-1350
250	280	0	9	-20	-14	-5	0	-56	-94	-158	-218	-315	-385	-475	-580	-710	-920	-1200	-1550
280	315	1	12	-20	-14	-5	0	-56	-98	-170	-240	-350	-425	-525	-650	-790	-1000	-1300	-1700
315	355	0	11	-21	-16	-5	0	-62	-108	-190	-268	-390	-475	-590	-730	-900	-1150	-1500	-1900
355	400	2	13	-21	-16	-5	0	-62	-114	-208	-294	-435	-530	-660	-820	-1000	-1300	-1650	-2100
400	450	48	59	-23	-17	-6	0	-68	-126	-232	-330	-490	-595	-740	-920	-1100	-1450	-1850	-2400
450	500	25	25	-23	-17	-6	0	-68	-132	-252	-360	-540	-660	-820	-1000	-1250	-1600	-2100	-2600
Over	Up to (Incl.)	M7	M8	>M8	N7	N8	>N8	P	R	S	T	U	V	X	Y	Z	ZA	ZB	ZC
Over	Up to (Incl.)	Fundamental Deviation (Es )

Important Note: For Fundamental deviations (P to Z) For ITn = 6 & 7 refer to table below..

Over	Up to (Incl.)	Fundamental Deviation (Es )
Over	Up to (Incl.)	P6	P7	R6	R7	S6	S7	T6	T7	U6	U7	V6	V7	X6	X7	Y6	Y7	Z6	Z7
	3	-6	-6	-10	-10	-14	-14			-18	-18			-20	-20			-26	-26
3	6	-9	-8	-12	-11	-16	-15			-20	-19			-25	-24			-32	-31
6	10	-12	-9	-16	-13	-20	-17			-25	-22			-31	-28			-39	-36
10	14	-15	-11	-20	-16	-25	-21			-30	-26			-37	-33			-47	-43
14	18	-15	-11	-20	-16	-25	-21			-30	-26	-36	-32	-42	-38			-57	-53
18	24	-18	-14	-24	-20	-31	-27			-37	-33	-43	-39	-50	-46	-59	-55	-69	-65
24	30	-18	-14	-24	-20	-31	-27	-37	-33	-44	-40	-51	-47	-60	-56	-71	-67	-84	-80
30	40	-21	-17	-29	-25	-38	-34	-43	-39	-55	-51	-63	-59	-75	-71	-89	-85	-107	-103
40	50	-21	-17	-29	-25	-38	-34	-49	-45	-65	-61	-76	-72	-92	-88	-109	-105	-131	-127
50	65	-26	-21	-35	-30	-47	-42	-60	-55	-81	-76	-96	-91	-116	-111	-138	-133	-166	-161
65	80	-26	-21	-37	-32	-53	-48	-69	-64	-96	-91	-114	-109	-140	-135	-168	-163	-204	-199
80	100	-30	-24	-44	-38	-64	-58	-84	-78	-117	-111	-139	-133	-171	-165	-207	-201	-251	-245
100	120	-30	-24	-47	-41	-72	-66	-97	-91	-137	-131	-165	-159	-203	-197	-247	-241	-303	-297
120	140	-36	-28	-56	-48	-85	-77	-115	-107	-163	-155	-195	-187	-241	-233	-293	-285	-358	-350
140	160	-36	-28	-58	-50	-93	-85	-127	-119	-183	-175	-221	-213	-273	-265	-333	-325	-408	-400
160	180	-36	-28	-61	-53	-101	-93	-139	-131	-203	-195	-245	-237	-303	-295	-373	-365	-458	-450
180	200	-41	-33	-68	-60	-113	-105	-157	-149	-227	-219	-275	-267	-331	-323	-416	-408	-511	-503
200	225	-41	-33	-71	-63	-121	-113	-171	-163	-249	-241	-301	-293	-376	-368	-461	-453	-566	-558
225	250	-41	-33	-75	-67	-131	-123	-187	-179	-275	-267	-331	-323	-416	-408	-511	-503	-631	-623
250	280	-47	-36	-85	-74	-149	-138	-209	-198	-306	-295	-376	-365	-466	-455	-571	-560	-701	-690
280	315	-47	-36	-89	-78	-161	-150	-231	-220	-341	-330	-416	-405	-516	-505	-641	-630	-781	-770
315	355	-51	-41	-97	-87	-179	-169	-257	-247	-379	-369	-464	-454	-579	-569	-719	-709	-889	-879
355	400	-51	-41	-103	-93	-197	-187	-283	-273	-424	-414	-519	-509	-649	-639	-809	-799	-989	-979
400	450	-55	-45	-113	-103	-219	-209	-317	-307	-477	-467	-582	-572	-727	-717	-907	-897	-1087	-1077
450	500	-55	-45	-119	-109	-239	-229	-347	-337	-527	-517	-647	-637	-807	-797	-987	-977	-1237	-1227
Over	Up to (Incl.)	P7	P7	R6	R7	S6	S7	T6	T7	U6	U7	V6	V7	X6	X7	Y6	Y7	Z6	Z7
Over	Up to (Incl.)	Fundamental Deviation (Es )

Drive Couplings

A coupling is used to connect two in-line shafts to allow one shaft (driver) to drive the second shaft(driven) at the same speed. A coupling can be rigid or, more normally, it can be flexible allowing relative radial, axial or angular movement of the two shafts. Unlike the clutch the coupling transmission is not designed to engage-disengage as a normal operation

Rigid Coupling

Flange locked onto each shaft. One flange with recess and the other with matching spigot. Flanges bolted together to form rigid coupling with no tolerance for relative radial, angular or axial movement of the shafts.

Muff Coupling

Long cylindrical coupling bored and keyed to fit over both shafts. Split axially and clamped over both shafts with recessed bolts. Rigid coupling for transmitting high torques at high speeds.

Beam Coupling

Single piece cylindrical coupling with a hole bored through it entire length. Each end bored to suite the relevant shaft. The helical slot is machined in the coupling in the central region. The reduces the coupling stiffness. The coupling is positive with some flexibility.

Pin Coupling

As rigid coupling but with no recess and spigot and the Bolts replaced by pins with rubber bushes. Design allows certain flexibility.

Flexible Rubber Disc Couping

As rigid coupling except that a thick rubber disc bonded between steel plates is located between the flanges. The plates are bolted to the adjacent coupling flanges.

Spider Coupling

Both half of the couplings have three shaped lugs . When the coupling halves are fitted together the lugs on one half fit inside the spaces between the lugs on the other side. A Rubber insert with six legs fits within the spaces between the lugs. The drive is by the lugs transmitting the torque through the rubber spider spacer... This coupling is only used for low power drives.

Bibby Coupling

The outer flanges of the two half couplings are serrated. A spring fits into the serrations connecting the two halves.

Chain Coupling

Flanges replaced a sprocket on each shaft. The coupling is by a duplex chain wrapped over both adjacent coupling.

Gear Coupling

Gear Coupling

Both coupling halves have a raised rim machined as an external gear. The sleeve which couples the two shafts comprises two halves bolted together, each half having a machine internal gear. This coupling requires lubrication. The coupling is capable of high speeds and high power capacity.

MetaFlex Coupling

Coupling halves connected via stainless steel diaphragms (discs). High speed high torque capability with good dynamic balance. Single coupling will accommodate angular and radial misalignment and fitted in pairs also allows lateral misalignment.

Fluid Coupling

Based on both coupling halves having vanes within a housing (case) containing viscous fluid which rotates with the driving shaft. The rotation is transmitted from one side (Driving) to the other (Secondary) via the viscous fluid. The coupling provides a soft start.

Universal Coupling

Coupling which allows large angle between drive halves(20-30^o). Generally based on a yoke mounted on each shaft . Between to yokes is mounted a trunnion cross. Needle bearings are used at the bearing points between the cross and the yokes. These type or units are used in pairs on carden shafts. Uses widely on rear wheel drive vehicle propshafts.

Universal Coupling / Uni-Joint

Simplest type of coupling which allows large angle between drive halves. Each side of coupling includes protruding pins. The halves of the coupling are fastened in a pivotting assembly. At all angles up to about 40^o the pins interlock with each other and rotation on one half forces the other half to rotate. Low power use only . Not smooth. Not reliable. Really only suitable for remote manual operations.

Roller Chains
Roller chain drives are designed using chain manufacturers' information. The manufacturer can complete the design process or provide sufficient information to allow the design process to be completed. The notes below are provided to enable a rough draft design.

Introduction

The roller chain is used to transmit motion between rotating shafts via sprockets mounted on the shafts. Roller chains are generally manufactured from high specification steels and are therefore capable of transmitting high torques within compact space envelopes. Compared to belt drives the chain drives can transmit higher powers and can be used for drives with larger shaft centre distance separations. In European /ISO standards the chains are normally rated on a standard 15000 hours life. Service factors on the drive and driven wheels are used to adjust the rating for non-standard conditions.

Chain Details

Typical Chain Arrangement

Relevant Standards

BS ISO 10823:1996 ..Guidance on the selection of roller chain drives.
BS 228:1994 ISO 606:1994..Specification for short-pitch transmission precision roller chains and chain wheels

Chain Description

Roller /Transmission chains are identified using three measurements

The pitch - centre distance between rollers (p)
The width between the inner plates (w)
The roller outside diameters (d _r)

Chains manufactured to British/ISO standards can be supplied as single strand (SIMPLEX), double strands (DUPLEX), or triple strands (TRIPLEX)..

Duplex Chain

The range of pitch sizes can vary between 4mm, (0.158 inch) to 114.3mm, (4.500 inch). The European/ISO chain standards have large pin diameter compared to the US standards, especially for the larger pitch sizes. This results in better wear resistance due to the greater bearing area.

The ISO standard has a simple form of part numbering, for example: 1/2 inch pitch duplex chain would be 06B-3. The first two digits are the pitch size in 1/16?s of an inch, therefore 06 = 6/16 or 3/8 inch. The letter ?B? indicates European Standard. The suffix 3 indicates the number of strands in the chain, in this case a triplex chain.

Chain Wheels /Sprockets

Chain wheels can be produced with a minimum of about 9 teeth but in practise the minimum number of teeth is normally restricted to about 19 teeth.  For special applications requiring smoother drives a the smallest sprocket should not have less than 23 teeth.

The maximum number of teeth on the larger wheel should not exceed 150 and generally the number of teeth is restricted to 114 providing a normal maximum ratio of about 6:1

The angle of contact of the chain and the smallest wheel should exceed 120^o.  This provides practical limitation on the size of the larger wheel or results increased center distance separation.  Larger wheel diameters tend to result in reduced chain life.

It is good practice on low ratio chain drives to ensure that the number of teeth on both wheel when added do not exceed 50.  A 1:1 drive should therefore have a maximum of 25 teeth on each wheel.

The large sprockets on high ratio drives are generally made of cast iron because the teeth have reduced chain engagements over time with consequent reduced fatigue and wear. For more arduous service conditions the larger sprockets may be made from cast steel or steel plate. The smaller sprockets when highly loade are generally made from steel type which allow the body to be heat treated for toughness while the teeth are hardened to resist wear e.g. case hardened. Heat treatment is generally required when :

The speed is above 0,7 time max speed when fully loaded
The speed is above 0,5 time max speed when fully loaded, under medium impulsive load
When the load is highly impulsive

Typical maximum speed are listed in the chain properties table below

For lower duties sprockets are generally machined from steel bar stock.

Idler sprockets

When the drive and driven sprockets centres are fixed it may be desireable to include idlers sprockets to take up the slack in the chain. Idler sprockets should preferably be located against the slack side of the chain within the chain envelope - diverting the chain outwards. Idler sprockets are subject to continuous impact from the chain and are subject to wear if only small sprockets are used and if the chain speed is high.

Chain System Design

A large number of roller chains are designed to provide a power transmission between two sprockets with minimum/no regular lubrication and under conditions of high levels of contamination. My bicycle chain drive has worked successfully for over 25 years ( including one replacement of the wheel sprocket and one replacement of the chain). The bicycle is used at least three journeys per week for an average journey time of about 20 minutes.
Motor cycle chain drives work in similar operating conditions...

Industrial chain drives are generally designed to operate in enclosed cases with installed lubrication systems.

Chains rarely fail because they do not have sufficient tensile strength. They most often fail in wear or fatigue. In practice sprocket teeth wear allowing the chains to jump the teeth. Manufacturers specify the chains based on the following parameters

15000 hours life
Single strand
ISO proportions
Service factor (Application factor) =1
Recommended lubrication
Maximum elongation 3%
Horizontal Shafts
Two 19 tooth sprockets
Sprocket centres = 40 pitches

Chain systems are designed with correction coefficients to compensate from the difference from these design conditions
It is important when designing chain drives to ensure good alignment of the sprocket shafts. It is also important to minimise chain slackness and if the centres can not be adjusted then it may be necessary to use idler sprockets.

Chain Lubrication

Chain drive lubrication provides similar benefits to bearing journal lubrication. The benefits include reduced friction, cooling, impact resistance at higher chain speeds. The chain supplier generally provides recommendations for the lubrication requirements for each chain drive. If suitable lubrication is not provided the then capacity of the chain drive is reduced.

There are four basic types of chain lubrication..

Manual /Drip lubrication..In manual lubrication oil is generously applied to the chain drive about every 8 operating hours. In drip lubrication oil is continuously dripped on the chain centre line.
Bath/Disc lubrication..In bath lubrication the lower strand of the chain runs through a sump containing oil. The oil level should be above the lowest pitch line of the chain when it is operating normally. Excessive immersion can result in turbulence of the oil bath. Disc lubrication is based on a disc attached to one sprocket which is immersed in an oil bath. As the disc rotates it picks up oil and deposit it onto the chain. A trough is often used to direct the oil oil to the optimum point on the chain. A peripheral disc speed of between 3 and 40 m/s
Oil stream lubrication...This is normally a continuous stream of filtered oil circulated by a pump. The oil should be spread evenly across the width of the slack side chain
Oil Mist lubrication...This is used for high speed chain drive and is based on the chain case being filled with a oil mist.

There is continuous development in chain drives and self-lubricated chains are available which do not require continuous lubrication and have similar performance to lubricated chain drives.

Plastic chains are also available which do not require lubrication. Plastic chain drives obviously have much reduced operating characteristics compared to steel chains.

Non-lubricated chains are essential for applications requiring controlled environments e.g.Paper, packaging, electronics, white and brown goods manufacture.

Nomenclature

B _a = Chain Bearing Area (mm²)
B _s = Chain Bearing stress (MPa)
C = Centre Distance (m)
D₁ = Drive sprocket Pitch Diameter (m)
D₂ = Driven sprocket Pitch Diameter (m)
d _r = Chain Roller Outside diameter (m)
f _a = application factor
F_t = Torque developed tensile force in chain (N)
F_c = Centrifugal tensile force in chain (N)
f _t = tooth factor
K_x (x = 1 to 8) = Correction Factors
L = Length of chain in pitches
T ₁= torque on driver pulley (Nm)
T ₂= torque on driven pulley (Nm)
S_d = Dynamic Factor of Safety

S_s = Static Factor of Safety
P ₁ = Driver power Transferred (kW)
P ₂ = Driven power Transferred (kW)= ηP1
m = Mass of chain / m (kg/m)
n ₁= Driver sprocket rotational speed (rpm = m^-1)
n ₂= Driven sprocket rotational speed (rpm = m^-1)
p = Pitch of chain (m)
v = Chain velocity (m/s>
w = Chain width between inner plates
η = Efficiency (normally about 98%)
z ₁= Driver sprocket - Number of teeth
z ₂= Driven sprocket - Number of teeth

Roller Chain Design Process

Specify the Drive speed , Driven speed and the power to be transferred
Identify the operating characteristics of the drive and driven shafts (smooth, rough, shock
Select the approximate shaft centre distance
Calculate the speed ratio using table of standard sprockets (minimum No of teeth normally 19
Calculate the appropriate design factors
Calculate the design power
Select a chain which has a higher power capacity than the design power.
This will involve some iteration
Confirm that there is sufficient safety on the tensile strength of the chain and the wear/fatigue strength of the bushing.
C omplete the detail design of the sprocket shaft systems, guards, lubrication system etc

Torque on driver /driven pulley

If the input power = P₁(kW) then the torque (Nm)=

T₁ = P₁ .9,549 / n

T₂ = P₁.η 9,549 / n

.

Chain Velocity

The chain velocity (v)is calculated as follows

v = D₁.π.n₁ /60 (m/s) = D₂.π.n₂ /60...(m/s)

Design Power For Chain

The design power is calculated as follows

P_d = P.K₁.K₂.K₃.K₄.K₅.K₆.K₇.

K₁. = Coefficient for teeth different to 19
K₂. = Coefficient for Transmission Ratio
K₃. = Application (Service) Factor
K₄. = Centre Distance Coefficient
K₅. = Lubrication Coefficient
K₆. = Temperature Coefficient
K₇. = Service Life Coefficient

Tooth Factor (K₁)

Normally the drive is a reduction drive and the driver sprocket is the smallest. This is normally selected as a 19 tooth wheel unless a high speed smoother drive is required then a 23 (or higher) tooth wheel is selected.

If a driver sprocket with z₁ teeth is select than a tooth factor f_t is used.

Tooth factor f_t = 19 / z₁

Ratio Factor (K₂)

This allows for the difference in ratio from the 3:1 ratio normally used in determining the design power for chain drives...

Ratio	1:1	1:2	1:3	1,4	1,6
K2	1,25	1,11	1,0	0,94	0,89

Table of Application Factor ..K₃

Driver Characteristics

Smooth Running, Electric Motors, IC engines with hydraulic couplings Some shock Loading..IC engines , Electric motors with frequent stops/starts Heavy shock Loading IC engines with less than six cylinders

Driven machine characteristics

Smooth running .Office Machines,Generators 1,0 1,1 1,2

Light duty ..Fans, pumps, compressors,printing machines, uniformly loaded conveyors, machine Tools 1,2 1,3 1,4

Moderate shock..concrete mixing , non-uniformly loaded conveyors, mixers. 1,4 1,5 1,7

Heavy shock loading.. Planars, presses, drilling rigs. 1,6 1,7 1,9

Centre Distance Factor (K₄)
This allows for chain designs with sprocket centre distances other than optimum= C/p = 40.
K₄

C/p	20	40	60	80	=<160
1,2	1	0,9	0,85	0,7

Lubrication Factor (K₅)

This factor involves some judgement and the notes below are provided for guidance.

If the chain is correctly lubrication and includes the recommended maintenance.
For self lubricated chain used correctly.
For very low duty chains with reasonable lubrication:
A factor of 1 applies.
If the chain is low - medium duty, provided with the recommended filtered lubrication but with average maintenance.
A service factor of say 1,25 applies
If the chain is medium to high duty, provided with lubrication with average maintenance.
A service factor approx 2 - 3 applies.
For a medium-high duty chain with no lubrication.
A service factor approx 5 will apply.

Temperature Factor (K₆)

For all normal duties at normal ambient temperatures (0-80^o C) the temperature factor will be 1. For cases when a higher ambient temperature is normal a temperature factor will be needed as table below.

Deg.C	0-80	80-150	150-250
K₆	1,0	1,1	1,2

Service Life (K₇)
The power capacity of a chain is based on a 8 hour per day operating cycle and K₇ = 1 . For more arduous operating a different value of K₇ is appropriate.
.

Operating time /day hours	0-8	8-16	16-24
K₇	1,0	1,1	1,2

Chain Power Capacities

The design power as evaluated by the above process should be less than the lower of the chain capacity associated with the link or the bush as calculated below...Note: Manufacturers and suppliers generally simplify this process by providing tables or charts to make this process more convenient. A typical graph is shown below Chain Power graph....

Link- Power capacity

Bushing - Power capacity

These capacities are modified by the strand factor S_f. If there are two strands then the power capacity is increased by a S _f = 1,7. If there are three strands then the power is increased by a strand factor S _f =2,5.

P_i = chain pitch in inches..

Chain	K_a	K_b	Chain	K_a	K_b	Chain	K_a	K_b	Chain	K_a	K_b	Chain	K_a	K_b
05B	0.0046	17	10B	0.0042	17	20B	0.0046	17	32B	0.0046	17	56B	0.0038	7
06B	0.0046	17	12B	0.0044	17	24B	0.0046	17	40B	0.0032	17	64B	0.0039	5
08B	0.0048	17	16B	0.0046	17	28B	0.0046	17	48B	0.0035	12	72B	0.004	2

Tensile Load on Chain

The dynamic load on a chain includes for the tensile load for transmitting the power and the centrifugal load resulting from the chain rotating on the sprocket.

For normal speed applications only the direct load due to the tensile load is relevant.

F_t = P ₁*1000 / v..(N)

Note this is derived in principle in the webpage for flat belt drives.. Flat belts

The tension resulting for the centrifugal force =

F_c = M_m.v²..(N)

The resultant total tensile force taken by the chain =

F_r = F_t + F_c

The value F_r is divided into the breaking strength of the chain F_b to obtain the static Factor of Safety of the Chain

S_s = F_r /F _b

This value of S_s is divided by the Application factor K₃ to arrive at the dynamic Factor of Safety

S_d = F_r / (F_b * K₃)

The graphs below show (very approximately) the range of recommended static and dynamic safety factors.

Chain Bearing Stress

The bearing stress is the resultant tensile stress / the bearing area B_a. Values for the bearing area are provided in the chain properties table below...

B _s= T_r / B _a

The calculated bearing stress should be less than the acceptable bearing pressure

The acceptable bearing pressure = Specific Pressure .K₉.K₈.

K₉ = 1 For ISO chains
K₈ = The friction coefficient / K₃ K₃ = Service_application factor (K₃) -see below

The specific pressure and the friction coefficents are obtained from the charts below:

Pitch Diameter

The pitch diameter of a chain sprocket can be obtained from the formula

D = P / sin (π / z)

Table below identifies the pitch diam for sprockets with chain pitch of 25,4mm . for other chain pitches the diameter will be proportional e.g for 12,7mm chain pitch the diameters will be half the tabled values.

Table of pitch diameters for 25,4mm pitch chain sprockets

No Teeth	Pitch Dia	No Teeth	Pitch Dia	No Teeth	Pitch Dia	No Teeth	Pitch Dia	No Teeth	Pitch Dia	No Teeth	Pitch Dia
	mm		mm		mm		mm		mm		mm
10	82.20	30	243.00	50	404.52	70	566.15	90	727.80	110	889.48
11	90.16	31	251.07	51	412.60	71	574.23	91	735.89	111	897.56
12	98.14	32	259.14	52	420.68	72	582.31	92	743.97	112	905.65
13	106.14	33	267.21	53	428.76	73	590.39	93	752.05	113	913.73
14	114.15	34	275.28	54	436.84	74	598.48	94	760.14	114	921.81
15	122.17	35	283.36	55	444.92	75	606.56	95	768.22	115	929.90
16	130.20	36	291.43	56	453.00	76	614.64	96	776.31	116	937.98
17	138.23	37	299.51	57	461.08	77	622.72	97	784.39	117	946.07
18	146.27	38	307.58	58	469.16	78	630.81	98	792.47	118	954.15
19	154.32	39	315.66	59	477.24	79	638.89	99	800.56	119	962.24
20	162.37	40	323.74	60	485.33	80	646.97	100	808.64	120	970.32
21	170.42	41	331.81	61	493.41	81	655.05	101	816.72	121	978.40
22	178.48	42	339.89	62	501.49	82	663.14	102	824.81	122	986.49
23	186.54	43	347.97	63	509.57	83	671.22	103	832.89	123	994.57
24	194.60	44	356.05	64	517.65	84	679.30	104	840.98	124	1002.66
25	202.66	45	364.12	65	525.73	85	687.39	105	849.06	125	1010.74
26	210.72	46	372.20	66	533.82	86	695.47	106	857.14	126	1018.82
27	218.79	47	380.28	67	541.90	87	703.55	107	865.23	127	1026.91
28	226.86	48	388.36	68	549.98	88	711.64	108	873.31	128	1034.99
29	234.93	49	396.44	69	558.06	89	719.72	109	881.39	129	1043.08

Chain Centre Distance

There are practical limitations for the minimum distance between the chain sprocket centres to prevent interference of the sprocket teeth. To provide a reasonable chain operating life is necessary to ensure good spacing and a minimum wrap of 120^o. The drive layout will determine the actual centre distance. A recommended value is about 40 time the chain pitch.

When the chain centre distance can be adjusted to suit the chain length then the length (L) of the chain in (pitches) can be used to determine the centre distance between sprockets (C) using the following formula

Chain Length

The length of the driving chain is normally required in numbers of double pitches because a complete link includes the inner and the outer link which covers two pitches.
The Chain length (L) in pitches (p) is given (to sufficient practical accuracy ) by the formula

Chain Properties

Values are from BS 228 , ISO 606
The normal maximum velocity relates to sprockets with 17-25 teeth.

Pitch	Normal max vel.	Chain Identity	Breaking Force (F _b)	Mass/m (m)	Chain Brg Area (B_a)	Chain Identity	Breaking Force (F _b)	Mass/m (m)	Chain Brg Area (B_a)	Chain Identity	Breaking Force (F _b)	Mass/m (m)	Chain Brg Area (B_a)
Pitch	(RPM)	Chain Identity	(N)	kg/m	mm²		(N)	kg/m	mm²		(N)	kg/m	mm²
8,00	5000	05B - 1	5000	0.2	11	05B - 2	7800	0.4	22	05B - 3	11100	0.5	33
9.525	4200	06B - 1	9000	0.4	28	06B - 2	16900	0.8	56	06B - 3	24900	1.2	84
12.7	3750	08B - 1	18000	0.7	50	08B - 2	32000	1.3	101	08B - 3	47500	2	151
15.875	2750	10B - 1	22400	0.9	67	10B - 2	44500	1.8	134	10B - 3	66700	2.8	202
19.05	2000	12B - 1	29000	1.2	89	12B - 2	57800	2.5	179	12B - 3	86700	3.8	268
25.4	1500	16B - 1	60000	2.6	210	16B - 2	110000	5.2	421	16B - 3	165000	7.7	631
31.75	1200	20B - 1	95000	3.8	296	20B - 2	170000	7.5	591	20B - 3	250000	11.2	887
38.1	900	24B - 1	160000	7	554	24B - 2	280000	13.9	1109	24B - 3	425000	20.7	1663
44.45	700	28B - 1	200000	9.1	739	28B - 2	360000	18	1479	28B - 3	530000	27	2218
50.8	550	32B - 1	250000	9.7	810	32B - 2	450000	19	1621	32B - 3	670000	28.3	2431
63.5	450	40B - 1	380000	16.8	1275	40B - 2	630000	33.5	2550	40B - 3	950000	43.3	3825
76.2	300	48B - 1	560000	25.9	2061	48B - 2	1000000	48.6	4123	48B - 3	1500000	72.5	6184
88.9	-	56B - 1	850000	35	2791	56B - 2	1600000	70	5582	56B - 3	2350000	105	8373
101.6	-	64B - 1	1120000	60	3625	64B - 2	2000000	120	7250	64B - 3	3100000	180	10875
114.3	-	72B - 1	1400000	80	4618	72B - 2	2500000	160	9234	72B - 3	4000000	240	13850

material material aluminium aluminium aluminum aluminum extrusion extrusion stainless mild steel

Springs

Introduction

A Spring is an engineering component which when deflected by a force tends to return to its unloaded shape. Ideally the energy input to cause the deflection is usefully recovered.   Springs are mechanical components designed to store mechanical energy, working on the principle of flexible deformation of material.   Springs constitute one of the most widely used group of components in mechanical engineering.  When designing and/or selecting metal springs consideration should be given to material, manufacturing process, heat treatment, dynamic properties, elastic properties, strength, fatigue, shock, stability, surging etc etc.

Spring types include:
Metal Springs

Helical Compression Springs

Helical Extension Springs

Helical Torsion Springs

Coil Springs

Disc Springs

Leaf Springs

Spiral Springs
Air Springs

Elastomer Springs

Spring Materials

Solid springs made from elastomers are not covered on this page.   This page covers materials used for making metal springs which mainly include helical compression, tensile, and torsion springs.  Leaf springs and disc spring materials properties may be identified in the more general notes.  The notes also concentrate more on the carbon steel and alloy steel grades rather than the non-ferrous grades.  Future updates will include more comprehensive information

A wide range of materials are available for the manufacture of metal springs including

Carbon steels

Alloy steels

Corrosion resisting steels

Corrosion resisting steels

Phosphor bronze

Spring brass

Beryllium copper

Nickel alloy steels

Titanium alloy steels

Springs are manufactured by hot or cold working processes.   The process depends on the section of the material, the spring index (C= D/d) and the properties required.

Pre-hardened wire should not be used if D/d < 4 or if d >6mm>

Reference Standard Notes
BS EN 10270-1:2001 ..Steel wire for mechanical springs. Patented cold drawn unalloyed spring steel wire
BS EN 10270-2:2001 ..Steel wire for mechanical springs. Oil hardened and tempered spring steel wire
BS EN 10270-3:2001 ..Steel wire for mechanical springs. Stainless spring steel wire

BS EN 10270-1:2001 ..Steel wire for mechanical springs. Patented cold drawn unalloyed spring steel wire
Wire designated within this standard is allocated one of a number of grades.

SL Grade is low tensile strength on static duties

SM Grade is medium tensile strength on static duties

SH Grade is high tensile strength on static duties

DM Grade is medium tensile strength on dynamic duties

DH Grade is high tensile strength on dynamic duties

A typical wire designation would be "Spring wire BS EN 10270-1-SH -3,60 ph.
Spring wire grade SH with a nominal diameter of 3,6mm phosphated.
The grade would have a tensile strength (according to the standard) R_m = 1700-1970MPa

BS EN 10270-2:2001 ..Steel wire for mechanical springs. Oil hardened and tempered spring steel wire
Wire designated within this standard is allocated one of nine grades.

Low Tensile grades - FDC (Static)... TDC(Medium Fatigue)...VDC (High Fatigue)

Medium tensile tensile grades- FDCrV (Static)... TDCrV(Medium Fatigue)...VDCrV (High Fatigue)

High tensile tensile grades- FDSiCr (Static)... TDSiCr(Medium Fatigue)...VDSiCr (High Fatigue)

The FD,FDCrV, and FDSiCr (Static) Grades have a size range of 0,5 to 17,00mm
The TDC,FDCrV, and TDSiCr (Medium Fatigue) Grades have a size range of 0,5 to 10,00mm
The VDC,VDCrV, and VDSiCr (High Fatigue) Grades have a size range of 0,5 to 10,00mm

A typical wire designation would be "Spring wire BS EN 10270-2-VDC -3,60 ".
Spring wire grade VDC with a nominal diameter of 3,6mm .
The grade would have a tensile strength (according to the standard) R_m = 1550-1650MPa

BS EN 10270-3:2001 ..Steel wire for mechanical springs. Stainless spring steel wire
This standard includes information on three steel grades 1,4310 ( with a normal tensile strength (NS) and a high tensile strength (HS)) , 1,4401, and 1,4568.

A typical designation according to this standard would be "Spring Wire BS EN 10270-2 - 1.4310 - NS -3,60 Ni coated
Steel designation number 1,4310 with nominal strength level. Nominal dia 3,6mm . Nickel coated
This steel has a nominal tensile R_m = 1500 MPa

Music Wire
This is the most widely used of all spring materials for small springs because it is the toughest.   It has the highest strength tensile and can withstand higher stresses under repeated loading conditions than any other spring material.  It can be obtained in diameters from 0,12 to 3mm.  It has a usable temperature range from 0 to 120^oC
Oil-tempered Wire.  Music wire will contract under heat, and can be plated.
This is a general purpose spring material used for spings where the cost of music wire is prohibitive and for sizes outside the range of music wire.  This material is not suitable for shock or impact loading.  This material is available in diameters from 3 to 12mm.   The temperature range for this material is 0 to 180 ^oC..Will not generally change dimensions under heat.   Can be plated.   Also available in square and rectangular sections.

Hard-drawn wire
This is the cheapest general purpose spring steel and is should only be used where life, accuracy and deflection are not too important.  This material is available in sizes 0,8mm to 12mm.  It has an operating range 0 to 120^oC
Chrome Vanadium wire
This is the most popular alloy spring steel for improved stress, fatigue, long endurance life conditions as compared to high carbon steel materials.  This material is also suitable for impact and shock loading conditions.  Is available in annealed and tempered sizes from 0,8mm to 12mm.  It can be used for temperatures up to 220 ^oC.   Will not generally change dimensions under heat. Can be plated.
Chrome-silicon wire
This an excellent spring material for highly-stressed springs requiring long life and/or shock loading resistance.  It is available in diameters 0,8mm to 12mmm and can be used from temperatures up to 250^oC.   Will not generally change dimensions under heat. Can be plated.

Martensitic Stainless steel wire
This is a corrosion, resisting steel which is unsuitable for sub-zero conditions.
Austentic Stainless steel wire
A good corrosion, acid, heat resisting steel with good strength and moderate temperatures.  Has low stress relaxation.
Spring Brass
This is a low cost material which is convenient to form.  It is a high conductivity material.  This material has poor mechanical properties.  This metal is frequently used in electrical components because of its good electrical properties and resistance to corrosion.
Phosphor Bronze
Popular alloy .Withstands repeated flexures. This metal is frequently used in electrical components because of its good electrical properties and resistance to corrosion.   Suitable to use in sub-zero temperatures.   They are much more costly than the more common stocks and cannot be plated.   Generally will not change dimensions under heat.
Beryllium Copper
High elastic and fatigue strength.  Hardenable.  They are much more costly than the more common stocks and cannot be plated.   Generally will not change dimensions under heat.
Nickel base alloys
These alloys are corrosion resistant.  They can withstand a wide temperature fluctuation.   The materials are suitable to use in precise instruments because of their non-magnetic characteristic. They also poses a high electrical resistance and should not be used as an electrical conductors.
Titanium
Used mainly in aerospace industry because of its extremely light weight and high strength.  This material is very expensive,  It is dangerous to work as titanium wire will shatter explosively under stress if its surface is scored.  Size range 0,8 to 12mm.   Generally will not change dimensions under heat.   Cannot be plated.

Spring Material Strength Values
Important Note..It is important to note that it is best to obtain springs from specialists suppliers who can provide the correct material for the specific application.  If springs are being designed for specific applications then strength values should be obtained from the relevant standards as identified above.  Care should be taken to include for fatigue and adverse operating conditions.   The notes on this page are for rough spring designs.

The material structure , the manufacturing process, and the heat treatment all have an influence on the strength of the spring material.  The strength of spring materials vary significantly with the wire size such that the strength of a selected spring material cannot be determined without knowing the wire size.  The standards identified all list the material strengths against the wire sizes.

The tensile strength versus the wire diameter is almost a straight line when plotted on log-log paper .   The equation for this line is..

S_ut = A / d^m

The table below provides some typical values for the above variables..

Material Diameter Range(mm) Exponent m A (MPa)

Music Wire 0,1 to 6,5 0,145 2211

Oil-Tempered 0,5 to 12 0,187 1855

Hard Drawn 0,7 to 12 0,190 1783

Chrome_Vanadium 0,8 to 12 0,168 2005

Chrome_Silicon 1,6 to 10 0,108 1975

302-Stainless 0,3 to 2,5, 0,146 1867

302-Stainless 2,5 to 5 0,263 2065

302-Stainless 5 to 10 0,478 2911

Phos-Bros 0,1 to 0,6 0 1000

Phos-Bros 0,6 to 2,0 0,028 913

Phos-Bros 2,0 to 7,5 0,064 932

In calculating the spring parameters the torsional yield strength (S _ys ) is used.   The relationship between the torsional yield strength and the ultimate strength S_ut can be approximated with a range as follows.

0,35 S_ut =< S _ys =< 0,52 S_ut

Music wire and hard drawn steel wire have an approximate relationship S _ys = 0,45 S_ut
Valve spring, CR_Va, CR-Si, Hardened and Tempered Carbon steel wires have an approximate relationship S _ys >= 0,50 S_ut
Many Non-ferrous materials have an approximate relationship S _ys >= 0,35 S_ut

Modulus Of Rigitity values

Typical Values for The modulus of Rigidity for different Spring materials are listed below

Material Modulus of Rigitity = G

- (x 10 ³ N/mm ² )

Carbon Steel 78,6

316 Stainless 68,9

Brass 34,5

Phos Bros 41,4

Monel 65,5

Iconel 72,4

Berylium copper 50,0

Helical Springs

A helical spring is a spiral wound wire with a constant coil diameter and uniform pitch.   The most common form of helical spring is the compression spring but tension springs are also widely used. .   Helical springs are generally made from round wire... it is comparatively rare for springs to be made from square or rectangular sections.  The strength of the steel used is one of the most important criteria to consider in designing springs.  Most helical springs are mass produced by specialists organisations.  It is not recommended that springs are made specifically for applications if off-the-shelf springs can be obtained to the job.

Compression Springs

Tension Springs

Nomenclature

C = Spring Index D/d
d = wire diameter (m)
D = Spring diameter = (D_i+D_o)/2 (m)
D_i = Spring inside diameter (m)
D_o = Spring outside diameter (m)
D_il = Spring inside diameter (loaded ) (m)
E = Young's Modulus (N/m²)
F = Axial Force (N)
F_i = Initial Axial Force (N)
     (close coiled tension spring)
G = Modulus of Rigidity (N/m²)
K _d = Traverse Shear Factor = (C + 0,5)/C
K _W = Wahl Factor = (4C-1)/(4C-4)+ (0,615/C)
L = length (m)
L ₀ = Free Length (m)
L _s = Solid Length (m)
n _t = Total number of coils
n = Number of active coils
p = pitch (m)
y = distance from neutral axis to outer fibre of wire (m)
τ = shear stress (N/m²)
τ _i = initial spring stress (N/m²)
τ _max = Max shear stress (N/m²)
θ = Deflection (radians)
δ = linear deflection (mm)

Note: metres (m) have been shown as the units of length in all of the variables above for consistency.   In most practical calculations milli-metres will be more convenient.

Spring Index

The spring index (C) for helical springs in a measure of coil curvature ..

For most helical springs C is between 3 and 12

Spring Rate

Generally springs are designed to have a deflection proportional to the applied load (or torque -for torsion springs).   The "Spring Rate" is the Load per unit deflection.... Rate (N/mm) = F(N) / δ _e(deflection=mm)

Spring Stress Values

For General purpose springs a maximum stress value of 40% of the steel tensile stress may be used. However the stress levels are related to the duty and material condition (ref to relevant Code/standard).

Compression Springs- Formulae

a)   Stress

A typical compression spring is shown below

Consider a compression spring under an axial force F.   If a section through a single wire is taken it can be seen that, to maintain equilibrium of forces, the wire is transmits a pure shear load F and also to a torque of Fr.

The stress in the wire due to the applied load =

This equation is simplified by using a traverse shear distribution factor K _d = (C+0,5)/C.... The above equation now becomes.

The curvature of the helical spring actually results in higher shear stresses on the inner surfaces of the spring than indicated by the formula above.  A curvature correction factor has been determined ( attributed to A.M.Wahl). This (Wahl) factor K _w is shown as follows.

This factor includes the traverse shear distribution factor K _d.. The formula for maximum shear stress now becomes.

A table relating K_W to C is provided below

C 3 4 5 6 7 8 9 10 11 12 13 14 15 16

K_w 1,58 1,4 1,31 1,25 1,21 1,18 1,16 1,14 1,13 1,12 1,11 1,1 1,1 1,09

b)   Deflection

The spring axial deflection is obtained as follows.
The force deflection relationship is most conventiently obtained using Castigliano's theorem. Which is stated as ... When forces act on elastic systems subject to small displacements, the displacement corresponding to any force collinear with the force is equal to the partial derivative to the total strain energy with respect to that force.

For the helical spring the strain energy includes that due to shear and that due to torsion.

Replacing T= FD/2, l = πDn, A = πd² /4 The formula becomes.

Using Castiglianos theorem to find the total strain energy....

Substituting the spring index C for D/d The formula becomes....

In practice the term (1 + 0,5/C²) which approximates to 1 can be ignored

c)  Spring Rate

The spring rate = Axial Force /Axial deflection

In practice the term (C² /(C² + 0,5)) which approximates to 1 can be ignored

Compression Spring End Designs

The figure below shows various end designs with different handing.   Each end design can be associated with any end design.  The plain ends are not desirable for springs which are highly loaded or for precise duties.

The table below shows some equations affected by the end designs...

Note: The results from these equations is not necessarily integers and the equations are not accurate.   The springmaking process involves a degree of variation...

Term Plain Plain and Ground Closed Closed and Ground

End Coils (n _e ) 0 1 2 2

Total Coils (n _t ) n n+1 n+2 n+2

Free Length (L ₀ ) pn+d p(n+1) pn +3d pn +2d

Solid Length (L _s ) d(n _t +1) dn _t d(n _t +1 dn _t

Pitch(p ) (L ₀-d)/n L ₀/(n +1) (L ₀-3d)/n (L ₀-2d)/n

Helical Extension Springs

The formulae provided for the compression springs generally also apply to extension springs.
An important design consideration for helical extensions springs is the shape of the ends which transfers the load to the the spring body.  These must be designed to transfer the load with minimum local stress concentration values caused by sharp bends.   The figures below show some end designs.. The third design C) design has relatively low stress concentration factors.

Extension Spring Initial Tension

An Extension spring is sometimes tightly wound such that it is prestressed with an initial stress τ _i . This results in the spring having a property of an initial tension which must be exceeded before any deflection can take place.   When the load exceeds the initial tension the spring behaves according the the formulae above.  This relationship is illustrated in the figure below

>

The initial tension load can be calculated from the formula.... T _i = π τ _i d ³/ ( 8 D)

Best range of of Initial Stress (τ _i) for a spring related to the Spring Index C = (D/d)

C = D/d Best Initial Tension Stress range = τ _i

(N/mm ² )

3 140 205

4 120 185

5 110 165

6 95 150

7 90 140

8 80 125

9 70 110

10 60 100

11 55 90

12 45 85

13 40 75

14 35 65

15 30 60

16 25 55

If the coils in a tension spring are not tightly wound, there is no initial tension and the relevant equations are identical to those for the spring under compression as identified above.

The equations for tension springs with initial tension are provided below

Helical Compression Springs (Rectangular Wire)

Spring Rate and Stress

Rate (N/mm) = K ₂ G b t ³/ (n D ³) Stress (N/mm ²) = K _W.K ₁F D /( b t ² )

D = Mean Diameter of spring(mm)

b = Largest section dimension(mm)

t = Smallest Section dimension(mm)

n = Number of Active turns

F = Axial Force on Spring

K ₁ = Shape Factor (see table)

K ₂ = Shape Factor (see table)

K _W = Wahl Factor (see table)

C = Spring Index = D/(radial dimension = b or t)

b/t 1.0 1.5 1.75 2.0 2.5 3.0 4.0 6.0 8.0 10.0

K ₁ 2.41 2.16 2.09 2.04 1.94 1.87 1.77 1.67 1.63 1.60

K ₂ 0.18 0.25 0.272 0.292 0.317 0.335 0.385 0.381 0.391 0.399

Conical Helical Compression Springs

These are helical springs with coils progressively change in diameter to give increasing stiffness with increasing load.  This type of spring has the advantage that its compressed height can be relatively small.  A major user of conical springs is the upholstery industry for beds and settees.

D₁ = Smaller Diameter

D₂ = Larger Diameter

Allowable Force on Spring...
F_a = allowable force (N)..τ = allowable shear stress (N/m²)

Stiffness of Spring...

Disc Springs

A disc spring is a conical shell spring which is loaded along its axis.  Disc springs can used as single or multiple units. When stacked in multiple units they can be stacked in series to give a low stiffness value or in parallel to give a higher stiffness value. By varying the size and the stacking arrangements an extremely wide variation in operating parameters can be achieved.

Parallel Stacked springs (n springs)..For a given force the spring deflection will be (1/n) x the deflection of a single spring.  The stress experienced by each spring will be 1/n the stress experienced by the single spring. (friction must be considered when loading is constantly changing )

Series Stacked Springs (n springs)..For a given force the spring deflection will be n x the deflection of a single spring.  Each spring will experience the same stress as that for a single spring

Series & parallel Stacked springs (n series + n parallel )...For a stack of springs n in parallel and n in series. The deflection for a given force will be the same as for one spring.   The springs will only experience 1/n of the stress of one spring.

Disc Spring are generally standardized according to DIN 2092 Calculations or DIN 2093 (dimensions /quality).
Din 2093 differentiate spring in three groups:

Group 1: Disc Spring thickness t < 1.25 mm Cold Formed

Group 2: Disc Spring thickness 1.25 <= t < 6 mm Cold Formed with inner/outer rings machined and inner edges rounded

Group 3: Disc Spring thickness 6 <= t < 16 mm Hot Formed, All surfaces machined and inner/outer edges rounded, bearing flats

Nomenclature

F = Axial Force (N)

h = Unloaded cone height (mm)

h' = Adjusted unloaded cone height (mm) = H - t'

H = Unloaded Total height (mm)

D = Outside Diameter (mm)

d = Inside Diameter (mm)

t = Thickness (mm)

t' = Adjusted thickness to allow for contact surfaces(mm)

u = Poissens Ratio (mm)

E = Youngs Modulus N/mm²

K₁ = Shape Factor (see formula below)

K₂ = Shape Factor (see formula below)

K₃ = Shape Factor (see formula below)

K₄ = Thickness Compensation Factor (See notes below)

δ Ratio OD/ID= D/d

Factors
The factors are calculated as follows.

D/d 1.5 1.6 1.7 1.8 1.9 2 2.2 2.4 2.6 2.8 3 3.5 3.6 3.8 4

K1 0.5248 0.5735 0.6131 0.6455 0.6722 0.6943 0.7281 0.7518 0.7684 0.78 0.788 0.7979 0.7987 0.7995 0.7994

K2 1.0982 1.1239 1.1488 1.1731 1.1967 1.2198 1.2643 1.307 1.3482 1.3879 1.4263 1.5178 1.5354 1.5699 1.6037

K3 1.1776 1.1776 1.1776 1.1776 1.1776 1.1776 1.1776 1.1776 1.1776 1.1776 1.1776 1.1776 1.1776 1.1776 1.1776

Springs With Contact Surfaces

Some of the springs in group 2 and all of the springs in group 3 are manufactured with contact surfaces to enable better load bearing.  These flats provide improved contact between springs and they also reduce the outside diameter.

A consequence of the altered geometry is that higher forces are generated.  To compensate for this undesirable effect the thickness of the spring is reduced from t to t'.
The normal ratio of t/t' is about 94% to 96%.   With this reduction the spring force at 75% deflection is about the same as a disk spring with no contact surfaces.

A factor K₄ is provided to allow for the different operating characteristics for disc springs with contact surfaces.

If the disc spring has no contact surfaces the K₄ = 1

Spring Force

For Goup 1 and Group 2 disc springs without contact surfaces (see note below) K₄ = 1

The force at a given disc spring deflection is obtained by the formula below.   This is for springs with no contact surfaces

For springs Group 3 disc spring with contact surfaces the formula below is more accurate.

When considering springs with contact surfaces.   Use the factor K₄ as calculated below and use t' instead of t and use h' = H - t'

Spring Stresses

The stresses in a disc spring at four critical locations 1,2,3,4 see sketch for positions are shown below _-ve values are compressive stresses and +ve values are tensile stresses)

Spiral / Clock Springs

A spiral spring consists of a strip or wire wound in a flat spiral .     This is subject to a torque to produce an angular deflection.     A typical spiral spring is a clock spring

Nomenclature

D = Outside diameter of spring (m)
b = Width of spring strip (m)
d = Inside diameter of spring (m)
t = thickness of spring strip (m)
n = Number of turns of spring
k = Spring rate = M /θ Nm/rad.
E = Young's Modulus (N/m²)
M = Moment/torque on spring = F.D / 2(Nm)
L = Length of strip (m)
G = Modulus of Rigidity (N/m²)
I = Second moment of intertia of spring strip (m⁴)
F = Force to deflect spring N
y = distance from neutral axis to outer fibre of wire/strip = y/2 (m)
θ = Deflection (radians)
α = Tensile/compressive stress resulting from spring deflection (N/m²

Note: metres (m) have been shown as the units of length in all of the variables above for consistency.   In most practical calculations milli-metres will be more convenient.

Spring Rate
The spring rate k is defined on this webpage as the torque (Nm)per unit angular deflection (θ ).

Spiral Spring Formulae

length of Strip

Spring Rate

Spring sureface stress

Leaf Springs

Leaf Springs are widely used in the automobile and railway industries for suspension applications.  The simplest variation is the single beam spring.  The more normal application is the laminated (multiple) leaf spring which provides a more efficient stress distribution..

Single Leaf Springs have the following characteristics.

They are suitable for low and medium load forces

They have reasonably linear working characteristics

They have relatively low spring constant

They are long items with relatively low cross section

They are relatively low cost items

Laminated leaf springs have the following characteristics

They are suitable for higher loading forces

They have theoretically linear working characteristics (friction between the leaves causes hysteretic pattern of the working curve)

Compared to single leaf springs they have relatively high spring constants (stiffness)

Laminated spings have high space requirements compared to single leaf springs

They require regular maintenance (lubrication and cleanness)

Nomenclature

E = Young's Modulus (N/m²)
F = applied Force (N)
t = thickness of leaf (m)
b = width (m)
L = length (m)
t = thickness of leaf (m)
δ = deflection (m)
σ = Bending stess <σ(N/mm²
k = spring rate (stiffness) F/δ (N/m

Single leaf springs

There are two primary variations the cantilever spring and the simply supported beam..

Cantilever spring

Simply supported spring

Multiple leaf springs

Considering a cantilever type leaf spring, the stress distribution is related to the distance from the load point.

σ = 6Fx / bt²

If (x/b) is constant along beam, of constant thickness t, then the stress level will be constant and the most efficient spring will result.  If x/b is constant then a triangular shaped spring results.   The multileaf spring is designed to provide a constant stress level along the spring length as it is designed to be equivalent to a triangular spring as shown below.

For this spring the maximum stress( which ideally is constant along the spring) is and the stiffness are as follows..

These equations apply to the quarter elliptic spring as shown below

The relevant equations for the semi-elliptic spring as shown below are

Torsion Springs

Springs used to apply torque or store rotational energy are generally called torsion or double torsion springs.   Torque by definition is a force that produces rotation.   A torsion spring exerts a force (torque) in a circular arc, and the arms rotate about the central axis.   The stress is in bending, not in torsion.   It is customary to specify torque with deflection or with the arms at a definite position.

Torsion bar

The torsion bar is the simplest form or torsion beam.   It comprises a solid or hollow bar which is stressed in torsion within its elastic limit.

Nomenclature

P = Force on lever arm (m).
T = Torque resulting from force = PR (Nm)
R = lever radius (m)
D= bar outside diameter (m)
d= bar inside diameter (m)
L = bar length (m)
k_L = Linear Spring rate || Force based = P / δ (Nm) ....Torque Based = T / δ (N)
k_A = Angular Spring rate || Forced based = P / θ (N)....Torque Based = T / θ (Nm)
G = Modulus of Rigidity (N/mm²)(m)
τ = Allowable shear stress
θ = Deflection (radians)
δ = Linear Deflection= θ. R(m )

Solid Bar

Force Based Torque Based

Deflection δ = 32PR² L /( π GD ⁴ )

Angular Deflection θ = 32PR L /( π GD ⁴ )

Linear spring rate k _L = P / δ = πG D⁴ /(32R ² L)

Angular spring rate k _A = P / θ =πG D⁴ /(32RL)

Maximum Load P_max = π D³ τ /(16 R)

Deflection δ = 32TRL /( π GD ⁴ )

Angular Deflection θ = 32TL /( π GD ⁴ )

Linear spring rate k _L = T / δ = πG D⁴ /(32RL)

Angular spring rate k _A = T/ θ = πG D⁴ /(32L)

Maximum Torque T_max = π D³ τ /16

Hollow Bar
Solid Bar

Force Based Torque Based

Deflection δ = 32PR² L /( π G(D ⁴ - d ⁴ )

Angular Deflection θ = 32PRL /( π G(D ⁴ - d ⁴ )

Linear Spring Rate k _L = P / δ= πG (D ⁴ - d ⁴) /(32R ² L)

Angular Spring Rate k _A = P / θ = πG (D ⁴ - d ⁴) /(32RL)

Maximum Load P_max = π(D ⁴ - d ⁴) τ /(16DR)

Deflection δ = 32TR L /( π G(D ⁴ - d ⁴ )

Angular Deflection θ = 32TL /( π G(D ⁴ - d ⁴ )

Linear Spring Rate k _L = T / δ= πG (D ⁴ - d ⁴) /(32RL)

Angular Spring Rate k _A = T / θ = πG (D ⁴ - d ⁴) /(32L)

Maximum Torque T_max = π(D ⁴ - d ⁴) τ /(16D)

A typical torsion helical spring is shown below.  There are a wide variety of coil end configurations to suit different applications and a torsion spring is usually positioned on a shaft.  The coils are usually close wound as are tension springs but they generally do not have any initial tension unlike tension springs.

The primary stress induced in torsion spring is a bending stress in the wire .  This is not the case for the tension and compression helical springs for which the primary stress is a torsional (shear) stress.  During forming residual stresses are built up in the winding process.  These residual stresses are in the same direction but of opposite sign to the working stresses resulting when the spring is loaded causing the coils to tighten.  Torsion springs are stronger as a result and they are often designed to work at, or above the yield strength.

Nomenclature

C = Spring Index D/d
d = wire diameter (m)
D = Spring diameter (m)
D_i = Spring inside diameter (m)
D_il = Spring inside diameter (loaded ) (m)
E = Young's Modulus (N/m²)
I = Moment of Inertia of wire(m⁴ )
F = applied Force (N)
G = Modulus of Rigidity (N/m²)(m)
k_a = Angular spring rate (stiffness) M /θ (Nm /radian)
L = length (m)
M = Moment (Torque) = RF (Nm)
n = Number of active coils
y = distance from neutral axis to outer fibre of wire (m)
τ = Allowable shear stress (N/m² )
θ = Deflection (radians)
σ = Bending stress (N/m² )

Note: metres (m) have been shown as the units of length in all of the variables above for consistency.   In most practical calculations milli-metres will be more convenient.

Torsion Spring Formulae

The spring stress concentration factors K_i =

The maximum bending stress is at the inner fibre of the coil and equals

The angular spring rate k_a =

Torsion springs are often used over shafts.  It is important that the spring inside diameter, when fully loaded is no t equal to, or less than the shaft diameter.  If this happens the spring will fail.   The inside diameter of the loaded tension spring is

Spring Fatigue

Note: Components which are subject to continuously cyclic loading often fail prematurely as a result of fatigue.  The worst fatigue loading regimes are loads which continuously reverse from negative (compressive) loading to positive (tensile) loading in a cyclic manner. Reference fatigue loading notes  Fatigue Index

As springs are often used under continuously fluctuating loading conditions it is necessary to consider fatigue loading and stress concentration factors.  Helical springs are never used under conditions of load reversals.   They are either normally in tension or normally in compression.  In addition springs are often prestressed as part of the forming process or/and preloaded, thus preventing the stress from being zero.  These factors mitigate, to some extent, the fatigue loading conditions.

All spring subject to continuous fluctuating load are candidates for fatigue failure. Typical springs are

Valve springs in automobile /aeroplane engines

Vehicle suspension springs.

Springs in press tools

Nomenclature

F_m = Mean axial force on spring (N)
F_a= Amplitude of axial force waveform spring (N)
n_s = Factor of Safety
S_se= Torsional endurance limit MPa
S_sl= Torsional strength at 10² cycles MPa
S_sy= Torsional yield strength MPa
S_sf= Torsional fatigue strength MPa
τ _m = Mean shear stress on spring ( MPa)
τ _a = Amplitude of shear stress waveform ( MPa)

Fatigue Notes

The normal shear stress condition experienced be a spring subject to continuous fluctuating loading is as shown below

The force amplitude and mean value are calculated as

The resulting alternating and mean stresses are

For springs the safety factor for torsional endurance life is

Experimental results have proved that for spring steel the torsional endurance limit is not directly related to size, tensile strength, or material for wires under 10mm diameter.   The resulting value from experiments has been determined as

S'_se = 310 MPa for unpeened springs and 465 MPa for peened springs

S'_sa = 241 MPa for unpeened springs and 398 MPa for peened springs

S'_sm = 379 MPa for unpeened springs and 534 MPa for peened springs

These values include all modifying factors except for the reliability factor. ref Fatigue modifying factors That is S_e = C_rS'_e

For springs subject to low cyclic /static loading the safety factor for torsional yielding is

It is generally safe to use a torsional yield strength of 40% of the ultimate tensile strength i.e S_syof 0,4S_ut ref notes Spring Materials

If the spring applications between 10³ and 10⁶ cycles of variation a modified torsional shear strength ( S_fs )can be used to determine the safety margin.

S_sf is the modified shear fatigue strength. This can be determined approximately if the endurance limit( S'_se ) and the fatigue strength at 10³ cycles ( S'_sl ) are available ref High cycle fatigue strength

Goodmans failure criterion..

The fatigue design of springs generally involves one of a number of failure criterions, as shown below.

Goodmans failure criterion..
The intersect equation for the Goodmans criterion is

The relevant factor of safety is calculated as follows

Spring Stability

It is necessary to check that relatively long compression springs are not are risk of buckling.  If buckling is a problem it is necessary to incorporate some method of guiding the spring by placing it in a hole or on a suitable rod.

A longitudinal spring which is subject to rapid cycling may be at risk of surging.  This is when the pulses of compression surge along the spring and back.  This could continue and magnify if the natural material frequency of the spring is near the frequency of repeated loading.

Nomenclature

C'₁ elastic constant = E /(2(E-G))
C'₂ elastic constant = 2 π² (E-G) /(2G+E))
E = Elastic Modulus (Pa)
d = wire diameter (m)
D = Spring diameter (m)
f_n= lowest natural frequency (cycles/second)
n_a = Number of active coils
G = Shear Modulus (Pa)
L₀ = Free Length of spring (m)
y_crit = critical deflection for onset of buckling (m)
α - constant depending on spring end conditions -see table
λ_eff = effective slenderness ratio = α L₀ /D
δ = spring material density (kg/m²)

Buckling

Just as a column will buckle when the load becomes too large a long compression spring may buckle when the deflection exceeds a certain value.   The critical deflection is given by the following equation.

Table showing α for different end conditions

End Conditions α

Spring between two flat parallel surfaces 0,5

Spring on one flat surfaces with other end hinged 0,707

Both ends hinged (pivoted) 1

One end clamped and the other free 2

Absolute stability
Absolute stability occurs when C'₂ /λ_eff² is greater than unity. The condition for absolute stability is therefore.

Surging

For more detailed notes refer to Surging of Springs

The equation for the lowest natural frequency of a compression spring located between two flat plates is..

Forcing frequencies near the above lowest natural frequency and at whole multiples (2,4,6...) of this frequency.

Helical Spring Surge / Natural Frequency

If a spring which is subject to a vibratory motion which is close to its natural frequency the spring can start to surge.    This situation is very undesireable because the life of the spring can be reduced as excessive internal stresses can result.     The operating characteristics of the spring are also seriously affected.     For most springs subject to low frequency vibrations surging is not a problem.    However for high frequency vibrating applications it is necessary to to ensure, in the design stage, that the spring natural frequency is 15 to 20 or more times the maximum operating vibration frequency of the spring.

Nomenclature

C = Spring Index D/d
d = wire diameter (m)
D = Spring diameter = (D_i+D_o)/2 (m)
D_i = Spring inside diameter (m)
D_o = Spring outside diameter (m)
D_il = Spring inside diameter (loaded ) (m)
E = Young's Modulus (N/m²)
F = Axial Force (N)
F_i = Initial Axial Force (N)
     (close coiled tension spring)
G = Modulus of Rigidity (N/m²)
K _d = Traverse Shear Factor = (C + 0,5)/C
K _W = Wahl Factor = [(4C-1)/(4C+5)}]+ (0,615/C)
L = length (m)
L ₀ = Free Length (m)
L _s = Solid Length (m)
k = Spring rate = M / δ (N/m.)
M = mass supported by spring (kg)
n _t = Total number of coils
n_a = Number of active coils
p = pitch (m)
u = motion of spring element(m)
y = distance from neutral axis to outer fibre of wire (m)
σ = Tensile/compressive stress (N/m² )
ω_n = Natural angular frequency of spring (rads/s)
f_n = Natural frequency of spring (cycles/s)
τ = shear stress (N/m²)
τ _i = initial spring stress (N/m²)
τ _max = Max shear stress (N/m²)
θ = Deflection (radians)
δ = linear deflection (mm)

Note: metres (m) have been shown as the units of length in all of the variables above for consistency.   In most practical calculations milli-metres will be more convenient.

Spring Rate
Generally springs are designed to have a deflection proportional to the applied load (or torque -for torsion springs).   The "Spring Rate" is the Load per unit deflection.... Rate (N/mm) = F(N) / δ _e(deflection=mm)

Natural Frequency of a Loaded spring System
Consider a mass M supported on a weightless spring with a spring rate k is illustrated below

This system has a natural frequency as shown below.

This frequency is the lowest natural frequency and is the most important natural frequency.     The spring , however not weightless and thus it has vibration characteristics of its own.    Vibration effects within the spring and the associated frequencies are found by examinging a small element of the spring in harmonic motion.

Surge / Natural frequency of spring

For more detailed notes refer to webpage Simple Harmonic Motion

Consider a spring below subject to a vibration amplitude u at a circular frequency ω

Consider a small element of the spring comprising dn coils.     The density of the spring material is ρ and the length of the wire in the element is π.D.dn.     If the wire diameter is d then the element has a mass

For a vibration amplitude u at a circular frequency ω and at any axial location from coil n = 0 to coil n = n_a the inertial force amplitude is as follows.

The spring rate k for a helical spring is derived on webpage Helical Springs as ..

In practice the term (1 + 0,5/C²) which approximates to 1 can be ignored

For the purpose of this analysis this is transformed to

This equation represents the force for n_a active coils .

The spring force amplitude per active coil resulting from a spring deflection per coil du/dn is

Equating the forces on the spring element to zero the equation below is derived.

This equation is solved using the equation.

u = A sin(c.n) + B cos(c.n)

A and B are arbitrary constants.
For a spring fixed at one end and free at the other the boundary condition occurs at

u = 0 occurs at coil 0
and the condition du/dn = 0 is applied at n = n_a

Resulting in the solution B = 0 and cos(c.n_a) = 0 from which

c.n_a = a.π / 2               ( a= 1,3,5 .....)

This results in a set of natural(surge)frequencies as follows..

For the important fundamental natural frequency, this can be simplified to..

Import Note : The above analysis relates to a spring with one end against a flat surface and the other end free.     The fixed.. fixed case i.e a spring located between and in full contact with two plates results in similar solutions where a = 2, 4, 6......This is also applicable to a spring with one end fixed and against a plate and the other end driven with a sin-wave motion.     The fundamental surge frequency resulting from these scenarios are