Tab 2.3.3
Tab. 2.3.2 (2.3.14)
233
6.1 Shaft with shoulder d = 42mm, Gcr(d) = 2 I d = 0,0476 mm" , ncr (d) = 1,022
For no surface treatment is Ky = 1. Therefore: (2.3.17) (2.3.13)
Design factor KWKt , = 1,134
Fatigue notch factor 1,557 Kfb = - - - - , 1, 109 ·1, 022
6 Examples
1,374.
(2.3.10)
Tab. 2.3.4 (2.3.26)
For no surface treatment is Ky= 1. For steel and cast iron materials is Ks = 1. For materials except GG is KNL,E = 1. Therefore: Design factor KWK,b = 1,374 + 1 10,857 - 1 = 1,541
(2.3.1)
Fatigue strength reduction factor Analytically derived from the stress concentration factor. Stress concentration factor
rid = 0,119, r/t = 1,25, diD = 0,84, Kt,t = 1,283 . (5.2.8) Kt- Kf ratio
Tab. 2.3.3 Tab. 2.3.2
Component fatigue limit for completely reversed torsional stress resulting from the material fatigue limit for shear stress and the design factor for torsion: TWK,t = 233 MPa 11,224 = 190 MPa .
(2.4.1)
Rm = 895 MPa, aM = 0,00035, bM = -0,1 , Mcr = 0,213 , fw,'t = 0,577 , M't = 0,577 . 0,213 = 0,123 .
Tab. 2.4.2 (2.4.34) Tab. 2.2.1 (2.4.34)
Equivalent mean stress Sm = Sm,b = 0, T m = Tm,t =.50 MPa, fw,'t = 0,577 , q = Smv , = Smv , ,GH =
(2.4.31) Tab. 2.2.1 (2.4.29)
J3 . 50 MPa =
86,6 MPa , (2.4.29) (2.4.28) Tm,v = 0,577 . 86,6 MPa = 50 MPa . (2.4.30) Calculation for the type of overloading F2.
Tab. 2.2.1
(2.3.14) (2.3.17) (2.3.13)
Fatigue notch factor
Mean stress factor for bending Smin b v = 86,6 MPa- 150 MPa =- 63,4 MPa , Sma:,~,v = 86,6 MPa + 150 MPa = 236 MPa, Rb,v =- 0,267 , (2.4.27) Because of -
00
< -0,267
~
0 field II applies:
M cr = 0,213 , 1 KAK,b = 1+0,213.86,61150 =
°' 890 .
(2.4.10)
(2.3.10) Mean stress factor for torsion
. Roughness factor Rm = 895 MPa, Rz = 10 urn , fw,'t = 0,58 , aR,cr = 0,22, Rm,N,min = 400 MPa , KR,'t = 0,917 .
(2.4.1)
SWK,b = 403 MPa I 1,541 = 261 MPa .
°,
Rm is to be replaced by fw,.. Rm.
1,283 Kf,t = 1,097.1,031 = 1,134 .
Component fatigue limit for completely reversed bending stress resulting from the material fatigue limit for normal stress and the design factor for bending:
Mean stress sensitivity
The calculation is as for bending.
fw,'t = 0,577 , Rm = 895 MPa, fw,'t' R m = 517 MPa, n't (r) = 1,098 . d = 42 mm, G't(d) = 2 I d = 0,0476 mml, n't(d) = 1,031.
4 COMPONENT FATIGUE STRENGTH
Amplitude of the component fatigue limit for the given mean stress for bending and torsion
Design factor for torsion
diD = 0,84, t I r = 0,8 , r = 5mm G't (r) = 0,230 mm-I , aa = 0,5, bG = 2700 ,
(2.3.1)
Component fatigue limit for completely reversed bending and torsional stress
Roughness factor R m = 895 MPa , Rz = 10 urn , aR,cr = 0,22, Rm,N,min = 400 MPa, KR,cr = 0,857 .
+ 1 10,917 -1 = .1,224
Tab. 2.2.1 Tab. 2.3.4 (2.3.26)
T mm, . t,v = 50 MPa- 100 MPa =- 50 MPa, Tmax,t,v = 50 MPa + 100 MPa = 150 MPa , (2.4.27) Rt,v =- 0,333 ,
234
6.1 Shaft with shoulder Because of -1 < -0,333
s
°
Amplitude of the component variable amplitude fatigue strength for bending and for torsion, see above,
field II applies:
M, = 0,123 , 1
KAK,I
= 1+0,123.50/100 = 0,942 .
(2.4.10)
Residual stress factor for normal stress and for shear stress KE,cr = KE,'t = 1.
= 0,890 . 1 . 261 MPa = 233 MPa, (2.4.6) = 0,942 . 1 . 190 MPa = 179 MPa .
Component variable amplitude fatigue strength for bending and torsion Considering the component constant amplitude fatigue limit, N > No,cr and N > No,'t , and the S-N curve model I (horizontal for N > No,cr and N > No,'t) the variable amplitude fatigue strength factors for bending and for torsion is
(2.4.48)
The amplitude of the component variable amplitude fatigue strength results from the variable amplitude fatigue strength factor and from the amplitude of the component fatigue limit for bending and for torsion, SBKb = 1· 233 MPa = 233 MPa, TB~,t = 1 . 179 MPa = 179 MPa .
(2.4.41)
5 SAFETY FACTORS In general there is
In =
(2.5.1)
1,5 .
For moderate consequence of failure and regular inspection, however, there is jo = 1,2.
Tab. 2.5.1
For normal temperature there is KT,o
=
1,
(2.2.4)
and therefore jges
= 1,2.
Degrees of utilization Individual types of stress, bending and torsion,
150 = 0 773 a BK,b = 233/1,2 ' ,
The amplitude of the component fatigue limit results from the mean stress factor, the residual stress factor and the component fatigue limit for completely reversed bending and torsional stress:
. KBK,b = KBK,I = 1.
TSBK,b = 233 MPa, TBK,I = 179 MPa.
(2.4.5)
Amplitude of the component fatigue limit
SAK,b TAK,I
6 Examples
(2.5.4)
6 ASSESSMENT Largest stress amplitudes for bending and for torsion, see above, Sa,b,1 = Sa,b = 150 MPa , T a.t, I = Ta.t = 100 MPa.
a
BK,t
= 100
179/1 ,2
(2 6 1) ..
= 0 670 . '
Combined types of stress fw,'t = 1//3, q = 0, Sa = aBK,b = 0,773 , ta = aBK,t = 0,670 , aGH
= ~0,7732 +0,670 2
aBK,Sv = 1,023 .
Tab. 2.2.1 (2.6.7) (2.6.6)
1,023 ,
(2.6.5) (2.6.4)
The degree of utilization of the component fatigue limit is 102 %• The assessment of the fatigue limit is approximately achieved.
235
6.1 Shaft with shoulder
6 Examples
Complementary assessment 1:
Degrees of utilization
Amplitude of the component fatigue limit for the given mean stress - improved method of calculation according to Chapter 5.9
Individual types of stress, bending and torsion,
Related equivalent stress amplitude for bending and torsion Sa,b = 150 MPa, SWK,b = 262 MPa , Sa = 150 / 262 = 0,573 , Ta,l = 100 MPa, TWK,1 = 190 MPa, t a = 100 / 190 = 0,526 , 2
sa,v =JO,573 +0,526
2
(5.9.3)
= 0,778 .
(5.9.2)
J3
7.0,573 2 +12.0,526 2 2
0,573 +0,526
2
= 7,224
a
BK,1
Tab. 5.9.1
(2.6.1)
°
= 168/1,2 100 = 715 . '
Combined types of stress
/.J3,
Related equivalent mean stress for bending and torsion 1 8·0,573·0,526 B3 = - ' = 1028 ' 3.0,573 2 +(4/3).0,526 2 A4
aBK b = 150 = 0,732 , , 246/1,2
Tab. 2.2.1 (2.6.7) (2.6.6)
fw,'t = 1 q = 0, sa = aBK,b = 0,732 , ta = aBK,1 = 0,715 , 2
=JO,732 +0,715 aBK,Sv = 1,023 . aGH
2
= 1,023 ,
(2.6.5) (2.6.4)
Complementary assessment 2: Assessment applying a class of utilization
The constants B I , Al and AS are not needed as Sm =0.
= 50 MPa, TWK,1 = 190 MPa, t m = 50/190 = 0,263 , Sm = 0, Sms = (3/7) . 1,028' 0,263 = 0,116 ,
Tm,l
t ml
= (1
/E) .J7,224.0,263
sm,v = 0,87 . (0,116
+ 0,154)
2
=
(5.9.6)
= 0,154, 0,235 .
Common mean stress factor smin,v = 0,235 - 0,778 = - 0,543 , smax,v = 0,235 + 0,778 = 1,013 , Rcr,v - 0,543 / 1,013
(5.9.5) (5.9.18)
- 0,536
Differing from the constant amplitude loading conditions and the assessment of the fatigue limit considered above an assessment of a variable amplitude loading according to the class of utilization B5 is carried out here. In this case the class of utilization B5 may, for example, stand for binominally distributed amplitudes of the stress spectrum, a spectrum parameter p = 1/3 and a required total number of cyclesN = 107 , Table 5.7.2. The variable amplitude fatigue strength factor for the class of utilization B5 for bending and for torsion is KBK,b
Because of - 00 < -0,267 ~ 0 field II applies: Mcr = 0,213 , 1 KAK,v = 1+0,213.0,235/0,778 = 0,940. (2.4.13) Residual stress factor for normal stress and for shear stress KE,cr
= KE;t = 1.
(2.4.5)
The amplitude of the component fatigue limit results from the mean stress factor, the residual stress factor and the component fatigue limit for completely reversed bending and torsional stress: SAK,b = KAK,v . KE,cr . SWK,b = (2.4.6) = 0,940' I . 262 MPa = 246 MPa, TAK,1 = KAK,v . KE,'t . SWK,b = = 0,940 . 1 . 190 MPa = 168 MPa. Using these values instead of the values 233 MPa and 179 MPa somewhat different degrees of utilization are obtained.
= 1,26, KBK,1 = 1,15 .
Tab 5.7.1
The amplitudes of the component variable amplitude fatigue strength for bending and for torsion for bending and for torsion are (2.4.41) SBK,b = KBK,b . SAK,b = 1,26' 246 MPa = = 310 MPa, TBK,1 = KBK,I' TAK,1 = 1,15' 168 MPa = = 193 MPa. With these values instead of the values 246 MPa and 168 MPa the degrees of utilization are lower than those from above.
236
6.2 Shaft with V-belt drive
6.2 Shaft with V-belt drive
6 Examples
Surface: average roughness of the shaft: Rz = 25 J..Lm.
*1 lR62 EN .docl
Key words: Shaft, V-belt drive, rolled steel, assessment of the static strength, assessment of the fatigue limit, type of overloading F 1, combined types of stress.
Given
Loading: Maximum torque (rated torque), idealized as a fluctuating loading (load ratio RM = 0), figure 6.2.1:
M, = M max = 1 kNm.
(6.2.1)
Type of overloading: when overloaded in service the tension of the belt remains constant and with that the mean stress remains approximately constant as well (Type of overloading Fl). Safety requirements: according to the statements "severe consequences of failure; no regular inspections" .
Task: Assessment of the component static strength and assessment of the component fatigue limit. Method of calculation: Rod-shaped (ID) componerit. Assessment using nominal stresses, Chapter 1 and 2.
Material: St 60 after DIN EN 10 025. Dimensions: Length of shaft 1 = 400 mm, diameter of shaft d = 56 mm.
ASSESSMENT OF THE COMPONENT STATIC STRENGTH
Pulley in center position, bearing on the left horizontally fixed, torque input at the right end.
1 LOADINGS Specifying the loadings is not subject of the guideline. Therefore the loadings are outlined here in brief *2.
M, is the torque input to the shaft. The torque input results in a maximum of six loads F x , ... , M, at the pulley. The initial tension of the belt results in a maximum of two lateral forces F y , Fz . The sense of rotation and direction of input and output are not important for the loads.
ZCA.l--~-+-HlI.-----~Mf---t
The torque is transmitted by the initial tension and frictional contact of the belt. Therefore the loadings "torque" and "initial tension" are to be distinguished. The axial load of the shaft and the lateral moments at the pulley are zero: Z
(6.2.2)
Data ofthe drive Diameter of pulley No.1: d 1W = 200 mm, Diameter of pulley No 2: d2W = 100 mm, Center distance: e = 200 mm, Wedge angle: Ys = 35°, Friction coefficient between belt and pulley: J..L= 0,1.
Figure 6.2.1 Shaft and loads due to the torque M n at the V-belt drive. F I
-
F 2 = F,
*2
Calculation equations Loading "torque" Circumferential force transmitting the torque
Ft=Mn ' 2/d 1w , = 1000 kNmm . 2 I 200 mm
= 10 kN.
(6.2.3)
Inclination of the belt 1 Results of computation obtained by the PC-program "Welle" 2 Full presentation of the example s given in the manual of the PC-program "Welle".
sin a,
a,
= (d 1w - d2w) 12e, (6.2.4) = (200 - 100) mm I (2 . 200 mm) = 0,25 ,
= 14,48°.
237
6.2 Shaft with V-belt drive
6 Examples Mz - 2,349kNm A ..----'''--'---------~i1___7z
Resulting loads
r, = Fy = 0,
y~
(6.2.5)
..;.,::
o
Fz = + F, . sin a. = 10 kN . 0,25 = 2,5 kN , M, = M; = I kNm , My=Mz= O.
lI:l C>J
B«<::...-----------.J o
II
-:t.
ails b622
Load "initial tension"
Figure 6.2.2 Resulting moments at mid shaft.
Value of the groove angle
Ys = 35
0.
Effective coefficient of friction
(6.2.6)
ASSESSMENT OF THE COMPONENT STATIC STRENGTH
J.lth = ~ / sin(ys / 2) = 0,1 / sin(35012) = 0,3326 . _3
Arc of belt contact at the smaller pulley (No.2) (6.2.7)
~ = 180° - 2 ·1a.1 = 180° - 2· 14,48 = 151,0°. For the smaller pulley there is a. < 0 and ~ < 180 0. For the larger Pulley there is a. > 0 and ~ < 180°.
1 CHARACTERISTIC STRESSES The reference point for the assessment of the static strength is in the middle of the shaft (to the right of the pulley). Maximum bending moment (point B in Figure 6.2.2) Mb= M yz
Auxiliary variable
=~M~ +M;
=~0,250kNm2+ 2, 349kNm 2
m = exp (~th . ~ . n / 180°), . (6.2.8) = exp (0,3326 . 151,0°. n /180°) = 2,403.
(6.2.12) = 2,362 kNm.
Maximum torsion moment Necessary minimum initial tension for transmitting the maximum torque *4 2 m+l Fs p = M, ·-·--·cosa.
I Id
1w
m->I
Extreme maximum stresses *5
(6.2.9)
Smax ex b = 32 Mb / pd 3 (1.1.1) = '32' 2,362 kNm / (n . (56 mm)3 ) = 137 MPa , T max ex t = 16 M t / pd 3 = 't6'· 1 kNm / (rt . (56 mm)3 ) = 29,0 MPa.
2 2,403+1 = 1000 kNmm' 200mm 2,403-1 . cosine 14,47° = 23,49 kN. Resulting loads
F x = 0, Fy = + F sp = 23,49 kN, Fz = 0, M x= M y= M z= O.
(6.2.10)
2 RESULTING MOMENTS AT MID SHAFT
2 MATERIAL PROPERTIES Tensile strength and yield strength for the standard dimension Rm,N = 590 MPa, Rp,N = 335 MPa. Tab. 5.1.1 Technological size factor ad,m = 0,15 , ad,p = 0,3 , deff,N = 40 mm , deff = d = 56 mm , ~,m = 0,982, Kd,p = 0,960 .
From the loadings "maximum torque" (M; = 1 kNm and F z = 2,5 kN) and from the loading "initial tension" (Fy = 23,486 kN) the moments in the middle of the shaft (to the right of the pulley) are
M, = - 1 kNm, (6.2.11) My = + F, . I / 4 = 2,5 kN . 400 mm / 4 = + 0,250 kNm, . M z = - F y . I /4 = 23,49 kN . 400 mm / 4 = - 2,349 kNm . The bending moments My and M, are presented in Figure 6.2.2. The loading without torque (only initial tension) corresponds to the point A. The loading under maximum M, torque (including initial tension) corresponds to the point B.
Tab. 1.2.1 Tab. 1.2.3 (1.2.9)
Anisotropy factor K A = 1.
(1.2.17)
Tensile strength and yield strength of the component R m = 0,982 . 1 ·590 MPa = 579 MPa, Rp = 0,960 . 1 . 335 MPa = 321 MPa .
4
(1.2.1)
Practically anincreased initial tension value needs tobeconsidered.
5 From here on the numbers of equations, tables and figures are those of the guideline. 3
Torque and the pre-tension aretobedetermined for the smaller pulley
238
6.2 Shaft with V-belt drive
6 Examples
3 DESIGN PARAMETERS
6 ASSESSMENT
Design factor for bending
Maximum stresses for bending and for torsion, see above,
Section factor Rp,max = 1050 MPa , Tab. 1.3.1 Rp = 321 MPa, Kp,b = 1,70, Tab. 1.3.2 npl,b = MIN (J1050/321 ; 1,70)= 1,70, 1.3.9)
Smax,ex,b = 137,0 MPa, Tmax,ex,t'= 29,0 MPa. Component static strength for bending and for torsion, see above, SSK,b = 985 MPa, TSK,t = 447 MPa .
Design factor KSK,b = 1 / 1,70 = 0,588.
(1.3.1)
Degrees of utilization Individual types of stress, bending and torsion,
Design factor for torsion 137 = 0376 aSK,b = 985 I 2,70 ' ,
Section factor Rp,max = 1050 MPa , Tab. 1.3.1 Rp = 321 MPa, Kp,t = 1,33 , Tab. 1.3.2 npI,t = MIN (J1050/321 ; 1,33) = 1,33. 1.3.9) Design factor KSK,t = 1 / 1,33 = 0,752.
(1.3.1)
a = 29,0 = 0176. SK,t 447 I 2,70 ' Combined types of stress f't = 0,577 , q = 0, s = aSK,b = 0,376 , t = aSK t = 0,176,
~0,3762 +0,176 2
4 COMPONENT STATIC STRENGTH
aGH =
Component static strength for bending and torsion
aSK,Sv = 0,415 .
Component static strength for bending resulting from the tensile strength and design factor:
(1.6.1)
Tab. 1.2.5 (1.6.7) (1.6.6) = 0,415 ,
(1.6.5) (1.6.4)
The degree of utilization of the component static strength is 42 %. The assessment of the static strength is achieved.
fa = 1 , Tab. 1.2.5 (1.4.1) SSK,b = 579 MPa / 0,588 = 985 MPa . Component static strength for torsion resulting from the shear strength factor f't , the tensile strength and the design factor: f't = 0,577 , Tab. 1.2.5 TSK,t = 0,577 . 579 MPa / 0,752 = 445 MPa . (1.4.1)
For severe consequences of failure and high probability of the occurrence of the characteristic stress there is Tab. 1.5.1
For normal temperature there is KT,m = KT,p = 1,
1 CHARACTERISTIC STRESSES Again the reference point for the assessment of the fatigue strength is in the middle of the shaft (to the right of the pulley).
The mean value and amplitude of the maximum cyclic bending moment is to be determined. The mean value of the bending moment for the rotating shaft is Mm,b =
(1.2.26)
and in Eq. (1.5.4) the terms 3 and 4 have no relevance: Rp = 321 MPa, R m = 579 MPa, jges = MAX (2,0; 1,5. 579) 321 = MAX (2,0; 2,70) = 2,70.
OF THE COMPONENT FATIGUE STRENGTH
Cyclic bending moment
5 SAFETY FACTORS
jm = 2,0 , jp = 1,5.
ASSESSMENT
°
The maximum amplitude of the bending moment for the rotating shaft is, as for the assessment of the static strength, Ma,b = 2,362 kNm.
(1.5.4)
The yield strength Rp (the second term) is determining.
239
6.2 Shaft with V-belt drive The amplitude of the bending moment is varying within the limits defined by no torque input or maximum torque input, points A and B in Figure 6.2.2. The assumption of a continuous action of the maximum bending moment is an approach on the safe side.
6 Examples For no surface treatment is Ky = 1. For steel and cast iron materials is Ks = 1. For materials except GG is KNL,E = 1. Therefore: Design factor
+ 1/0,858 - 1
KWK,b = 0,979
Fluctuating torsion moment The mean value and amplitude of the torsion moment fluctuating between zero and maximum is
= 1,144
(2.3.1)
Design factor for torsion Fatigue notch factor
Mm,t = Ma,t =IMxl /2 = 0,5 kNm.
Stress concentration factor for the cylindrical shaft *6
It is valid for the non-rotating shaft too.
Kt,t = 1.
Characteristic stresses The cyclic stresses resulting from the bending moment and the torsion moment are
(2.1.1) Sm b = 0, , 3 Sa,b = 32 Ma,b / pd = 32 . 2,362 kNm / (n . (56 mmr' ) = 137 MPa Tm,t = 16 Mt,m / pd 3 = 16 . 0,5 kNm / (n . (56 mm)3 ) = 14,5 MPa , Ta,t = Tm,t == 14,5 MPa.
2 MATERIAL PARAMETERS
K,-K r ratios, no notchradius,
G, (r) = 0, n, (r) = 1 , = 56 mm, G, (d) = 2/ d = 0,0357 mm -1 , n, (d) = 1,027 .
d
(2.3.17) (2.3.13)
Fatigue notch factor Kf,t
= 1 / (1 . 1,027) = 0,974.
(2.3.10)
Roughness factor
Material fatigue limits for completely reversed normal stress, crW,zd , and shearstress, 'tw,s, : R m = 579 MPa, fw,cr = 0,45 , Tab. 2.2.1 crW,zd = 0,45 . 579 MPa = 261 MPa , (2.2.1) Tab. 2.2.1 f w " = 0,577 , 'tw,s = 0,577 . 261 MPa = 151 MPa . (2.2.1)
R m = 579 MPa , R z = 25 um , fw" = 0,58, aR,cr = 0,22, Rm,N,min = 400 MPa , KR" = 0,918 .
Tab. 2.2.1 Tab. 2.3.4 (2.3.26)
For no surface treatment is Ky = 1. For steel and cast iron materials is Ks = 1. For materials except GG is KNL,E = 1. Therefore: Design factor KWK,t = 0,974
3 DESIGN PARAMETERS
+
1/0,918 - 1 = 1,063
(2.3.1)
Design factor for bending Fatigue notch factor
4 COMPONENT FATIGUE STRENGTH
Stress concentration factor for the cylindrical shaft *6
Component fatigue limit for completely reversed bending and torsional stress
Kt,b = 1.
Component fatigue limit for completely reversed bending stress resulting from the material fatigue limit for normal stress and the design factor for bending,
K,-K r ratios, no notch radius,
G cr (r) = 0, ncr (r) = 1 , d = 56 mm, Gcr(d) = 2/ d = 0,0357 mm -1 , ncr (d) = 1,022 .
SWK,b = 261 MPa /1,144 = 228 MPa.
(2.3.17) (2.3.13)
Fatigue strength reduction factor Kf,b
= 1 / (1 . 1,022) = 0,979 .
(2.3.10)
(2.4.1)
Component fatigue limit for completely reversed torsional stress resulting from the material fatigue limit for shear stress and the design factor for torsion, TWK,t = 151 MPa / 1,063 = 142 MPa .
(2.4.1)
Roughness factor R m = 579 MPa, R z = 25 um , aR,cr = 0,22, Rm,N,min = 400 MPa, KR,cr = 0,858 .
Tab. 2.3.4 (2.3.26)
6 Assuming an unnotched cylindrical shaft is an unrealistic simplification for the present example. Actually a keyway or a press-fit between pulley andshaft, Chapter 5.3.3.4, would have to beconsidered.
240
6.2 Shaft with V-belt drive Amplitude of the component fatigue limit for the given mean stress for bending and torsion Mean stress sensitivity Rm = 579 MPa, aM = 0,00035 , bM = - 0,1 , Me; = 0,103 , fW;t = 0,577 , M't = 0,577 . 0,103 = 0,059 .
model I (horizontal for N > ND,e; and N > ND,'t ) the variable amplitude fatigue strength factors for bending and for torsion is KBK,b = KBK,t = 1.
Tab. 2.4.2 (2.4.34) Tab. 2.2.1 (2.4.34)
Equivalent mean stress
°,
Sm = Sm,b = (2.4.31) T m = Tm,t = 14,5 MPa , fw,'t = 0,577 , Tab. 2.2.1 q = (2.4.29) Sm,v = Sm,v,GH 14,5 MPa = 25,1 MPa, (2.4.29) (2.4.28) Tm,v = 0,577 ·25,1 MPa = 14,5 MPa . (2.4.30)
°,
6 Examples
=J3 .
Mean stress factor for bending
SBK b = 1 . 225 MPa = 225 MPa, TB~,t = 1 . 141 MPa = 141 MPa.
Because of - 1,115:::;; 0,110:::;; 0,907 field II applies:
(2.4.15)
Mean stress factor for torsion
5 SAFETY FACTORS For severe consequence of failure and no regular inspections there is
.io
(2.5.1), Tab. 2.5.1
1,5.
=
Jges
(2.2.5) (2.5.4)
1,5 .
=
6 ASSESSMENT Largest stress amplitudes for bending and for torsion, see above, Sa b l = Sa b = 137 MPa , Ta',t:! = T a:t = 14,5 MPa . Amplitude of the component variable amplitude fatigue strength for bending and for torsion, see above,
tm,t,v = Tm,v / TWK,t = 14,5/142 = 0,102 M, = 0,059, 1 / (1 + M't) = 1 / (1 + 0,059) = 0,944 .
SBK,b = 225 MPa, TBK,t = 141 MPa .
Because of 0:::;; 0,102:::;; 0,944 field II applies:
Degrees of utilization (2.4.15)
Residual stress factor
Individual types of stress, bending and torsion,
aBK,b
=
For normal stress and for shear stress there is KE,e; = KE,'t = 1.
(2.4.41)
KT,D =1,
Sm,b,v = Sm,v / SWK,b = 25,1 /228 = 0,110, Me; = 0,103 , - 1 / (1 - Me;) = - 1 / (1 - 0,103) = - 1,115 , 1 / (1 + Me;) = 1 / (1 + 0,103) = 0,907 .
KAK,'t = 1 - 0,059' 0,102 = 0,994.
The amplitude of the component variable amplitude fatigue strength results from the variable amplitude fatigue strength factor and from the amplitude of the component fatigue limit for bending and for torsion,
For normal temperature there is
Calculation for the type of overloading F1.
KAK,e; = 1 - 0,103 . 0,110 = 0,989 .
(2.4.48)
(2.4.5)
Amplitude of the component fatigue limit The amplitude of the component fatigue limit results from the mean stress factor, the residual stress factor and the component fatigue limit for completely reversed bending stress and torsional stress, SAK,b = 0,989 . 1 . 228 MPa = 225 MPa, (2.4.6) TAK,t = 0,994 . 1 . 142 MPa = 141 MPa .
a
t =
BK,
137 = 225/15 ,
°9~
(2 • 6 • 1)
' ~.J,
14,5 = 0,155 . 141/ 1,5
Combined types of stress Tab. 2.2.1 (2.6.7) (2.6.6)
fw,'t = 0,577 , q = 0, Sa = aBK,b = 0,913 , ta = aBK,t = 0,155 . aGH =
~0,9132 +0,155 2
=
0,926 ,
aBK,Sv = 0,926. Component variable amplitude fatigue strength for bending and torsion Variable amplitude fatigue strength factor
(2.6.5) (2.6.4)
The degree of utilization of the component fatigue limit is 93 %. The assessment of the fatigue limit is achieved. *7
Considering the component constant amplitude fatigue limit, N > ND,e; and N > ND,'t , and the S-N curve 7 See the objections in footnote 4 and 6, however.
241
6 Examples
6.3 Compressor flange made of grey cast iron
6.3 Compressor flange made of grey cast iron "'I 1R63 EN.docl
ASSESSMENT OF THE COMPONENT STATIC STRENGTH
1 CHARACTERISTIC STRESSES Key words: Grey cast iron GG-30, assessment of the static strength, assessment of the fatigue limit, local elastic stresses, type of overloading F2, elevated temperature, combined types of stresses GI and G2 .
*2 (3.1.6)
Maximum stresses
GI ,max ,ex = 33,6 MPa, G2,max ,ex = 11,2 MPa .
2 MATERIAL PROPERIES
Given
Stresses: Proportional, constant amplitude loading, locally elastic stresses in the directions 1 (longitudinal) and 2 (circumferential) at the reference point (node 99) of a block-shaped (3D) component, Figure 6.3.1, GI = GI,m ± GI,a = 15,0 MPa ± 18,6 MPa , G2 = G2,m ± G2,a = 5,0 MPa ± 6,2 MPa , G3 = 0. Stress amplitudes at the neighbouring point (node 98) in a distance s = 7,7 mm below the surface GI,a = ± 10,0 MPa, G2,a = ± 5,3 MPa.
Tensile strength for the standard dimension Rm,N
=
300 MPa .
Tab. 5.1.14
Technological size factor deff = 2s = 2 . 32 mm = 64 mm , Kd,m = 0,800 .
Tab. 3.2.3 (3.2.5)
Anisotropy factor
KA
=
(3.2.18)
1.
Tensile strength of the component Rm = 0,800' 1 . 300 MPa = 240 MPa:
(3.2.1)
Temperature factors
Flange
aT,m = 1,6 , Tab. 3.2.6 KT m = 0,769 , (3.2.31) Crr: = 25 , Tab. 3.2.7 aTt,m = - 1,46, bn,m = 2,36, CTt,m = - 0,90 , Pm = 10 - 4. (380 + 273) . (25 + 19 100.000) = = 1,959, (3.2.35)
111 91.
_ (-1,46+2,36.1,959-0,90.1,959 2 ) KTt,m -10 = 0,511 . 92
3 DESIGN PARAMETERS
87
6.~.1
Section factor for GG
Figure 6.3.1 Compressor flange made of grey cast iron.
Material: GG-30 according to DIN 1691 or DIN EN i,:,61.
Temperature and time: T
npl,O'I = n pl,O'2 = 1.
Constant KNL for the consideration of the non-linear elastic strain characteristic of GG for tension KNL = 1,05.
=
380°C, t
=
100.000 h.
Dimensions: Effective wall thickness at the reference point (node 99) s ;::: 32 mm.
Surface: Skin Type of overloading: : When overloaded in service the stress ratios remain constant (Type of overloading F2).
Safety requirement: according to the statements: "severe consequences of failure; no regular inspection", casting tested non-destructively.
(3.3.7)
Tab. 3.3.4
Design factor for directions 1 and 2 KSK,O'I = KSK,O'2 = 1/ (1'1,05) = 0,952.
(3.3.3)
4 COMPONENT STRENGTH Component strength resulting from the tensile strength and the design factor. For directions 1 and 2 there is fO' = 1 , Tab. 3.2. GI,SK = G2,SK = 240 MFa / 0,952 = 252 MPa . (3.4.3)
Task: Assessment of the static strength and assessment of the fatigue limit. Method of calculation: Block-shaped (3D) component. Assessment using local elastic stresses, Chapter 3 and 4
I Results of computation obtained by the PC-program "RIFESTPLUS" 2 Numbers of Equations, tables and figures arethose of the guideline.
242 6 Examples
6.3 Compressor flange made of grey cast iron
5 SAFETY FACTORS For severe consequences of failure and high probability of the occurrence of the maximum stress and for nondestructively tested castings there is
Jm = 2,5 , jmt = 1,9 .
Tab. 3.5.2
Because of the low elongation of GG the safety factor is to be increased by ~j : ~j =
0,5 , jm = 3,0 , jmt = 2,4 .
(3.5.2)
Jges
=
°.
(3.5.4)
= MAX (3,90; 4,70) = 4,70.
Material fatigue limit for completely reversed normal stress, CfW,zd , R m = 240 MPa, fw,cr = 0,30 , crW,zd = 0,30 . 240 MPa
The creep strength Rm,Tt is is determining, Figure 3.2.2.
= 72,0 MPa .
= 33,6 MPa, cr2,max,ex
Tab. 4.2.2 (4.2.8)
aT,D = 1,0, KT,D = 0,856 .
Maximum stresses, see above, =
11,2 MPa .
3 DESIGN PARAMETERS
Component strength values, see above,
(4.3.16)
Kt-Kf ratios
crl,SK = cr2,SK, = 252 MPa.
Gal Individual types of stress, directions 1 and 2
Go ?:
33,6
I_.(I_IO,OJ 18,6
__ 7,7mm
Degrees of utilizaion
I (I - 6,2 5.3J
-
- 7,7mm'
=
0,0600 0,0189 mm- l ,
aa
= - 0,05, bG = 3200 , nal = 1,179, n cr2 = 1,056.
= 0,209 .
aG H
0,85 , 0,759 , aSK,crl = 0,627 , aSK.cr2 = 0,209 ,
=
~. ((0,627 -0,209)2 +0,209
Tab. 4.3.2 (4.3.10)
Roughness factor
Combined types of stress f, = q = 81 = S2 =
Tab. 4.2.1 (4.2.1 )
Temperature factor
6 ASSESSMENT
aSK,cr2 = 252/4,70
(4.1.1 )
2 MATERIAL PROPERTIES
MAX(~._2LJ 0,769'0,511
crLmax.ex
1 CHARACTERISTIC STRESSES
crl.a = 18,6 MPa, 01,m = 15,0 MPa . cr2,a = 6,2 MPa, 02.m = 5,0 MPa , cr3,a = 0 , 03.m =
KTt,m = 0,511 .
For GG the terms 2 and 4 of Eq. (3.5.4) are not relevant. Therefore .
ASSESSMENT OF THE FATIGUE STRENGTH
Constant amplitude cyclic stresses
Temperature factors, see above, KT,m = 0,769,
Note: For the assessment of the component static strength the presented computation for the combined types of stress applies to both proportional stresses and non-proportio-nal stresses o l and cr2, see footnote 1 in Chapter 3.6.
Tab. 3.2.5 (3.6.23) (3.6.22)
R m = 240 MPa, Rz = 200 um , aR,cr = 0,06, Rm,N.min = 100 MPa, KR,cr = 0,906 , Surface treatment factor,
K v = 1. 2
+0,627
2
)
(4.3.28)
Coating factor
= 0,553 , (3.6.21) a NH = MAX (0,627; 0,209) = 0,627 , aSK.sY = 0,759 . 0,626 + (I - 0,759) . 0,553 = = 0,608 . (3.6.20)
The degree of utilization of the component static strength is 61 %. The assessment of the static strength is achieved.
Ch.4.3.3 Tab. 4.3.4 (4.3.26)
*3
(4.3.29)
KKS = 1. Constant KNL.E KNL,E
=
1,025 .
Tah. 4.3 (
3 Even though the present assessment ofthe creep strength may he the safe side, see Chapter 3.1.0 on.Elevated temperature'
tar on
243 6.3 Compressor flange made of grey cast iron Design factors Kf = 1,
6 Examples
Tab. 4.3.1 1
KWK,al = 1,1\9 -(1+i-( 0,:06 -1))
1·1·1,025
= 0,913,
(4.3.3) 1
KWK,a2
= 1,;56 -( 1+ i -( 0,:06 -1)) 1·1·1,025
= 1,019.
crl,AK = 0,713 . 1 . 79 MPa = 56 MPa , cr2,AK = 0,713 . 1 ·71 MPa = 50 MPa .
Component variable amplitude fatigue strength for bending and torsion Considering the component constant amplitude fatigue limit, N > No,a and N > NO;t , and the S-N curve model I (horizontal for N > NO,a and N > No,'t) the variable amplitude fatigue strength factors for normal stress is KBK,al = KBK,a2 = 1.
4 COMPONENT FATIGUE STRENGTH Component fatigue limit for completely reversed normal stresses Component fatigue limit for completely reversed normal stress resulting from the material fatigue limit for normal stress and the design factors: crl,WK = 72,0 MPa / 0,913 = 79 MPa, (4.4.3) cr2,WK = 72,0 MPa / 1,019 = 71 MPa. Amplitude of the component fatigue limit for the given mean stress Mean stress sensitivity Tab. 4.4.2 (4.4.34)
According to Chapter 4.4.2.2 the individual mean stresses are to be applied instead of an equivalent mean stress. Calculation for the type of overloading F2. Mean stress factor for direction 1
(4.4.26)
crl,min = 15 MPa - 18,6 MPa = - 3,6 MPa , crl,max = 15 MPa MPa + 18,6 = 33,6 MPa, Ral = - 3,6 / 33,6 = - 0,107 . Because of Ma
CX)
< -0,107
~
° field II applies:
= 0,5 ,
KAK,al =
1 1+0,5.15/18,6
(4.4.47)
The amplitude of the component variable amplitude fatigue strength results from the variable amplitude fatigue strength factor and from the amplitude of the component fatigue limit for normal stress crl,BK = 1·56 MPa = 56 MPa, cr2,BK = 1 . 50 MPa = 50 MPa.
(4.4.45)
5 SAFETY FACTORS For severe consequence of failure, for no regular inspection and for castings tested non-destructively
.io =
aM = 0, b M = 0,5 , M a = 0,5 .
(4.4.8)
Tab. 4.5.2
1,9 .
Because of the low elongation of GG the safety factor is to be increased by ~j : ~j
= 0,5
,
(4.5.2)
.in = 2,4. Temperature factor, see above, KT,O = 0,856 . Total safety factor jges =
2,4/0,856 = 2,80 .
(4.5.4)
6 ASSESSMENT
= 0,713 .
(4.4.10)
Stress amplitudes, see above cra,l = 18,6 MPa , cra,2 = 6,2 MPa .
Mean stress factor for direction 2 The stress ratios of both directions agree:
R a 2 = Ral = - 0,107 , KAK,a2 = 0,713 .
Amplitudes of the component variable amplitude fatigue strength, see above, crl,BK = 56 MPa, cr2,BK = 50 MPa .
Residual stress factor Degrees of utilization
For normal stress there is KE,a = 1.
(4.4.5)
Individual types of stress, direction 1 and 2
5:~;680 ,
Amplitude of the component fatigue limit
aBK,al =
The amplitude of the component fatigue limit results from the mean stress factor, the residual stress factor and the component fatigue limit for completely reversed normal stress:
aBK,a2 = 50
~':,80
= 0,929 , =
0,346 .
(4.6.17)
244 6.3 Compressor flange made of grey cast iron
6 Examples
Combined types of stress fw,'t = 0,85 , q = 0,759 , Sl,a = aBK,crl S2,a = a BK,cr2
a GH
= 0,929, = 0,346 ,
. Tab. 4.2.1 (4.6.23) (4.6.22)
= ±.((0,929-0,346)2 +0,346 2 +0,929 2)
= 0,813, (4.6.21) a NH = MAX (0,929; 0,346) = 0,929 , aBK,sv = 0,759 . 0,929 + (1 - 0,759) . 0,813 = = 0,901. (4.6.20)
The degree of utilization of the component fatigue strength is 90 %. The assessment of the fatigue limit is achieved.
Note: Different from an assessment of the static strength the presented calculation for combined types of stress applies in the case of proportional stresses only. For non-proportional stresses o l and cr2 the rules of superposition in Chapter 4.6 are not applicable and the procedure proposed in Chapter 5.10 is to be applied instead.
........
245 6 Examples
6.4 Welded notched component
6.4 Welded notched component
Safety requirement: According to the statements "with moderate consequences of failure; regular inspections".
Key words: Milled steel, welded notched component, assessment of the static strength, assessment of the variable amplitude fatigue strength, type of overloading F2, class of utilization, nominal stress, structural stress, effective notch stress.
Task: Assessment of the static strength and assessment
'--~-64---EN""'.""'do--'q
Given:
Stresses: Cyclic variable amplitude axial stresses, Figure 6.4.1. Characteristic nominal stresses, determined elementary: Szd = Sm,zd ± Sa,zd = 150 MPa ± 75 MPa . Characteristic structural stresses at the edge of the weld seam, determined by finite element analysis without observing the weld seam, (stress concentration factor Kt,cr = O'max / Szd = 2,5), 0'
= O'm ± O'a = 375 MPa ± 187,5 MPa.
Characteristic effective notch stresses at the edge of the weld determined by finite element analysis in considering the weld seam with an effective notch radius r = 1 mm, (stress concentration factor Kt,crK = O'K,max / Szd = 5,6), O'K = O'K,m ± O'K,a = 840 MPa ± 420 MPa . The respective characteristic stress amplitudes refer to the largest amplitude of the stress spectrum which co-responds to the class of utilization B2. S1:d
F
of the variable amplitude fatigue strength. Method of calculation: Rod-shaped (ID) component. The assessment of the static strength is to be carried out using nominal stresses and structural stresses (an assessment of the static strength using effective notch stresses is not possible). The assessment of the variable amplitude fatigue strength is to be carried out using nominal stresses, structural stresses and effective notch stresses. The degrees of utilization obtained with nominal stresses, with structural stresses and with effective notch stresses should agree .
ASSESSMENT OF THE COMPONENT STATIC STRENGTH For welded components the assessment of the component static strength is in general to be carried out separately for the toe section (toe of the weld) and for the throat section (root of the weld). In the present case the assessment of the throat section (root of the weld) is sufficient. An equivalent stress is to be computed for the throat section.
Calculation using nominal stresses 1 CHARACTERISTIC STRESSES
F
~ -t--+-t-~!l--t---~-f-,-........-
*1 (1.1.1)
Maximum nominal stress Smax ,ex,zd = 150 MPa
+ 75
MPa = 225 MPa . (1.1.2)
Maximum equivalent nominal stress Smax,ex,wv,zd
=
Sl.,zd
= Smax,ex,zd = 225 MPa .
2 MATERIAL PROPERTIES
6.4..1
Figure 6.4.1 Welded notched component
Tensile strength and yield strength of the plate material R m = 610 MPa, R p = 500 MPa. Tab. 5.1.2 The values apply to the component. The technological size factor is not relevant, Eq. (1.2.15).
Material: StE 500, DIN 17 102. Dimensions: Width of plate B = 375 mm Thickness of plate s = 20 mm, Radius of cut-out r = 100 mm. Weld seam: Full penetration butt weld, as welded, toe angle S; 30 0, tested non-destructively, low residual stresses. Type of overloading: When overloaded in service the stress ratios remain constant (type of overloading F2).
3 DESIGN PARAMETERS Section factor npl,zd = 1 .
(1.3.9)
Weld factor Ow
Ow
=
1,0 .
Tab. 1.3.3
1 Numbers of equations, tables and figures arethose of the guideline.
246
6 Examples
6.4 Welded notched component Design factor
(3.1.2)
Maximum equivalent structural stress
KSK,zd = 1 / (1 . 1 ) = 1 .
(1.3.4)
O'max,ex,wv = O'-L = 562,5 MPa .
2 MATERIAL PROPERTIES
4 COMPONENT STATIC STRENGTH
Tensile strength and yield stress of the plate material
Nominal value of the component static strength resulting from the tensile strength and the design factor: fer = 1 , Tab. 1.2.5 SSK,zd = 1 ·610 MPa / I = 610 MPa . (1.4.1)
Rm = 610 MPa, R,
=
500 MPa .
Tab. 5.1.2
The values apply to the component. The technological size factor is not relevant, Eq. (3.2.15).
3 DESIGN PARAMETERS
5 SAFETY FACTORS
Plastic notch factor Tab. 1.5.1
For moderate consequences of failure, however, there is Jm = 1,75 , jp = 1,3 .
Tab. 1.5.1
For normal temperature there is KT,m = KT,p = I ,
(1.2.26)
Rm = 610 MPa, Rp = 500 MPa
= MAX (1,75; 1,3· :~~)
(3.3.13)
Section factor E = 2,1 . 105 MPa, Tab. 3.3.1 5 % = 0,05 , Rp = 500 MPa, 5 npl." = MIN 2-, 1-.1-0-.-0-,0-5-/5-0-0; 2,5) = 2,5 . 8 ertr =
and in Eq. (1.5.4) the terms 3 and 4 have no relevance:
. jges
Kp,er = Kt,a = 2,5 .
(1.5.4)
(Jr-
(3.3.9) The section factor is limited by the plastic notch factor . Constant KNL (3.3.16)
KNL = I.
= MAX (1,75; 1,59) = 1,75. Due to a high yield stress, Rp / R m > 0,75 , the tensile strength is deciding.
Weld factor Ow Ow = 1,0.
Tab. 3.3.3
Design factor KSK,er = 1 / (2,5 . 1 . 1,0) = 0,400 .
6 ASSESSMENT
(3.3.4)
Maximum nominal stress, see above, Smax,ex,wv,zd = 225 MPa .
4 COMPONENT STATIC STRENGTH
Component static strength, nominal value, see above, SSK,zd
= 610 MPa
.
Local value of the component static strength resulting from the tensile strength and the design factor: fa = 1 , Tab. 3.2.5 O'SK = 1 ·610 MPa / 0,400 = 1525 MPa. (3.4.4)
Degree of utilization aSK d = .z
225 = 0,646 . 610/1,75
(1.6.1)
The degree of utilization of the component static strength is 65 %. The assessment of the static strength is achieved.
5 SAFETY FACTORS Safety factors according to Chapter 3, as before: jges
= 1,75.
6 ASSESSMENT
.Calculation using structural stresses
1 CHARACTERISTIC STRESSES Maximum structural stress at the inside edge of the weld (3.1.1) O'max,ex = 375 MPa + 187,5 MPa = 562,5 MPa .
Maximum structural stress, see above, O'max,ex,wv = 562,5 MPa . Component structural static strength, see above, O'SK = 1525 MPa .
(3.5.4)
247
6 Examples
6.4 Welded notched component Fatigue class
Degree of utilization 562,5 aSK,cr = 1525/1,75 = 0,646 .
(3.6.1)
FAT 40. Thickness factor
The degree of utilization of the component static strength is 65 % (as with nominal stresses). The assessment of the static strength is achieved.
(2.3.33) Surface treatment factor (2.3.28)
Kv = 1. Constant KNL,E ASSESSMENT
KNL,E
OF THE COMPONENT FATIGUE STRENGTH
For welded components the assessment of the fatigue strength is in general to be carried out separately for the toe section (toe of the weld) and for the throat section (root of the weld). Furthermore the assessment of the fatigue strength for welded components is in general to be carried out separately for the base material (with rolling skin) and for the weld. *2. The less favorable case is deciding. Because of the comparatively low fatigue properties of welded components the weld is normally deciding. In this present case an assessment of the fatigue strength for the root of the weld is sufficient.
For the weld section the (largest) amplitude and the mean value of the characteristic nominal stress is
75 MPa, Sm,zd
=
150 MPa .
(2.1.1)
Independent of the type of steel the weld specific fatigue limit for completely reversed nominal stress is
= 92 MPa .
(2.2.3)
3 DESIGN PARAMETERS The fatigue class (FAT) is required to derive the design factor after Eq. (2.3.4). However the welded notched component considered is not contained in Table 5.4.1 and therefore an assessment using nominal stresses is not possible, on principle. Nevertheless in order to demonstrate the respective way of calculation, a (re-calculated) fatigue class is supposed to be approximately valid for the calculation below:
Because ofdifferent slopes the SoN curves ofnon-welded and ofwelded components may overlap for amplitudes exceeding the fatigue limits
2
(2.3.4)
4 COMPONENT FATIGUE STRENGTH Component fatigue limit for completely reversed stress Component fatigue limit for completely reversed nominal stress resulting from the weld specific fatigue limit for normal stress and the design factor: (2.4.1)
SWK,zd = 92 MPa / 5,63 = 16,3 MPa.
M cr
= 0,3 .
Tab. 2.4.1
Calculation for the type of overloading F2. '.
Stress ratio Smin,zd = 150 MPa - 75 MPa = 75 MPa , (2.4.26) Smax,zd = 150 MPa + 75 MPa = 225 MPa , Rzd = 75 /225 = + 0,333 .
2 MATERIAL PROPERTIES
crW,W
KWK,zd = 225 / (40 . 1 . 1 . 1) = 5,63 .
Mean stress sensitivity, low residual stresses,
1 CHARACTERISTIC STRESSES
=
Design factor
Amplitude of the component fatigue limit for the given mean stress
Calculation using nominal stresses
Sa,zd, I
(2.3.32)
1.
=
Because of 0 < Rzd < 0,5 field III applies: Mean stress factor 1+0,3/3 1+0,3 KAK,zd = 0 3 150 = 0,705 . 1+-'-·3 75
(2.4.12)
Residual stress factor, low residual stresses, KE,cr
=
1,54 .
Tab. 2.4.1
Amplitude of the component fatigue limit The nominal stress amplitude of the component fatigue limit results from the mean stress factor, the residual stress factor and the component fatigue limit for completely reversed nominal stress: (2.4.6) SAK,zd = 0,705 . 1,54' 16,3 MPa = 17,7 MPa.
248
6.4 Welded notched component
6 Examples
Component variable amplitude fatigue strength
Calculation using structural stresses
Variable amplitude fatigue strength factor
1 CHARACTERISTIC STRESSES
For the class of utilization B2 fatigue strength factor is
At the inside edge of the weld the (largest) amplitude and the mean value of the characteristic stress is
*3
the variable amplitude
KBK,zd = 6,30 .
Tab. (5.7.1)
The nominal stress amplitude of the component variable amplitude fatigue strength results from the variable amplitude fatigue strength factor and the amplitude of the component fatigue limit: SBK,zd = 6,30' 17,7 MPa= 112 MPa.
(2.4.41)
cra,l = 187,5 MPa, crm = 375 MPa .
(4.1.2)
2 MATERIAL PROPERTIES Independent of the type of steel the weld specific fatigue limit for completely reversed normal stress is (4.2.3)
crw,w = 92 MPa.
5 SAFETY FACTOR
.in
=
1,5 .
(2.5.1)
For moderate consequences of failure and regular inspection there is, however,
.io
=
1,2.
Tab. 2.5.1
For normal temperature there is KT,D = 1,
(2.2.5)
Therefore jges
= 1,2.
(2.5.4)
3 DESIGN PARAMETERS Fatigue class (FAT) The fatigue class (FAT) for structural stresses of a butt weld with a toe angle s 30 0 is required to derive the design factor after Eq. (2.3.4) for the welded notched component. It is FAT 100.
Tab. 5.4.3, Nr.212
Further factors of influence according to Chapter 4 1 as before, ft = 1 , Kv = 1 , KNL,E = 1 , Design factor
6 ASSESSMENT Characteristic (largest) nominal stress amplitude, see above, Sa,zd,l = 75 MPa .
SBK,zd = 112 MPa. Degree of utilization
,
(4.3.4)
4 COMPONENT FATIGUE STRENGTH
Nominal value of the amplitude of the component variable amplitude fatigue strength, see above,
75 aBK,zd = 112 / I 2 = 0,804 .
KWK,cr = 225 / (100 . 1 . 1 . 1) = 2,25 .
Component fatigue limit for completely reversed stress Component fatigue limit for completely reversed structural stress resulting from the weld specific fatigue limit for normal stress and the design factor: crWK = 92 MPa /2,25 = 40,9 MPa .
(2.6.1)
The degree of utilization of the component variable amplitude fatigue strength is 80 %. The assessment of the fatigue strength is achieved.
(4.4.1)
Amplitude of the component fatigue limit for the given mean stress Mean stress factor according to Chapter 4, as before, KAK,cr = 0,705 ,
(4.4.12)
Residual stress factor according to Chapter 4, as before, KE,cr = 1,54 ,
Tab. 4.4.1
Amplitude of the component fatigue limit The structural stress amplitude of the component fatigue limit results from the mean stress factor, the residual stress factor and the component fatigue limit for completely reversed structural stress: (4.4.6) 3 See Chapter 5.7, Footnote 3.
crAK = 0,705 . 1,54' 40,9 MPa = 44,4 MPa .
249
6 Examples
6.4 Welded notched component Component variable amplitude fatigue strength
4 COMPONENT FATIGUE STRENGTH
Variable amplitude fatigue strength factor according to Chapter 4, as before,
Component fatigue limit for completely reversed stress
KBK,zd = 6,30 .
Tab. (5.7.1)
The structural stress amplitude of the component variable amplitude fatigue strength results from the variable amplitude fatigue strength factor and the amplitude of the component fatigue limit: crBK = 6,30 . 44,4 MPa = 280 MPa .
(4.4.41)
5 SAFETY FACTOR
crK,WK = 92 MPa /1 = 92 MPa .
= 1,2 .
(4.5.4)
(4.4.1)
Amplitude of the component fatigue limit for the given mean stress Mean stress factor according to Chapter 4, as before, KAK,cr = 0,705 ,
Safety factor according to Chapter 4, as before, jges
Component fatigue limit for completely reversed effective notch stress resulting from the weld specific fatigue limit for normal stress and the design factor:
(4.4.12)
c(
Residual stress factor according to Chapter 4, as before, KE,cr = 1,54 , Tab. 4.4.1 Amplitude of the component fatigue limit
6 ASSESSMENT Characteristic (largest) structural stress amplitude, see above, cra,l = 187,5 MPa . Structural stress amplitude of the component variable amplitude fatigue strength, see above,
Degree of utilization
=
187,5 280[1,2
Component variable amplitude fatigue strength Variable amplitude fatigue strength factor according to Chapter 4, as before, KBK,cr = 6,30 , Tab. (5.7.1) '7
crBK = 280 MPa .
aBK cr ,
The effective notch stress amplitude of the component fatigue limit results from the mean stress factor, the residual stress factor and the component fatigue limit for completely reversed effective notch stress: (4.4.6) crK,AK = 0,705 . 1,54' 92 MPa = 99,9 MPa .
= 0,804 .
(4.6.1)
The degree of utilization of the component variable amplitude fatigue strength is 80 % (as with nominal stresses). The assessment of the fatigue strength is achieved.
The effective notch stress amplitude of the component variable amplitude fatigue strength results from the variable amplitude fatigue strength factor and the amplitude of the component fatigue limit: crK,BK = 6,30 . 99,9 MPa = 629 MPa.
(4.4.41)
5 SAFETY FACTOR Safety factor according to Chapter 4, as before,
Ca.lcu.lation using effective notch stresses
jges
= 1,2 .
(4.5.4)
1 CHARACTERISTIC STRESSES At the inside edge of the weld the (largest) amplitude and the mean value of the characteristic stress is crK,a,l = 420 MPa, crK,m = 840 MPa .
(4.1.3)
6 ASSESSMENT Characteristic (largest) effective notch stress amplitude, see above, crK,a,l = 420 MPa .
2 MATERIAL PROPERTIES Independently of the type of steel the weld specific fatigue limit for completely reversed normal stress is crw,w
= 92 MPa .
(4.2.3)
crK,BK = 629 MPa . Degree of utilization 420 aBK,crK = 629/12 , = 0,802 .
3 DESIGN PARAMETERS Factors of influence according to Chapter 4, as before, Kv = 1, KNL,E = 1 . Design factor KWK,crK = 1 / (1 . 1) = 1 .
Effective notch stress amplitude of the component variable amplitude fatigue strength, see above,
(4.3.8)
(4.6.1)
The degree of utilization of the component variable amplitude fatigue strength is 80 % (as with structural stresses. The assessment of the fatigue strength is achieved.
250
6.5 Shaft subject to two independent loads
6.5 Cantilever subject to two independent loads
1R65 EN.dog
6 Examples Safety requirements: According to the statements: "with severe consequences of failure; no regular inspections".
Key words: Cantilever, two independent random loads, assessment of the static strength, assessment of the variable amplitude fatigue strength, combined stresses.
Task: Assessment of the static strength and assessment of the variable amplitude fatigue strength for cross section x = O.
Given
Method of calculation: Rod-shaped (lD) component. Assessment to be carried out using nominal stresses.
Stresses: In cross-section x = 0 due to two independent random loads with binomially distributed amplitudes, figure 6.5.1. In load case G1 the load Fy,Gl acts with a total number of cycles N Gl = 105 which are binominally distributed with a spectrum parameter PGl = O. The characteristic mean value and amplitude are: Fm,y,Gl = 1 kN, Fa,y,GI =
± 23,5 kN.
In load case G2 the the load Fy,G2 acts with a total number of cycles NGl = 107 which are binominally distributed with a spectrum parameter PGl = 0,5. The characteristic mean value and amplitude are: Fm,y,G2
ANALYSIS OF THE STATIC LOADING Load case GI Mm,z,GI = 0,10 kNm, Ma,z,GI = ± 2,35 kNm, Mm,x,Gl = 0,10 kNm, Ma,x,GI = ± 2,35 kNm. Load case G2 M m,z,G2 =- 0,15 kNm, Ma,z,GI = ± 2,50 kNm, M m,x,G2 = 0,15 kNm, Ma,x,GI = ± 2,50 kNm. Superimposed loads, case UI
= - 1,5 kN, Fa,y,G2 = ± 25 kN.
The loads Fy,Gl and Fy,G2 vary non-proportionally (as they observe different load spectra!) and they may act both simultaneously or separately. At the cross-section at x = 0 bending moments Mz,Gl , M z,G2 and torsion moments Mx,Gl, M x,G2 with their mean values (index m) and amplitudes (index a) result from Fy,Gl , Fy,G2 and the lever arms I = 0,1 m.
The highest bending moment (absolute value) results if the amplitudes Fa,y,Gl and Fa,y,G2 act unidirectionally (in direction of the resulting mean value Fm,y,OI = Fm,y,Gl + Fm,y,G2, that is downwards in Figure 5.6.1): Mm,z,OI = (0,1- 0,15) kNm =- 0,05 kNm, Ma,z,OI = (- 2,35- 2,5) kNm =- 4,85 kNm, Mmin,z,OI = (- 0,05- 4,85) kNm =- 4,90 kNm. The related torsion moment is: Mm,x,OI = (0,1 + 0,15) kNm = 0,25 kNm, Ma,x,OI = (- 2,35 + 2,5) kNm = 0,15 kNm, Mmax,x,OI = (0,25 + 0,15) kNm = 0,40 kNm.
y z
Superimposed loads, case U2 The highest torsion moment results if the amplitudes Fa,y,Gl, and Fa,y,G2 act opposingly (in direction of the resulting mean value Mm,y,02 = Mm,y,GI + Mm,y,G2 as shown in Figure 6.5.1): Bile b65
Figure 6.5.1 Cantilever subject to two independent loads.
Mm,x,02 = (0,1 + 0,15) kNm = 0,25 kNm, Ma,x,02 = (+ 2,35 + 2,5) kNm = 4,85 kNm, M max,x,02 = (0,25 + 4,85) kNm = 5,10 kNm. The related bending moment is:
Material: St 60, DIN EN 10 025. Dimensions: Diameter of cantilever at x = 0: d = 57,5 mm, Length oflever arms: 1= 0,1 m. Surface: Average roughness R, = 25 um. Type of overloading: When overloaded in the service the stress ratios remain constant (type of overloading F2).
Mm,z,02 = (0,1- 0,15) kNm =- 0,05 kNm, Ma,z,02 = (2,35- 2,5) kNm =- 0,15 kNm, Mmin,z,02 = (- 0,05- 0,15) kNm =- 0,20 kNm.
251
6.5 Shaft subject to two independent loads ASSESSMENT OF THE STATIC STRENGTH
6 Examples Kp,b = 1,70, Tab. 1.3.2 npl,b = MIN (J1050/321 ; 1,70) = 1,70. 1.3.9)
The assessment of the static strength is to be carried out for each of the superimposed loads, cases VI and V2. The highest degree of utilization is determining.
Design factor
1 CHARACTERISTIC STRESSES
Design factor for torsion
Superimposed loads, case VI
Section factor
The maximum bending stress and the related torsional stress at the reference section with d = 57,5 mm is: Smax,b,-ol
=
32
IMmin,Z,OII I pd3
= 32 . 4,90 kNm I [TC . (57,5 mm)3] = 263 MPa, T max t -01 = 16 M max x -01 I pd3 = 16 : 0,40 kNm I (57,5 mm)3 ] = 10,7 MPa.
[n' .'
KSK,b = 1 I 1,70 = 0,588.
(1.3.1)
Rp,max = 1050 MPa, Tab. 1.3.1 Rp = 321 MPa, Kp,t = 1,33, Tab. 1.3.2 npl,t = MIN (J1050/321 ; 1,33 =1,33. (1.3.9) Design factor KSK,t
= 1 11,33 =
0,752.
(1.3.1)
4 COMPONENT STATIC STRENGTH Superimposed loads, case V2 The maximum torsional stress and the related bending stress at the reference section with d = 57,5 mm is: Smax,b,-02 = 32
IM min,z,0 2I I pd3
= 32· 0,20 kNm I [TC . (57,5 mmj-'] = 10,7 MPa. Tmax t -02 = 16 Mmax x -02 I pd 3 = 16: 5,10 kNm I .'(57,5 mm)3] = 137 MPa,
[n'
2 MATERIAL PROPERTIES
Tab. 5.1.1
Technological size factor ad,m = 0,15, ad,p = 0,3, deff,N = 40 mm, deff = d = 57,5 mm, Kd,m = 0,980, Kd,p = 0,956.
1, Tab. 1.2.5 SSK,b = 1 . 579 MPa I 0,588 = 985 MPa. (1.4.1)
Nominal value of the component static torsional strength resulting from the tensile strength and the design factor f't = 0,577, Tab. 1.2.5, (1.4.1) TSK,t = 0,577 ·579 MPa I 0,752 = 445 MPa.
For severe consequences of failure and a low probability for the simultaneous occurrence of the peak values of the stress spectra there is
jm = 1,8, jp = 1,35. Tab. 1.2.1 Tab. 1.2.3 (1.2.9)
Anisotropy factor K A = 1.
r, =
5 SAFETY FACTORS
Tensile strength and yield strength according to standard Rm,N = 590 MPa, Rp,N = 335 MPa.
Nominal value of the component static bending strength resulting from the tensile strength and the design factor
For normal temperature there is KT,m = KT,p = 1.
Tensile strength and yield strength of the component R m = 0,980 . 1 ·590 MPa = 579 MPa, (1.2.1)
R, = 0,956 . 1 . 335 MPa = 321 MPa.
(1.2.26)
and in Eq. (1.5.4) the terms 3 and 4 have no relevance: Rm = 579 MPa,
(1.2.17)
Tab. 1.5.1
Re = 321 MPa,
= MAX (1,8;1,35. 579) 321 = MAX (1,8; 2,44) = 2,44.
jges
(1.5.4)
The second term, the yield strength R, , is determining.
3 DESIGN PARAMETERS
6 ASSESSMENT
Design factor for bending
Superimposed loads, case VI
Section factor
Maximum stress in bending and related stress torsion, see above,
Rp,max = 1050 MPa, Rp = 321 MPa,
Tab. 1.3.1
Smax,b,-ol = 263 MPa, Tmax,t,-ol = 10,7 MPa.
ill
252
6.5 Shaft subject to two independent loads Component static strength, in bending and in torsion, see above, SSK,b = 985 MPa, TSK,I = 445 MPa. Degrees of utilization
6 Examples ASSESSMENT OF THE FATIGUE STRENGTH
The assessment of the fatigue strength is to be carried out according to Chapter 5.10: For the two nonproportional load cases Gl and G2 the individual degrees of utilization are to be determined and added.
Individual types of stress, bending or torsion, 263 a b= = 0651 SK, 985/ 2,44 ' , 10,7 a K = = 0059. S ,I 445/2 ,44 '
(1.6.1)
Load case Gl Bending and torsion moments (step 1 of the spectrum):
Combined types of stress f't = 0,577, q = 0, s = aSK,b = 0,651, t = aSK,1 = 0,059, aGH =
Tab. 1.2.5 (1.6.7) (1.6.6)
~0,6512 +0,059 2 =
0,654,
(1.6.5) (1.6.4)
aSK,sv = aSK,OI = 0,654.
Superimposed loads, case
Nominal stresses: Sm,b,1 = Sm,b,GI,1 = 32 Mm,z,Gl / pd 3 (2.1.1) = 32· 0,10 kNm/[n . (57,5 mm)3 ] = 5,4 MPa, Sa,b,1 = Sa,b,GI,1 = 32 Ma,z, GI / pd3 = 32 . 2,35 kNm / [n . (57,5 mm)3 ] = 126 MPa,
i6 .
ih
Smax,b,02 = 10,7 MPa, T max,I,U2 = 137 MPa. Component static strength in bending and in torsion, see above,
Ta,I,1 = Ta,I,GI,1 = 16 Ma,x, GI / pd 3 = 16 . 2,35 kNm / [n . (57,5 mm)3 ] = 63 MPa.
Load case G2 Bending and torsion moments: (step 1 of the spectrum): M m,z,G2 =- 0,15 kNm, Ma,z,Gl = ± 2,50 kNm, M m,x,G2 = 0,15 kNm, Ma,x,Gl = ± 2,50 kNm.
SSK,b = 985 MPa, TSK,I = 445 MPa. Degrees of utilization
Nominal stresses:
Individual types of stress, bending or torsion, = 0027 10,7 aSK,b = 985/2,44 ' ,
Sm,b,1 = Sm,b,G2,1 = 32IMm,z,Gl/ / pd3 (1.6.1)
=
(2.1.1)
32· 0,15 kNrn/[n . (57,5 mm)3 ] = 8,0 MPa,
Sa,b,1 = Sa,b,G2,1 = 32 Ma,z, GI / pd3 = 32· 2,50 kNm / [n . (57,5 mm)3 ] = 134 MPa,
137 aSK,1 = 445/2,44 = 0,751.
Tm,I,1 = T m,I,G2,1 = 16 Mm,x, GI / pd 3 = 16· 0,15 kNm / [n . (57,5 mm)3 ] = 4,0 MPa,
Combined stress
~0,0272 +0,751 2 =
Mm,z,GI = 0,10 kNm, Ma,z,GI = ± 2,35 kNm, Mm,x,Gl = 0,10 kNm, Ma,x,GI = ± 2,35 kNm.
T m I I = T m I GI I = 16 M m x GI / pd 3 = 0,10 kN~ / [n . (57,5 ~)3 ] = 2,7 MPa,
Maximum stress in torsion and related bending stress, see above,
aGH =
1 CHARACTERISTIC STRESSES
0,751,
aSK,sv = aSK,02 = 0,751.
(1.6.5) (1.6.4)
Tm,I,1 = Ta,I,G2,1 = 16 Ma,x, Gl / pd 3 = 16· 2,50 kNm / [n . (57,5 mm)3 ] = 67 MPa. The negative sign of the mean stress for bending is not important here.
Definite degree of utilization aSK = MAX (aSK,0 I, aSK,02 )
2 MATERIAL PROPERTIES
= MAX (0,654; 0,751) = 0,751.
Material fatigue limits for fully reversed normal stress, crW,zd, and shear stress, LW,S :
The degree of utilization of the component static strength is 75 %. The assessment of the static strength is achieved.
Rm = 579 MPa, fw,« = 0,45, crW,zd = 0,45 . 579 MPa = 261 MPa, fw,'t = 0,577, LW,S = 0,577 . 261 MPa = 151 MPa.
Tab. 2.2.1 (2.2.1) Tab. 2.2.1 (2.2.1)
253
6.5 Shaft subject to two independent loads
6 Examples
3 DESIGN PARAMETERS
4 COMPONENT FATIGUE STRENGTH
Design factor for bendin..g
Component fatigue limit for completely reversed bending and torsional stress:
Fatigue notch factor
Component fatigue limit for completely reversed bending stress resulting from the material fatigue limit for normal stress and the design factor for bending:
Stress concentration factor of a plain shaft *1, Kt,b = 1.
SWK,b = 260 MPa /1,145 = 228 MPa.
KcKf ratios, no notch radius, Gcr(r) = 0, ncr (r) = 1, d = 57,5 mm, Gcr(d) = 2/ d = 0,0357 mm -I , ncr (d) = 1,021.
(2.3.17) (2.3.13)
Fatigue notch factor
Component fatigue limit for completely reversed torsional stress resulting from the material fatigue limit for shear stress and the design factor for torsion: TWK,t = 150 MPa /1,064 = 141 MPa.
(2.4.1)
Load case Gl
Kf,b = 1 / (1 . 1,021) = 0,979.
(2.3.10)
Roughness factor Rill = 579 MPa, Rz = 25 urn, aR,cr = 0,22, Rm,N,min = 400 MPa, KR,cr = 0,858.
(2.4.1)
Component fatigue limit for the given mean stress for bending and torsion The calculation for type of overloading F2.
Tab. 2.3.4 (2.3.26)
For no surface treatment is KV = 1. For materials except GG KNL,E = 1. Therefore: Design factor KWK,b = 0,979 + 1/0,858 - 1 = 1,145
(2.3.1)
Mean stress sensitivity R m = 579 MPa, aM = 0,00035, bM = - 0,1, M, = 0,103, fw,~ = 0,577 M~ = 0,577 . 0,103 = 0,059.
Tab. 2.4.2 (2.4.34) Tab. 2.2.1 (2.4.34)
Equivalent mean stress Sm = Sm,b,GI = 5,4 MPa, Tm,t,GI = 2,7 MPa, fw,~ = 0,577, q = 0,
Design factor for torsion Fatigue notch factor Stress concentration factor of a plain shaft *1,
KcKf ratios, no notch radius, G~ (r) = 0, v~(r) = 1, d = 56 mm, G~ (d) = 2/ d = 0,0357 mm -I , n, (d) = 1,026.
Tab. 2.2.1 (2.4.29)
Sm,v,l= Sm,v,GH =~5,42+3.2,72 MPa = = 7,1 MPa, (2.24.29) (2.4.28) Tm,v,1 = 0,577 . 7,1 MPa = 4,1 MPa. (2.4.30)
Kt,t = 1.
Stress ratio for bending (2.3.17) (2.3.13)
Fatigue notch factor Kf,t = 1 / (1 . 1,026) = 0,975.
(2.3.10)
Roughness factor R m = 579 MPa, R z = 25 urn, fw,~ = 0,577 aR,cr = 0,22, Rm,N,min = 400 MPa, KR,~ = 0,918.
(2.4.31)
r., =
Rs,v = (Sm,v,1 - Sa,b,Gl,I) / (Sm,v,1 + Sa,b,Gl,l) =(7,1 126) / (7,1 + 126) = - 0,89. Mean stress factor for bending; because of - 00 :;;; Rs,v :;;; field II applies:
°
KAK,b = 1/(1 + Sm,v,1 / Sa,b,GI,I) = 1/(1 + 7,1 / 126) = 0,947. Tab. 2.2.1 Tab. 2.3.4 (2.3.26)
(2.4.10)
Stress ratio for torsion RT,v = (Tm,v,1 - Ta,t,Gl,I) /(Tm,v,1 + Ta,t,GI,I) =(4,1 63) / (4,1 + 63) = 0,88.
For no surface treatment is KV = 1. Therefore Mean stress factor for torsion; field II applies: because of -1:;;; RT,v :;;;
°
Design factor KWK,t = 0,974 + 1 /0,918 - 1 = 1,064
(2.3.1)
I This isa simplifying assumption made for the present example that would not bevalid for a real component, however!
KAK,t = 1 / (1 + Tm,v,1 / Ta,t,Gl,1 ) = 1/(1 + 4,1 /63) = 0,939.
(2.4.10)
Residual stress factor for normal stress and for shear stress KE,cr = KE,~ = 1. (2.4.5)
254
6.5 Shaft subject to two independent loads Amplitude of the component fatigue limit The amplitude of the component fatigue limit results from the mean stress factor, residual stress factor and the component fatigue limit for completely reversed bending or torsional stress: SAK,b = 0,947 . 1 . 228 MPa = 216 MPa, (2.4.6) TAK,t = 0,939 . 1 . 141 MPa = 132 MPa. Component variable amplitude fatigue strength for bending and torsion Variable amplitude fatigue strength factors for bending and torsion KBK,b = 3,84, KBK,t = 2,88.
(2.4.51)
The variable amplitude fatigue strength factors follows from the elementary version of Miner's rule and a critical damage sum D M = 0,3 , Chapter 2.4.3.1. The component variable amplitude fatigue strength results from the variable amplitude fatigue strength factor and from the mean stress dependent amplitude of the component fatigue limit for bending or for torsion: SBK b = 3,84' 216 MPa = 829 MPa, (2.4.41) TB~,t = 2,88' 132 MPa = 380 MPa.
SBK,b ::;; 0,75 . R, . ~,b = 0,75' 321 MPa '1,70 = 409 MPa, (2.4.42) TBK,t ::;; 0,75' ft . R, . ~,t = 0,75' 0,577 . 321 MPa '1,33 = 185 MPa. Relevant component variable amplitude fatigue strength SBK b Gl = MIN (829,409) MPa = 409 MPa, TB~,b',Gl = MIN (380, 185) MPa = 185 MPa.
Load case G2
SBK b = 1,00' 211 MPa = 211 MPa, TB~,t = 1,04' 129 MPa = 134 MPa.
(2.4.41)
5 SAFETY FACTOR For severe consequences of failure and no regular inspections
.io = 1,5.
(2.5.1)
For normal temperature is KT,D = 1,
(2.2.4)
= 1,5.
(2.5.4)
jges
6 ASSESSMENT Load case Gl Characteristic stress amplitudes for bending and for torsion, see above, Sa,b,l = Sa,b,Gl,l = 126 MPa, Ta,t,l = Ta,t,Gl,l = 63 MPa.
SBK,b,Gl = 409 MPa, TBK,t,Gl = 185 MPa. Degrees of utilization Individual types of stress, bending or torsion, 126 409/1,5 = 0462 ' , 63 = 0511. a K = B,t 185/15 ' ,
aBK,b
(2.6.1)
=
Combined types of stress
Component fatigue limit for the given mean stress for bending and torsion An analog calculation results in
(2.4.6)
Component variable amplitude fatigue strength for bending and torsion Variable amplitude fatigue strength factors for bending and torsion KBK,b = 1,00, KBK,t = 1,04.
factor and from the mean stress dependent amplitude of the component fatigue limit for bending or for torsion:
Amplitude of the component variable amplitude fatigue strength for bending and for torsion, see above,
Limitation of the amplitudes
SAK,b = 211 MPa, TAK,t = 129 MPa.
6 Examples
(2.4.51)
The variable amplitude fatigue strength factor follows from the elementary version of Miner's rule and a critical damage sum DM = 0,3 , Chapter 2.4.3.1. The component variable amplitude fatigue strength results from the variable amplitude fatigue strength
fw,"t = 0,577, q = 0, Sa = aBK,b = 0,462, ta = aBK,t = 0,511,
Tab. 2.2.1 (2.6.7) (2.6.6)
aGH = ~0,4622 +0,511 2 = 0,689, aBK,Sv = aBK,Gl = 0,689.
(2.6.5) (2.6.4)
Load case G2 Characteristic stress amplitudes for bending and for torsion, see above, Sa,b,l = Sa,b,G2,l = 134 MPa, Ta,t,l = Ta,t,G2,l = 67 MPa. Amplitude of the component variable amplitude fatigue strength for bending and for torsion, see above, SBK,b,G2 = 211 MPa, TBK,t,G2 = 134 MPa.
255
6.5 Shaft subject to two independent loads Degrees of utilization Individual types of stress, bending or torsion, 134
aBK,b
= 211/1,5
a
=
BK,t
= 0953
'
67
=
134/1,5
,
(2.6.1)
0750. '
Combined types of stress
fw,'t = 0,577, q
Tab. 2.2.1
= 0,
Sa
= aBK,b = 0,953,
ta
= aBK,t = 0,750,
= ~0,9532 +0,750 2 = 1,213, aBK,Sv = aBK,G2 = 1,213.
a GH
(2.6.7) (2.6.6) (2.6.5) (2.6.4)
Total degree of utilization from the combined effect of load case G1 and G2 aBK
=
aBK,GI
= 0,689 +
+
aBK,G2
1,213
= 1,902
(5.10.3)
The cyclical degree of utilization of the component is 190 %. The assessment of the variable amplitude fatigue strength is not achieved.
6 Examples
6.6 Component made of a wrought aluminium alloy
256
6 Examples
6.6 Component made of a wrought aluminium alloy *1 lRi66.docl
ASSESSMENT
Key words: Wrought aluminium alloy, assessment of the static strength, assessment of the fatigue strength for finite life, local elastic stresses, type of overloading F2, combined types of stress, (JI and (J2 .
1 CHARACTERISTIC STRESSES Maximum stresses (3 . 1.6) (JI ,max,ex = 238 MPa, (J2, max ,ex = 58,4 MPa .
Given:
2 MATERIAL PROPERTIES
Stresses: Proportional, constant amplitude, local elastic stresses in the directions 1 (longitudinal direction) and 2 (circumferential direction) at the reference point of a block-shaped (3D) component, Figure 6.6.1,
AlZn4,5Mgl T651, Tab. 5.1.22, pp. 151. R, = 350 MPa, R, = 280 MPa, As = 9 %.
(JI = (JI,m± (JI,a = 119 MPa ± 119 MPa, (J2 = (J2,m ± (J2,a = 29,2 MPa ± 29,2 MPa , (J3 = O. Stress amplitudes at the neighbouring point in a distance s = 1,254 mm below the surface (JI,a = ± 54,1 MPa,
(J2,a = ± 22,5 MPa . 4
Number of cycles: N = 5 . 10 (R, = 0).
OF THE COMPONENT STATIC STRENGTH
*2
Tensile strength and yield strength for the standard dimension Rm,N = 350 MPa , Tab. 5.1.14 Rp,N = 280 MPa . No technological size factor Kd,m = 1.
(3.2.5)
Anisotropy factor KA = 1.
(3.2.18)
Tensile strength and yield strength for the component (3.2.1) R m = 1 . 1 . 300 MPa = 350 MPa, Rp = 1 . 1 . 280 MPa = 280 MPa .
3 DESIGN PARAMETERS Section factor E = 0,70 'lOs MPa ,
Tab. 3.3.1
£crtr=2%, = 280 MPa, ~,cr = 1,7, ~
Figure 6.6.1 Aluminum component with pivot.
0,70.10 5.0,02 npl,crl = MIN ( 280 ; 1,7)
Material: AIZn4,5Mgl , T651, aging wrought aluminium alloy Temperature: T = 50°C (normal temperature). Dimensions: Effective diameter at the reference point: defT = 18 mm. Surface: average roughness
R..
= 10 urn,
Type of overloading: When overloaded in service the stress ratios remain constant (Type of overloading F2). Safety requirements: according to the statements: "with moderate consequences of failure; no regular inspection" .
= MIN (2,24; 1,7) = 1,7 .
(3.3.9)
For n pl,cr2 the same value is assumed: n pl,cr2 = 1,7 . Design factor for directions 1 and 2 KSK,crl = KSK,a2 = 1 / 1,7 = 0,588
(3.3.3)
4 COMPONENT STATIC STRENGTH The component strength results from the tensile strength and the design factor. For directions 1 and 2: fa = 1, Tab. 3.2.5, (3.4.3) (JI,SK = (J2,SK = 1 . 350 MPa / 0,588 = 595 MPa .
Task: Assessment of the static strength and of the constant amplitude fatigue strength for finite life (N = 5 . 104 cycles).
5 SAFETY FACTORS
Method of calculation: Block-shaped (3D) component. Assessment using local elastic stresses, Chapter 3 and 4. For T = 50°C temperature factors do not need to be considered (KT,m ~ ... = 1).
For moderate consequences of failure and a high probability of occurrence of the maximum (constant amplitude) stress there is jm = 1,75, jp = 1,30. Tab. 3.5.1
I
Results of computation obtained by the pc- program "RIFESTPLUS".
2
The numbers of equations, tables and figures are those of the
guideline.
6.6 Component made of a wrought aluminium alloy For an elongation As < 12,5 % (here A s = 9 %) the safety factors are to be increased by ~j:
~j = 0,5 --h/50 = 0,08, jm = 1,83, jp = 1,38 .
(3.5.2) Tab. 3.5.2
For normal temperature the terms 3 and 4 of Eq. (3.5.4) are not relevant.
jg~s =
MAX (1,83: 1,38. 350) 280
6 Examples ASSESSMENT OF THE COMPONENT FATIGUE STRENGTH
1 CHARACTERISTIC STRESSES Proportional constant amplitude stresses O"I,a,1 = 119 MPa, 0"1,01,1 = 119 MPa, (4.1.1) 0"2,a,1 = 29,2 MPa, 0"2,01,1 = 29,2 MPa .
(3.5.4)
2 MATERIAL PROPERTIES
= MAX (1,83, 1,73) = 1,83.
Material fatigue limit for completely reversed normal stress, O"W,zd, for N = 106
The tensile strength R m is deciding.
Rm = 350 MPa, Tab. 4.2.1 fw,a = 0,30 , (4.2.1) O"W,zd = 0,30 . 350 MPa = 105 MPa .
6 ASSESSMENT Maximum stresses, see above, O"I,max,ex = 238 MPa,
257
0"2,max,ex = 58,4 MPa .
Component strength values, see above,
3 DESIGN PARAMETERS K,-Krratios
O"I,SK = 0"2,SK = 595 MPa.
=
1 .(1- 54,1) = 0,435 mm -I, 1,254mm 119
Ga 2 =
1 .(1- 22,5) = 0,183 mm -I, 1,254mm 29,2
Gal
Degrees of utilization Individual types of stress, direction 1 and 2 aSK I =
238 = 0,732, 595/1,83
aSK,a2 =
59~~':'83
.o
(3.6.17)
As = 9 % , q = 0,5, before (3.6.23), Tab. 3.6.1 (3.6.22) Sl = aSK c l = 0,732, S2 = aSK:a2 = 0,179 ,
L((0,732-0,179)2 +0,179 2 +0,732 2 )
2 = 0,660, aNH = MAX (0,732, 0,179) = 0,732,
aSK,sv = 0,5 . 0,732
aa
Tab. 4.3.2 (4.3.14)
= 0,05, b G = 850 , nal = 1,228, na2 = 1,148.
Roughness factor R z = 10 um, aR,a = 0,22, Rm,N,min = 133 MPa, KR,a = 0,841 .
= 0,179 .
Combined types of stress
aGH =
(4.3.16)
(3.6.21) (3.6.20)
+ (1 - 0,5) . 0,660 = 0,696.
The degree of utilization of the component static strength is 70 %. The assessment of the static strength is achieved. This result of the assessment is only exact, however, if the stresses 0"1 and 0"2 are proportional, even if, on principle, it is supposed for the assessment of the component static strength that different types of stress observe their extreme value simultaneously, that is, as if they were proportional. But with non-proportional stresses 0"1 and 0"2 it would be of importance that the individual degree of utilization aSK o l = 0,732 is higher than the degree of utilization 'of the combined types of stress aSK,sv = 0,696 derived.
Surface treatment factor Kv = 1, Ks = 1 .
and
Tab. 4.3.4 (4.3.26)
coating factor (4.3.28) (4.3.30)
A universal value forKf after table 4.3.1 would be: Kf = 2. Tab. 4.3.1 Instead a value forKf derived from the stress concentration factor of a substitute structure after Chapter 5.12 shall be applied here: G a I = 0,435 mm -I , r=2/G a l =4,6mm, derr = b = 18 mm, K = 1OO,066-0,36.1g(rlb) = 1 904 t
"
(5.12.2) (5 12 1)
.
•
nal = 1,228 , I(f,1 = 1,904 I 1,228= 1,550". Accordingly K f ,2 = 1,214. Design factors KWK,al
= 1,;28 {1+
(4.3.3)
1,5~0 {0,~41 -1)) = 0,913 ,
KWK a2 = _1_.(1+!.(_1__ 1)) = 1,007. , 1,056 1 0,906
6.6 Component made of a wrought aluminium alloy
258
6 Examples
.io
4 COMPONENT FATIGUE STRENGTH Component fatigue limit for completely reversed stress The component fatigue limit for completely reversed normal stress results from the material fatigue limit for completely reversed normal stress and the design factors: (4.4.3)
O"I.WK = 105 MPa / 0,913 = 115 MPa , O"2.',VK = 105 MPa / 1,007 = 104 MPa .
Amplitude of the component fatigue limit for the given ""- .
mean stress
Mean stress sensitivity
Tab. 4.5.2
1,3.
=
For an elongation As < 12,5 % (here As safety factors are to be increased by ~j:
~j = 0,5 -.Jg/50 = 0,08;
.io
9 %) the
.
(4.5.2)
1,38.
=
Therefore the total safety factor is
= 1,38.
jges
(4.5.4)
6 ASSESSMENT Characteristic stress amplitudes, see above O"I,a,1 = 119 MPa, 0"2,a,1 = 29,2 MPa.
aM = 1,0, bM = - 0,04, M cr = 0,31 .
Tab. 4.4.2 (4.4.34)
Amplitudes of the component variable amplitude fatigue strength, see above,
According to Chapter 4.4.2.2 no equivalent mean stress is to be applied; the individual mean stresses are to be applied instead. The type of overloading is F2.
Degrees of utilization
Mean stress factor for the direction 1 and 2.
Individual types of stress, direction 1 and 2
Stress ratios Rcrl = R cr 2 = O.
O"I,BK = 160 MPa, 0"2,BK = 145 MPa .
aBK
.c
KAK,crl = KAK,cr2 = 1+ 0,31 = 0,763 . Residual stress factor KE,cr = KE;t = 1 .
(4.4.10)
(4.4.5)
Amplitude of the component fatigue limit The amplitude of the component fatigue limit results from the mean stress factor, the residual stress factor and the component fatigue limit for completely reversed stress, O"I,AK = 0,763 . 1 . 115 MPa = 88 MPa , 0"2,AK = 0,763 . 1 . 104 MPa = 80 MPa .
(4.4.8)
a
I
=
119 160 11,38
= 1,027 ,
(4.6.17)
229,2 = 0,277 . BK.o - 145/1,38
Combined types of stress As = 9 % , q = 0,5, SI,a = aBK,crl = 1,027 , S2,a = aBK cr2 = 0,277 ,
aGH
Tab. 4.6.1 (4.6.22)
= L((1,027-0,277)Z +0,277 2 +1,027 2 )
2 = 0,921 , (4.6.21) aNH = MAX (1,027, 0,277) = 1,027 , (4.6.20) aBK,crv = 0,5 . 1,027 + (1 - 0,5) . 0,921 = 0,974
Component variable amplitude fatigue strength Variable amplitude fatigue strength factor For the constant amplitude stresses of concern with a required number of cycles N = N = 5 . 104, where N < No,cr= 106 , and a component constant amplitude S-N curve model I or model II the variable amplitude fatigue strength factor is (4.4.48) KBK,crl = KBK,cr2 = (10 6 / 5 ' 104 )
(I/S)
= 1,821.
The amplitude of the component variable amplitude fatigue strength results from the variable amplitude fatigue strength factor and the amplitude of the component fatigue limit: O"I,BK = 1,82' 88 MPa = 160 MPa , 0"2,BK = 1,82' 80 MPa = 145 MPa .
(4.4.45)
5 SAFETY FACTORS For moderate consequences of failure and no regular inspection there is
The degree of utilization of the component variable amplitude fatigue strength is 97 %. The assessment of the fatigue strength is achieved. This result of the assessment of the fatigue strength is valid only because the stresses 0"1 and 0"2 are proportional and definitely occur in parallel. Obviously the degree of utilization of the individual type of stress 0"1 is higher than the degree of utilization of the combined types of stress 0"1 and 0"2: aBK,crl = 1,027 is higher than aBK,sv = 0,974.
In the case that - different from the present example 0"1 and 0"2 could occur independent of each other, the higher individual degree of utilization aBK o l of 103 % would be deciding, a result that supports the demand of always considering the degrees of the individual types of stress as well.
259
7 Symbols and basic formulas lRi7 EN.dog
7.1 Abbreviations B
F FAT GS
GG GGG GT GTS GTW
class of utilization (B-7, ... , B 10) type of overloading (FI, F2, F3, F4) fatigue class (for welded components) cast steel and heat treatable cast steel, for general purposes cast iron with lamellar graphite (grey cast iron) nodular cast iron malleable cast iron GT, black heart (non-decarburized) GT, white heart (decarburized)
7 Symbols and basic formulas
7.2 Indices a b eff ex ges i j
amplitude bending effective extreme total number of a step of a stepped stress spectrum number of the last step of a stepped stress spectrum (smallest amplitude), number of all steps mean value maximum value minimum value shear torsion equivalent value equivalent value (assessment of the static strength of welded components) axial direction of rod-shaped (ID) components first direction of shell-shaped (2D) components first lateral direction of rod-shaped' (ID) components second direction of shell-shaped (2D) components second lateral direction of rod-shaped (ID) components tension, compression, or tension-compression component fatigue limit component variable amplitude fatigue strength local stress (effective notch stress) component static strength welded component fatigue limit for completely reversed stress number of the first step of a stepped stress spectrum (largest amplitude) index of the first principle stress index of the second principle stress index of the third principle stress, normal to the surface (pointing into the interior of the component) normal stress shear stress
m max
mm s v
wv
x
x I Applies to Rm . For Rp the m index is to be replaced by p.
y
2 Applies to normal stress. For shear stress c is to be replaced by "to
y
3 Applies to nominal stress. For local stress (also structural stress or effective notch stress) S is to be replaced by cr.
z
4 These symbols apply to nominal stress (tension-compression). Szd , Sa,zd , Sm,zd , Smax,zd , Smin,zd , SAK,zd , SBK,zd ' SSK,zd , SWK,zd , KAK,zd , KBK,zd , KSK,zd , KWK,zd , Rzd , Yzd . ... to other kinds of nominal stress Sb ' T s , T t ' Sx , Sy , T, .... ... to local stress 0',
(!a , ... , oAK ,... , KAK,cr ,... , Rcr , Vcr ,
t , ax ,
cry , 0'1 ,0'2,0'3
zd
AK BK K SK W WK
5 Swv,zd applies to nominal stress (tension-compression). Also Swv,b ' Twv,s, Twv,t, Swv.x . Swv,y, T wv . For local stress crwv , "tw v , crwv,x ' crwv,y . 6 Kt,zd is applies to nominal stress (tension-compression). Also Kt,b ' Kt,s , Kt,t .
I 2 3
Similarly for Kf . 7 aBK,zd applies to nominal stress (tension-compression). Also aBl(,b , aBK,s , aBK.,t , aBK,x , aBK.,y·
cr 1:
For local stress aBK,cr , aBK.,"t , aBK,crx , aBK.,cry , aBK,crl ' aBK,cr2 , aBK.,cr3 . Similarly for aSK and asK.,wv.
7.3 Lower case characters
8 npl,b applies to nominal stress in bending, npl,t in torsion. For local stress npl,cr and npl,"t . 9 Kp,b applies to nominal stress in bending, Kp,t in torsion. For local stress Kp,cr and Kpl,"t. 10 Fatigue strength value. II Type of loading tension or compression zd, also types of loading, b, s, t, X, y.
aBK,Sv
au, . aM,
aR,cr
.
Constant for ~ *1 degree of utilization (for the assessment of the fatigue strength) aBK for combined stresses *3 Constants for ncr '" Constants for M cr Constant for KR,cr
*7
260 aSK.wv.zd aSK for welded components (equivalent stress for a particular type of loading) *7 degree of utilization aSK.zd (for the assessment of the static strength) *7 aSK for combined stresses *3 aSK.Sv aSK,Swv aSK for welded components (for the combined types of loading) *3 constant for KT,m *1 aT,m constant for KT,D aT,D aTt.m ... constants for KTt,m *1 width b diameter d effective diameter of the semi-finished deff product or the raw casting deff for R m *' deff,m deff m for R m N *1 deff,N,m dia~eter of a' longitudinal hole dL diameter of the reference component dp (the test specimen) diameter of a transverse hole diameter of the material test specimen, do = 7,5 rom thickness factor (for welded components) fatigue limit factor for completely reversed stress shear fatigue strength factor compression strength factor shear strength factor number of cycles in step i of the stress spectrum safety factor J total safety factor Jges j for R m and R m,T jm j for Rm,Tt jmt j for ~ and ~,T ~p j for ~,Tt ~pt j for fJw and/or T:w JD partial safety factor for allowable defects in JF castings slope of the constant amplitude S-N curve for kD,cr N > ND,cr *2 slope of the constant amplitude S-N curve for N::; ND,cr *2 n,1 number of cycles in step i of the stress spectrum section factor for static bending *8 KcKf ratio for the fatigue strength *2 ncr due to the related stress gradient from bending or torsion of the cross section *2 ncr (r) ncr due to the related stress gradient caused by a notch of the component in question *2 ncr due to the related stress gradient caused ncr (r p ) by a notch of the reference component (the test specimen) *2 spectrum parameter p q constant for aSK,Sv .. , notch radius of the component in question r rp notch radius of the reference component (the test specimen)
7 Symbols and basic formulas
s
wall thickness of the component time of operation at temperature T depth of a notch damage potential of the stress spectrum *4
Vzd
7.4 Upper case characters A, A 3 , As B Cm
D
DM F
,
<
elongation elongation width Larsen-Miller-constants for KTt,m ... diameter critical damage sum axial load (tension, compression, or tension-compression) related stress gradient *2
total number of cycles in a stress spectrum, H=Hj=L:hi technological size factor Kd Kd for R m and/or fJw Kd,m Kd for Rp ~d,p constant, estimate for Kf Kr fatigue notch factor *6, applying to the Kf,zd component in question, dimensions deff and r Kf,zd (d p ) Kf of the reference component (the test specimen), dimensions d p and r p *6 Kp,b plastic notch factor *9 ~,Zd stress concentration factor *6 KA anisotropy factor KAK zd mean stress factor *4 KBK'Zd variable amplitude fatigue strength factor *4 , *2 KE,cr residual stress factor (welded components) KNL constant allowing for the non-linear stress-strain behavior of GG in the case of static loading constant allowing for the non-linear stress-strain behavior of GG in the case of fatigue loading Roughness factor *2 design factor (static strength) *4 temperature factor KT for Rm,T KT for ~,T KT for fJw KT for Rm,Tt KT for ~,Tt surface treatment factor, . design factor (fatigue strength) *4 bending moment torsion moment mean stress sensitivity *2 number of cycles after the component constant amplitude S-N curve N total number of cycles after the component variable~mplitude S-N curve (fatigue life curve), N = L: ni Reference number of cycles Nc (for the S-N curve of welded components)
261 Number of cycles at knee point of the component constant amplitude S-N curve type I or type II *2 N D,cr,II number of cycles at second knee point of the component constant amplitude S-N curve type II *2 a surface (area) of the considered section of a component Larsen-Miller-parameter for the creep Pm,Pp strength probability of survival compression strength yield strength in compression yield strength tensile strength, "component property according standard" for deff "component actual value" of R m Rm ,1 property according standard of R m for ~,N deff,N,m constant for Kp,cr strength at elevated temperature T creep strength at temperature T and time t "component specified value according to drawing" ofRm yield stress, a generalization of R, or ~O,2 ' "component property according standard" for deff' constant for npl,b (bending) and Rp,max npl,t (torsion) "component actual value" of Rp property according standard of Rp for deff,N,p yield strength at elevated temperature T 1 %-creep stress at temperature T and time t 0.2 % proof stress shear strength yield stress in shear average roughness of the surface stress ratio *4 Rzd of step 1 of the stress spectrum *4 stress amplitude *4 equivalent stress amplitude for a number of cycles ND,cr *4 Sa zd of step i of the stress spectrum *4 Sa,zd,i Sa'zd of step 1 (largest amplitude) of the Sa,zd,1 st;ess spectrum *4 Sm zd mean stress *4 Sm.z ' d', 1Sm zd of step i of*4the stress spectrum *4 , Sm,zd,1 Sm,zd for Sa,zd,1 Smax,zd maximum stress of the stress spectrum *4 . Smax,ex,zd ex treme maximum stress *4 Smax ex wv zd extreme maximum stress of Swv zd *5 *11 Smin ~d ' ~nimum stress of the stress spectru~ *4 Smin:ex,zd extreme minimum stress *4 . Smin,ex,wv,zd ex treme reme mi nummum stress 0 f Swv,zd *5*11 Sm v equivalent mean stress *3 Sw~,Zd equivalent stress (welded component) *5 *11 SAK zd amplitude of the component fatigue limit , for Sm.z d or for Sm.v *4 *10
7 Symbols and basic formulas SAK,II,zd amplitude of the component endurance limit for Sm.z d or for Sm.v *4 *10 amplitude (largest amplitude of the spectrum) of the component variable amplitude fatigue strength for Sm zd or for Sm v *4 *10 component static strength *4'*10 SSK,zd SWK,zd component fatigue limit for completely reversed stress *4 *10 normal stress parallel to the weld seam *3 *11 SII,zd normal stress normal to the weld seam? *11 S.l,zd temperature T shear stress parallel to the weld seam *3 *ll TII,zd shear stress normal to the weld seam *3 *11 T.l,zd V Volume of the considered section of a component
7.5 Greek alphabetic characters weld factor limit value of total strain local stress (non-welded component) or 0' structural stress (welded component) O'K effective notch stress (welded component) *2 O'Sch,zd fatigue limit for zero-tension stress O'Sch,zd,N property according standard of O'Sch,zd for deff,N,m material fatigue limit for completely reversed normal stress "component property according standard" for deff,m O'w for bending b O'W,b O'W,b,N property according standard of O'wb for deff,N,m ' weld specific fatigue limit (normal stress) O'W,W O'w for tension-compression zd O'W,zd O'W,zd,N property according standard of O'w zd for deff,N,m ' O'W,zd,T O'w zd for elevated temperature 0'1 first principle stress *4 stress amplitude at the surface *2 O'I,a 'C material fatigue limit for completely reversed w shear stress "component property according standard" for deff,m 'Cw for shear s property according standard of 'Cw s for deffN ' , ,m 'Cw,s for elevated temperature 'CW,s,T 'Cw for t torsion 'Cw,t property according standard of 'Cw t 'CW,t,N ' for deff,N,m weld specific fatigue limit (shear stress) slope angle of a changing cross section increase of j for non-ductile materials Sertr
262
7 Symbolsand basic formulas
7.6 Basic formulas Assessment of the component static strength The characteristic service stress values are or
Smax,ex
or
Tmax,ex
O"max,ex
or
'tmax,ex
The component static strength values are SSK = f, .R, .npl .(KNL ) or
TTJ< = f t '.Rn npl' (KNL ) O"SK = f, ..Rn . ~l . (KNL ) 'tTJ< = ft '.Rn ~]. (KNL)
The denotations S and T apply when using nominal stresses, the denotations 0" and 't when using local stresses.
Different values ofthe section factor ~I apply to the individual types ofstress. Moreover the sectionfactors used with nominal or local stresses are to be determined in a different way.
The degrees of utilization are derived as:
or
aSK = I Smax,ex / ( SSK . jerf) I ~ 1 aSK = I Tmax,ex/(TsK . jere> I ~1 aSK = I Smax,ex / ( SSK . jerf) I s 1 aSK = I Tmax,ex/(TSK'jerf)1 ~1
Assessment of the component fatigue strength The characteristic service stress values are s., or Ta,J or or
Sa,1 , SBK, Ta,] , TBK or O"BK, O"BK, 'ta,] , 'tBK, respectively, refer to the largest amplitude ofthe stress spectrum ofconcern.
The component fatitgue strength values are for notched components
or
SBK = {fwo' n, 'K AK . KE,a 'KBK'K y '(KNL) } / TBK= {fwt . fwo' .Rn 'K AK ' KE,t 'KBK'K y '(KNL)
{x, / ( no(r) . no(d) ) + 1 / KR,o(RJ -1 } } / {x, / [llr(r) 'llr(d) ] + 1 / KRiRJ -1 } O"BK = {fwo' n, 'K AK . KE,a 'KBK 'K y '(KNL) } / {I + [ 1/ Kf ] . [1 / KR,iRJ -1] . [1/ (llcr(r,d)] } 'tBK = {fwt . fwo·.Rn 'K AK ' KE,t 'KBK'K y '(KNL) } / {I + [ 1 / Kf ] . [1/ KR,lRJ -1] . [1/ (llr(r,d)] }
for welded components: SBK = (JW,zd' KAK ' KE,a' KBK' (Ky ) ' (KNL) (FAT / FATO,O") ':4 TBK = 'tw,s . KAK . KE,t . KBK' (Ky ) . (KNL) . (FAT / FATO,'t) . :4 or (for structural stresses or effective notch stresses) O"BK = (Jw,w' KAK . KE,a . KBK. (Ko) . (KNL) . (FAT / FATO,O") . :4 'tBK = 'tw,w . KAK ' KE,t . KBK' (Ky ) . (KNL) . (FAT / FATO,'t) . :4 The degrees of utilization are derived as:
or
aBK = 1 s., / (SBK' jere> I ~1 aBK = I T a,1 / (TBK' jerf)1 ~1 aBK = I O"a, I / ( O"BK . jerf) 1 ~ 1 aBK = 1'ta,1 / ( 'tBK . jerf ) I ~ 1
In the case ofmultiaxial stresses a total degree ofutilization is computedfrom the individual degrees ofutilization ofnormal or shear stress by means ofan interactionformula.
263
9 Subject index
8 Subject index
component static strength 19, 33, 73, 89 - local failure 19,73 - global failure 19,73
aluminium alloy 2·m. 48f. 78ff. 103f. 13If. 142ff. 193 - aging - 27f. 81. 83 - ductil - 90 - non-aging - 27f. 81. 83. - non-ductil - 91
component fatigue strength 57, 113
anisotropy facto.' 26. 80 assessment - assessment of the fatigue strength -- using local stresses 13,97, 127 -- using nominal stresses 12,41,70 - assessment of the static strength - - using local stresses 12, 19. 73, 93 - - using nominal stresses II, 19, 36 - method of assessment 11 - procedure of calculation 10 binomial frequency distribution 44, 100 - see standard stress spectrum calculation of stress see types of stress case hardening steel 25,79,136, 193 cast aluminium alloys 172ff, 193 - for general application 175 - for high pressure die casting 174, 177 - for investment casting 174, 176, 177 - for permanent mold casting 173, 175, 177 - for sand casting 172, 175, 176, 177 cast iron materials 25ff, 79ff, 140ff, 193 - ductil cast iron 90 - non-ductil cast iron 91 - properties (material tables) 140ff cast iron with lamellar graphite 25, 79, 140 cast steel 25, 79, 139 castings - non-destructively tested 68, 125 - not non-destructively tested 68, 125 characteristic service stress - assessment of the fatigue strength 41,97 - assessment of the static strength 19, 73 - probability of occurrence 34f, 90f class of utilization 43,45,66, 100, 102, 122, 218ff, 235 coating factor 55f, llOf coefficient of variation (static strength) 23, 77 compoment variable amplitude fatigue strength 63ff, 119ff
component, kinds of - 13 - block-shaped (3D) 15, 75, 96 - block-shaped (3D), welded 16, 75 - rod-shaped (lD) 13,20, 36, 73, 93 - rod-shaped (lD), welded 14,20,37, 74, 94 - shell-shaped (2D) 14, 20, 38, 73, 94 .. shell-shaped (2D), welded 15, 20, 38, 74, 95 component fatigue limit 57ff, 113ff - according to mean stress 58ff, 114ff, 223ff - see fatigue limit - see material fatigue limit component constant amplitude S-N curve - see constant amplitude S-N curve compression strength 27,81 compresion strength factor 26, 80 compression yield strength 27, 81 consequence offailure 34,68, 90, 125,212 constant amplitude S-N curve 63,66 F, 119, 123f, 212 - for non-welded components 66f, 123f - for welded components (model I) 66f. 123f - lower boundary of the number of cycles 66, 123 - model I and model II 64ff, 66f, 120ff, 123f - non-welded aluminum alloys (model II) 66f, 123f - non-welded steel and cast iron materials (model II) 66f, 123f - parameters 42,67, 99, 124,212 - see endurance limit - see fatigue limit constant amplitude stress 42, 63f, 99, 120f constant K NL 88 constant K NL E 56, III constant q 37ff, 7lf, 94ff, 128ff - aluminium alloy 7lf, 128ff - component, welded 7lf, 128ff - component, surface treated 7lf, 128ff conversion factor, statistical- 229 corrosion 9 1% creep limit 11, 19, 27ff, 35, 73, 81ff, 92 creep strength 11, 19, 27ff, 35, 73, 81ff, 92 damage potential 42,44,46, 99, 101, 102 damage sum, critical 63,65, 119, 121
264
damage-equivalent stress amplitude 45,66, 102, 122 degree of utilization - assessment of the fatigue strength 12, 13,70,127, 226 - assessment of the static strength 11, 13,36,93 design factor 30, 50f, 85, 106f design parameters - assessment of the fatigue strength 5Off, 106ff - assessment of the static strength 30ff, 85ff diameter, effective - 22, 24, 26, 76, 78, 80 direction of 0'3 99, 106, 109 ductility 12, 37, 71, 94, 128, see elongation effective notch stress 73, 97f, 210, 245ff effective diameter 22, 24, 26, 76, 78, 80 elastic limit load 78 elongation 68f, 125f, 140f, 143ff endurance limit 58,63,64, 66f, 114, 119, 120, 123f, 131, equivalent maximum stress 61f, 117f equivalent mean stress 61f, 117f, 223f equivalent minimum stress 61f, 117f equivalent stress amplitude 44f, 71[, 100, 102, 128f, 223 equivalent stress ratio 61f, 117f examples 231ff - cantilever subject to two independent loads (nominal stress) 250ff - component made of a wrought aluminium alloy (local stress) 256ff - compressor flange made of grey cast iron (local stress) 241 ff - shaft with shoulder (nominal stress) 231ff - shaft with V-belt drive (nominal stress) 236ff - welded notched component (nominal, structural and local stresses) 245ff experimentally determined strength value 227 - mean value 227f - sample size 227f - standard deviation 227ff - statistical conversion factor 228
9 Subject index
fatigue classes (of welded components) 56, II If, 195ff - effective notch stress 112, 196 - nominal stress 56, 195, 197ff - nominal shear stress 207 - structural stress 112, 195, 208ff - structural shear stress 195 fatigue life - computed - 63, 119 - required - 42,63, 99, 119 fatigue life curve 45,63, 102, 119 fatigue limit - according to mean stress 58ff, 114ff - component - 57, 113 - material- 47f, 103f - model I and model II 64ff, 67, 120ff, 124 - number of cycles at knee point 42, 67, 99, 124, 212 - specific values for welds 47, 103 fatigue notch factor 51ff, 187ff - cast iron and aluminum material 193f - derived from stress concentration factor 51, 187 - determined by the user 194 - definition 53 - experimentally determined 52, 189ff - of superimposed notches 54 - round and flat structural details 187ff fatigue strength - see constant amplitude S-N curve - see variable amplitude S-N curve fatigue strength factor - for normal stress 48, 104 - for shear stress 48, 62, 71, 104, 118, 128 fine grain structural steel 25, 79, 133 fluctuating stress spectrum 42 f, 99 f frequency, influence of - 47, 103 Gassnerlinie see fatigue life curve GG see cast iron with lamellar graphite GGG see nodular cast iron GS see cast steel GT see malleable cast iron guideline, field of application 9 Haigh diagram 59,63, 115, 119,212,214,217
exponential frequency distribution 44, 100,219ff - see standard stress spectrum
high quality castings 34, 68, 90, 125
FAT see fatigue classes
individual mean stress 6lf, 117f
hot-spot stress see structural stress
inspection 68, 125,212 kind of material 9,24, 78, 131ff
265
~-Kf
-
ratio 5lf, 108f for normal stress 51, 108 for shear stress 52, 109 for superimposed notches 109 for components being surface treated 52, 108 in the direction of (J 3 106
largest amplitude in stress spectrum 42, 99 Larsen-Miller parameter 29, 83 Larsen-Miller constant 29, 83 malleable cast iron 25, 79, 141 material - ductil 34, 68f, 90, 125f - non-ductil 34, 69, 90f, 126 - brittle 9 material fatigue limit 47f,67, 103f, 124, 131ff - aluminum 67, 124, 131f, 142 - bending 131ff - shear 131ff - steel 47f,67, 103f, 124, 131ff - tension-compression 47, 103, 131ff - torsion 131ff material properties - according to standards 22, 76 - assessment of the fatigue strength 47, 103 - assessment of the static strength 22, 76 material tables 131ff - cast aluminium alloys 172ff - cast iron materials 139ff - steel 132ff - wrought aluminium alloys 143ff material test specimen 22, 76, 131
9 Subject index
modulus of elasticity 86 neighbouring point 14, 15, 109 multiaxial stress 16, 37ff, 41, 7lf, 94ff, 97, 128ff, 222ff, 226, 250ff Neuber's formula 86 nitriding steel 25, 79, 136 nodular cast iron 25, 79, 140 nominal stress 10,11,12,178,181,190,210, 245ff non-destructive testing - of castings 128 - of welds 197 non-proportional stress 17,41,97,226,250 normal stress criterion 37ff, 7lf, 94ff, 128ff notch radius, effective
~
15,211, 214f, 245
number of cycles - at knee point of S-N curve 42,67, 99, 124, 212 - at lower boundary of S-N curve 9,41, 64f, 97, 123f - see constant amplitude S-N curve - see fatigue life - see total number of cycles - see variable amplitude S-N curve parameters of the stress spectrum 41ff, 97ff plastc limit load 87 plastic notch factor 31, 86f principle stress 15, 75, 96, 99, 113, 129 probability of occurrence - of the characteristic stress 34f, 90f
maximum stress 20f, 61, 74f, 117
probability of survival 22f, 34, 47, 68, 76f, 90,103, 125, 131, 227
mean stress 41ff, 58, 59f, 6lf, 97ff, 99, 114ff, 223ff - fields of mean stress 58ff, 114ff
procedure of calculation 10
mean stress factor 58, 59ff, 114ff, 210, 213f, 225
proportional stress 16,41, 97
mean stress sensitivity 59ff, 62, 115ff, 118, 213f
0.2 % proof stress see yield strength
mean stress spectrum 42f, 99
quenched and tempered steel 25, 79, 134, 135
mean value from experiments 227ff
radius, effective notch
method of calculation - variable amplitude fatigue strength 53, 104 - class of utilization 43, 45, 66, 100, 102, 122, 218ff, 235 -constant amplitude stress 63f, 119f - Haibach version of Miner's rule 65, 121 - Miner's rule, elementary version 45,64, IOlf, 120 - Miner's rule, consistent version 45,65, 101, 121 - Miner'rule, modified version 65, 121
raintlow matrix 216
Mises (v.Mises) criterion 37ff, 71f, 94ff, 128ff
~
15,211,214
raintlow cycle counting procedure 43, 100 reference point 10, 13, 14, 15, 36, 93 required total number of cycles 44, 45f, 64ff, 100, 102f, 119ff, 218ff residual stress factor 58,62, 114, 118,210, 213f residual stress 12, 13, 62, 118
266
RIFESTPLUS (PC program) 241,254,271 root of weld 74,85,89, 98, 107,209 roughness, average - 54f, 109f roughness factor 50, 54f, 106, 109f round specimen see material test specimen rule of sign see superposition S-N curve - see constant amplitude S-N curve - see variable amplitude S-N curve safety factor 4ff, 68ff, 90ff, 125ff, 212 - assessment of the fatigue strength 68, 125 - assessment of the static strength 34, 90 - cast aluminium alloy 34, 69, 91, 126 - cast iron material 34, 68f, 90f, 125f - consequence offailure 34,68, 90, 125,212 - creep limit 34f, 90f - creep strength 34f, 90f - inspection 68, 125,212 - non-destructive testing 68, 125 - non-ductile materials 69,91,126 - steel 34, 68, 90, 125 - tensile strength 34f, 90f - total safety factor 35, 69, 92, 126 - wrought aluminium alloy 34,69,91, 126 - yield strength 34f, 90f sample size 227ff section factor 30ff, 85ff semi-finished product 23, 77, 131ff sequence of assessment l lff service life see fatigue life service stress 10 - see characteristic service stress - see parameters of the stress spectrum
9 Subject index
standard stress spectrum 44f, 100f - binomial frequency distribution 44, 100 -exponential frequency distribution 44, 100 static strength see component static strength steel 24ff, 78ff, 131ff - austenitic 31,58,66,67,86,114,123,124,137 - ferritic or martensitic 137 - normalized 25, 79, 138 - quenched and tempered 25, 79 - stainless 25, 79 straight line distribution see exponential frequency distribution strength at elevated temperature 11, 19, 27, 48, 62, 73, 81, 104, 118 strength hypothesis 37, 71f, 94, 128ff stress - allowable - l lf, 36, 70, 93, 127 - characteristic service - 19,41,73,97 - extreme 11, 12, 36, 93 - geometrical - see structural stress - local, elastic - 12, 97 - multiaxial>- 16, 36, 93, 223ff, 226, 235 - non-proportional- 17,41, 97, 226 - proportional- 16; 41, 97 - synchronous - 17,41,97, 223ff, 236 - uniaxial - 16 - see type of stress stress amplitude 41ff, 97ff, 223ff - damage-equivalent stress amplitude 45, 66, 102, 122 - equivalent stress amplitude 44f, 7lf, 100, 102, 128f, 223 - largest amplitude in stress spectrum 42, 99 stress cycle 41, 97
shape of stress spectrum 42, 99
stress gradient - related 52f, 109 - in direction of stress 15, 85, 108f - normal to direction of stress 15,85,106, 108f
shear strength 27, 81
stress history 42, 99
shear strength factor 27, 37, 81, 94
stress ratio 41, 97 - adjusting the stress ratio ofa spectrum 43,100,216
size factor, technological -
24, 78
spectrum see stress spectrum
stress ratio spectrum 42f, 99
spectrum parameter p 44, 101, 219ff
stress spectrum - adjusting the stress ratio 43, 100, 216 - damage potential 42,44, 99, 101 - deriving a stepped spectrum 43, 100,217 - determination of the parameters 43ff, 100ff - parameters of a spectrum 42, 99 - spectrum parameter p 44, 101, 219ff - stress ratio spectrum 42f, 99 - total number of cycles of the spectrum 42,64ff - see standard stress spectrum
stair case tests 229 standard deviation 227ff standard stress spectrum 44, 100 - binomial distribution 44, 101, 219ff - exponential distribution 44,101, 219ff - spectrum parameter p 44,.101, 219ff
267
stress concentration factor 51, 178ff, 187 - for a substitute structure 87, 106,230 - for flat specimen 181ff - for round specimen 178ff structural steel, non-alloyed 25, 79, 132 structural stress 10, 73, 97f, 210, 245ff superposition 19, 36ff, 41, 70ff, 73, 93, 95f, 97, 127, 129f
9 Subject index
type of stress - block-shaped (3D) component 15f - combined ~ 12,37,39,71,72,94,95,96,128,129 - individual ~ l3ff, 36, 38, 71, 93, 94, 96, 128, 129 - rod-shaped (lD) component 13f - shell-shaped (2D) component 14f type of material 24, 78, 131ff ultimate strength see tensile strength uniaxial stress 16
surface hardening 31, 52, 55, 62, 66, 71, 72, 86, 108, 110, 11~ 123, 128, 129,222
variable amplitude fatigue strength 63ff, 119ff
surface roughness see roughness, average ~
variable amplitude S-N curve 45, 63ff, 102, 119ff
surface treatment factor 55f, 11Of, 194
variable amplitude fatigue strength factor 63ff, 119ff, 218, 221
symbols 5, 257ff synchronous stress 17,41,97, 223ff, 236
weld factor Uw 31,87
technological size factor 24, 78
weld imperfections 47, 103
temperature - elevated ~ 19,27,48,62, 73, 81, 104, 118 -Iow~ 27,48,81,104 - normal~ 27,48,81,104
weld seam 210
temperature factor - fatigue strength 48, 104 - static strength 27,81 - short-term strength value 27f, 82f - long-term strength value 28f, 83f
WELLE (pC programm) 231,236,250,271
temperature range 9,27,48,81, 104 tensile strength 22ff, 35, 47, 76ff, 92, 13lff - value according to standard 23f, 77f - component value according to the drawing 23, 77 - actual componentr value 24, 77 test piece (of castings) 23, 77, 131
welded components - comments about the fatigue strength 209ff - specific values of the fatigue strength 47, 103
wrought aluminium alloys 143ff - for extruded section 158ff, 168 - for forgings 169,170,171 - for plates 143ff - for rods / bars 154ff, 157ff - for sheets 143ff, 151ff - for strips 143ff, 151ff - for tubes 154ff yield stress see yield strength
throat section 20f, 42, 209
yield strength 22ff, 35, 76ff, 92, 131ff - value according to standard 23f, 77f - component value according to the drawing 23, 77 - actual componentr value 24, 77
toe of weld 74, 85, 89, 98, 107,209
zfP see non-destructive testing
thickness factor 50f, 56, 107, 112
toe section 20, 42, 209 total safety factor 35, 69, 92, 126 total number of cycles 9, 4lf, 97, 99 - computed fatigue life 63, 119 - of a stress spectrum 42, 64ff - required ~ 42,63, 99, 119 type of overloading 58, - type of overloading Fl - type of overloading F2 - type of overloading F3 - typeofoverioadingF4
115,225 60, 116 59f, 115f 60, 116f 61,117
IiZII IMA Materialforschung und
Geschattsfuhrer: Prof. Dr.-Ing. Wilhelm Hanel· Tel.: +49 351 8837-322' Fax: +49351 8804313 Address: Wilhelmine-Reichard-Ring 4 . Postfachadresse: Postfach 80 01 44 . 01101 Dresden
Anwendungstechnik GmbH
PC-programs RIFESTPLUS and WELLE for the strength assessment of components in mechanical engineering.
Assessment of the static strength and of the fatigue strength (fatigue limit and variable amplitude fatigue strength) of non-welded components made of steel or cast iron materials or of wrought or cast aluminum alloys.
RIFESTPLUS
is intended for shell-shaped (20) and for block-shaped (3D) components
made of steel or cast iron materials or of wrought or cast aluminum alloys the local elastic stresses of which have been determined by a finite element analysis or by a strain gauge analysis, for example. (A German and an English version of RIFESTPLUS is available.)
Shell-shaped (2D) component (on the left), block-shaped (30) component (on the right), local elastic stress amplitudes o, . WELLE
is intendedfor axles and shafts F
made of steel or cast iron materials,
F
F
subject to forces and moments, including those from gears. The calculation is based on nominal stresses. (A German version of WELLE is available only.)
Shaft with bearings and forces F
The two programs are based on the
FKM-Guideline "Analytical Strength Assessment of Components in Mechanical Engineering", Frankfurt: VDMA-Verlag, 4 th edition, 2002 (in German).
Personal contacts:
Dr.-Ing. B. Hanel
Tel. +49 351 8837 324 E-Mail: haenel@ima-dresden.de
Dipl.-Ing. G.-R. Bitterlich
Tel. +49351 8837 383 E-Mail: bitterlich@ima-dresden.de
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ISBN 3-8163-0425-7