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Low-cycle fatigue in rotating cantilever under bending I: Theoretical analysis
Int. J. Fatigue Vol. 18, No. 6, pp. 401-412, 1996 Copyright © 1996 Elsevier Science Limited Printed in Great Britain. All rights reserved 0142--I 123/96/$15.00
ELSEVIER
PII:S0142.1123(96)00021-7
Low-cycle fatigue in rotating cantilever under bending I: Theoretical analysis M.M. Megahed, A.M. Eleiche*t and N.M. Abd-Allah** Mechanical Design and Production Department, Faculty of Engineering, Cairo University, Egypt *Mechanical Engineering Department, Faculty of Engineering, UAE University, United Arab Emirates **Maadi Co. for Engineering Industry, Cairo, Egypt
Rotating bending fatigue of a cantilever beam is examined analytically. New relationships are derived to determine the surface strain amplitude, at the mid-point of the tested zone, as a function of the applied nominal bending stress, the vertical and horizontal deflections, and Young's modulus. A numerical methodology is also derived to calculate the rotating bending fatigue behaviour of a material from its cyclic stress-strain response and strain-life curve, and vice versa (Keywords: cantilever b e a m ; rotating bending; fatigue analysis; hysteresis angle; vertical and horizontal deflections; cyclic stress-strain response; strain-life curve)
The stress versus cycles-to-failure curve may be obtained under conditions of uniaxial push-pull, torsion, flexural bending or rotating bending. The resulting curves are known to depend on the loading mode. Figure 1 depicts examples of differing bending and axial fatigue properties that may be due to: (a) liability of the elastic bending relation, Sb =MR/I (routinely used to calculate stress), to increasing error at high stresses, (b) the difference in the stress gradient; and (c) the cross-sectional area of the material experiencing the applied stresses; the axially loaded specimen experiences a uniform stress across the whole section, while a rotating bending specimen experiences a steep stress gradient from the outer surface to the inner surface of the specimen. Furthermore, there are differences in the fatigue crack propagation modes between bending and axial loading. However, the major effect is due to the fact that nominal bending stresses are considerably higher than actual values owing to plastic deformation. Therefore, if the bending problem is treated as an elasto-plastic problem, the obtained fatigue data are expected to be the same when compared with those from axial fatigue.
its axial strain-cycling fatigue data. Every technique stipulates a certain procedure to calculate the bending moment required to produce a prescribed surface strain. From the prescribed surface strain amplitude, fatigue life can be established and similarly nominal bending stress from the calculated bending moment. Therefore, a curve can be plotted between the calculated nominal bending stresses and the corresponding fatigue life (S-N). Considering flexural bending, three main techniques have been reported in the literature: the elasto-plastic solution by Manson', the closed-form solution by Manson and Muralidharan 2, and the reference stress method by Megahed 3. The three techniques calculate, in a different manner, the nominal bending stress, Sb, corresponding to a prescribed surface strain. Figure 1 depicts the predicted S-N curves obtained by the three techniques. Comparisons between the values of Sb obtained by the three different techniques show good correlation for beams with rectangular and circular cross-sections.
FLEXURAL AND ROTATING BENDING FROM AXIAL FATIGUE
For a circular cross-section under rotating bending, Manson I showed that, under cyclic plasticity, the crosssection bends about an axis making an angle ~b with the loading axis. This angle is designated as the hysteresis angle, with the result that the cross-section deflects in both the vertical and horizontal directions. Taking the hysteresis angle into account, Manson's experimental work in rotating bending was based on the calcu-
PROBLEM STATEMENT
Few techniques have been reported to predict the flexural and rotating bending behaviour of a metal from
t On leave from Cairo University, Egypt.
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Cycles to Failure, Nf Figure 1 Prediction of flexural bending fatigue from axial fatigue, and comparison with experiment
lation of S b for an assumed value of surface strain, and hence fatigue life Art-. In fact, neither the actual imposed surface strain nor the hysteresis angle were measured. Under such conditions, the cyclic stressstrain response under rotating bending had not been truly investigated. In the experimental work of Kawamoto et al. 4 under rotating bending, they considered the applied nominal bending stress as the true stress. Furthermore, they considered the horizontal and vertical deflections to represent the plastic and total strain amplitudes respectively, without presenting sufficient evidence to support these assumptions. They also considered the relation between the deflection and strain under cyclic loading to be the same as in static flexural bending for a virgin specimen. Consequently, the effect of cyclic stressstrain response and the hysteresis angle were neglected. The main objective of the present work is therefore to develop proper analytical procedures that permit the prediction of rotating bending fatigue behaviour of a material from its cyclic stress-strain response and strain-life curve, and vice versa. THEORETICAL ANALYSIS The problem of bending fatigue is treated here as an elasto-plastic problem. The specimen to be used in the tests by Eleiche et al. 5 will be a cantilever beam with variable circular cross-section designed to guarantee a
zone of uniform surface strain under cyclic loading. The approach we use is as follows. First, a material with prescribed axial fatigue data and with known stress-strain curve is considered. Under both flexural and rotating bending, it is required to predict, for the tested specimen: (a) the nominal stress-life curve, and (b) the pattern of deflection. Next, it is shown how to determine the cyclic stress-strain response for a material with prescribed rotating bending fatigue data.
Bending fatigue from axial fatigue It is intended here to develop a detailed analysis of the stress-strain relations under rotating bending before determining the corresponding nominal bending stresslife curve. Hysteresis angle in rotating bending. Figure 2a shows the cross-section of a round specimen subjected to flexural bending with surface strain amplitude e~. All points at the same distance from the deflection axis c~c~ (strain axis), such as A and B, will have exactly the same strain and stress levels. Consequently, these points have the same hysteresis loop as shown in Figure 2c. Note that the loading axis ZZ in this case coincides with the deflection axis c~c~. Figure 3a shows the same cross-section of the round specimen, now subjected to rotating bending with surface strain amplitude e~. The two elements A and B are at the same distance from the strain axis c~c~
Low-cycle fatigue in rotating cantilever under bending I • --
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Figure 2 Determination of stress distribution in flexural bending from axial strain cyclic properties (deflection axis). Therefore, the two elements are at the same strain level, e, as shown in Figure 3b. On the other hand, the two elements now share the same hysteresis loop also, but are not, at any given moment in time, at the same stress level, owing to rotation, as shown in Figure 3c. Point A is moving towards a higher stress level, whereas point B is moving towards a lower one. Each point will ultimately reach a maximum strain level era, when they achieve the maximum vertical distance R from the strain axis. Because of rotation, the stress-strain relation for all points along a circle with radius r is represented by one hysteresis loop. Therefore the section will deflect about the axis c~c~, which makes an angle ~ with the loading axis ZZ. The angle ~b is designated the hysteresis angle to indicate the presence of hysteresis loops on the radii of the section. When the behaviour is elastic, as in high-cycle fatigue, the hysteresis loops degenerate to an elastic line. Therefore the hysteresis angle vanishes. In this case, the rotating bending problem can be treated as a flexural bending problem, wherein the loading and strain axes always coincide.
Stress-strain analysis in rotating bending. Considering low-cycle fatigue tests under rotating bending, a test specimen may be developed 4's such that the tested zone has constant surface strain amplitude e~. Consider
a cross-section in the tested zone with outer radius R, as shown in Figure 3a, subjected to rotating bending. Since in bending it is commonly assumed that plane sections remain plane, the strain will vary linearly across the section, as shown in Figure 3b. The strain em at distance r from the deflection axis a a (strain axis) is
Points A and B, at the same height from the deflection axis, rotate on the circle with radius r. Therefore the stress-strain relation of these points is represented by one hysteresis loop with strain amplitude era, as shown in Figure 3c. To obtain the value of stress at point A (O'A) or at point B (o'B), the shape of the hysteresis loop must first be defined. According to the Masing rule ~, the shape of the settled hysteresis loop branches can be obtained by magnifying the relation of the stress-strain curve by a factor of 2. Since the cyclic stress-strain curve is described by e =
+ \K'/
~2)
where E is the modulus of elasticity, n' is the cyclic strain-hardening exponent and K' is the cyclic strength
404
M.M. Megahed et al.
p
"~
(a)
(b)
8A* (3
~B*Om i/
2am (~
(c)
/ J
Cyclic ~re~rmn
~" T
curve ~m
28m Figure3 Determination of stress distribution in rotating bending from axial strain cyclic properties coefficient, the following relation for the shape of a hysteresis loop is obtained by applying the Masing rule:
Points (1) and (2) in Figure 3c, which lie on the cyclic stress-strain curve, are considered to be the origins of the upper and the lower branches of the settled hysteresis loop corresponding to the strain amplitude em. The upper branch is described by s t r e s s t A R and strain EAR as follows:
ga*=(O'-~-) +21-11n'(tARIlln" \ K' /
(3)
The lower branch is described by stress t~* and strain eB* as follows:
\ K' }
(4)
Stresses tA and t u can then be determined from this settled hysteresis loop with strain range 2era, as shown in Figure 3c, from
t A = O'A* -- t m
(5)
o'B = ~rB* - t m
(6)
Therefore stresses tA*, tB* and t m should be determined first prior to calculating tA and tB. O"m is determined as follows. Points (1) and (2) in Figure 3c lie on the cyclic stress-strain curve described by Equation (2). Hence
tm (tml vw gm = T ' t - \Kt / Considering Equation (1) and making the necessary manipulations yields O.mV, + (K,)t/w ~r,,- (K,)Vw r
= 0
(7)
This is a non-linear equation in tim, which can be solved for all genetic radii 0 ~< r <~ R by the NewtonRaphson method. The determination of t A R and t n * then proceeds as follows. Referring again to Figure 3a, points A and B are defined by the radius r and the angle 0. The angle 0 is measured clockwise, in the direction of rotation,
Low-cycle fatigue in rotating cantilever under bending I from the negative direction of the strain axis aa. Figure 3a shows that points A and B are at the strain level e. Strain relationships can be determined as follows.
value of 49, the bending moments about the axes ZZ and YY can be rewritten as follows: Mzz=COS 49~A O ' J dA r sin 0+sin 49(.]A (TOar
e = em sin O; e g * = e +em = em(1 + sin 0); ea* = e m -
e = em (1 -
sin O)
eA* = (1 + sin 0)
es
(8)
ea* = (1 + sin 0) (R) e~
(9)
Substituting Equation (8) into Equation (3) yields
COS0
0+sin
-
(21-"" K') I;~"(1 + sin 0) ~ e~ = 0
Mzz = M,,~ cos 49 + Moo sin 49
(14a)
Mrv = Mt3t3cos 49 + M
(14b)
sin 49
Note that the bending moment about the YY axis vanishes; hence
(10)
f~/2fR O-A r2 sin 0 dr dO dO
Similarly, substituting Equation (9) into Equation (4) yields
JO
o-B r2 sin 0 dr dO
+ 0
7r/2
o-B r2 sin 0 dr dO
+ J~-
Solving Equations (10) and (11) by using the NewtonRaphson method, O-A* and o-B* can be determined at any point defined by r and 0, under applied surface strain es. Once O'A*, O-B* and o-m are determined, the stresses o-A and o-B can be determined from Equations (5) and (6). The complete stress distribution can now be described with respect to the deflection axis a a , whose direction has not been yet determined with respect to the loading axis ZZ. Hence the hysteresis angle 49 between these two axes must first be determined prior to establishing the final stress distribution with respect to the loading axis.
+
dO
f fR 3 ~-/2
M~ =
Determination of the hysteresis angle. Consider a typical element of area dA located at angle 0 and radius r, as in Figure 3a, where 0 is measured clockwise from the negative direction of the deflection axis a a . The spatial stress distribution O-(O-Aor o-B) has been already determined as outlined above, i.e. the stress is known as a funtion of 0, r and e~. Hence it is possible to determine bending moments about the axes a a , /3/3, ZZ and YY. Thus:
Me=f o-da
(12a)
gA
o-dA rsin(O+ 49)
Mrv = ~ o- dA r cos (0 + 49)
JO
+
o-R r2 cos 0 dr dO rr/2
l)
+
o-B r2 cos 0 dr dO Jrr
J0
+
O-A r2 cos 0 dr dO
Referring to the stress-strain relations as shown in Figure 3c and making the necessary manipulations, Equations (16a) and (16b) can be rewritten as follows: M~ = 2
O-Ar2 sin 0 dr dO \JO
JO
+
o-~ r 2 sin 0 dr JI)
At a specified cyclic loading condition, and at a fixed
(17a)
JO
M~ = 2
O-A 1"2COS 0 dr dO JO
oBr2 cos 0 dr dO JO
JA
(16b)
o
(12b)
(16a)
O-A r2 COS 0 dr dO JO
\JO
Mzz = (
O-Ar2 sin 0 dr dO
0
~7r/2
O;
(15)
Values of M ~ and Mt3t3 are obtained from M,~ =
M.o=f o-dA
(13b)
Substituting Equations (12a) and (12b) in Equation (1 3a) and (1 3b) yields
49 = tan-' (Mt3~-t3)
(r)
(13a)
49fo-darCOSa 49fo-dArsinOA
Mrr=cos
Considering Equation (1), then the final expressions for eg* and ea* are given by
405
(17b)
JO
The above double integrals are determined numerically, by using Simpson's rule for double integration, for any given values of the three material constants K', n' and E, the surface strain e~ and the radius R. Once M ....
M.M. Megahed et al.
406
and Mt3t3 have been obtained, the hysteresis angle 4> can be determined from Equation (15). Then the bending moment Mzz, required for the prescribed surface strain es, is obtained from Equation (14a), with the corresponding elastically calculated surface bending stress, Sb, given by 4 Mzz Sb --
Deflection-strain analysis in rotating bending Introduction. Figure 8a shows, in schematic form, the
,rrR 3
Life can be obtained from the strain-life relation under axial conditions. A curve representing the relation between nominal stress Sb and cycles to faiulre Nf can then be plotted.
Numerical example. In this section, a specified material, SAE 950X-150 HB (S.A.E.7), with known cyclic stress-strain curve and strain-life relation, is considered. The cyclic strain-hardening exponent n' and the cyclic strength coefficient K' of this material are 0.143 and 896 MPa respectively. The behaviour of this material under rotating bending fatigue is obtained by applying the analytical procedure mentioned above. Table 1 summarizes the predicted behaviour of this material under rotating bending. Rotating versus flexural bending fatigue. It is worth noticing material flexural rotation,
that a solution of the same problem for SAE 950X-150Hb can be established under bending fatigue by ignoring the effect of i.e. 4>= 0. Figure 4 compares the obtained results in rotating bending with those of flexural bending for the same material. In high-cycle fatigue, both rotating and flexural bending give approximately the same results. In low-cycle fatigue under flexural bending, the nominal bending stress required to maintain a prescribed surface strain is higher than that under rotating bending at the same surface strain. Therefore low-cycle fatigue tests under flexural bending will give higher lives than those under rotating bending at the same nominal bending stress. Figures 5 and 6 show the predicted stress distribution, at mid-life, over the tested-zone surface in both rotating and flexural bending, at a surface strain of 0.01. Under this condition of equal surface strain, the predicted life is the same for both rotating and flexural bending, but the required nominal bending stress in Table 1 Prediction of rotating bending from axial fatigue information for material SAE 950X-150 HB Axial fatigue data (n'--0.143, K'=896 MPa)
Test no. 0 1 2 3 4 5 6 7 8
e 0.0012 0.0015 0.002 0.0025 0.005 0.0075 0.01 0.015 0.02
flexural bending is higher than that in rotating bending by about 9%. Note that the angle (0 + 4>) representing the horizontal axis in Figure 6 is measured from the loading axis ZZ of the specimen (cf. Figure 3a). Figure 7 depicts the stress-strain relation under the same condition in both rotating and flexural bending.
~r (MPa) 241 265 303 324 386 424 448 479 502
Prediction of rotating bending fatigued data
2Nf
Sh (MPa)
q5 (degrees)
1 130000 289 000 70 200 29 000 4040 1700 950 450 300
244 287.69 351.22 397.05 514.98 571.18 607.53 655.68 688.56
0 3.649 8.22 12.631 26.844 34.069 38.51 43.87 47.088
shape of the tapered test specimen proposed for use in rotating bending fatigue tests 4'5. The specimen is designed so that the nominal stress in the outer fibres of the tapered zone II can be regarded as uniform. This is achieved by adjusting the diameter of the tapered zone to be a rational function of the third degree of the lengthwise coordinate 4,8. The specimen is fixed as a cantilever, and is free to deflect in both the vertical and horizontal directions. Under low-cycle fatigue conditions, it is reasonable to assume that zones I and III will behave elastically, while zone II is strained beyond the proportional limit of the material. Therefore the contributions of zones I and III are deflections 61 and 6ii1 respectively in the vertical direction YY only. Figure 9a shows that the tested zone II will deflect about its strain axis a~, which makes a hysteresis angle 4~ with the loading axis ZZ. Therefore the contribution of zone II is deflection 6., which has two components 6i~v and 6Hh in the vertical and horizontal directions respectively. Thus 8,h = 61i sin ~h; 6,v = ~11 c o s IJ); ~ii = (82Ih + 81Iv) 1/2 (18) The total vertical deflection &, shown in Figure 9b, is the cumulative effect of the deformations of zones I, II and II. Thus 8 v ~ 81 + 811v + 8111 ~ 81 + 811 " COS (~ + 6111; ~h = 811h 7-- 8i I s i n d~
(19)
Figure 8a shows that the deflections 8v and 6h are measured at points a and b respectively. Therefore the experimental hysteresis angle ~bcxp can be obtained as follows: tan ~bexp= 6d(6, - 81 - 6110
(20)
It is now intended to relate the deflections of the specimen to: (a) surface strain amplitude, e,, in zone II; ( b ) a x i a l distribution of bending moment; and (c) dimensions of the specimen. The problem will be treated for both the elastic and elasto-plastic behaviour of the material.
Deflection of a beam bent by transverse loads. Figure 10 shows a beam bent by transverse loads. Hence, for small deformation, the total deflection of point B relative to the tangent at A, i.e. BB', is denoted by 6. and can be defined using the following equation:
6=f~[dslp(X)]dX
(21)
Equation (21) is used to determine the deflection of beams in the general case, particularly when elastoplastic behaviour takes place 9'l°. The bending moment M(X) varies along the length of the beam, as shown in Figure 10. Therefore the
Low-cycle fatigue in rotating cantilever under bending I
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407
Elasto-plastic s o l u t i o n in r o t s t l n s bendins Elasto-plastic s o l u t i o n in f l e x u r a l bendin8 Closed-form s o l u t i o n in f l e x u r a l bending Reference-stress s o l u t i o n in f l e x u r a l bendinl
N~.N,
600 o.
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400
SAE 9 5 0 X - 1 5 0
200
l
I
I l Illil
10
1
I
I
I
lib
I
I I Ill
103
I
I I I Illl
10 q Cycles to failure,
Figure 4
" ~ ' - ~ . . _
1
I
[ t t~.t
l05
106
Nr
Prediction of nominal bending stress-life fatigue curves
Rota~.._.~"
Y
y
~
7-200 MPa
\~
\
Z
Z
Z
--
z
Loading
\
Y
(a) Flexural bending
Y
(b) Rotating bending
F i g u r e 5 Polar plot of predicted stress distribution under surface strain of 0.01 for material SAE 950X-150HB, at mid-life, in: (al flexural bending; (b) rotating bending
408
M.M. Megahed et al.
400;
!=o
\ /
Rotating bendAn~
~
0
i
/
..
o -200
"
\
~
1
1
1
/
-400.
Figure 6 Prediction of stress distribution under surface strain of 0.01 for material SAE 950X-150HB, at mid-life o (MVt)
(a) the bending moment M at the mid-point of zone II, and which is required to maintain the prescribed e~; and (b) the hysteresis angle ~b. Zones I and III are assumed to strain elastically. Thus, referring to Figure 8b, Equation (23) is used to determine deflections 8~ and 6m: 31 =
-0.01 //
]
I
-200
IIll
/- /
/
/
/
/
/
0.01
f
x~ o
L2
--Rotating bending
Figure 7 Stress-strain relation under surface strain amplitude of 0.01 for material SAE 950X-150HB curvature llp/(X) also varies along the beam under elastic conditions ~J. Thus
1
M(X) (22)
where I(X) is the second moment of area about the deflection axis as a function of the length coordinate X. Substitution of Equation (22) in Equation (21) yields
(R I -
(24)
DI/2) 4
~II1 =
f x3 X2
[4 M(X+LO] d X (25) / L2 E (R2 - ~/R 2 - (Xl - X) 2 + Dl/2) 4
The elastic-plastic behaviour takes place in zone II. Therefore Equation (21) is used to determine 6,. Noting that Up(X) = es(X)/R(X), Equation (21) can be rewritten as follows: 8. =
f
X2~s(X) ,
o~
0-~
(26)
Xl / ~ t A )
where e~(X) and R(X) are the surface strain and outer radius respectively, as functions of the length coordinate X. From the design of the test specimen, es(X) can be obtained as a function of surface strain at midpoint of zone II, G, as follows:
G(X+L1)(D2/2)3
(B M(X) 8 = jAE. ~
E
dX
Flexural bending
-4OO
p(X) - E.I(X)
4 M(X+LO / ~/R 2 - (XI-X) 2 +
. XdX
(23)
Equation (23) is used to determine the deflection of beams when the material behaves elastically.
Prediction of specimen deflection under rotating bending. Figure 8b shows the dimensions of the test specimen proposed to be used in rotating bending tests. It is intended to predict the vertical and horizontal deflections 6v and 6, respectively at section CC, where deflections are measured. The given data are: (a) dimensions of the test specimen; (b) surface strain amplitude e~ at mid-point of zone II; and (c) straincycling properties of the specimen material, n' and K', under axial conditions. From the knowledge of n', K' and es, the analytical procedure described above can be applied to obtain:
G(X) =
L2[R(X)] 3
(27)
Substitution of Equation (27) in Equation (26) and making the necessary manipulations yields
f x2 ~s(X+Ll )(D2/2)3 hl = XI L2{D1/2+ ( X - X 1 ) (D3-D1)/[2(X2-XI)] } 4x
dX
(28) Deflections 6~, 6, and 6m given by Equations (24), (28) and (25) can be readily evaluated numerically using Simpson's rule. The predicted vertical and horizontal deflections can then be calculated by applying Equation (19) as follows: 3v = 61 +
6n
COS 6 +
~ni; ~l~ =
blb = 6n sin ~b (19 bis)
The total deflection 6,, at section
CC, is thus given by
Low-cycle fatigue in rotating cantilever under bending I
409
Section at which deflections are
Y//////////////////////////A~.j-o~
--,-- C
"///////////////////////////2~ '
--~-C
--III
II
measured
Section cc
I----.
I 1
I I
J Load
Bending
moment
I I I
I I
Nominal surface stress
(a)
d
t5-t
~2"//////////////////////////~ I=
:ii
17.~
=-=
~1• I
II
X-
L1= 3 ~
= X2=48.2 X3=6E,
• L
=
I
°
L3=102
j ~Ioocl P
(b) Figure 8
Schematic of rotating bending and test specimen configuration and dimensions
6, = ~/(6~, + 6~)
(29)
Numerical example. Consider again material SAE 950X-150 HB. The specimen dimensions for rotating bending are assumed to be as in Figure 8b. The predicted deflection values are obtained by applying the analytical procedures outlined above. Table 2 summarizes the obtained results, while Figure 11 shows the relation between deflection and surface strain amplitude e~. Figure 12 depicts the relation between e, and deflection of specimen zones (I+III) and II. The following relationships are obtained: 6. - 0.002 895
(mm)
(30)
61 + 6m -
136.27 Sb E
(ram)
(31)
where Sb is the maximum nominal bending stress at the mid-point of zone II, and e, is the maximum surface strain at the same point. It is noteworthy to mention that the constants 0.002 895 and 136.27 depend on material type, specimen configuration, dimensions, fixation and measurement location.
Determination of cyclic stress-strain response .from rotating bending fatigue data So far, the procedure adopted to predict the rotating bending fatigue data for a material from the knowledge of its cyclic stress-strain response (n" and K') and modulus of elasticity E has been illustrated. It is intended here to carry out the reverse procedure: i.e.
M.M. Megahed et al.
410 Y
Y
,e
#
i
/\ zJ
,/
\
I
Loading axis
--)z-
\/ #
Y (a)
(b)
Figure 9
(a) The attributed horizontal and vertical deflections of the tapered zone 11. (b) The total recorded vertical deflections at points a and b, respectively (cf. Figure 8a)
Tr/2lR
I I
Cx)/a0 I A
/
F~(n',K',E,e,,R,r,O)dr dO
+ jo
I
(32)
JO
MS~ = Mzz sin 4~exp=
F3(n',R',E s.R,r,O)dr dO JO JO
"~ --~
dO
-'1 Xd0
F4(n',K',E,s,,R,r,O)dr dO
+ JO
x
dX
I
Bending moment
Figure 10
Equations (32) and (33) represent two non-linear equations involving double integration, where all quantities are known except n' and K'. These equations can be recast as follows: F(n',K') = 0;
Deflection of beam under transverse loads
to determine the cyclic stress-strain response for a material from the knowledge of its rotating bending fatigue data, namely: (a) the applied nominal bending stress, Sb (thus bending moment Mzz); (b) the corresponding induced surface strain amplitude, e~, and (c) the corresponding experimental hysteresis angle,
F(no + dn, Ko + dK) = O;
G(no + dn, Ko + dK) = O
Expanding by Taylor series, and truncating after the first-order terms, yields F(no,Ko) + dn
Procedure f o r calculating cyclic stress-strain response. Making use of Equations (5), (6), (7), (10)
G(n~,Ko) + dn
+ dK 0
Tr/2¢R
M ~ = Mzz cos ~exp =
Fl(n',K',E,s.R,r,O)dr dO JO
Table 2
G(n',K') = 0
These two equations can be solved by a series expansion technique ~2. Suppose that (no, Ko) is an approximate solution, and let dn and dK be corrections to be determined. Therefore
(~exp'
and (11), which provide the values of stress at any given point (r,O) in the tested section, the equilibrium Equations (17a) and (17b) can be rewritten as follows:
(33)
JO
0
=0
(34)
=0
(35)
0
+ dK
OK
0
These two linear equations in dn and dK can be solved to give the next approximation. The final values of n' and K' can be determined by carrying out an iteration technique until the values of dn and dK become negligible. Input data consists of the applied nominal bend-
jo
Prediction of specimen deflection under rotating bending fatigue for material SAE 950X-150HB, at mid-life
Surface strain amplitude, s~ Nominal bending stress, Sb (MPa) Hysteresis angle, q5 (degrees) Horizontal deflection, 8h (mm) Vertical deflection, 8v (mm)
0.0012 244 0 0 0.58
0.0015 288 3.6 0.03 0.71
0.002 351 8.2 0.1 0.92
0.0025 397 12.6 0.19 I. I I
0.004 481 22.4 0.53 1.6
0.005 515 26.8 0.78 1.89
0.0065 552 31.7 1.18 2.28
Low-cycle fatigue in rotating cantilever under bending I
//
//
/
• 006
/
/
/
/
/
/
/
=ff
• 004
/
411
//
/
// /
// /
/
/
/
t..
/ /
== r.~
/
/ -/
• 002
' Total deflection, iit
/
If 0
....
Vertical deflection, 8,,
.....
Horizontal deflection, 5,,
1
1
1
2
Deflection (ram)
Figure 11 Prediction of specimen deflection in rotating bending for material SAE 950X-150HB
=~ .oo4
Deflection, 8n
'~ .002
. / / / 0
Deflection, (81 I 3.
+ 8m)
l 2
_--
Deflection (nun) Figure 12
Deflection of test specimen zones I1 and (I+I11) versus surface strain, in rotating bending for material SAE 950X-150HB
ing stress Sh, the corresponding surface strain amplitude e~, the corresponding experimental hysteresis angle &exp, and the modulus of elasticity E. An initial guess (no and Ko) is made at the two unknowns n' and K'. After carrying out the iteration procedure, the final values of n' and K', and hence the corresponding true stress ~r and surface plastic strain amplitude e~, are determined for the test specimen. Repeating this procedure for other specimens as well, the cyclic stress-strain response of the tested material can be calculated.
Application example. From the information of the predicted rotating bending fatigue data for material
SAE 950X-150 HB, shown in Table 2, it is intended to determine the cyclic stress-strain response (n' and K') for the same material. The analytical procedure detailed above was used. Table 3 lists the calculated values of n' and K' corresponding to each prescribed testing condition (Sb, e~, 4)~xp). It is noted that all prescribed test conditions give exactly the same values of n" (0.143) and K' (896MPa), which are the same original values of the considered material, measured under reversed push-pull condition. Consequently, experimental output data of one test specimen in rotating bending fatigue may give the cyclic stress-strain response (n' and K') for the elasto-plastically deformed material, especially when the applied conditions induce
M.M. Megahed et al.
412 Table 3
Determination of cyclic stress-strain response from rotating bending data for material SAE 950X-150 HB Rotating bending fatigue data
Calculated cyclic stress-strain response
Test no.
Sb (MPa)
e~
4~ (degrees)
n'
K' (MPa)
tr (MPa)
1 2 3 4 5 6 7 8
287.69 351.22 297.05 514.05 571.18 607.53 655.68 688.56
0.0015 0.002 0.0025 0.005 0.0075 0.01 0.015 0.02
3.649 8.22 12.631 26.844 34.069 38.51 43.87 47.088
0.144 0.143 0.143 0.143 0.143 0.143 0.143 0.143
903 899 897 896 896 896 896 896
265 303 327 391 424 448 479 502
an appreciable amount of cyclic plasticity (experimentally exhibited by a high value of ~bexp). These results indicate that the numerical procedure proposed herein can be used with confidence to determine the cyclic stress-strain response from well-controlled experimental data collected from rotating bending tests, as shown in a companion paper5.
CONCLUSIONS
A complete and fresh analytical examination of rotating bending fatigue led to the development of a new relationship to determine the surface strain amplitude es, at the mid-point of the tested zone of a tapered rotating cantilever specimen, as function of: the applied nominal bending stress Sb; the recorded values of vertical deflection, 6v, and horizontal deflection, 6h; and the modulus of elasticity E. The theoretical analysis also led to the derivation of a numerical methodology whereby the cyclic stress-strain response could be determined from the measured rotating bending data,
0.000 194 0.000 5 0.000 89 0.003 07 0.005 4 0.007 79 0.001 263 0.017 52
namely: the nominal bending stress Sb, vertical deflection 6v, and horizontal deflection 6h. REFERENCES 1 2 3 4 5 6 7 8 9
10
11 12
Manson, S.S. Exp. Mech. 1965, 5, 193 Manson, S.S. and Muralidharan, U. Fatigue Fract. Eng. Mater. Struct. 1987, 9, 343 Megahed, M.M. Fatigue Fract. Eng. Mater. Struct. 1990, 13, 361 Kawamoto, M., Nakagawa, T. and Ida, A. Bull. JSME 1966, 9 Eleiche, A.M., Megahed, M.M. and Abd-Allah, N.M. Int. J. Fatigue 1996, 18 Masing, G. In 'Proceedings 2nd International Congress for Applied Mechanics, Zurich', 1926 Society of Automotive Engineers, 'Fatigue Design Handbook', 2nd edn Beer, F.P. and Johnston, E.R. Jr 'Mechanics of Materials', McGraw-Hill, New York, 1992 Megahed, M.M. and Leckie, F.A. In 'Cyclic Hardening Analysis of Beams under Cyclic Loading', Second Cairo University MDP Conference, Egypt, 1982 Timoshenko, S. 'Elements of Strength of Materials', Van Nostrand Reinhold, New York, 1970 Chakrabarty, J. 'Theory of Plasticity', McGraw-Hill, New York, 1987 Froberg, C.E. 'Introduction to Numerical Analysis', AddisonWesley, London, 1970
ELSEVIER
PII:S0142.1123(96)00021-7
Low-cycle fatigue in rotating cantilever under bending I: Theoretical analysis M.M. Megahed, A.M. Eleiche*t and N.M. Abd-Allah** Mechanical Design and Production Department, Faculty of Engineering, Cairo University, Egypt *Mechanical Engineering Department, Faculty of Engineering, UAE University, United Arab Emirates **Maadi Co. for Engineering Industry, Cairo, Egypt
Rotating bending fatigue of a cantilever beam is examined analytically. New relationships are derived to determine the surface strain amplitude, at the mid-point of the tested zone, as a function of the applied nominal bending stress, the vertical and horizontal deflections, and Young's modulus. A numerical methodology is also derived to calculate the rotating bending fatigue behaviour of a material from its cyclic stress-strain response and strain-life curve, and vice versa (Keywords: cantilever b e a m ; rotating bending; fatigue analysis; hysteresis angle; vertical and horizontal deflections; cyclic stress-strain response; strain-life curve)
The stress versus cycles-to-failure curve may be obtained under conditions of uniaxial push-pull, torsion, flexural bending or rotating bending. The resulting curves are known to depend on the loading mode. Figure 1 depicts examples of differing bending and axial fatigue properties that may be due to: (a) liability of the elastic bending relation, Sb =MR/I (routinely used to calculate stress), to increasing error at high stresses, (b) the difference in the stress gradient; and (c) the cross-sectional area of the material experiencing the applied stresses; the axially loaded specimen experiences a uniform stress across the whole section, while a rotating bending specimen experiences a steep stress gradient from the outer surface to the inner surface of the specimen. Furthermore, there are differences in the fatigue crack propagation modes between bending and axial loading. However, the major effect is due to the fact that nominal bending stresses are considerably higher than actual values owing to plastic deformation. Therefore, if the bending problem is treated as an elasto-plastic problem, the obtained fatigue data are expected to be the same when compared with those from axial fatigue.
its axial strain-cycling fatigue data. Every technique stipulates a certain procedure to calculate the bending moment required to produce a prescribed surface strain. From the prescribed surface strain amplitude, fatigue life can be established and similarly nominal bending stress from the calculated bending moment. Therefore, a curve can be plotted between the calculated nominal bending stresses and the corresponding fatigue life (S-N). Considering flexural bending, three main techniques have been reported in the literature: the elasto-plastic solution by Manson', the closed-form solution by Manson and Muralidharan 2, and the reference stress method by Megahed 3. The three techniques calculate, in a different manner, the nominal bending stress, Sb, corresponding to a prescribed surface strain. Figure 1 depicts the predicted S-N curves obtained by the three techniques. Comparisons between the values of Sb obtained by the three different techniques show good correlation for beams with rectangular and circular cross-sections.
FLEXURAL AND ROTATING BENDING FROM AXIAL FATIGUE
For a circular cross-section under rotating bending, Manson I showed that, under cyclic plasticity, the crosssection bends about an axis making an angle ~b with the loading axis. This angle is designated as the hysteresis angle, with the result that the cross-section deflects in both the vertical and horizontal directions. Taking the hysteresis angle into account, Manson's experimental work in rotating bending was based on the calcu-
PROBLEM STATEMENT
Few techniques have been reported to predict the flexural and rotating bending behaviour of a metal from
t On leave from Cairo University, Egypt.
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Cycles to Failure, Nf Figure 1 Prediction of flexural bending fatigue from axial fatigue, and comparison with experiment
lation of S b for an assumed value of surface strain, and hence fatigue life Art-. In fact, neither the actual imposed surface strain nor the hysteresis angle were measured. Under such conditions, the cyclic stressstrain response under rotating bending had not been truly investigated. In the experimental work of Kawamoto et al. 4 under rotating bending, they considered the applied nominal bending stress as the true stress. Furthermore, they considered the horizontal and vertical deflections to represent the plastic and total strain amplitudes respectively, without presenting sufficient evidence to support these assumptions. They also considered the relation between the deflection and strain under cyclic loading to be the same as in static flexural bending for a virgin specimen. Consequently, the effect of cyclic stressstrain response and the hysteresis angle were neglected. The main objective of the present work is therefore to develop proper analytical procedures that permit the prediction of rotating bending fatigue behaviour of a material from its cyclic stress-strain response and strain-life curve, and vice versa. THEORETICAL ANALYSIS The problem of bending fatigue is treated here as an elasto-plastic problem. The specimen to be used in the tests by Eleiche et al. 5 will be a cantilever beam with variable circular cross-section designed to guarantee a
zone of uniform surface strain under cyclic loading. The approach we use is as follows. First, a material with prescribed axial fatigue data and with known stress-strain curve is considered. Under both flexural and rotating bending, it is required to predict, for the tested specimen: (a) the nominal stress-life curve, and (b) the pattern of deflection. Next, it is shown how to determine the cyclic stress-strain response for a material with prescribed rotating bending fatigue data.
Bending fatigue from axial fatigue It is intended here to develop a detailed analysis of the stress-strain relations under rotating bending before determining the corresponding nominal bending stresslife curve. Hysteresis angle in rotating bending. Figure 2a shows the cross-section of a round specimen subjected to flexural bending with surface strain amplitude e~. All points at the same distance from the deflection axis c~c~ (strain axis), such as A and B, will have exactly the same strain and stress levels. Consequently, these points have the same hysteresis loop as shown in Figure 2c. Note that the loading axis ZZ in this case coincides with the deflection axis c~c~. Figure 3a shows the same cross-section of the round specimen, now subjected to rotating bending with surface strain amplitude e~. The two elements A and B are at the same distance from the strain axis c~c~
Low-cycle fatigue in rotating cantilever under bending I • --
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e
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,
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Figure 2 Determination of stress distribution in flexural bending from axial strain cyclic properties (deflection axis). Therefore, the two elements are at the same strain level, e, as shown in Figure 3b. On the other hand, the two elements now share the same hysteresis loop also, but are not, at any given moment in time, at the same stress level, owing to rotation, as shown in Figure 3c. Point A is moving towards a higher stress level, whereas point B is moving towards a lower one. Each point will ultimately reach a maximum strain level era, when they achieve the maximum vertical distance R from the strain axis. Because of rotation, the stress-strain relation for all points along a circle with radius r is represented by one hysteresis loop. Therefore the section will deflect about the axis c~c~, which makes an angle ~ with the loading axis ZZ. The angle ~b is designated the hysteresis angle to indicate the presence of hysteresis loops on the radii of the section. When the behaviour is elastic, as in high-cycle fatigue, the hysteresis loops degenerate to an elastic line. Therefore the hysteresis angle vanishes. In this case, the rotating bending problem can be treated as a flexural bending problem, wherein the loading and strain axes always coincide.
Stress-strain analysis in rotating bending. Considering low-cycle fatigue tests under rotating bending, a test specimen may be developed 4's such that the tested zone has constant surface strain amplitude e~. Consider
a cross-section in the tested zone with outer radius R, as shown in Figure 3a, subjected to rotating bending. Since in bending it is commonly assumed that plane sections remain plane, the strain will vary linearly across the section, as shown in Figure 3b. The strain em at distance r from the deflection axis a a (strain axis) is
Points A and B, at the same height from the deflection axis, rotate on the circle with radius r. Therefore the stress-strain relation of these points is represented by one hysteresis loop with strain amplitude era, as shown in Figure 3c. To obtain the value of stress at point A (O'A) or at point B (o'B), the shape of the hysteresis loop must first be defined. According to the Masing rule ~, the shape of the settled hysteresis loop branches can be obtained by magnifying the relation of the stress-strain curve by a factor of 2. Since the cyclic stress-strain curve is described by e =
+ \K'/
~2)
where E is the modulus of elasticity, n' is the cyclic strain-hardening exponent and K' is the cyclic strength
404
M.M. Megahed et al.
p
"~
(a)
(b)
8A* (3
~B*Om i/
2am (~
(c)
/ J
Cyclic ~re~rmn
~" T
curve ~m
28m Figure3 Determination of stress distribution in rotating bending from axial strain cyclic properties coefficient, the following relation for the shape of a hysteresis loop is obtained by applying the Masing rule:
Points (1) and (2) in Figure 3c, which lie on the cyclic stress-strain curve, are considered to be the origins of the upper and the lower branches of the settled hysteresis loop corresponding to the strain amplitude em. The upper branch is described by s t r e s s t A R and strain EAR as follows:
ga*=(O'-~-) +21-11n'(tARIlln" \ K' /
(3)
The lower branch is described by stress t~* and strain eB* as follows:
\ K' }
(4)
Stresses tA and t u can then be determined from this settled hysteresis loop with strain range 2era, as shown in Figure 3c, from
t A = O'A* -- t m
(5)
o'B = ~rB* - t m
(6)
Therefore stresses tA*, tB* and t m should be determined first prior to calculating tA and tB. O"m is determined as follows. Points (1) and (2) in Figure 3c lie on the cyclic stress-strain curve described by Equation (2). Hence
tm (tml vw gm = T ' t - \Kt / Considering Equation (1) and making the necessary manipulations yields O.mV, + (K,)t/w ~r,,- (K,)Vw r
= 0
(7)
This is a non-linear equation in tim, which can be solved for all genetic radii 0 ~< r <~ R by the NewtonRaphson method. The determination of t A R and t n * then proceeds as follows. Referring again to Figure 3a, points A and B are defined by the radius r and the angle 0. The angle 0 is measured clockwise, in the direction of rotation,
Low-cycle fatigue in rotating cantilever under bending I from the negative direction of the strain axis aa. Figure 3a shows that points A and B are at the strain level e. Strain relationships can be determined as follows.
value of 49, the bending moments about the axes ZZ and YY can be rewritten as follows: Mzz=COS 49~A O ' J dA r sin 0+sin 49(.]A (TOar
e = em sin O; e g * = e +em = em(1 + sin 0); ea* = e m -
e = em (1 -
sin O)
eA* = (1 + sin 0)
es
(8)
ea* = (1 + sin 0) (R) e~
(9)
Substituting Equation (8) into Equation (3) yields
COS0
0+sin
-
(21-"" K') I;~"(1 + sin 0) ~ e~ = 0
Mzz = M,,~ cos 49 + Moo sin 49
(14a)
Mrv = Mt3t3cos 49 + M
(14b)
sin 49
Note that the bending moment about the YY axis vanishes; hence
(10)
f~/2fR O-A r2 sin 0 dr dO dO
Similarly, substituting Equation (9) into Equation (4) yields
JO
o-B r2 sin 0 dr dO
+ 0
7r/2
o-B r2 sin 0 dr dO
+ J~-
Solving Equations (10) and (11) by using the NewtonRaphson method, O-A* and o-B* can be determined at any point defined by r and 0, under applied surface strain es. Once O'A*, O-B* and o-m are determined, the stresses o-A and o-B can be determined from Equations (5) and (6). The complete stress distribution can now be described with respect to the deflection axis a a , whose direction has not been yet determined with respect to the loading axis ZZ. Hence the hysteresis angle 49 between these two axes must first be determined prior to establishing the final stress distribution with respect to the loading axis.
+
dO
f fR 3 ~-/2
M~ =
Determination of the hysteresis angle. Consider a typical element of area dA located at angle 0 and radius r, as in Figure 3a, where 0 is measured clockwise from the negative direction of the deflection axis a a . The spatial stress distribution O-(O-Aor o-B) has been already determined as outlined above, i.e. the stress is known as a funtion of 0, r and e~. Hence it is possible to determine bending moments about the axes a a , /3/3, ZZ and YY. Thus:
Me=f o-da
(12a)
gA
o-dA rsin(O+ 49)
Mrv = ~ o- dA r cos (0 + 49)
JO
+
o-R r2 cos 0 dr dO rr/2
l)
+
o-B r2 cos 0 dr dO Jrr
J0
+
O-A r2 cos 0 dr dO
Referring to the stress-strain relations as shown in Figure 3c and making the necessary manipulations, Equations (16a) and (16b) can be rewritten as follows: M~ = 2
O-Ar2 sin 0 dr dO \JO
JO
+
o-~ r 2 sin 0 dr JI)
At a specified cyclic loading condition, and at a fixed
(17a)
JO
M~ = 2
O-A 1"2COS 0 dr dO JO
oBr2 cos 0 dr dO JO
JA
(16b)
o
(12b)
(16a)
O-A r2 COS 0 dr dO JO
\JO
Mzz = (
O-Ar2 sin 0 dr dO
0
~7r/2
O;
(15)
Values of M ~ and Mt3t3 are obtained from M,~ =
M.o=f o-dA
(13b)
Substituting Equations (12a) and (12b) in Equation (1 3a) and (1 3b) yields
49 = tan-' (Mt3~-t3)
(r)
(13a)
49fo-darCOSa 49fo-dArsinOA
Mrr=cos
Considering Equation (1), then the final expressions for eg* and ea* are given by
405
(17b)
JO
The above double integrals are determined numerically, by using Simpson's rule for double integration, for any given values of the three material constants K', n' and E, the surface strain e~ and the radius R. Once M ....
M.M. Megahed et al.
406
and Mt3t3 have been obtained, the hysteresis angle 4> can be determined from Equation (15). Then the bending moment Mzz, required for the prescribed surface strain es, is obtained from Equation (14a), with the corresponding elastically calculated surface bending stress, Sb, given by 4 Mzz Sb --
Deflection-strain analysis in rotating bending Introduction. Figure 8a shows, in schematic form, the
,rrR 3
Life can be obtained from the strain-life relation under axial conditions. A curve representing the relation between nominal stress Sb and cycles to faiulre Nf can then be plotted.
Numerical example. In this section, a specified material, SAE 950X-150 HB (S.A.E.7), with known cyclic stress-strain curve and strain-life relation, is considered. The cyclic strain-hardening exponent n' and the cyclic strength coefficient K' of this material are 0.143 and 896 MPa respectively. The behaviour of this material under rotating bending fatigue is obtained by applying the analytical procedure mentioned above. Table 1 summarizes the predicted behaviour of this material under rotating bending. Rotating versus flexural bending fatigue. It is worth noticing material flexural rotation,
that a solution of the same problem for SAE 950X-150Hb can be established under bending fatigue by ignoring the effect of i.e. 4>= 0. Figure 4 compares the obtained results in rotating bending with those of flexural bending for the same material. In high-cycle fatigue, both rotating and flexural bending give approximately the same results. In low-cycle fatigue under flexural bending, the nominal bending stress required to maintain a prescribed surface strain is higher than that under rotating bending at the same surface strain. Therefore low-cycle fatigue tests under flexural bending will give higher lives than those under rotating bending at the same nominal bending stress. Figures 5 and 6 show the predicted stress distribution, at mid-life, over the tested-zone surface in both rotating and flexural bending, at a surface strain of 0.01. Under this condition of equal surface strain, the predicted life is the same for both rotating and flexural bending, but the required nominal bending stress in Table 1 Prediction of rotating bending from axial fatigue information for material SAE 950X-150 HB Axial fatigue data (n'--0.143, K'=896 MPa)
Test no. 0 1 2 3 4 5 6 7 8
e 0.0012 0.0015 0.002 0.0025 0.005 0.0075 0.01 0.015 0.02
flexural bending is higher than that in rotating bending by about 9%. Note that the angle (0 + 4>) representing the horizontal axis in Figure 6 is measured from the loading axis ZZ of the specimen (cf. Figure 3a). Figure 7 depicts the stress-strain relation under the same condition in both rotating and flexural bending.
~r (MPa) 241 265 303 324 386 424 448 479 502
Prediction of rotating bending fatigued data
2Nf
Sh (MPa)
q5 (degrees)
1 130000 289 000 70 200 29 000 4040 1700 950 450 300
244 287.69 351.22 397.05 514.98 571.18 607.53 655.68 688.56
0 3.649 8.22 12.631 26.844 34.069 38.51 43.87 47.088
shape of the tapered test specimen proposed for use in rotating bending fatigue tests 4'5. The specimen is designed so that the nominal stress in the outer fibres of the tapered zone II can be regarded as uniform. This is achieved by adjusting the diameter of the tapered zone to be a rational function of the third degree of the lengthwise coordinate 4,8. The specimen is fixed as a cantilever, and is free to deflect in both the vertical and horizontal directions. Under low-cycle fatigue conditions, it is reasonable to assume that zones I and III will behave elastically, while zone II is strained beyond the proportional limit of the material. Therefore the contributions of zones I and III are deflections 61 and 6ii1 respectively in the vertical direction YY only. Figure 9a shows that the tested zone II will deflect about its strain axis a~, which makes a hysteresis angle 4~ with the loading axis ZZ. Therefore the contribution of zone II is deflection 6., which has two components 6i~v and 6Hh in the vertical and horizontal directions respectively. Thus 8,h = 61i sin ~h; 6,v = ~11 c o s IJ); ~ii = (82Ih + 81Iv) 1/2 (18) The total vertical deflection &, shown in Figure 9b, is the cumulative effect of the deformations of zones I, II and II. Thus 8 v ~ 81 + 811v + 8111 ~ 81 + 811 " COS (~ + 6111; ~h = 811h 7-- 8i I s i n d~
(19)
Figure 8a shows that the deflections 8v and 6h are measured at points a and b respectively. Therefore the experimental hysteresis angle ~bcxp can be obtained as follows: tan ~bexp= 6d(6, - 81 - 6110
(20)
It is now intended to relate the deflections of the specimen to: (a) surface strain amplitude, e,, in zone II; ( b ) a x i a l distribution of bending moment; and (c) dimensions of the specimen. The problem will be treated for both the elastic and elasto-plastic behaviour of the material.
Deflection of a beam bent by transverse loads. Figure 10 shows a beam bent by transverse loads. Hence, for small deformation, the total deflection of point B relative to the tangent at A, i.e. BB', is denoted by 6. and can be defined using the following equation:
6=f~[dslp(X)]dX
(21)
Equation (21) is used to determine the deflection of beams in the general case, particularly when elastoplastic behaviour takes place 9'l°. The bending moment M(X) varies along the length of the beam, as shown in Figure 10. Therefore the
Low-cycle fatigue in rotating cantilever under bending I
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• ~.
407
Elasto-plastic s o l u t i o n in r o t s t l n s bendins Elasto-plastic s o l u t i o n in f l e x u r a l bendin8 Closed-form s o l u t i o n in f l e x u r a l bending Reference-stress s o l u t i o n in f l e x u r a l bendinl
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400
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l05
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Prediction of nominal bending stress-life fatigue curves
Rota~.._.~"
Y
y
~
7-200 MPa
\~
\
Z
Z
Z
--
z
Loading
\
Y
(a) Flexural bending
Y
(b) Rotating bending
F i g u r e 5 Polar plot of predicted stress distribution under surface strain of 0.01 for material SAE 950X-150HB, at mid-life, in: (al flexural bending; (b) rotating bending
408
M.M. Megahed et al.
400;
!=o
\ /
Rotating bendAn~
~
0
i
/
..
o -200
"
\
~
1
1
1
/
-400.
Figure 6 Prediction of stress distribution under surface strain of 0.01 for material SAE 950X-150HB, at mid-life o (MVt)
(a) the bending moment M at the mid-point of zone II, and which is required to maintain the prescribed e~; and (b) the hysteresis angle ~b. Zones I and III are assumed to strain elastically. Thus, referring to Figure 8b, Equation (23) is used to determine deflections 8~ and 6m: 31 =
-0.01 //
]
I
-200
IIll
/- /
/
/
/
/
/
0.01
f
x~ o
L2
--Rotating bending
Figure 7 Stress-strain relation under surface strain amplitude of 0.01 for material SAE 950X-150HB curvature llp/(X) also varies along the beam under elastic conditions ~J. Thus
1
M(X) (22)
where I(X) is the second moment of area about the deflection axis as a function of the length coordinate X. Substitution of Equation (22) in Equation (21) yields
(R I -
(24)
DI/2) 4
~II1 =
f x3 X2
[4 M(X+LO] d X (25) / L2 E (R2 - ~/R 2 - (Xl - X) 2 + Dl/2) 4
The elastic-plastic behaviour takes place in zone II. Therefore Equation (21) is used to determine 6,. Noting that Up(X) = es(X)/R(X), Equation (21) can be rewritten as follows: 8. =
f
X2~s(X) ,
o~
0-~
(26)
Xl / ~ t A )
where e~(X) and R(X) are the surface strain and outer radius respectively, as functions of the length coordinate X. From the design of the test specimen, es(X) can be obtained as a function of surface strain at midpoint of zone II, G, as follows:
G(X+L1)(D2/2)3
(B M(X) 8 = jAE. ~
E
dX
Flexural bending
-4OO
p(X) - E.I(X)
4 M(X+LO / ~/R 2 - (XI-X) 2 +
. XdX
(23)
Equation (23) is used to determine the deflection of beams when the material behaves elastically.
Prediction of specimen deflection under rotating bending. Figure 8b shows the dimensions of the test specimen proposed to be used in rotating bending tests. It is intended to predict the vertical and horizontal deflections 6v and 6, respectively at section CC, where deflections are measured. The given data are: (a) dimensions of the test specimen; (b) surface strain amplitude e~ at mid-point of zone II; and (c) straincycling properties of the specimen material, n' and K', under axial conditions. From the knowledge of n', K' and es, the analytical procedure described above can be applied to obtain:
G(X) =
L2[R(X)] 3
(27)
Substitution of Equation (27) in Equation (26) and making the necessary manipulations yields
f x2 ~s(X+Ll )(D2/2)3 hl = XI L2{D1/2+ ( X - X 1 ) (D3-D1)/[2(X2-XI)] } 4x
dX
(28) Deflections 6~, 6, and 6m given by Equations (24), (28) and (25) can be readily evaluated numerically using Simpson's rule. The predicted vertical and horizontal deflections can then be calculated by applying Equation (19) as follows: 3v = 61 +
6n
COS 6 +
~ni; ~l~ =
blb = 6n sin ~b (19 bis)
The total deflection 6,, at section
CC, is thus given by
Low-cycle fatigue in rotating cantilever under bending I
409
Section at which deflections are
Y//////////////////////////A~.j-o~
--,-- C
"///////////////////////////2~ '
--~-C
--III
II
measured
Section cc
I----.
I 1
I I
J Load
Bending
moment
I I I
I I
Nominal surface stress
(a)
d
t5-t
~2"//////////////////////////~ I=
:ii
17.~
=-=
~1• I
II
X-
L1= 3 ~
= X2=48.2 X3=6E,
• L
=
I
°
L3=102
j ~Ioocl P
(b) Figure 8
Schematic of rotating bending and test specimen configuration and dimensions
6, = ~/(6~, + 6~)
(29)
Numerical example. Consider again material SAE 950X-150 HB. The specimen dimensions for rotating bending are assumed to be as in Figure 8b. The predicted deflection values are obtained by applying the analytical procedures outlined above. Table 2 summarizes the obtained results, while Figure 11 shows the relation between deflection and surface strain amplitude e~. Figure 12 depicts the relation between e, and deflection of specimen zones (I+III) and II. The following relationships are obtained: 6. - 0.002 895
(mm)
(30)
61 + 6m -
136.27 Sb E
(ram)
(31)
where Sb is the maximum nominal bending stress at the mid-point of zone II, and e, is the maximum surface strain at the same point. It is noteworthy to mention that the constants 0.002 895 and 136.27 depend on material type, specimen configuration, dimensions, fixation and measurement location.
Determination of cyclic stress-strain response .from rotating bending fatigue data So far, the procedure adopted to predict the rotating bending fatigue data for a material from the knowledge of its cyclic stress-strain response (n" and K') and modulus of elasticity E has been illustrated. It is intended here to carry out the reverse procedure: i.e.
M.M. Megahed et al.
410 Y
Y
,e
#
i
/\ zJ
,/
\
I
Loading axis
--)z-
\/ #
Y (a)
(b)
Figure 9
(a) The attributed horizontal and vertical deflections of the tapered zone 11. (b) The total recorded vertical deflections at points a and b, respectively (cf. Figure 8a)
Tr/2lR
I I
Cx)/a0 I A
/
F~(n',K',E,e,,R,r,O)dr dO
+ jo
I
(32)
JO
MS~ = Mzz sin 4~exp=
F3(n',R',E s.R,r,O)dr dO JO JO
"~ --~
dO
-'1 Xd0
F4(n',K',E,s,,R,r,O)dr dO
+ JO
x
dX
I
Bending moment
Figure 10
Equations (32) and (33) represent two non-linear equations involving double integration, where all quantities are known except n' and K'. These equations can be recast as follows: F(n',K') = 0;
Deflection of beam under transverse loads
to determine the cyclic stress-strain response for a material from the knowledge of its rotating bending fatigue data, namely: (a) the applied nominal bending stress, Sb (thus bending moment Mzz); (b) the corresponding induced surface strain amplitude, e~, and (c) the corresponding experimental hysteresis angle,
F(no + dn, Ko + dK) = O;
G(no + dn, Ko + dK) = O
Expanding by Taylor series, and truncating after the first-order terms, yields F(no,Ko) + dn
Procedure f o r calculating cyclic stress-strain response. Making use of Equations (5), (6), (7), (10)
G(n~,Ko) + dn
+ dK 0
Tr/2¢R
M ~ = Mzz cos ~exp =
Fl(n',K',E,s.R,r,O)dr dO JO
Table 2
G(n',K') = 0
These two equations can be solved by a series expansion technique ~2. Suppose that (no, Ko) is an approximate solution, and let dn and dK be corrections to be determined. Therefore
(~exp'
and (11), which provide the values of stress at any given point (r,O) in the tested section, the equilibrium Equations (17a) and (17b) can be rewritten as follows:
(33)
JO
0
=0
(34)
=0
(35)
0
+ dK
OK
0
These two linear equations in dn and dK can be solved to give the next approximation. The final values of n' and K' can be determined by carrying out an iteration technique until the values of dn and dK become negligible. Input data consists of the applied nominal bend-
jo
Prediction of specimen deflection under rotating bending fatigue for material SAE 950X-150HB, at mid-life
Surface strain amplitude, s~ Nominal bending stress, Sb (MPa) Hysteresis angle, q5 (degrees) Horizontal deflection, 8h (mm) Vertical deflection, 8v (mm)
0.0012 244 0 0 0.58
0.0015 288 3.6 0.03 0.71
0.002 351 8.2 0.1 0.92
0.0025 397 12.6 0.19 I. I I
0.004 481 22.4 0.53 1.6
0.005 515 26.8 0.78 1.89
0.0065 552 31.7 1.18 2.28
Low-cycle fatigue in rotating cantilever under bending I
//
//
/
• 006
/
/
/
/
/
/
/
=ff
• 004
/
411
//
/
// /
// /
/
/
/
t..
/ /
== r.~
/
/ -/
• 002
' Total deflection, iit
/
If 0
....
Vertical deflection, 8,,
.....
Horizontal deflection, 5,,
1
1
1
2
Deflection (ram)
Figure 11 Prediction of specimen deflection in rotating bending for material SAE 950X-150HB
=~ .oo4
Deflection, 8n
'~ .002
. / / / 0
Deflection, (81 I 3.
+ 8m)
l 2
_--
Deflection (nun) Figure 12
Deflection of test specimen zones I1 and (I+I11) versus surface strain, in rotating bending for material SAE 950X-150HB
ing stress Sh, the corresponding surface strain amplitude e~, the corresponding experimental hysteresis angle &exp, and the modulus of elasticity E. An initial guess (no and Ko) is made at the two unknowns n' and K'. After carrying out the iteration procedure, the final values of n' and K', and hence the corresponding true stress ~r and surface plastic strain amplitude e~, are determined for the test specimen. Repeating this procedure for other specimens as well, the cyclic stress-strain response of the tested material can be calculated.
Application example. From the information of the predicted rotating bending fatigue data for material
SAE 950X-150 HB, shown in Table 2, it is intended to determine the cyclic stress-strain response (n' and K') for the same material. The analytical procedure detailed above was used. Table 3 lists the calculated values of n' and K' corresponding to each prescribed testing condition (Sb, e~, 4)~xp). It is noted that all prescribed test conditions give exactly the same values of n" (0.143) and K' (896MPa), which are the same original values of the considered material, measured under reversed push-pull condition. Consequently, experimental output data of one test specimen in rotating bending fatigue may give the cyclic stress-strain response (n' and K') for the elasto-plastically deformed material, especially when the applied conditions induce
M.M. Megahed et al.
412 Table 3
Determination of cyclic stress-strain response from rotating bending data for material SAE 950X-150 HB Rotating bending fatigue data
Calculated cyclic stress-strain response
Test no.
Sb (MPa)
e~
4~ (degrees)
n'
K' (MPa)
tr (MPa)
1 2 3 4 5 6 7 8
287.69 351.22 297.05 514.05 571.18 607.53 655.68 688.56
0.0015 0.002 0.0025 0.005 0.0075 0.01 0.015 0.02
3.649 8.22 12.631 26.844 34.069 38.51 43.87 47.088
0.144 0.143 0.143 0.143 0.143 0.143 0.143 0.143
903 899 897 896 896 896 896 896
265 303 327 391 424 448 479 502
an appreciable amount of cyclic plasticity (experimentally exhibited by a high value of ~bexp). These results indicate that the numerical procedure proposed herein can be used with confidence to determine the cyclic stress-strain response from well-controlled experimental data collected from rotating bending tests, as shown in a companion paper5.
CONCLUSIONS
A complete and fresh analytical examination of rotating bending fatigue led to the development of a new relationship to determine the surface strain amplitude es, at the mid-point of the tested zone of a tapered rotating cantilever specimen, as function of: the applied nominal bending stress Sb; the recorded values of vertical deflection, 6v, and horizontal deflection, 6h; and the modulus of elasticity E. The theoretical analysis also led to the derivation of a numerical methodology whereby the cyclic stress-strain response could be determined from the measured rotating bending data,
0.000 194 0.000 5 0.000 89 0.003 07 0.005 4 0.007 79 0.001 263 0.017 52
namely: the nominal bending stress Sb, vertical deflection 6v, and horizontal deflection 6h. REFERENCES 1 2 3 4 5 6 7 8 9
10
11 12
Manson, S.S. Exp. Mech. 1965, 5, 193 Manson, S.S. and Muralidharan, U. Fatigue Fract. Eng. Mater. Struct. 1987, 9, 343 Megahed, M.M. Fatigue Fract. Eng. Mater. Struct. 1990, 13, 361 Kawamoto, M., Nakagawa, T. and Ida, A. Bull. JSME 1966, 9 Eleiche, A.M., Megahed, M.M. and Abd-Allah, N.M. Int. J. Fatigue 1996, 18 Masing, G. In 'Proceedings 2nd International Congress for Applied Mechanics, Zurich', 1926 Society of Automotive Engineers, 'Fatigue Design Handbook', 2nd edn Beer, F.P. and Johnston, E.R. Jr 'Mechanics of Materials', McGraw-Hill, New York, 1992 Megahed, M.M. and Leckie, F.A. In 'Cyclic Hardening Analysis of Beams under Cyclic Loading', Second Cairo University MDP Conference, Egypt, 1982 Timoshenko, S. 'Elements of Strength of Materials', Van Nostrand Reinhold, New York, 1970 Chakrabarty, J. 'Theory of Plasticity', McGraw-Hill, New York, 1987 Froberg, C.E. 'Introduction to Numerical Analysis', AddisonWesley, London, 1970