DISCUSSION A-169
Beam No. 18. This was a flat-topped box beam with four was an accelerated bending which eventually became so severe
simple-boam-type bulkheads with simply supported ends. I t is that the beam would carry no additional load. Fig. 1 of this
therefore the type of beam for which the theory should deter- discussion is an experimental curve which shows the buckling
mine the buckling load quite accurately. The following con- deflection ("humping" with respect to the spars) versus axial
stants are given: stress. It is questionable whether the predicted buckling stress
indicated on this curve coincides with actual buckling. How-
I = 0.028 in. 4 bulkhead ever, it is clear that the beam carried considerable load in
L = 8 in., io = 29 in., r = 0.0895 in. excess of buckling. This is undoubtedly due to the fact that
te = average thickness of plate-stringer combination = 0.168 buckling occurred early, and that the spar flanges were capable
in. of picking up the load when the compression panels buckled.
By Equation [7] of the paper, the bulkhead spring coefficient is Beam No. 17. Beam No. 17 was like beam No. 18, but it had
full-depth bulkheads with three vertical struts. N o diagonal
EI (bulkhead) ,r4 X 10.5 X 106 X 0.028 trussing was provided and therefore the bulkheads had very
C = - = 40.3 psi
294 little shear stiffness. The bulkhead coefficient, based upon the
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Hence combined stiffness of the upper and lower bulkhead chord mem-
bers, has been calculated by the theory of Equation [7] of the
CD CL3 40.3 X
1.46 paper to be 59.6 psi. Hence, by the theory of Shou-Ngo Tu, 4
EI El, 10.5 X 106 X 0.168 X 0.0895 2 the buckling stress should have been 4100 psi.
The theory of Shou-Ngo Tu 4 then yields An experimental deflection curve for the panels of this beam is
shown in Fig. 1. Here, again, the theoretical buckling stress was
S„ = 3500 psi greatly exceeded, partly because of the capacity of the spar
flanges to pick up load, and partly because the bulkheads were
r = radius of gyration
supported by the tension surface.
Ser = buckling stress
Beam No. 11. This beam was similar to beam No. 18, but the
Actually, the compression surface of this beam did not buckle in compression surface was cambered. The ratio of beam width to
the Euler sense. Rather, the behavior of the compression panels radius of camber was 0.41. Since the theory takes no account
* "Column With Equal-Spaced Elastic Supports," by Shou-Ngo Tu, of camber the same computation procedures apply as in the ex-
Journal of the Aeronautical Sciences, vol. 11, no. 1, Jan., 1944, p. 07. ample of beam No. 18. These calculations yield S„ — 3200 psi.
Fig. 1 shows an experimental deflection curve for this beam.
It is seen that, the beam carried a much liigher stress than the
predicted buckling stress (3200 psi). A comparison of the curves
KT2.6 MOD
for beam No. 11 and beam No. 18 shows that camber has a de-
AN & BOUT
cidedly beneficial effect.
TREE: E N O U G H The general conclusion to bo drawn from the three tests cited
T O PIVOT
is that the theory establishes a conservative procedure for calcu-
lating the requisite stiffness of wing bulkheads.
AUTHOR'S CLOSURE
The author believes that the comments of Messrs. K. R . Jack-
man and C. Conaway on his paper Parallel Columns With Com-
mon Lateral Supports are correct. The tests which they discuss
had not been performed at the time the original paper was writ-
£ .12.5X1X1
ANGLES Y2-50l ten, and therefore a discussion of the test data could not be in-
cluded in it.
Cumulative Damage In Fatigue1
R . B . B L A N D 2 AND A . A . P U T N A M . 3 Of the four assumptions
T Y P S'PAC.
1/4- X upon which the author bases his development of the damage con-
2.4S-T&4- STR.
051 2 - f S - T S e cept, the first two should be put in more general form. In regard
SKIN
to the first assumption, rather than specify a sinusoidal loading
cycle, it only appears necessary to specify a "simple" cycle, i.e.,
one in which da/dt changes sign only twice during a cycle. N o
work can be recalled by the writers in which it has been shown
that the shape of a simple cycle is important. Furthermore, if
the shape is an important factor, rate of loading and creep effects
would be significant and thus the cycle frequency would also be
an important variable.
In regard to the second assumption, the derivation of
N
1 By M. A. Miner, published in the September, 1945, issue of the
JOURNAL OF APPLIED MECHANICS, Trans. A.S.M.E., vol. 6 7 , p. A - 1 5 9 .
2. A G 8 IO 12 14- IG> 2 Structural Engineer, National Advisory Committee for Aero-
COMPRESSIVE STRE.55 (PSI X IO"') nautics, Washington, D. C.
3 Mechanical Engineer, National Advisory Committee for Aero-
FIG. 1 BEAM DEFLECTION VERSUS STRESS nautics, Washington, D. C. Jun. A.S.M.E.
Copyright © 1946 by ASME
A-170 J O U R N A L OF APPPLIED M E C H A N I C S SEPTEMBER, 194
presented implies that the work per cycle is constant for each to be expected if the concept of cumulative damage is not exactly
magnitude of cycle. This assumption is not necessary for the correct.
derivation. All that is required is that (a) the rate of work input Again, if it is assumed that a stabilization of the work input
as a function of the fraction of total cycles to failure at any one associated with a certain magnitude of cycle takes place in about
magnitude of cycle be independent of the cycle size, and (6) that 100 cycles, it would be expected that the concept of cumulative
the material does not care how the work is put in. This results damage would not be seriously in error for tests such as these
in an equation for the rate of work input of the form where one magnitude of cycle is run for about 104 times before a
different magnitude of cycle is run. However, it does not follow
that this law should hold when the number of the same magnitude
o f cycles in sequence is onfy of the order of 102, 10 l , or even 10°.
The data presented, plus the data of Stiekley which is mentioned
(6 different magnitudes of cycles), show an increased deviation of
rather than the special case the results from those predicted on the basis of cumulative dam-
age with increasing number of different magnitudes of cycles.
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dm
= IF If failure is the result of strain hardening as outlined by the
d
60 theory of Orowan, 4 a family of curves, on a plot of per cent damage
at the second stress to the per cent cycle ratio at the first stress,
and would allow the rate of work input and the damage to re- would result, one curve for each combination of stress and cycle
main small until just before failure, thus explaining the absence ratio. These curves would be centered about the curve deter-
of significant effects of fatigue-stressing on other properties such n
mined by ) = 1. However, a small number of tests which
as tensile strength and impact resistance.
The third assumption is not pertinent to the concept of dam- give only one or two points on a curve would give a pattern of
age and is apparently only to be used to justify an approximate points which would not reveal the curves (especially with the
method of relating various stress ratios. It so happens that one scatter inherent in fatigue tests) but which would appear random
of the writers was formerly associated with a major airplane manu-
facturer in this same field of work and had occasion to develop E—n = 1. In other words, the number of
methods of fatigue analysis that could be applied to parts still N
on the drawing board. Due to the lack of adequate material variables in these tests is so large and the number of tests is so
data, it was necessary to attempt to predict fatigue strength at small that from them no conclusions can be drawn as to the cor-
various range ratios and cycle numbers from fatigue-strength rectness of any theory.
data for completely reversed stress. It was subsequently found The concept of cumulative damage does give a method of mak-
that the results obtained from the use of the Goodman diagram ing an estimate of life of structures with discrete types of load-
for this purpose were so seriously in error that the entire project ing, and what may be more important at the present state of
had to be discarded. knowledge, it gives a basis for comparing various structures under
As an example of the error involved in this approximation, it is the same loading. However, caution must be used in extending
possible to compare the extreme range of test values given in Fig. the theory to many of the problems common to industry. For
S example, the loads on an airplane wing, caused by atmospheric
turbulence, occur in a more or less random manner about a fairly
1(6) of the author's paper, with the derived value. At — = 0.5 constant mean load. If the fatigue life of a structural member
the possible R values range from 0.65 to 0.2 and the corresponding in the wing is to be determined by using this concept, should the
deviations from the mean load be paired in pairs of equal absolute
minimum stress values range from 0.32 SU to 0.1 SU; at — = 0.3 but opposite numerical magnitude to form cycles about the mean
the possible R values range from 0.2 to —1 and the minimum load, or considered each separately, or grouped statistically to
stress values range from 0.06 <S„ to —0.3 SU (compression). The form cycles of various amplitudes and mean stresses similar to
writers would not consider this range of values to be a good ap- actual sequence of loads? Large factors will separate the lives
proximation, and still wider deviations are to be found when so obtained and there is no reason to believe that any one of
applying the Goodman diagram at lower cycle numbers. The these methods is least in error.
writers believe that due to the inherent inaccuracies of the "ap- AUTHOR'S CLOSURE
proximation" the Goodman diagram is not usable at all for this
Messrs. Bland and Putnam's comments regarding the more
type of work, where margins greater than 10 to 15 per cent, on
general nature of both the cycle of loading and the broader form
stress, are considered excessive. In fact, if data are available to
of the expression for damage are appreciated. The remarks are
define the actual shape of the Goodman diagram so that the
justified by experimental information available at present. Their
straight-line approximation need not be used, then those data
comments concerning the lack of accuracy of the modified Good-
can define the entire fatigue surface and the Goodman diagram
man diagram are correct; however, it should be pointed out that
is superfluous. Parenthetically, the writers would like to know
no use of this method was made in the development or experi-
if the derived curve in Fig. 1(b), likewise represents the other
mental confirmation of the cumulative-damage concept. The
extreme range of test values.
Goodman diagram was introduced to provide an approximate
The confirmation of the concept of cumulative damage is based
method in cases where suitable SN-curves are not available. The
upon 22 tests in which a large number of variables were present.
author heartily concurs that this approach is at best a poor sub-
For this reason it is difficult to determine if any trends are pres-
stitute for experimental data.
ent due to the magnitude of cycles, sequence of cycles, or range
ratio. However, statistical tests were applied to the data in With regard to any possible correlation between difference in
which only two magnitudes of cycles were used, with the result •v-v ii
ranges of stress and the absolute value of 2_, the following
that a correlation was found between the difference in ranges of
ENn 1. Such a correlation is
4 "Theory of the Fatigue of Metals," by E. Orowan, Proceedings
of the Royal Society of London, series A, vol. 171, 1939, p. 79.
DISCUSSION A-17I
add itional tests have been completed using steel specimens, and the author has conducted cxperiments at very high loads (95 to
they do not show a signifi cant trend. 100 per cent of t he ul timate strength for 24S-T a lclad sheet
Seven I'Otating-beam specimens (flash-welded 4130 steel specimens) , obtaining resul ts that are not subject to rational
tubes) were tested at various numbers of stress magnitudes analysis owing to scatter. However, si nce the st.rtlCtural alumi-
num a lloys normally fail at from 1000 to 10,000 cycles of loading
ra nging from two to t.e n. The average value for L ~'\\'as 1.11; (varying with stress ratio and norma l scattcr) at 95 per cent of
the ultimate tensi le strength, t here is only this narrow region at
for a specimen I'lIl1 at ten different st ress values, L ~ = 1.06 ; very hi gh stresses within which t he cumula tive damage is not
for nine levels, 0.86; and for fiv e level., 1.20. adequate for present ana lysis purposes. Of course for a prac-
tical casc, material with various stress concentrations must be
N ine flat sheet specimens of S.A.E. 4130 steel were tested at
considercd, but here again the u ual margin of safety found in a
from two to four (lifferent tress levels a nd at various stress-ratio
structure will normally obviate stresses of such a magnitude as to
n
values. The average va lue was L
N = 0.94, the maximum resul t in fatiguc fai lures in less than 1000 cycles.
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The use of the cum ul ativc-damage concept provides con-
1.'13 (at two levels), a nd the minimum 0.69 (also at two sidera bly greater accuracy in the ana lysis of the effects of gust
levels). loads on aircraft structures than is warranted by prcsently a vaila-
It is noted that t he over-all average for t he original 22 a lumi- ble information on the frequency of occurrence of atmospheri c
num-:Illoy specimens plus ' t hese 16 steel specimen ' re: ul ts in distmbances. Also, it is the ta k of the analyst and t he ex-
n
L N deviating by on ly 1.5 pel' cent from the thcoretical va lue perimenter to detcrm ine the propel' method for handling gust
loadings, a problem independent from that of the damage ques-
of 1. tion .
To cst.ab li. h thc effect. of a numbcr of different· magnitudes of The general comments rega rding conclusions draw n from a
stress, another experimentcr hn s testcd an nluminum-alloy . mall number of tcsts with a Inrge number of variables arc valid
specimen for 28 excursions of a repen t.ing stre s pnttern which to the extent that a prceise a nalysis is not possible ai present by
contnined four diffel'Cnt mngnitudc. of. tre ·s, nil nt n constant use of the cumulativc-damage concept nor is that the in tent.
value of mean t·rcss. The present status of a ll fatigue-annlysis methods, and experi-
ments, is such that any significnnt trend which may be shown to
The failure 0 'e\ll'rcd at II L N- on ly. 1 pel' cent different from
It
exist is very useful in spite of limi ted accm acy. The need is
t he theoreLica l valuc. 1\0 : i ~nificant co rrelation with the num- great for furthcr experimentation on genera l fatigue problems
ber of magnitude: of stress is appnrent for t he data presented and the damage question in particular.
on either a lwninum or stcel specimens. The author wishes to acknowledge an original theoret ical
The sugges t.ion that s t.abiliz nt.ion of work input occ urs nt some approach to the problem of repeated loadings of ail'craft struc-
relatively small number of cycles, say, 100, offers interest.ing tmes in which a damage criterion was previously proposed by
possibilities for mod ification if the cum ulntive-damage co ncept P. H. Dcnkc, of t he Douglas Aircraft Company, Santa
does not acc ura tely represent this region of loading. However, l\Ionica, Ca lif.
