Fatigue Resistance and Allowable Stress of Ductile ...

15

Fatigue Resistance and Allowable Stress of Ductile Materials

under Combined Alternating Stresses.

By Kazuo Terazawa, Kogakusiti, Member.

Professor of the Yokohama Technical College.

Synopsis,

Considering the effects due to mean stress and due to variable stress on the fatigue

failure of ductile materials separately, the shear strain energy law (v. Mises-Hencky's

law) has been applied to the fatigue test results for bending and twisting. The con-

clusion is obtained that, if the range of variable shear strain energy has a smaller

value than a certain constant value, and if the maximum shear strain energy during

one cycle of stresses does not exceed the shear strain energy corresponding to the

statical yield point, a ductile material can sustain an infinite number of cycles of

combined stresses, with neither fatigue failure nor statical yielding. In the light of

this conclusion, a method, which gives the allowable stress for ductile materials under

combined alternating stresses, has been derived.

I. Introduction.

For a ductile material subjected to combined statical stresses, it is generally

accepted(1) that the condition which causes the statical failure is given accurately

by the shear strain energy law and approximately by the maximum shear stress

law. Denoting each of the principal stresses by ƒÐ1•„ƒÐ2•„ƒÐ3, the latter law is written

as

(1.1)

and the former one as

(1.2)The above statements are confined to the case of statical stress, but, in practice,

(1) V. Lode: Der Einfluss der mittleren Hauptspannung auf das Fliessen der Metalle; Forschungsarbeiten V. D. I., Heft 303, (.1928),G. Cook: The Yield Point and Initial Stage of Plastic Strain in Mild Steel subjected to Uni-form and Non-Uniform Stress Distribution; Phil. Trans., Series A, vol. 230 (1931), p. 103,G. I. Taylor & H. Quinney: The Plastic Distorsion of Metals; Phil. Trans., Series A, vol. 230(7931), p. 323,W. A. Scoble: The Strength and Behaviour of Ductile Materials under Combined Stress; Phil.Mag, vol. 12 (1006), p . 533,and Others.

16 K. Terazawa:

the structure- and machine-parts are often subjected to alternating stress. That is

to say, they are often required to stand for combined actions of compound variable

stresses and compound statical stresses. It seems very important, therefore, to find

the condition which causes the failure of material under such a state of stress, as

a guide for a reasonable estimation of the allowable stress. With regard to this

subject, many investigations have been made and valuable results published. Most

of these investigations are confined to such limited cases, in which a material is

subjected to combined actions of reversed bending and reversed or statical torsion,

so that their results are not immediately applicable to the general case of practical

design work.

The present writer intends to attack this problem by using the results obtained

by previous investigators, and to deduce a law of fatigue failure which is applicable

to the general design work. In the first part of this paper, the fatigue failure of

ductile materials under combined alternating stresses will be discussed, and, in the

latter part, using the results obtained in the former part, the method of estimating

the allowable stress will be derived and some examples of practical interest will be

illustrated.

II. On the Law of Fatigue Failure for Ductile Materials

under Combined Alternating Stresses.

The fatigue properties of materials are generally tested under simple straining

actions, such as alternating bending or twisting; the results of these tests are

generally shown in a well-known fatigue diagram. Fig. 1 shows a fatigue-diagram

which represents a relation between the safe range of stress and the mean stress.

In this figure, or or T and or are mean stress and variable stress of an alter-

nating stress, and being normal stresses, and r and shear stresses respectively,

hence the maximum stress and the minimum stress are given by or T + and

σ-ξ or т-ζ, according as whether the stress is a normal one or a shear one. The

statical yield points for normal and shear stresses are represented by ƒÐs, and „„s.

A material may withstand an infinite number of cycles of any stress whose

maximum and minimum values are indicated by points which lie inside the area

ABUDE without fatigue failure. It is generally accepted that the statical yielding

failure may occur when the maximum value of an alternating stress exceeds the

statical yield point as or T. And the statical yield point is indicated by a point

lying on the straight line BC in Fig. 1; the equation of this line is given as ƒÐ+ƒÄ

Fatigue Resistance and Allowable Stress of Ductile Materials &c. 17

=σs or T+ζ=7s. Hence,

an alternating stress os-

cillating between two

extreme values, which are

indicated by points lying

inside the area ABCDE,

will cause neither statical

yielding nor fatigue fai-

lure of a material after

an infinite number of

cycles.

Structure- and ma-

chine-parts are required

to withstand all kinds of

alternating stresses with-

out statical and fatigue

failure. And the condi-

tion for statical failure

is more acculately given

by the shear strain energy

law (1.2). That is to say,

the statical yielding due

to alternating stresses

will occur when the

maximum shear strain

energy during one cycle

of stresses exceeds the shear strain energy corresponding to statical yield point.

Hence, the present treatment is confined to such a fatigue failure preceding statical

failure; that is, the fatigue failure under the alternating stresses, corresponding to

which the maximum shear strain energy does not exceed the shear strain energy

for statical yielding, will be discussed in this paper. For instance, if the case of

simple alternating stress is concerned, the fatigue failure due to the alternating

stress, the extreme values of which corresponds to points lying on the curves AB

and ED in Fig. 1, will be our subject.

Fig. 1. Fatigue Diagram.

18 K. Terazawa:

1. The Fatigue Failure due to Combined Reversed Stresses.

The reversed stress means an alternating stress whose mean stress has zero

value. With regard to the law of failure for ductile materials subjected to combined

reversed stresses, a large number of investigations have been previously made, but the

investigators have not always arrived at the same conclusion. Hence, by collecting

the various results obtained by previous investigators together, the writer intends

to deduce a law of fatigue failure, which is applicable for practical design works.

The method for attacking this subject, in this paper, is similar to that adopted

by the previous investigators. A bar under bending moment is subjected to an

uni-axial stress, and torsional moment produces a state of combined stresses, i. e.,

bi-axial stress, in which two of the principal stresses are equal in magnitude and

opposite in sign. By comparing the fatigue limit for reversed flexure with that for

reversed torsion, a condition which causes the fatigue failure under combined

Table 1.

Fatigue Resistance and Allowable Stress of Ductile Materials &c. 19

reversed stresses may be known, as will be described later.

It is a wellknown fact that, owing to the effect due to stress-distribution on

the statical failure, the yield point of a ductile material under bending has a large

value than that under direct stress.(1) In the case of fatigue failure, too, the effect

due to stress-distribution on a fatigue resistance can not be disregarded, as a fatigue

limit for flexure has a larger value than that for direct stress. In the both cases

of bending and twisting, however, as the stress is distributed in a similar manner,

the fatigue limit for flexure may be immediately compared with that for torsion,

disregarding the effect due to stress-distribution.

Now, putting the results obtained by the previous investigators together, the

reversed torsional fatigue limit -17. and the reversed bending fatigue limit br of

ductile material, are as shown in Table 1. Some of the test-results quoted in this

table are those obtained by Turner, (2) Stanton and Batson,(3) Gough,(4) and others are

seen in the " Dauerfestigkeits-Schaubilder ".(5)

Plotting these results in the Ābr - Ātr plane, they are shown graphically in Fig. 2.

By this figure, the following relation may be obtained.

(2.1)To apply the maximum shear stress law to the present case, substituting and

ξ3 forσ1 and σ3 in the expression (1.1), this law is written as

(2.2)

where each of ƒÌ1•„ƒÌ2•„ƒÌ3 indicates the variable principal stress. The value of ratio

of fatigue limit for reversed torsion to that for reversed bending is given by (2.2)

as 0.5, 1. e.,

(2.3)(1) F . Nakanishi: On the Yielding of Mild Steel; Jour. Soc. Mech. Engrs. in Japan, vol. 31, no.

130 (1928), p. 39,

F. Nakanishi: Strength of M id Steel Beams under Uniform Bending; Jour. Soc. Mech. Engrs.

in Japan, vol. 32, no. 144 (1929), p. 171,

F. Nakanishi: Strength of Mild Steel Beams under Uniform Bending (2nd Report); Jour. Soc.

Mech. Engrs, in Japan, vol. 34, no. 165 (1931), p. 21,

etc.(2) L. B. Turner: The Strength of Steels in Compound Stress, and Endurance under Repetition

of Stress; Engng. vol. 92 (1911), p. 115, 133, 246, 305.(3) T. F. Stanton & R. G. Batson: On the Fatigue Resistance of Mild Steel under Various Con-

dition of Stress Distribution; Engng. vol. 102 (1916), p. 269.

(4) H. J. Gough: The Strength of Metals under Combined Alternating Stresses; Engng. vol. 140

(1935), p. 511 & 565.(5) Fachausschuss fiir Maschinenelemente bei V. D. I.: Dauerfestigkeits-Schaubilder; Beilage zur

Z. V. D. I. Bd. 77 (1933).

20 K. Terazawa :

Comparing (2.3) with (2.1), it will be seen that the difference between the

value of the ratio computed by the law and that obtained experimentally amounts

to about twenty per cent. This amount of error is in the same order as that in

the case where this law is applied to ductile material under statical stress, and in

this kind of problem the error within this amount may be considered as being

practically permissible. Hence, Turner, Stanton and Batson, and others had suggestedthat maximum shear strt ss law is a safe side approximation for ductile materials

under combined reversed stresses.

On the other hand, if the shear strain energy law holds good also in this case

as in the case of statical stress, the failure of a ductile matel ial under combined

reversed stresses should be caused according to the following condition.

(2.4)

and, from this condition, the following relation is given.

(2.5)

The difference between (2.5) and (2.1) is so small in its amount that it can be

neglected safely. According to the recent experimental investigation on ductile

materials under various kinds of combinations of reversed torsion and reversed

bending, H. J. Gough(1) has suggested that the shear strain energy law is the most

(1) H. J. Gough : loc. cit.

Fig. 2. Relation between Reversed and Twisting Fatigue Limits.

Fatigue Resistance and Allowable Stress of Ductile Materials &c . 21

preferable for these cases. Thence, we may now conclude that the shear strainneergy law is a very reasonable one, while the maximum shear stress law is only a

safe side approximation for ductile materials subjected to combined reversed stresses.

2. The Fatigue Failure due to Combined Alternating Stresses .

In this article, the fatigue failure of ductile materials caused by combined alter-

nating stresses, which have finite values in their mean stresses, will be discussed.

An alternating stress is to bJ completely defined when the variable stress and

the mean stress are given. The mean stress acts statically and the variable stress

is one half of the range of varying stress, that•is, these two stresses will act in a

different manner. Therefore, the effect due to the mean stress on the fatigue failure

should be distinguished from that due to the variable stress. , In the following

treatment, on this point of view, the results of fatigue tests on a material under

bending will b compared with those on the same material under torsion.

(2. a) The Maximum Shear Stress Law. The bending fatigue limit and twisti

ing one are denoted by σb+ξband Tt+ζt; σband 7t represent mean stresses and ξb

and　 ζt,　are　 variable　 strcsses.

As a mean stress is to be distinguished from a variable stress in their effects

on a fatigue failure, when the values of the maximum shear stresses corresponding

to mean stresses are same in both cases of alternating bending and torsion , the corresponding variable maximum shear stress should be equal in its magnitude in

these two cases, if it is assumed that the maximum shear stress law holds well .

That is, when the condition

(2.6)

is fulfilled, the following relation should be given.

(2.7)

If we calculate the values of ratio ofƒÄt to ƒÌb), satisfying the above condition

(2.6), from the test-data,(1 then they are as shown in Table 2. And the average

values of them for each material are given as

(2.8)Comparing (2.7) with (2.8), the difference between the theoretical value of the

ratio and the experimental one is found to be twenty per cent in its amount .

(1) Numerical data used in this calculation are found in the "Dauerfestigkeits -Schaubilder", Beilage

zur Z. V. D. I. Bd. 77 (1933).

22 K. Terazawa:

Hence, it may be said that the maximum shear stress law does not hold very well

in this case.

(2. b) The Strain Energy Law. In a material subjected to an alternating

stress, the strain energy varies, also, with the cycle of stress, and this alternating

energy will be considered as consisting of two kinds of energies, statical and vari-

able. The statical strain energy is the one corresponding to the mean stress and

expressed as a function of the mean stress alone. The range of variable strain

energy denotes the value of strain energy varying in one cycle of stress. The more

rigorous definition for it will be described later.

The strain energy is divided into two parts, one due to the change in volume

and the other due to the distortion. For the present purpose, only the latter part,

-.e., the shear strain energy, is necessary to be considered. The shear strain energy

Table 2.

Fatigue Resistance and Allowable Stress of Ductile Materials &c.23

per unit volume is generally represented by the following expression.

where and E are Poisson's ratio and Young's modulus, and C= The

quantity W, to which the shear strain energy is proportional, is given by

(2.9)Denoting the quantities W's which are proportional to the statical shear strain

energy (shear strain energy corresponding to mean stresses) and the range of vari-

able shear strain energy, which correspond to torsional and flexural fatigue limits ,by Wtm, Wbm, and Wb, respectively, we get the following expressions according

to (2.9), i. e.,

(2.10)

(2.11)

or(2.12)

or(2.13)

In this case too, each of the effects due to the statical shear strain energy and

the range of variable shear strain energy on the fatigue failure should be considered

separately as was already mentioned in the above. If we assume that the shear

strain energy law holds well for the present kind of failure, the range of variable

shear strain energy for the fatigue limit should have the same value in both cases

of flexure and torsion when the corresponding statical shear strain energies are equal

in their magnitudes. That is, when

(2.14)

it should follow that

(2.15)

According to the statements (2.10)•`(2.i3), and the condition (2.14) being ful-

filled, the values of ratio of Wtv to Wbv are evaluated from the test data(' and the

results are shown in Table 3. The average values of them for each mater:al are

obtained as

(2.16)

A small difference which amounts to nearly ten per cent is found between the

(1) Numerical data used in this calculation are seen in the " Dauerfestigkeits-Schaubilde. ", Beilage

zur Z. V. D. I. Bd. 77 (1933).

24

theoretical (2.15) and the experimental (2.16) values, and this difference is a smaller

percentage than that found between (2.6) and (2.7). Therefore, the shear strainenergy law seems to hold more acculately than the maximum shear stress law, forthe fatigue failure of ductile material under combined stresses. Accordingly, we

may conclude that an infinite number of cycles of combined stresses are required

to cause a fatigue failure of a ductile material, when the range of variable shear

strain energy has a certain limiting value and the maximum shear strain energy

during one cycle does not exceed the shear strain energy for statical yielding.

Table 3.

K. Terazawa:

Fatigue Resistance and Allowable Stress of Ductile Materials &c. 25

3. The Limiting Value of the Range of Variable Shear Strain Energy.

In the preceding articles it has been shown that the fatigue failure may occur

after an infinite number of cycles of stresses if the range of variable shear strain energy corresponding to combined alternating stresses possesses a certain limiting

value. Now, we will proceed to obtain the relation between the limiting range of variable shear strain energy and the corresponding statical shear strain energy.

Representing the quantities, IV's, which are proportional to the shear strain energy corresponding to the statical yield point and to the range of variable shear

strain energy for the reversed fatigue limit, by Ws, and Wr, the quantities Wbm.Wvs,

Wtm/Wts, Wbv/Wbr and Wtv/tvi Wtr, in which the s4ffix b and t mean bending and torsion, are calculated and the results are graphically shown in Fig. 3 and Fig. 4

Some of the test-data, from which the above quantities are calculated, were taken

f corn Gough's book(1) and Nishihara's paper,(2) and others are seen in the " Dauer-

festigkeits-Schaubilder ".(3) These figures show as if there might exist a certain

(1) H. J. Gough: The Fatigue of Metals, 1926, p. 72-74.

(2) T. Nishihara and others: Dauerversuche der Stahle fi .ir Zug-. Druck- und Biegungsbeanspruch-

ungen; Jour. Soc. Mech. Engrs. in Japan, vol. 36 (1933), p. 673. (3) Dauerfestigkeits-Schaubilder: loc. cit.

Fig. 3. Relation between W and Wm (by Dauerfestigkeits-Schaubilder).

26 K. Terazawa:

relation which connects the limiting value of the range of variable shear strain

energy with the corresponding statical shear strain energy.

Jasper'" suggested that the range of variable strain energy corresponding to the

fatigue limit for simple alternating stress possesses a certain constant value which

is independent of the statical strain energy. In the cases of flexure and of torsion,

the mathematical representation of the strain energy is in a form similar to that

of the shear strain energy. Hence, we will now follow the Jasper's suggestion and

assume that the relation between the limiting value of the range of variable shear

strain energy and the corresponding statical shear strain energy is representedby

the straight line AB in Figs. 3 and 4. That is to say, if the range of variableshear

strain energy has a certain constant value which is independent of the staticalshear

strain energy, and if the maximum shear strain energy during one cycle does not

exceed the shear strain energy for statical yielding, an infinite number of cycles

of combined stresses will be required to cause fatigue failure previous to statical

yielding. Some years ago, Ono(2) made a fatigue test on various kinds of steels under

combined statical torsion and reversed flexure, and his results are reproduced in

Table 4. In order to ascertain whether the above conclusion is reasonable or not,

(1) T. M. Jasper: The Value of Energy Relation in the Testing of Ferrous Metalsat Varying

Range of Stress and at Intermediate and High Temperatures, Phil. Mag . vol. 46 (1923), p. 6C9. (2) A . Ono : Fatigue of Steel under Combined Bending and Torsion; Memories of the Coll. En;.

Kyushu Imp. Univ. vol. 2 (1920-1922), p. 117.

Fig . 4. Relation between Wv and Wm (by Gough's book and Nishihara's paper).

Fatigue Resistance and Allowable Stress of Ductile Materialpi Aye 27

these results are plotted in

the Wv Wm plane as shown

in Fig. 5. In this figure,

it can be seen that the

present conclusion is a

reasonable one and a close

approximation for ductile

materials.

Now, representing the

quantity, which is pro-

portional to the range of

variable shear strain

energy corresponding to

fatigue limit for any combined alternating stress, by Wv the equation of the straight

line AB in these figures is given by

(2.17)where Wr is the quantity which is proportional to the range of variable shear

strain energy for reversed fatigue limit. The result , which was obtained in Article 1 of this Section, shows that Wr, and also the range of variable shear strain energy , corresponding to reversed fatigue limit has always a constant value for any combined

reversed stresses. Therefore, the condition for fatigue failure is given by theabove

equation (2.17). And the constant value of TV, is computed, by putting o-b=0 and

6,= ebr in (2.12), as follows

(2.18)

Table 4.

Fig. 5. Relation between Wv and Wm (by Ono's exp .).

28 K. Terazawa:

From (2.17) and (2.18), we get

(2.19)

In the above treatment, only a case of combined stresses, in which stressdistri-

butions are linearly varying, was discussed. In a case of combined stresses which

are distributed uniformly, taking the effect due to stress-distribution on a fatigue

resistance into account, the above statement (2.19) should be rewritten in the follow-

in form.

(2.20)

where er is the fatigue limit for reversed direct stress.

Therefore, it can be concluded that, if the range of variable shear strain energy

has a smaller value than a certain constant value, i. e., the limiting range ofvari-

able shear strain energy for simple reversed stress, and if the maximum shear

strain energy does not exceed the shear strain energy for statical yielding, a ductile

material can sustain an infinite number of cycles of combined stresses, with neither

fatigue failure nor statical yielding.

It must be noticed that the above treatment concerns only with the limited

case in which all of alternating stress-components have an equal frequency and are

in the same phase ; and other cases are reserved to a future investigation.

III.•@ The Range of Variable Shear Strain Energy.

In the case of three- or two-dimensional alternating stresses, it is not generally

so simple to express the range of variable shear strain energy in terms of stresses

as in the cases treated in the above. In this section, the general expression of the

range of variable shear strain energy corresponding to combined stresses, which

oscillate in the same frequency and the same phase, will be obtained.

1. Three-Dimensional Alterrnating Stresses.

Let K, Y and Z be normal stresses and X Y, YZ and ZY shear stresses, then

the quantity Jr, which is proportional to the shear strain energy, is given by

(3.1)

In the case of alternating stress, all stress-components at any instant are repro-

sented as are shown in Fig. 6. In this figure, cr and T are mean normal stress

and mean shear stress, and and ' are variable normal stress and variable shear

stress, respectively. And T denotes a half period of stress-cycle and the time is

represented by t. According to (3.1), the quantity W, which is proportional to the

shear strain energy at any time t, is expressed as follows :

Fatigue Res:stance and Allowable Stress of Ductile Materials &c. 29

(3.2)

Now putting

(3.3)

the above expression (3.2) is given in the following simple form.

(3.4)

The extreme values which the quantity TV takes during a h.1 if cycle of stresses

are obtained by using the expression (3.4). And the roots of the following equation,

Fig. 6. Alternating Stress Components. (3-dimensional).

30 K. Terazawa:

will give the time at which the quantity TV together with the shear strain energy

takes the extreme values. Solving this equation with respect to t, we get

(3.5)

Substituting these roots in the expression (3.4), the extreme values of TV are

obtained as follows : -

(3.6)

And these extreme values take the maximum or minimum value according as

whether ƒ¿2 is positive or negative and I ad is larger or smaller than ƒ¿3, as shown

in the following.

(1) When ƒ¿2=0, or when ƒ¿2•‚0 and ƒ¿3 •„ •bƒ¿2•b ,

W becomes maximum at t=0,

W becomes minimum at cos t=-

and W becomes maximum at t= T.

(2) When ƒ¿2 •„0 and ƒ¿3 •…•bƒ¿2•b,

W becomes maximum at t= 0

and W becomes minimum at t = T.

(3) When ƒ¿2 •ƒ0 and ƒ¿3 I •bƒ¿2•b

W becomes minimum at t=0

and W becomes maximum at t = T.

Accordingly, the quantity Wv, which is proportional to the range of variable

shear strain energy, is obtained as shown in the following.

(1) When ƒ¿2 = 0, or when ƒ¿2•‚0 and ƒ¿3 •„•bƒ¿2•b ,

(3.7)

(2) When ƒ¿2•‚0 and ƒ¿3•‚•bƒ¿2•b ,

Fatigue Resistance and Allowable Stress of Ductile Materials &c. 31

(3.8)

By the statements (3.3) and (3.7) or (3.8), we can obtain the general expression

of Wy, so that, accordingly, the range of variable shear strain energy can be ex.-

pressed as a function of stress-components, mean and variable.

2. Two-Dimensional Alternating Stresses.

In the case of two-dimensional alternating stresses, putting

in Fig. 6, stress-components at any time are given. Hence, corresponding to (3.3),

if we put

(3.9)

the quantity W, which is proportional to the shear strain energy at any instant,

can be written as follows:

(3.10)

The expression (3.10) is similar in form to that of (3.4), and the extreme values of

the quantity W are obtained in a similar manner to that in the preceding case.

Hence, by substituting gb ƒÀi for at in the expression (3.6), the extreme values of TV in

this case are obtained as in the following.

(3.11)

And the quantity WV, which is proportional to the range of variable shear strain

energy, is obtained by substituting for at in the expression (3.7) and (3.8) as in

the following.

( 1 ) When ƒÀ2=0, or when ƒÀ2=0 and ƒÀ3•„•bƒÀ2•b,

(3.12)

( 2 ) When ƒÀ•‚0 and 1821,

(3.13)

By using (3.3) and (3.12) or (3 13) we can represent the quantity W, and also

the range of variable shear strain energy, for the two-dimensional case as a function

32 K.Terazawa:

of stress-components, mean and variable.

Accordingly, by using the results obtained in this section, the condition (2.17),

i. e., (2.19) or (2.20), which causes the fatigue failure of ductile material under

combined stresses can be represented as a function of mean and variable stress-

components.

IV. The Allowable Stress of Ductile Materials Subjected toCombined Alternating Stresses.

The allowable stress must be estimated so as to make the structure- or machine-

part stand for the statical yielding and the fatigue failure, and the conditions for it will be given by (2.17) and (1.2), as

(4.1)and (4.2)

where Wmax. and Ws mean the quantities which are proportional to the maximum

shear strain energy during one cycle of stress and to the shear strain energy cor-

responding to the statical yield point, respectively.

1. A Single Condition for Estimation of Allowable Stress.

As the above two condition (4.1) and (4.2) are mutually independent, a so-called

trial method is to be used, in order to determine the safe dimensions of structure-

or machine-part, in which acting stresses must satisfy each of these conditions.

Although the application of the trial method to this case is not so difficult, yet it

is very troublesome. Therefore, it seems very convenient if we can replace the above

two conditions by a single one. For this purpose, by representing these conditions

(4.1) and (4.2) graphically in the Wv-W. plane, we want to determine the safe

domain, in which the ordinate of every point indicates the value of TV, causing

neither statical yielding nor fatigue failure.

To represent the condition (4.2) in the Wv-W. plane, it is necessary to find a

relation among Try, W. and Wmax.. And, to be safe in any case of combined

stresses, this relation has to be the one which gives the minimum value of Wv of

all Wv's corresponding to a given value of Wm,, for all eases when Wmax. is (qual to

the constant value Ws. It seems very difficult to deduce the above relation from

the treatment of the general case. Therefore, the present writer will attack this

subject from another point of view, as is described in the following.

During one cycle of stresses, the statical shear strain energy corresponding to

mean stresses is always smaller than the maximum shear strain energy and always

Fatigue Resistance and Allowable Stress of Ductile Materials &c. 33

larger than the minimum shear strain energy, in their magnitude, i. e.,

Hence, the difference between Wmax, and W. is alwayssmaller than the true value of

117, in all cases. Accordingly, as a safe side approximation for the above relation,

the following expression can be given.

(4.3)Substituting (4.2) in (4.3), the following expression is obtained.

(4.4)

Accordingly, the condition (4.2) can be replaced by the above statement (4.4).

For representing the conditions (4.1) and (4.4) in the Wv - Wm plane so as to

be able to determine the safe domain in which the ordinate of every point indicates

TIT,, causing neither fatigue failure nor statical yielding for all ductile material, the

value of the ratio of reversed fatigue limit to the statical yield point has to be

given. The reversed fatigue limits and the statical yield points are seen in Table 5, and these values are plotted as are shown in Figs.7 a and 7 b. Figs. 7 a and

7 b correspond to bending and direct stress, respectively. By the aid of these figures,

the value of the above ratio can be obtained as follows:-

(4.5)

Table 5.

34 K. Terazawa:

and

(4.6)

Using (4.5) and (4.6), if we represent the conditions (4.1) and (4.4) in the W;

―Ｗm plane, they are shown by straight lines RA and BS in Figs. 8 and 9. Figs.

8 and 9 correspond to the case of uniform stress-distribution and to that of linearly

varying stress-distribution respectively. The ordinate of a point lying in the area

Fig. 7a. Direct stress.

Fig. 7b. Bending stress.

Fig. 7, Ratio of Reversed Fatigue Limit to Yield Point.

Fatigue Resistance and Allowable Stress of Ductile Materials &c . 35

Fig. 8. Safe Domain for Combined Alternating Stresses (uniform stress distribution) .

Fig. 9. Safe Domain for Combined Alternating Stresses . (linearly variable stress-distribution).

36 K. Terazawa :

ORA S in these figures corresponds to the range of variable shear strain energy

which causes neither fatigue failure nor statical yielding. That is, the area which

is bounded by the co-ordinate axes and the two straight lines, RA and AS,

represents the safe domain. This safe domain can be approximately replaced by a

quadrant of an ellipse ORES as seen in these figures. Therefore, the above two conditions (4.1) and (4.2), which are mutually independent, may be replaced by a

single condition as follows :-

(4.7)

In this statement, Wr and

Ws are constants for a given

material, and Wr, is indicated

by 4ξ2r or 4ƒÌbr and Ws by

2σs or 2σbs, accordingas

whether the stress-distribu-

tion is unlform or linearly

varying.

If the expression (4.7) is

applied to the case of simple

alternating stress, it is re-

presented by the curve ASB in Fig. 10. In this figure,

it can be seen that the curve

ASB is a closer approxima-

tion for the actual fatigue

di ag am than the assumed dia-

gram ASB which is indicat-ed by broken lines and is

commonly adopted in the

practical design work.(1)

(1) F. R. Fischer: Vorschlag zur Festlegung der zulassigen Beanspruchungen in Maschinenbau;

Z. V. D. I. Bd. 76 (1932),

H. Helold: Wechselfestigkeit metalischer Werkstoffe; (1935), S. 240,

C. R. Soderberg : Working Stress; Jour. App. Mech. vol. 2 (1935), p. 106.

Fig. 10. Fatigue Diagram (0.13% C steel).

37

2. Estimation of Allowable Stress.

For estimating the allowable stress, the factor of safety is generally used.

Recently, various discussions and investigations(1) have been made on the factor of

safety. By the results of these investigations, it may be shown that a factor of

safety for statical stress should have a different value from that for reversed stress,

even if the internal and external conditions, to which the structure-or machine-

parts in service are subjected, are same. For example, ROtscher had estimated the value of the so-called "Sicherheitgrundzahl" for the ductile material as follows: -

For statical stress,

and for reversed stress,

For an intermediate case between the above two, the factor of safety is obtained by

an interpolation from those for the two extreme cases.

Now representing the factors of safety for statical stress and reversed stress by n,

and nr, respectively, the formula which gives the allowable stress for materia 1 under

combined alternating stresses is obtained from (4.7) as is shown in the following.

(4.8)

where for uniform stress-distribution,

or for linearly varying stress-distribution,

for uniform stress-distribution,

or for linearly varying stress-distribution.

In the formula (4.8), W, is expressed in terms of at or fit by the aids of the

(1) F. Witscher: Sicherkeit und Beanspruchung bei der Berechnung von Maschinenteilen; Ma-

schinenbau, Bd. 9 (1930), S. 225, B. Garlepp: Zu:assige Spannungen und Dauerfestigkeit im Kran- und Veria'ebrrc:.enbau;

Maschinenbau, Bd. 10 (1931), S. 86. A. Thum : Zur Frage der Sicherheit in der Konstruktionslehre ; Z. V. D. I. Bd. 75, nr. 23 (1931).

Zulassige Spannungen der in Maschinenbau verwendeten Werkstoffe; Maschinenbau, Bd. 10 (1931), Heft 3.

38 K. Terazawa:

results of Section III and Wm is equal to al or 01, according as whether the

stress-state is three- or two-dimensional. Hence, the formula (4.8) is represented in

each case as in the following.

(1) For three-dimensional alternating stresses,

(i)　 whenα2=0　 or　 whenα2≠0andα3>― α2―,

(4.9)

(ii)　 whenα ≠O　 andα3≦ ―α2―

(4.10)

(2) For two-dimensional alternating stresses,

(i) whenβ2=0,or when β2≠0　 andβ3>― β2―,

(4.11)

(ii)　 whenβ2≠0　 andβ3≦ ―β2―,

Although the formula (4.8) can be applied to the case of a simple alternating

stress also, yet the allowable stress in this case is more easily obtained by means of

Soderberg's method.(1)

V. Examples.

As an example, the method of application of the formula (4.8) to a circulaJ

shaft subjected to the combined action of alternating bending and alternating torsion

will be illustrated in this

section. This case is one

of most interesting pro-

blems which we often meet

in practice. In this case

a shaft is subjected to

two-dimensional alternat-

ing stresses, and those

stress-components at any

instant is shown Fig.. 11.

In this figure, mean bend-

ing stress and variable

(1) C . R Soderberg: loc. cit.

Fig. 11. Stress Components under Combined

Alternating Bending and Twisting.

Fatigue Resistance and Allowable Stress of Ductile Mater!als &c. 39

bonding stress are denoted by ƒÐx and and mean stress and variable stress due

to twisting moment by Txy, and Āxy Representing bending moment and twisting

moment by the following notations,

Mb,: mean bending moment,

Mb: variable bending moment,

Me,: mean twisting moment,

Mt: variable twisting moment,

and the diameter of a shaft by d, the stresses will be given as follows:

(5.1)

(5.2)

From (5.1), (5.2) and (3.9), we get the following expressions.

(5.3)

Substituting (5.3) into the equation (4.11) or (4.12), the formula which gives

the allowable stress is obtained. As the stress-distribution varies linearly in this

case, the values of A,. and N., in the formula for the allowable stress, are to be

chosen as follows:

(5.4)

1. Particular Cases.

The case of a shaft under alternating torsion and alternating bending is divided

into two cases, one in which allowable stresses are given by (4.11) and the otherin

which allowable stresses are given by (4.12).

(1.a) The casc of β2=0, or β2≠0 and β3≧ β. When β2=0,or when

40 K. fierazawa:

$2 * 0 and 03.-32, by substituting (5.3) into (4.11), the following equation is

given,

and the diameter of the shaft is obtained as follows:—

(5.5)

By means of (5.5), the diameter of a shaft for some special cases will be obtained

as described hereafter.

(i) When a shaft is subjected to the combined action of reversed bending and

reversed torsion, the mean bending moment and the mean torsional moment are

zero, i. e.,

Hence, from the formula (5.5), we obtain,

(5.6)

(ii) When a shaft is subjected to the combined action of statical bending and

reversed torsion, the variable bending moment and the mean torsional moment are

Substituting these values into the formula (5.5), the diameter of the shaft is given

as follows :

(5.7)

(iii) When a shaft is subjected to the combined action of statical torsion and

reversed bending, the mean bending moment and the variable torsional moment are

Hence, from the formula (5.5), we get

(5.8)

(iv) In the case of a shaft subjected to the combined action of pulating

bending moment and pulsating torsional moment, in which the stresses oscillate

between zero value and the maximum, the relation of the mean moments to the

variable moments are given, as

Fatigue Resistance and Allowable Stress of Drctle Materials &e. 41

Substituting these relation in (5.5), we get

(5.9)

(1. b) The case of A*0 and ƒÀ3•…ƒÀ2. When ƒÀ2•‚0 and ƒÀ3•…ƒÀ2, substituting (5.3)

in (4.12), we get

and the diameter d of a shaft is obtained as follows:—

(5.10)

By the aid of (5.10), the allowable stresses for special cases are similarly

obtained as those in the preceding case.

2. Compgrison of Bailey's Method' with the Author's One.

R. W. Bailey had previously put forward a method(1) for estimating the allow-

able stress in the case of two-dimensional alternating stresses. In his paper, he

assumed that the maximum shear stress law holds well in the case of combined

alternating stresses and the fatigue diagram is represented by a quadratic curve.

And his treatment was confined to ,a special case such that a material, the breaking

strength of which is equal to four times as much as reversed fatigue limit, is sub-

jected to the combined action of reversed bending and alternating torsion. Repre-senting the statical shear strength by fo, Bailey's formula is written as follows:

(5.11)

where K is a coefficient and its value is determined by the following two quantities,

By Bailey's assumption, the ratio of fo to the reversed shear fatigue limit ris

and (5.11) becomes

(5.12)

In Bailey's treatment, the effect due to stress-distribution on the fatigue failure

was not taken into account. Hence, in the following comparison, this effect will be

(1) R. W. Bailey: Behaviour of Ductile Materials under Variab'e Shear Stress, with Special Re-ference to Shaft subjected to Bending and Twisting; Engineering, vol. 104 (1917), p. 81.

42 K. Terazawa:

disregarded; that is, r and as will be adopted instead of tbr and abs. For the sake

of simplicity, we assume that the values of factors or safety, nr and n, are unit, i.e.,

and the ratio of ter to Ts iS

On these assumptions, the comparison of Bailey's method with the author's one

will be described in the following.

(1) When a circular shaft is subjected to the combined action of reversed

bending and reversed torsion, the mean bending moment and the mean torsional

moment are

If we assume that Mt, Mb, the values of 8 and 11 are

8,2 and /1=0.5,

and the value of K is given as

Accordingly. from (5.12). the diameter d' of a shaft is given as

while, by the author's method, from (5.7), the diameter d of a shaft becomes

The ratio of d to d' is obtained as follows:—

(ii) When a circular shaft is subjected to the combined action of reversed bending and alternating torsion, the mean bending moment is

And if we assume that Mt=3Mts and Mb=4Mts, the values of 8 and become

ƒÂ= 1.5 and ƒÊ=0.5.

Hence, the value of K is given as

Substituting this value in Bailey's formula (5.12), the diameter d' becomes

Fatirue Resistance and Allowable Stress of Duelle Materials &c. 43

while, according to the author's method, from (5.5), the diameter d is obtainedas

The ratio of d to d' is given as follows:—

(iii) In the case of a shaft subjected to the combined action of reversed bend-

ing and pulsating torsion, the following relation is given.

MbS 0 and Mts=Mt.

If we assume that Mb=2114, the values of 8 and ,(.6 are

8,1 and 11, = 5,

and that of K becomes

Accordingly, from Bailey's formula (5.12), the diameter d' is given as

while, from the author's formula (5.5), the diameter d becomes

The ratio of d to d' is obtained as follows:—

By the above comparison, it can be seen that the two methods give the nearly

equal results for the above particular cases. However, it should be noticed that

Bailey's formula (5.11) or (5.12) is confined to the limited case, while, by author's

method, the allowable stress for any case can be given.

Summary and Conclusion.

Considering the effects due to mean stress and due to variable stress on the

fatigue failure of ductile material separately, the fatigue test data for bending were

compared with those for torsion. By these comparison, it was found that the

shear strain energy law (v. Mises-Hencky's law) holds more acculately than the

maximum shear stress law (Guest-Mohr's law) for the case of 3ombined alternating

stresses. And a conclusion is obtained that, if the range of variable shear strain

energy has a smaller value than the limiting range of variable shear strain energy

44 K. Terazawa :

for simple reversed stress, and when the maximum shear strain energy during one

cycle does not exceed the shear strain energy for statical yielding, a ductile material

can sustain an infinite number of cycles of combined stresses, without fatigue failure

preceding statical yielding. In the light of the above conclusion, a method which gives the allowable

stresses for a ductile material under combined alternating stresses was derived . And

as an example, a circular shaft which is subjected to combined action of alternating

bending and alternating torsion, has been treated.

In conclusion, the present writer wishes to acknowledge his gratitude to Prof . Tu. Inokuty for his unfailing encouragement as well as his kind advice . Many

thanks are also due to Dr. K. Kido, for the good-will shown by him during the

course of the present work.

July, 1936.

Department of Shipbuilding and Aeronautical Engineering,

The Technical College, Yokohama.

討 論

○座 長(重 光 蕨君)唯 今の御講演に就て御質問叉は御討論があ りました ら御述べを願ぴます。

○井 口常 雄君 寺澤君の非常に努力されました、叉design上 の基礎 となるfatigueの 場合のcom-

bind altemating stressesに封 し御講演なされた事に樹し私は非常に嬉 しく思ふのであります。これ

に就て一言申上げたいが時間があ りません故、叉前刷 を四五 日前に頂きまして未だ牛分程 しか讃んで

居ませんので…… こんな事からして一二の黒占に就て一寸お伺ひしたいと思ぴます。

此のpaperの 基礎になる事ですが、21頁 の上から数へて8行 目か ら11行 目にある、繰返 し應の

力 の ときmeanが0で な い場 合 の材 料 の 破壌 に 封 す る 考 へ 方 を.mean stressの 働 きのe挽ctと

variable stressの 働 くe伽ctと の 關 係 が 何 か 文 章 か ら言 ふ と、破 壌 のmechanismに 於 て全 く別D

檬 に も見 え ま す。 其 の 下 の(2a)の 庭 の第4行 目及 第5行 目 に も見 え ます が 、 この事 に就 て 著者 の

御 考 を伺 ひ 度 い の で す 。 時 間 が あ りませ んか ら之 れ に關 し私 の 意 見 を先 き に述 べ ます 。分 け て考へ る

事 は差 支 へ な い が 破 壌 に封 し其 のmechanismに 全 然 相 違 が あ る とい ふ こ とに 何 か根 擦 が あ るか の様

に 申 され た の は言 過 ぎで はな い か と思 ひ ます 。5(23頁 の(2.12)range of variable strain energy

を見 る と(2.12)の 上 の 方 帥 ちstressが+か ら0に な り、0か ら一 に攣 る場 合 のenergyの 攣 化 は

energyは 自乗 で利 くか ら 、+か ら0に な り、0か ら又+に な りま す 。夫 れ を鼓 で は寄 せ て居 られ る

Fatigue Resistance and AllowaMe Stress of Ductile Materials &c. 45

のは ど ういふ謬 で す か。first termだ けで よい檬 に思 ひ ます 。3、4、5圖 を見 る と、著 者 は皆 の や つ

た 實験 をplotし てAB直 線 で 置 換 へ て 居 られ るが 、之 れ は差 支 へ な い が 多 くの 黙 は これ よ り上 に

出て ゐ る。 是等 の實 験 中 に(2.12)の 下 の式 に當 嵌 ま る場 合 が どれ だ け含 まれ て 居 りま す か 、之 れ が

(2.12)の 上 の方 の 式 で+か ら一 に變 化 す る時 のdamが 多 い とす る と.(2.12)式 のsecond termを

止 めて しまふ とABが 本 當 に近 くな る様 な 氣 が します 。 之 れ も一 寸 伺 ぴ 劣 々意 見 を述 べ ます 。

○ 寺 澤 一 雄 君 一 言 申上 げ ます 。mean stressとvariable stressのeffectが 違 ふ と云 ふ の は 、質

的 にで はな く量 的 に違 ふ と云 ふ意 味 で あ ります 。 例 へ ばstatieal stressの 値 が10の 時failureを

起 す檬 な材 料 で もvariable stressを 受 け る 時 は其 の値 が5叉 は4で もfailureを 起 す と云 ふ意 味

で あ ります 。(2.12)式 の 方 はstressの 一週 期 中 に 兎 に 角 攣 郵 す るshearing strain energyを 問 題

に しま した が.今 のお 勤 め に從 ひ ま して も う一 度 能 く此 の 黙 を考 慮致 して 見 た い と思 ひ ます。 種 々御

批 評蛇 に御 教 示 を下 さい ま して有 難 うご ざい ま した 。

○ 座 長 重 光 蕨 君)其 他 に御 質 問 御 意 見 は あ りませ ん か 。 御 座 い ませ ん けれ ば 、此 の 席 を汚 させ

て 頂 い て居 ります 爲 に 一言 申上 げ ます 。最 初 に井 口博 士 の言 はれ た事 が其 儘 本 協 會 と して 申上 げ る事

にな ら う と思 ぴ ます 。此 の様 な論 文 に就 て は質 問 も討 論 も多 くあ る事 と思 ひ ます が 、時 間 が あ りませ

んか ら書 いた もの で協 會 へ 御 逡 附 を願 ぴ協 會 が 之 を御 取 次 して著 者 の 御 同 答 を得 て 之 を會誌 に載 せ る

事 に致 しま す。 御 力 作 に樹 し拍 手 を以 て感 謝 の 意 を表 した い と思 ひ ます 。(一 同拍 手)