In the present paper, using the method of solution due to Howland and some known stress expressions of an infinite solid under the forces and moment acting at a point within, two satisfactory solutions of the following problems of an infinite strip are given.(1) Four equal and opposite forces acting on an infinite strip are applied at any points a very small distance d apart.(2) Two equal and opposite moments acting on an infinite strip are applied at two points any distance 2η apart. In order to ascertain the correctness of the calculations carried out in this paper and moreover, to show the theory deduced here definitely, various numerical calculations are performed. From the numerical calculations mentioned above, many interesting and important characteristic properties implied in the problem in question are pointed out and discussed.

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The fatigue strength of material is closely connected with the stress concentration due to a notch and the value of maximum stress in a specimen with several notches is in general somewhat diminished by the interference of notches. To investigate the interference, we consider the stress distribution using the next transformation [numerical formula] where z=x+iy, w=u+iv. C_m are chosen according to the boundary form. The stress function F is composed of fundamental stress function F₀ and additional stress function F_a, and each function is given by φ₀+xφ₁, where φ₀ and φ₁ are harmonic functions. The results of calculation show that the concentrated stress is decreased by the interference of notches comparing with the case single notch. The fact is available for engineering problems to increase the fatigue strength of structure.

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In Report No. 2, the case wherein a concentrated load does not act on the boundary of a twodimensional elastic body was analyzed. In the present paper, the study is furthered by considering the stresses for the case in which a concentrated load acts on the boundary of a semi-infinite plate. As one example, the stress concentration occuring in a corner is considered.

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In Report No. 2, the stress concentration factor was calculated for a hole having a bounary whose mapping function could be given by a rational function. In the present paper, the case in which the mapping function is not a rational function is treated ; by successive approximations the solution of the fundamental equation (1) was solved. As examples, the stress concentration factors for a hole bounded by two circular arcs intersecting at right angles with each other as shown in Fig. 1 and those for a circular hole having a crack as shown in Fig. 2 were derived theoretically.

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The writers made the studies on the general solution of the stresses in a ring under radial loads acting on the inner and outer boundaries, with the use of the general solutions of the two-dimensional elastic problems refered to polar coordinates. Numerical calculations are carried out for three, four and five inner radial loads, when the ratio of the radius to the inner outer radius of a ring is 1 : 2. The relation between the maximum circumferential normal stress and the number of loads is shown by making use of above mentioned restlts. For the case of two or three radisl loads acting on the outer boundary, numerical calcuations are carried out. And comparing these results with that of G. Bell's studies, we have found tha there are some errores in G. Bell's studies.

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We have solved the shearing stress distribution on the cross-section of a cantilever subjected to a terminal transverse load which is parallel to one end fo the principal axis of the cross-section, by using the following complex variables, [numerical formula] From above the functional relation, the ring region of t plane hss been transformed to the regular polygon having a circle in its center on the Z plane, and this shape of transformation becomes the cross-section of the cantilever.

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When a round rod having a longitudinal grove subject to the twisting moment, the shearing stress distribution on the cross-section and the relation between the twisting moment and specific angle of torsion will vary with the groove form. Author has discussed using a function of x+iy=c coth (ξ+iη) as curvilinear co-ordinate, and above mentioned relation has derived from its results.

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This paper deals with the problem of the infinite thin plates with the hypotrochoidal hole subjected to 1) transverse bending and 2) twisting. In the first and second parts, using the Poisson-Kirchhoff's theory, the finite expressions of stress resultants were obtained and their distributions were computed especially along the boundary of a near rectangular hole with rounded corners. In the third part, the stress resultants of the same plates having a reinforced part around the hole were considered. The results of these problem were shown graphically and the stress concentration was plotted as a function of ρ₀/l or ρ₀/h where ρ₀ is the radius of the curvature of rounded corners, l, the width of the hole and h, the length of a diagonal.

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In this paper, the torsion problem of a prismatic cylinder with sharp notches in the direction of the axis is treated. As an example, the solution for a cylinder with a star shaped cross section is obtained including numerical result, and the theoretical result is confirmed by an experiment using test pieces cut out precisely.

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In the preceding paper, we obtained solutions for a circular plate of an aeolotropic material, subjected to uniform lateral load. In this paper, we introduce the general solution for a rectangular plate with clamped edges where the load is distributed over the surface of the plate in any way. Numerical solution for a simple example is also given.

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In the preceding paper, the form of the corrugation was assumed to be composed of circular arcs. In this paper, we treat the problem of a corrugated plate in which the form of the corrugation is defined by [numerical formula] where r is the radius of curvature, θ is the angle between the vertical axis and the normal to the curve, a and k are constants which determine the magnitude and the form of the corrugation respectively. Two fundamental cases are solved, viz., (1) equal vertical load acts at each crest of the corrugation, (2) equal tangential load acts at each crest of the corrugation.

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There are many papers on the stress distributions of rotating disks whose sectional forms being symmetric with respect to the perpendicular plane to the rotating axis. But the authors are not aware of a paper on the rotating disk whose sectional forms are a symmetric to the above-said perpendicular plane. The authors, therefore, tried to solve the problem on the rotating disk whose sectional form is conical with uniform thickness, even though it is a very special form. And they obtained the following results : - (1) The value of stress at the center of the disk without central hole is not always greater than the value at the outer side. (2) The stress at the center may possibly become the compressive stress in some cases. (3) In the special case, the value of stress at the center is equal to that at the outer side, and it is lower, by 30%, than the value of stress which occurs at the center of the rotating plane-disk with uniform thickness.

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The report is a sequel of the last announcement and it treated also the numerical calculation of the buckling load of the thin deep beam. The cross sections of the beam have two portions of variable and constant ones, and are symmetrical to the middle of the span. The variable cross sections change gradually according to the next law : [numerical formula] B and C are the bending and torsional rigidities of the cross sections, and a, b are constants, and m, n are real numbers greater than unity. By using the conditions at the beam ends and the connecting sections of constant and variable portion, we can find the buckling moment of such a beam. The buckling moment is taken as following formula : [numerical formula] Where B₀, C₀ are the rigidities of fundamental cross sections. The ccefficient μ depends upon the shape of beam.

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In these papers, writer deals with the lateral buckling of beam with cross scction which discontinuosly changes by steps. The differential equation of equilibrium for the small angle of twist is used as follows : -[numerical formula] B and C are the flexural and torsional rigidity of the beam sections. The angle and its first derivative θ' can be found by integration, and takes the form as the next expression for the portion of the beam. [numerical formula] The boundary conditions have two kinds in this case. The one is that of beam ends, and the other is that of the continuation of the seams in the portions. At the k th sections and k+1 th sections, the next relations are formed. [numerical formula] Elliminating the integration constants, we can determine the critical moment expressed as follows : -[numerical formula] where μ is the coefficient depending upon the kind of load, the state of the beam sections, and number of their steps. It can be found by trial-error-method in general. As the numerical examples, the critical moment and critical load of the beam with two steps in the sections are determined.

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Using the energy method, the authors have determined the approximate critical load, which acts on the middle point of the span of beam with narrow cross section, when the beam is subjected to an axial load W, the end conditions being simply supported. The approximate critical load [numerical factor] where m is a numerical factor, the value of which depends on [numerical formula] and l is length of a beam, B₁ is the flexural rigidity of the beam, and C means the torsional rigidity.

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Concerning the buckling load of thin circular plate of variable thickness, little has been investigated. This is due to the reason that the differential equation becomes so complicated that it can't be easily solved. From this point of view, the author proposes a new slope-deflection method. That is, the circular plate of variable thickness is substituted by the connection of circular ring plates and a center circular plate, thickness of which differs from each other in stepwise, and he applies the slope-deflection formula to the continuity condition at the connecting circles of two circular ring plates and also to the boundary condition. This method is superior to the other in the following points : a) The solution is always possible. b) The solution is simplified, compared with A. Willer's method, because the determinant necessary to get the critical value is so much simple that there is no difficulty in its solution.

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Author intend to propose the equation of the laminated spring : [numerical formula] in which, L : 2l : span, in mm, L_m : span when spring becomes horizontal, in mm, effective length of the main spring, in mm, c : camber at no-load, in mm, δ : any vertical deflection, in mm, as a formula for the variation of the span, which meets with the experimental investigati on as shown in Fig. 4. The above formula is developed from the theory of the spring of uniform strength which was discussed by the author lately.

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The deflection w of the welded plates were measured and the curvatures were determined corresponding to ∂²w/∂x², which were obtained by graphical differentiation of deflection curves. The strains were found from the curvature (ε_x=-z ∂²w/∂x² etc.) and the corresponding stresses in the plastic scope wire determined. The elastic residual strains and stresses were measured and the max. intensity was 20 kg/mm², in the case where both edges were welded. The distribution of the residual strains in the plate sections of the welded and permanently deformed portion have been analysed by the bending theory corresponding to similar conditions.

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The moment-deflection curve of a mild steel beam remains nearly straight until the applied moment reaches as high as 1^.5 times of M_s, where M_s denotes the moment when the stress of outermost fiber becomes the equivalent yielding stress σ_s in tension test. To investigate the phenomena, the surface stress of uniformly bent bar of 0^.09% carbon steel was measured by means of X-ray and the stress distribution within the cross section was found utilizing residual stress measurement by corrosion method. As the results, we concluded that the deformation is perfectly elastic as far as the applied moment does not exceed M_s. And even beyond this limit, the surface stress remains under σ_s. Some analytical method is then introduced to obtain the stress distribution.

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The problem, concerning the deld of circular plate with round hole under internal pressure, is heated by Nadai in his famous book "Plasticity". ⁽⁴⁾ To deal with residual stresses by plug weld, we followed his calculations and found one or two mistakes in it. Nadai gave the following solution, [numerical formula] for the differential equation, [numerical formula] Solution (30) is clearly mistake and must be corrected as follows, [numerical formula] Therefore Nadai's equation (33) must be changed as equation (b) ; [numerical formula] The values of (b/a), S_r and S_t corresponding to the assumed value of θ=θ_a at inner hole were calculated from the above exact solution, and when the circular plate is under internal pressure, θ_t is between -60° and 30°, while, in Nadai's book Fig. 251, θ_a is between 30° and 120°.

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It has been said that material has proportional limit, elastic limit and yield point. But elastic limit is the beginning of molecular process in faults of lattice, and in technical experiment the measurement of such elastic limit is impossible and senseless. In precise technical experiment mild steel has yield point alone. Velocity of deformation of material is different from pulling speed and between elastic range and plastic. Since velocity of deformation increases at the beginning of slip, we can measure yield point by measureing the point at which velocity of deformation begins to increas. This can be measured by means of mirror type extensometer and the others. Therefore, we should adopt yield point alone and establish as follows ; "As yield point, we dicide the limit of stress, at which the first slip arises in material, or velocity of deformation of material begins to increase, in tensile test of definite constant elastic strain rate."

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By the author's theory on the rupture of brittle materials like cast iron, in cases where stress distributions are symmetrical, the rupture occurs when a kind of weighted mean value of the normal tensile stress over the whole section attains a definite value. For tensile load the weight is unity, and the criterion of rupture becomes F/A=const.=σ_B, where, F=the tensile load, A=min. sectional area, and σ_B=tensile strength under uniform stress. Test pieces with notches and holes were tested ; the above relation conforms very closely with experiments.

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1. Preface When the metallic materials were broken down due to fatigue under repeated stresses, it was previously published partially that the repeating speeds had influences upon the fatigue-curves, especially the endurance limits. But they were not investigated systematically. The causes lay, it was though, in the fact that no reliable high speed fatigue testing machine was available. In this research, some carbon steels from 0^.07% to 0^.6% carbon has been studied under the new Nishihara type high speed fatigue testing machine, which was designed as satisfying the purposes, and also under the old Woler type fatigue testing machine, and some other materials were also studied. 2. Effects of the repeating speeds We have studied on the range from 5, 000 rpm to 15, 000 rpm by the new Nishihara type high speed fatigue testing machine, and studied on the range from 50 rpm to 2, 000 rpm by the old Woler type fatigue testing machine. It was imperfectly known that the endurance limits were rising as the repeating speeds were rising. By this research, the facts that the endurance limits were rising as the repeating spceds were rising were confirmed, and these rates of the rising are different according to the contents of carbons or the internal crystal systems or the results of the heat treatment. When we consider all states of fatigue curve at the high stresses, the materials are fatigued sooner in the cases of repitition of high speed than of low speed, i. e. at the high stresses, the fatigue strengths are lower than at the low stresses, at the high speed repitition. But at the endurance limits, or at the lower stresses near such limits, the materials are fatigued sooner in the case of repitition of low speed than of high speed. As the cause that the fatigue strength is varied by such variations of the repeating speeds, we consider 2 causes. At the high stresses the facts that the fatigue strengths at the high speed repetitions are lower than at the low speed repetition, are considered as the next. The heat quantity which is accumlated in the material in unit time is much larger at high speed repetition than at lower speed repetition, and then the temperature of the material rises and the material is softened. So the strength becomes smaller. At the low stress near the endurance limits, the rising of temperature of the material is very much smaller, but on the contrary, the variations of the internal crystal system are larger at low speed repetition than at high speed repetition, so that the material at low speed repetition is fatigued sooner than at high speed repetition. 3. Conclusion To save the testing time of fatigue test, we perform the fatigue test under high speed repeating stresses. But by such a method, the phenomena, that the endurance limits were varied by the repeating speed, were discovered, and the facts that according to the internal crystal system of material, the rate of influences was different, was also discovered. we now believe that we must study another materials, ana discuss the effects of the repeating speed that influences the fatigue strength.

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Endurance tests were carried out for rolling contacts of roller on roller, ball on saddle type specimen, and ball on ball, lubricating with machine oil. The limit values of their pitting strength indicated by Hertz's maximum pressure increased remarkably with the decrease of cling factors. In the case of rolling contact of ball on ball, the contact surface deformed plastically marking a flat band, and the logarithmic diagram of the width of this contact area versus repetition numbers was in linear relation. From this fact we can practically estimate the variation of maximum pressures acting on the curved surfaces. The fatigue diagram for the constant equivalent shear stresses calculated for these varying maximum pressures coincided with the fatigue diagram in the case of the constant shear stresses for the contact of rollers considered as a problem of plane strain. Applying these results, we have tried to explain the life of ball bearings.

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Neuber's form factor is determined from the following relations : [numerical formula] where a is the form factor for the notch of any depth, and a_f, a_t are those for notches regarded as very shallow or very deep one, respectively. But the value of a calculated from the above equation is sometimes smaller than the value of notch factor β obtained from fatigue tests. Then the authors derived the following relation : [numerical formula] where r is the radius of a specimen, a is the radius of the specimen at the bottom of a notch and t is depth of the notch.

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Let the form factor be a and the notch factor be β, the author obtained the following relation from the results of experiments : [numerical formula] where ξ, c and n are constants (n>1). Generally β becomes large as ρ becomes small and reaches to the maximum value at a certain value of ρ and then becomes small and approaches to a definite value. These relations are satisfied with the above equation by an appropriate choise of the values of constants. The constant ξ can be considered to represent the sensitivity of the materials for a notch and is independent of the value of ρ.

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Up to the present, many reports have been published about the stress concentration when materials have notches. But on the effect of the notch angle of V-shaped notch, few studies have been made. So we made many fatigue tests on specimens having V-shaped notches of various notch angles (from 0° to 180°) and fillet radii. From the result of the fatigue tests we derived the relation between notch factor β and notch angle θ or fillet radius ρ as follow : [numerical formula] where β is the notch factor at any value of notch angle θ, and (β)^θ=0 is that at θ=0, and ρ is in mm.

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Sometimes shafts and axles have a collar for the purpose of attaching ball bearings and etc. Then the fatigue resistance of the shafts or axles are lowered by the collar, that is a ring form projection. In this report we have discussed on the notch effect of width and height of a collar from our experimental data. Experiments, that is, repeated bending tests and repeated torsional tests, have been made on specimens of 0.34% carbom steel with the collars of various width and height. As the chief results of this research, we have expressed the effect of width of a collar with a coefficient ξι, the value of which is independent of the height of it. Similarly the effect of height of a collar can be also expressed with a coefficient ξ_D, the value of which is independent of the width of it.

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Occasionally roller chain fails by fatigue, though abrasion generally seems to be the cause of the failure In this report we considered the life of roller chain from the results of fatigue tests on roller chain. The testing machine used was the Haigh's tension compression fatigue testing machine. The chains applied to the tests was those of the capacity 2, 000 kg for a static load. The results obtained were as follows : (1) Fatigue tests were made on chains having the roller links tempered at various temperature, and it was found that the tempering at 450°C is most suitable for chains used at heavy load. (2) Fatigue tests were made on chains having different forms of roller links, and it was found to be favourable to extend the breadth of the neck of a roller link. (3) Deformation due to fatigue was very small. (4) The value of the notch factor of roller chain is about 3. (5) From the test results the diagrams showing the life of roller chains for fatigue were made.

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Cast iron is sometimes used in machine parts without surface finishing. Then it is very important to know what effect the cast skin exerts on the fatigue resistance of cast iron. The materials applied to fatigue tests in this report were gray cast iron with tensile strength of 27 kg/mm². The summary of the results are as follows : (1) The fatigue limit of cast iron with cast-skin is about 10% lower than that with finished surface of small finishing allowance, and that with large allowance is about 6% lower than that with small allowance. (2) On the contrary the existence of cast-skin is profitable for the fatigue limit of cast iron, when there is corrosive action.

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Stress waves applied on machine parts are not always pure sinusoidal, but sometimes more complicated. In this report we considered the fatigue resistance of steel under the stress waves which are composed by superposition of the two stress waves, i. e., the wave with large amplitude and low frequency (the primary wave) and that with small amplitude and high frequencv (the secondary wave). As the results of experiments we found that the secondary stress waves exerts very bad effect on the fatigue resistance of steel, and the effect of the frequency and amplitude of the secondary stress wave on the fatigue resistance of steel was represented by an expression.

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We conclude, considering the former experimental data, that the fatigue failure happens when the tensile strain reaches the allowable limit. In this case, the Poisson's constant has not the usual valwe determined macrographically for the isotropic elastic body, but it seems to take another value m₀ under the plastic deformation of small grain by cyclic stresses. For plane and plastic deformation m₀ becomes unity because of no area change. In general m₀ takes greater value when the material is plastically less deformable. From this fact we can understand the fatigue failures under combined stresses and can explain various modes of the damage diagrams of different materials. And we get the damage diagram of 0^.07% C-steel whose m₀ is found to be 1^.25 from the combined bending and torsion fatigue tests, and we may confirm the applicability of this theory.

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The purpose of this paper is to investigate the physical meaning of the indentation hardness such as Brinell, Mayer-hardness, etc., and to find the relations between hardness and its measuring methods, i. e. 1) the felation between indenting tools and hardness, 2) the relation between the amount of plastic deformation and hardness, and 3) the relation among the number of hardness, the indenting load and the depth of indentation.

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1. There are various investingators of metalic abrasion, but never established any fundamental common principle, therefore results obtained are incomparable. 2. After laborious experiments, proposed Prof. Dr. Okoshi a general formula L=f (pν), with abrasion contour diagram expoessing hyperbola. 3. Differentiating this function, we get [numerical formula], and for special value of p and p, we realized [numerical formula] simultaneously, which means the unique ideal testing condition we attempted to determine. 4. Applying this principle to abrasion Tester, I have invented an abrasion meter and applied for a pattent (No. 3078). 5. Using this abrasion meter, we can express abrasion degree by means of "Special abrasion Number", as will be seen in the case of Hardness Testing.

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This paper deals with the calculation of the rolling contact of disks caused by elastic deformation. First, by the study on the mechanism of rolling contact, it is known that in the load transmitting wheel there always occurs the apperent slippage due to the elastic deformation of the wheels in and near the contact area at any small load and this brings about the loss of energy. Then the condition of the elastic contact of two bodies and the integral equations or distribution of normal and tangential forces are obtained. Under the assumtion of Coulomb's friction law, these equations are solved and force distribution is obtained as the functions of external forces and the slip ratio of two wheels or the loss of transmitting energy. The distributions of stress and strain are also calculated.

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In the former report, the mechanism of rolling contact was discussed and the distribution of force, stress, stain and the sliding velocity in the contact area, as well as the relation between the driving force and the rolling friction, were obtained, on the assumption that the sliding friction coefficient does not depend on the sliding velocity (Coulomb's friction law). But in the case of rolling, there arises very low sliding velocity in the contact area, so the friction law does not always follow Coulomb's law, but rather depends on the state of the surface of contact. So in this report, the problems of tolling contact are discussed for general case. For example, a certain form of friction coefficient beins assumed, the relation between the driving force and the slipping ratio, which is equall to the ratio of loss of energy, is calculated and it is in good agreement with Sachs' experiment. Also the distribution of force, stress and strain in the contact area are obtained. In the last section, the relation between the motion of a wheel and the side force is discussed.

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In the preceding paper the author obtained experimentally the values of coefficient of dry friction for five metallic materials under high pressures and they decrease generally with the increase of contact-pressure. In this paper the author obtains the same tendency for copper. Furthermore, the values under low pressures are measured to compare with them under high pressures. Then factors to have an effect on the values under low pressures are researched and recognized that roughness of contact surface and hardness are essentialy effective to them. On the relation between the values and roughness, taking the values as ordinate and roughness as abscissa, the graph for each combination of six materials becomes a smooth curve respectively and the values suddenly increase with increase of roughness where be about 10^-3mm. And the values decrease with increase of hardness number for every combination of materials and roughness.

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