THE HYPOTHESIS OF BROKEN SECTIONS AND ITS APPLICATION TO THE CALCULATION OF RODS OF COMPLEX CONFIGURATION*)

A. V. VERKHOVSKY

Professor, Doctor of Technical Sciences

Introduction

The problem of calculating rods of complex configuration with variable cross-section has a definite practical significance.

Rods of the type under consideration may be subdivided into straight-axis and curved-axis rods. A characteristic example of a straight-axis rod of variable cross-section is the tooth of a gear wheel; of a curved-axis rod—hooks, machine frames, and the like.

An extensive literature is devoted to the problem of calculating the teeth of gear wheels, but nevertheless it cannot be said that complete clarity has been achieved in this calculation. In the practice of gear wheels there are cases in which checking a tooth for strength gives stresses considerably exceeding the temporary resistance of the material from which the tooth is made, and yet the tooth in fact works and does not break. This leads to the idea of the deficiency of the existing methods for calculating the teeth of gear wheels, which probably arises chiefly as the result of two causes: 1) in calculating a tooth in bending, the formula known from the course of strength of materials is used, derived for the calculation of prismatic beams, whereas the tooth of a gear wheel is not a prismatic beam; 2) the concentration of stresses at the root of the tooth shank is taken into account very roughly and inaccurately.

Thus, already from the example considered, the necessity of creating a more exact method for calculating rods of variable cross-section becomes sufficiently evident, with more accurate allowance for the concentration of stresses at fillets.

In the theory of elasticity, problems on the calculation in bending of beams with variable cross-section have been solved only for the simplest cases. The problem of the bending of a wedge-shaped cantilever beam, loaded at the point by a concentrated force or by a uniformly distributed load along the face of the wedge, is solved analytically [1].

With regard to the calculation of bodies of more complex shape, Prof. Timoshenko writes: “In many cases plates acted upon by external forces lying in their plane have such a form that the analytical investigation of the distribution of stresses becomes difficult and it is necessary to resort to experimental investigation. The optical method has proved very suitable for this purpose” [1, 2].

Even the shape of a tooth of a gear wheel proves to be so complex that, in solving the problem of the distribution of stresses, one renounces an exact analytical solution and resorts to experimental investigation.

*) The work was presented at the scientific conference of the Mechanical Faculty of the Tomsk Polytechnic Institute in June 1943.

S. V. Serensen, in his book Strength of Metal and Calculation of Machine Parts, writes: “As was noted, the use of long thin rods in machine structures is very limited. In most cases the configuration of parts is geometrically rather complex, and all dimensions are commensurable: it is enough to take such typical elements as gear teeth, crankshafts, pins, cams, etc. For their calculation, even at the present time, elementary methods and formulas of strength of materials are most widely used, which do not reflect the actual distribution of stresses and give deviations from it reaching hundreds of percent.”

Here one may mention the works of Neuber [3] and Afanas’ev [4], in which analytical methods are given for determining stress concentration in grooves and fillets.

In Neuber’s work, on the basis of the theory of elasticity, a solution is given for the problem of determining stresses in a groove. The exact solution of the problem is restricted to the consideration of extremely shallow and extremely deep grooves having a parabolic outline.

In Afanas’ev’s work, empirical formulas are given for determining stress concentration in fillets and grooves of flat specimens. Assuming the law of stress distribution to be hyperbolic, by analogy with a curved bar Afanas’ev gives a transition formula for the stress-concentration factor in the case of a round specimen [4,5].

A general method for calculating rods of variable section has not been proposed up to the present time.

All that has been said suggests that the appearance of a sufficiently accurate and convenient method for practical application in calculating straight-axis and curved-axis rods of variable cross section would be highly desirable.

The present work sets itself the task of giving an approximate method for determining stresses in straight-axis rods of variable cross section with any form of outline of these rods, and even in the presence in them of fillets and grooves of small radius, and also of giving a solution for certain particular cases of rods of variable cross section with a curvilinear axis.

Part I. Bending of straight-axis rods of shaped outline

1. The hypothesis of broken sections and the hypothesis of cylindrical sections. General considerations

In calculating straight-axis rods of shaped outline for bending, the formula usually used is

\[ \sigma = \frac{M}{W}, \tag{1} \]

derived for prismatic rods. In this formula \(\sigma\) is the stress, \(M\) is the bending moment, and \(W\) is the section modulus.

Formula (1) provides for a linear law of stress distribution, and the stress diagram will look as shown in Fig. 1(a). The indicated stress diagram, as well as formula (1), are, for the given case, defective for two reasons: 1) the stress \(\sigma\) at point \(A\) is not directed along the tangent to the contour and, consequently, is not a principal stress; and 2) the rectilinear law of stress distribution is incorrect owing to the curvature of the rod contour. The reasons noted may sometimes lead to a very inaccurate result when determining the stress by formula (1).

The inaccuracy of formula (1) in its application to the calculation of straight rods of shaped outline is due to the fact that formula (1) is derived on the basis of the hypothesis of plane sections (Bernoulli’s hypothesis), whereas in bending of the rods under consideration the plane sections are considerably curved.

On the basis of the investigations that will be set out below, one may come to the conclusion that substantially more accurate results in determining the stresses in bending of the rods under consideration can be obtained on the basis of the hypothesis of broken, rather than plane, sections.

In Fig. 1(b), for a certain point \(A\) taken on the contour of the rod, a broken section \(ADB\) has been drawn. This section is drawn in such a way that the side \(AD\) is the normal to the upper contour, and the side \(DB\) is the normal to the lower contour. The same figure shows the stress diagram for this broken section. It will be shown further that the law of stress distribution in a broken section will in general be hyperbolic.

The hypothesis of broken sections, or, more precisely, the hypothesis of non-curving of broken sections, is not entirely exact.

However, comparison of the results obtained on the basis of the proposed hypothesis with experimental data indicates that it is sufficiently accurate for practical use.

Fig. 1

The following considerations may serve above all as justification for adopting the hypothesis of broken sections:

1) The normal stress at a point taken on the upper or lower surface of a bent rod will lie in the plane tangent to this surface, and consequently will be a principal stress, which was not obtained when the hypothesis of plane sections was applied [cf. Fig. 1(a) and 1(b)].

2) From a fundamental point of view, a certain curvature of a broken section in bending of rods of the type considered is not a fact that invalidates the hypothesis of broken sections. It is known that the application of the hypothesis of plane sections in the calculation of prismatic rods in bending is likewise conditional, since exact investigation shows that in bending even prismatic rods the plane sections do not remain plane, but are somewhat curved. Nevertheless, the hypothesis of plane sections is widely used in calculating prismatic rods (beams) in bending. Thus, the basic criterion for judging one hypothesis or another is the extent to which the given hypothesis gives sufficiently accurate results for solving practical problems.

3) Experiments show that in fracture of rods of variable cross-sectional height, if the rods are made of a brittle material, the initial crack forms in the direction of the normal to the contour of the rod, i.e., it coincides with the direction of the broken section. This circumstance also speaks in favor of the hypothesis of broken sections.

We also note that fairly accurate results, agreeing rather well with experiment, can be obtained by means of the hypothesis of cylindrical sections\(^{1}\) [see Fig. 1(c)]. Investigations carried out by the author—

\(^{1}\) The hypothesis of cylindrical sections assumes that, during deformation of the rod, cylindrical sections change according to a certain definite law, namely in such a way that at the boundary points the cylindrical section remains normal to the contour of the rod.

which showed that preference should be given to the hypothesis of broken sections as a hypothesis sufficiently accurate and more convenient for practical use. In what follows, a certain comparison of the indicated hypotheses will be made.

2. Application of the hypothesis of broken sections to the determination of normal stresses in bending of straight-axis rods of arbitrary curvilinear outline

Consider a cantilever straight-axis beam whose cross-sectional height varies according to an arbitrarily prescribed law (see Fig. 2). For the present we shall confine ourselves to considering a beam of rectangular cross section with constant width \(b\).

A certain generalization with respect to beams or rods of non-rectangular section will be made below.

Let the contour of the beam under consideration be given by some equation

\[ y=f(x) \tag{2} \]

in the system of coordinate axes indicated in Fig. 2. In this case, the straight-axis character of the beam, under the condition of rectangular cross sections, determines the symmetry of the upper and lower contours.

The beam is loaded at the end by a concentrated force \(P\). Suppose that it is required to find the stress at some point \(A\) of the beam contour, whose abscissa is \(x\).

Draw through point \(A\) the broken section \(ADB\) so that \(AD\) and \(DB\), respectively, are normals to the upper and lower outlines of the beam. Then take point \(A_1\), whose abscissa is \(x+dx\), and in an analogous manner draw through this point the broken section \(A_1D_1B_1\).

Fig. 2

Consider the fiber \(KF\) of material, enclosed between sections \(ADB\) and \(A_1D_1B_1\) and situated parallel to \(AA_1\). In bending of the beam, section \(ADB\) will rotate relative to section \(A_1D_1B_1\) through an angle \(d\varphi\), taking the position \(A_2DB_2\). The relative elongation of the fiber \(KF\) will be

\[ \varepsilon=\frac{FN}{KF}. \]

Determine \(FN\) and \(KF\).

\[ FN=d\varphi \cdot u, \]

where \(u=DF\) is the distance of the fiber under consideration from point \(D\). Further, from Fig. 2 we find

\[ KF=QD+(AA_1-QD)\frac{DF}{AD}. \]

Determine the values of the quantities entering the right-hand side. From Fig. 2,

\[ \frac{QD}{DD_1}=\frac{dx}{AA_1}. \]

Hence

\[ QD=\frac{dx}{AA_1}\cdot DD_1=\frac{dx}{AA_1}\,dx_0, \]

where \(dx_0=DD_1\).

\(AA_1\), as the length of an arc, may be represented as

\[ AA_1=\sqrt{dx^2+dy^2}=dx\sqrt{1+\left(\frac{dy}{dx}\right)^2}=dx\cdot q, \]

where

\[ q=\sqrt{1+\left(\frac{dy}{dx}\right)^2}. \]

Further, from Fig. 2,

\[ AD=\sqrt{y^2+\left(y\frac{dy}{dx}\right)^2} = y\sqrt{1+\left(\frac{dy}{dx}\right)^2}=yq, \]

\[ OD=x_0=x+y\frac{dy}{dx}. \]

Differentiating, we obtain

\[ dx_0=\left[1+y\frac{d^2y}{dx^2}+\left(\frac{dy}{dx}\right)^2\right]dx=m\,dx, \]

where

\[ m=1+y\frac{d^2y}{dx^2}+\left(\frac{dy}{dx}\right)^2. \]

Then

\[ QD=\frac{dx}{AA_1}\,dx_0 =\frac{dxm\,dx}{dx\cdot q} =\frac{m}{q}\,dx. \]

From the relations written above we find

\[ KF=\frac{m}{q}\,dx+\left(q\,dx-\frac{m}{q}\,dx\right)\frac{u}{yq} =dx\left[\frac{m}{q}+\left(q-\frac{m}{q}\right)\frac{u}{yq}\right], \]

or, taking into account that

\[ dx=\frac{dx_0}{m}, \]

we obtain

\[ KF=\frac{dx_0}{m}\left[\frac{m}{q}+\left(q-\frac{m}{q}\right)\frac{u}{yq}\right] =\frac{dx_0}{mq}\left[m+\left(q^2-m\right)\frac{u}{yq}\right]. \]

Then the relative elongation of the fiber under consideration will be expressed as:

\[ \varepsilon=\frac{FN}{KF} =\frac{dx\cdot u} {\dfrac{dx_0}{mq}\left[m+\left(q^2-m\right)\dfrac{u}{yq}\right]} = \frac{dx\cdot mqu} {dx_0\left[m+\left(q^2-m\right)\dfrac{u}{yq}\right]}. \]

The stress in the fiber \(KF\) will be

\[ \sigma=\varepsilon E= \frac{dx\cdot Emqu} {dx_0\left[m+\left(q^2-m\right)\dfrac{u}{yq}\right]}. \tag{3} \]

Taking into account that

\[ \frac{dx_0}{d\varphi}=\rho_0, \]

where \(\rho_0\) is the radius of curvature of the bent axis of the beam at point \(D\), we can rewrite the expression for \(\varepsilon\) as follows:

\[ \sigma=-\frac{Emqu}{\rho_0\left[m+\left(q^2-m\right)\frac{u}{yq}\right]}. \tag{4} \]

In this expression:

\(E\) is the modulus of elasticity,

\(u=DF\) is the distance of the fiber under consideration from point \(D\),

\(y\) is the ordinate of point \(A\),

\(\rho_0\) is the radius of curvature of the bent axis of the beam at point \(D\), and

\[ \begin{aligned} m&=1+y\frac{d^2y}{dx^2}+\left(\frac{dy}{dx}\right)^2,\\ q&=\sqrt{1+\left(\frac{dy}{dx}\right)^2}. \end{aligned} \tag{5} \]

Let us note that all quantities entering the right-hand side of expression (4) are, for the given broken section \(ADB\), constant, with the exception of the quantity \(u\). Thus, the stress at any point of the section under consideration is a function of the distance of this point from point \(D\).

As is seen from expression (4), in the general case the stress will be distributed over the section \(ADB\) not according to a linear law, but according to a hyperbolic one.

Further, from the condition of equilibrium we may write

\[ M=2\int_{0}^{AD}\sigma b u\,du, \]

or

\[ M=2\int_{0}^{yq} \frac{Emqbu^2\,du} {\rho_0\left[m+\left(q^2-m\right)\frac{u}{yq}\right]}, \]

whence

\[ \frac{E}{\rho_0} = 2\,\frac{M} {\displaystyle\int_{0}^{yq} \frac{mqbu^2\,du} {m+\left(q^2-m\right)\frac{u}{yq}}}. \]

Substituting \(\dfrac{E}{\rho_0}\) into formula (4), we obtain the expression for the stress in the following form:

\[ \sigma= \frac{Mmqu} {\left[m+\left(q^2-m\right)\frac{u}{yq}\right]\cdot 2\cdot \displaystyle\int_{0}^{yq} \frac{mqbu^2\,du} {m+\left(q^2-m\right)\frac{u}{yq}}}. \tag{6} \]

Solving the integral that appears in the denominator of expression (6), and then taking \(u=AD=y\cdot q\), we obtain the stress at point \(A\)

\[ \sigma= \frac{M} {2by^{2}\left[\frac{1}{2n}-\frac{a}{n^{2}}+\frac{a^{2}}{n^{3}}\ln\left(\frac{a+n}{a}\right)\right](a+n)^{3}}, \tag{7} \]

where

\[ n=\frac{q^{2}-m}{q}; \qquad a=\frac{m}{q}, \]

or, taking into account the notation previously adopted for \(q\) and \(m\),

\[ n=-\frac{y\dfrac{d^{2}y}{dx^{2}}} {\sqrt{1+\left(\dfrac{dy}{dx}\right)^{2}}}; \qquad a= \frac{ 1+y\dfrac{d^{2}y}{dx^{2}}+\left(\dfrac{dy}{dx}\right)^{2} } {\sqrt{1+\left(\dfrac{dy}{dx}\right)^{2}}}. \tag{8} \]

\(M\)—the moment of the external force with respect to point \(D\)

\[ M=P\cdot OD, \]

\(b\)—the width of the transverse section of the beam,

\(y\)—the ordinate of point \(A\),

\(\dfrac{dy}{dx}\) and \(\dfrac{d^{2}y}{dx^{2}}\)—respectively represent the values of the first and second derivatives of the equation of the contour of the beam or rod at the point \(A\) under consideration (see eq. 2).

The expressions (8) for \(n\) and \(a\) may be represented otherwise. It is known that

\[ \frac{1}{\rho} = \pm \frac{\dfrac{d^{2}y}{dx^{2}}} {\left[1+\left(\dfrac{dy}{dx}\right)^{2}\right]^{3/2}}, \tag{9} \]

where \(\rho\) is the radius of curvature, in our case—the radius of curvature of the beam contour at point \(A\). The sign in expression (9) must be chosen in accordance with the sign of the second derivative.

Expression (9) may be rewritten as:

\[ \frac{1}{\rho} = \pm \frac{\dfrac{d^{2}y}{dx^{2}}} {\sqrt{1+\left(\dfrac{dy}{dx}\right)^{2}}\left[1+\left(\dfrac{dy}{dx}\right)^{2}\right]}, \]

whence

\[ \frac{\dfrac{d^{2}y}{dx^{2}}} {\sqrt{1+\left(\dfrac{dy}{dx}\right)^{2}}} = \pm \frac{1+\left(\dfrac{dy}{dx}\right)^{2}}{\rho} = \pm \frac{1+\operatorname{tg}^{2}\alpha}{\rho} = \pm \frac{1}{\rho\cos^{2}\alpha}, \]

where \(\alpha\) is the angle made by the tangent to the beam contour with the axis of abscissas for the point \(A\) under consideration.

Then expressions (8) may be transformed as follows:

\[ n=\pm \frac{\frac{y}{\rho}}{\cos^2\alpha};\quad a=\frac{\cos\alpha+\frac{y}{\rho}}{\cos^3\alpha}. \tag{10} \]

In the expressions obtained, the upper sign must be taken if the point \(A\) under consideration lies on the concave part of the contour; the lower sign should be taken if the point under consideration lies on the convex part of the contour.

In many cases it is more convenient to carry out the calculations using expressions (10), since \(y\), \(\rho\), and \(\alpha\), which enter these expressions for a given contour of a beam or rod, can readily be found graphically.

Formula (7), together with expressions (8) or (10), solves the stated problem of determining the stress at a certain point \(A\) of the contour of the beam under consideration.

From formula (7), derived for a straight beam or rod of arbitrary outline, solutions may be obtained for various particular cases; thus, for example:

1. For the case of a prismatic beam or rod,

\[ \frac{dy}{dx}=0;\quad \frac{d^2y}{dx^2}=0, \]

and consequently, on the basis of expressions (8),

\[ n=0;\quad a=1. \]

After substituting the obtained values into formula (7) and resolving the indeterminacy, we find

\[ \sigma=1.5\,\frac{M}{by^2}, \tag{11} \]

i.e., we obtain a formula valid for the calculation of prismatic beams. Taking into account that

\[ y=\frac{h}{2}, \]

where \(h\) is the height of the beam section, formula (11) may be given the more usual form:

\[ \sigma=\frac{M}{\frac{bh^2}{6}}, \]

taking into account that we are considering beams of rectangular section.

1. For the case of a wedge-shaped beam we shall have

\[ \frac{dy}{dx}=\operatorname{tg}\alpha;\quad \frac{d^2y}{dx^2}=0, \]

and consequently, according to expressions (10), we obtain

\[ n=0;\quad a=\sqrt{1+\operatorname{tg}^2\alpha}=\frac{1}{\cos\alpha}. \]

Substituting the expressions found into formula (7) and resolving the indeterminacy, we find

\[ \sigma=1.5\,\frac{M\cos^2\alpha}{by^2}. \]

If the angle \(a\) does not exceed \(20^\circ\), then the expression obtained differs from the exact calculation of a wedge-shaped beam, based on the theory of elasticity, by no more than 3%.

The limit of applicability of formula (7) is the case when, for the convex part of the contour, the center of curvature of the contour lies on the same side

Fig. 3

of the axis of abscissas as the contour itself (see Fig. 3), since in this case the quantity whose logarithm must be taken in formula (7) becomes negative.

3. Stress diagrams for normal stresses obtained according to the hypothesis of broken sections for various forms of straight-axis beams

According to formula (4), in Fig. 4 stress diagrams have been constructed for various sections of a cantilever beam having the form of a shaped plate. The plate has been chosen in the form indicated in Fig. 4 so that all possible cases could be represented on it.

Let us note the characteristic features of the individual diagrams shown in the figure under consideration.

1. The stress diagram constructed for section \(A_1 B_1\) corresponds to the prismatic part of the beam and gives a rectilinear law of stress distribution, which agrees with the usual notions concerning the bending of prismatic beams.

2. The diagram constructed for section \(A_2 D_2 B_2\) corresponds to the wedge-shaped part of the beam. The diagram obtained represents a linear law of stress distribution. The diagram under consideration does not fully coincide with the diagram obtained in the exact solution for a wedge-shaped beam. In this respect, the hypothesis of cylindrical sections would give a more accurate result. For a wedge-shaped beam loaded by a force applied at the vertex of the wedge, the hypothesis of cylindrical sections gives a result coinciding with the exact solution,

Fig. 4

\[ \sigma = \frac{P \cdot \sin \beta} {b \cdot r \left( \alpha - \frac{1}{2}\sin 2\alpha \right)} , \]

where \(\alpha\) is half the angle of taper of the wedge,

\(\varphi\) and \(r\) are the polar coordinates of any point of the wedge, if the origin of coordinates coincides with the vertex of the wedge and the polar axis with the axis of the wedge.

1. The stress diagram constructed for section \(A_3B_3\) corresponds to the convex part of the beam contour and represents the law of stress distribution in the form of certain hyperbolas. From the indicated circumstance it follows that, in sections corresponding to the convex part of the contour, the beam resists bending more intensively than in sections corresponding to the prismatic part.

2. The stress diagram constructed for section \(A_4B_4\) is characterized by a concentration of stresses at the concave parts of the contour. We shall use this circumstance later to determine the concentration of stresses at fillets and notches.

3. The stress diagram constructed for section \(A_5B_5\), which is characteristic in that the contour of the beam corresponding to this section has a center of curvature lying on the neutral axis, is represented in the form of two rectangles. This case is a limiting one; further it passes into the case shown in Fig. 3, for which the hypothesis of broken sections is inapplicable. Therefore it must be assumed that the stress diagram obtained for section \(A_5B_5\) is considerably inaccurate.

4. Determination of stress concentration by formula (7), derived from the hypothesis of broken sections, at fillets and notches. Comparison of the results obtained by formula (7) with experimental data

In Fig. 5 a comparison is reproduced of the results obtained by formula (7) with experimental data.

In Fig. 5(a) there is shown a straight-axis plate with two symmetrically arranged notches. The plate is subjected to pure bending. In the figure under consideration, the stress diagram for the central section is given, constructed by means of formulas (4) and (7).

Fig. 5

In Fig. 5(b) there is shown a plate having one notch. For the central section of this plate the stress diagram obtained by the optical method is shown \([1, p. 151]\).

Unfortunately, a complete comparison cannot be made between the two presented results, since in the first case the plate is taken with two notches (for the reason that formula (7) is applicable only to straight-axis beams), and in the second—with one.

However, it may be stated that in the parts adjoining the notches, the diagrams in both cases turned out to be sufficiently identical. In both cases the stress distribution in the considered parts of the diagrams has a hyperbolic character.

In Fig. 6 a comparative picture is given of the distribution of normal stresses along the contour of a notch.

In Fig. 6 the thick line shows the diagram constructed on the basis of experimental data obtained by the optical method \([1, p. 151]\). The thin line shows the stress diagram calculated by formula (7). The diagrams under consideration give comparatively good agreement. The greatest discrepancy does not exceed 4%.

In Fig. 7 are shown curves representing the stress-concentration coefficients as functions of the ratio \(\dfrac{\rho}{h}\). The curves were obtained for the plate shown in Fig. 5(a), subjected to pure bending. In Fig. 7, line (I) gives the curve obtained experimentally (see N. M. Belyaev, Course in Strength of Materials, 1939, p. 604), for \(H=h+2\rho\). Line (II) shows the curve of stress-concentration coefficients,

Fig. 6

Fig. 7

calculated by means of formula (7), from which the following expression may be obtained for the stress-concentration coefficient for a point lying at the bottom of the notch:

\[ K= \frac{1}{ -3\left[ \frac{1}{2c} + \frac{1+c}{c^2} + \frac{(1+c)^2}{c^3} \ln\left(\frac{1}{1+c}\right) \right] }, \]

where

\[ c=+\frac{y}{\rho}=+\frac{h}{2\rho}. \]

In the graph under consideration, Fig. 7, one can observe a very good agreement between the experimental and theoretical results, which indicates the sufficient accuracy of formula (7). The discrepancy between the experimental and theoretical curves, beginning with the ratio \(\dfrac{\rho}{h}\) equal to 0.1 (and greater), does not exceed 2–3%.

On the same graph, Fig. 7, a dashed line shows the curve of stress-concentration coefficients calculated by means of the hypothesis of cylindrical sections. The stress-concentration coefficients for a point lying at the bottom of the notch may be obtained, on the basis of the hypothesis of cylindrical sections, from the following formula, which we give without derivation:

\[ K= \frac{1}{ 6n\left[ \frac{1}{2}\sqrt{2n+1}\cdot \ln\left( \frac{\sqrt{2n+1}+1}{\sqrt{2n+1}-1} \right) -1 \right] }, \]

where

\[ n=\frac{\rho}{y}=\frac{2\rho}{h}. \]

Considering the graph in Fig. 7, one can state that line II, obtained on the basis of the hypothesis of broken sections, gives a smaller discrepancy with the experimental curve than the dashed line obtained on the basis of the hypothesis of cylindrical sections.

Table 1 gives a comparison of the stress-concentration coefficients obtained by various investigators for the case of bending of a flat specimen with lateral notches.

Table 1¹)

$\dfrac{\rho}{h}$	0.1	0.1	0.1	0.2	0.2	0.3
$\dfrac{t}{h}$	0.1	0.2	0.4	0.2	0.4	0.3
According to Buchmann	2.0	2.45	2.9	—	—	—
According to Frocht	1.85	1.93	2.04	1.58	1.6	1.43
According to Afanas’ev’s calculations	1.75	1.82	1.91	1.56	1.58	1.44
According to the hypothesis of broken sections	1.90	1.90	1.90	1.50	1.50	1.40

Let us proceed to consider the concentration of stresses at fillets. In Figs. 8(a) and 8(b) plates are shown, each having two symmetrically arranged fillets and subjected to pure bending. In the same figure are plotted the stress diagrams found by formula (7) for points on the contours of the notches.

At point $A$, according to formula (7), a jump in stresses is obtained. This circumstance is explained by the fact that at point $A$ the second derivative of the equation representing the contour of the plate changes discontinuously. But a discontinuous change of stress at point $A$ cannot correspond to reality, since it cannot be admitted that at one point there simultaneously exist, in one direction, two different normal stresses. It must be recognized that, owing to the additional curvature of the broken sections, an equalization of stresses will occur here along a certain line $FR$, indicated in Fig. 8 by a dashed line.

Fig. 8

In the literature there are experiments, carried out by the optical method, with plates of the indicated form. The stress diagrams constructed from the results of these experiments differ hardly at all from the corrected form of the theoretical diagram of Fig. 8.

The maximum stress is obtained at a certain point $A_1$, located at a certain angle to the vertical. From comparison of Figs. 8(a) and 8(b) it is seen that this angle will be the smaller, the greater the ratio $\dfrac{\rho}{h}$. Below, more specific values of the angle under consideration will be given.

¹) In compiling the table, the article by N. N. Afanas’ev [4] was used.

Figure 9 gives curves of the stress-concentration coefficient as a function of the ratio \(\rho/h\) for plates having the shape shown in Fig. 8. The thick line (I) shows the curve of stress-concentration coefficients obtained experimentally, [1, p. 150]. The thin line (II) is the curve of stress-concentration coefficients obtained by means of formula (7). The discrepancy between the curves under consideration is explained in part by the fact that, in constructing the theoretical curve, the maximum stress was assumed to occur at point \(A\), whereas in reality it will occur at point \(A_1\), lying at a certain angle to the vertical (see Fig. 8).

Fig. 9

If, in constructing the theoretical curve, one starts from the stress present at point \(A_1\), determining it by formula (7), then the theoretical curve almost coincides throughout with the experimental curve, provided that the following values of the angle \(\alpha\) are taken:

\[ \text{for } \frac{\rho}{h}=0.1 \qquad \alpha=20^\circ \]

\[ \text{for } \frac{\rho}{h}=0.2 \qquad \alpha=10^\circ \]

\[ \text{for } \frac{\rho}{h}>0.3 \qquad \alpha=0^\circ . \]

Therefore, in order to determine the greatest stress in the case of a cut-out, the calculation should be carried out by formula (7), taking the indicated values of the angle \(\alpha\).

5. Graph for determining the stress by formula (7)

An obstacle to the use of formula (7) may be its complexity. The calculation process can be greatly accelerated with the aid of the graph given below.

Formula (7) may be represented in the following form:

\[ \sigma=\frac{M\cdot B}{b\cdot y^2}, \tag{12} \]

where

\[ B= \frac{1}{ 2\left[ \frac{1}{2n} -\frac{a}{n^2} +\frac{a^2}{n^3}\ln\left(\frac{a+n}{a}\right) \right](a+n)^3 }. \tag{13} \]

From expressions (10) and (13) it follows that the quantity \(B\) is a function of the angle \(\alpha\) and of the ratio \(y/\rho\).

The graph (Fig. 10) gives curves representing \(B\) as a function of the ratio \(\dfrac{y}{\rho}\) for various values of the angle \(\alpha\).

Thus, if the quantities \(\dfrac{y}{\rho}\) and \(\alpha\) are known, then the quantity \(B\) is found from the graph very quickly, as will be shown in the example.

Fig. 10

The value of \(B\) thus found should then be substituted in formula (12), from which the stress \(\sigma\) is determined.

Suppose a cantilever beam is given, with variable height of the cross section (see Fig. 11), loaded by a force \(P\) applied at point \(O\).

Fig. 11

The force \(P = 1000\ \mathrm{kg}\). It is required to determine the stress at point \(A\).

Let the following quantities be known (or determined from the drawing):

\[ \begin{aligned} x &= 9.3\ \mathrm{cm} && \text{(abscissa of point \(A\))},\\ y &= 2\ \mathrm{cm} && \text{(ordinate of point \(A\))},\\ \rho &= 2\ \mathrm{cm} && \text{(radius of curvature of the contour at point \(A\))},\\ \alpha &= 20^\circ,\\ b &= 5\ \mathrm{cm} && \text{(width of the beam cross section)}. \end{aligned} \]

We draw through point \(A\) the broken section \(ADB\).
The lever arm of the force will be

\[ OD=x+y\tg\alpha=9.3+2\cdot 0.36\simeq 10\ \text{cm}. \]

Then the bending moment for section \(ADB\) is

\[ M=P\cdot OD=1000\cdot 10=10000\ \text{kg}\cdot\text{cm}. \]

Next we find

\[ \frac{y}{\rho}=\frac{2}{2}=1. \]

From the graph (Fig. 10) we determine the value of \(B\) as follows. On the lower horizontal axis of the graph, on which the ratios \(\frac{y}{\rho}\) are plotted, we find the point denoted by the number 1. Taking into account that we are determining the stress at point \(A\), which lies on the concavity of the contour, the point denoted by the number 1 must be sought on the indicated axis of the graph to the left of zero. Having found the point denoted by the number 1, we draw through it a vertical line up to its intersection with the curve \(\alpha=20^\circ\). The point obtained on the curve is carried horizontally to the right-hand vertical axis of the graph, on which we obtain the value of \(B\). In our case

\[ B=1.62. \]

After this, using formula (12), we determine the stress

\[ \sigma=\frac{M\cdot B}{by^2}=\frac{10000\cdot 1.62}{5\cdot 2^2}=810\ \frac{\text{kg}}{\text{cm}^2}. \]

Such will be the stress at point \(A\).

Let us note that if the beam were bent not under the action of the force \(P\), but under the action of a couple of forces, then the value of the moment of the couple of forces should be substituted in formula (12). In this case the finding of point \(D\) is no longer needed.

If it were required to determine the stress for a point lying on the convex portion of the contour, then, after computing the ratio \(\frac{y}{\rho}\), the corresponding point should be sought on the lower horizontal axis of the graph in Fig. 10 to the right of the zero point. If it were required to determine the stress for a point lying on a rectilinear portion of the contour, then the zero point should be taken on the lower horizontal axis of the graph.

On the left vertical axis of the graph are plotted the values of the stress-concentration coefficient.

In the example considered, the stress-concentration coefficient for point \(A\) is obtained from the graph as equal to 1.1.

Let us note that here the stress-concentration coefficient is understood as the ratio of the magnitude of the stress computed by formula (12) to the magnitude of the stress which would be obtained for the same point according to the hypothesis of plane sections.

6. Some examples of the application of the derived formulas to the calculation for bending of gear teeth

A characteristic example of a straight-axis beam of variable cross-sectional height is the tooth of a gear wheel.

Let us consider the tooth of a gear wheel shown in Fig. 12. To determine the stress at some point \(A\) of the tooth profile, we measure on the drawing (Fig. 12) the following quantities:

\[ \alpha;\ y;\ OD;\ \rho=AT. \]

Proceeding from these quantities, one can determine, by formula (12), the stress at point $A$, in the same way as was done in the preceding example. Having determined the stresses for a whole series of points of the profile, one can construct a stress diagram along the entire length of the profile. Such a diagram is shown in Fig. 12.

On the basis of the data given in item 4 (see Fig. 8), the theoretical diagram should be somewhat corrected near the point of inflection of the profile (point $A_0$). In Fig. 12 the correction has been made along the curve $FR$. From the diagram (Fig. 12) it is seen that the dangerous point of the tooth profile, in the sense of the greatest normal stress, in the present example is obtained on the fillet of the leg near point $A_0$.

Fig. 12                Fig. 13

In constructing the stress diagram (Fig. 12), it was assumed that the gear wheel has no rim, but is made in the form of a solid disk. In the case where the wheel has a rim of limited thickness, the stress diagram in the part lying near point $A_3$ will take on a somewhat different form.

In Fig. 13 a stress diagram is shown for points of the tooth profile belonging to a gear wheel having a rim of limited thickness.

On the segment of the profile $KA_2$, the stress diagram will retain its former form, coinciding with Fig. 12. Beginning from point $A_2$, the stress diagram, because of the deformation of the rim, will take on a somewhat different form, shown approximately in Fig. 13 by the dashed line.

What has been said is explained by the fact that formulas (7) or (12), by means of which the stress is calculated, may be applied only so long as the vertex of the broken section (point $D_2$) is within the body of the beam being bent. As soon as point $D_2$ goes beyond the limits of the rim, the stresses on the contour, beginning from point $A_2$, will be determined not by bending of the tooth, but by bending of the rim, which bending will occur in two separate sections $A_2D_2$ and $B_2D_2$.

7. Extension of the hypothesis of broken sections to the case where a straight-axis rod of any curvilinear outline has an I-section

In Fig. 14 there is shown a straight-axis beam of arbitrary outline, having an I-section. In the right part of the figure an unfolded broken section of the beam $ADB$ is shown.

The problem will consist in determining the stress at point $A$ of the beam under consideration.

The solution of the problem can be obtained from formula (6), which need only be somewhat transformed, since formula (6) was obtained for a rod of rectangular cross-section. In transforming formula (6) it is necessary to take into account that the width of the cross-section \(b\) will not be constant.

According to the notation adopted earlier and Fig. 14,

\[ y \cdot q = AD = u_a . \]

Then the integral entering into formula (6) can be represented in the form of the sum of two integrals with limits from 0 to \(u_0\) and from \(u_0\) to \(u_a\) (see

Fig. 14

Fig. 14). Since the stress is being sought at point \(A\), in the part lying outside the integral one should put

\[ u = u_a . \]

Now formula (6) can be given the following form:

\[ \sigma = \frac{M m q u_a} { \left[m + (q^2 - m)\right]\cdot 2 \left[ \displaystyle \int_{0}^{u_0} \frac{m q b_2 u^2\,du} {m + (q^2 - m)\dfrac{u}{u_a}} + \int_{u_0}^{u_a} \frac{m q b_1 u^2\,du} {m + (q^2 - m)\dfrac{u}{u_a}} \right] } . \tag{14} \]

The integrals in the expression obtained are integrals of rational functions, and their solution presents no difficulty.

Formula (14) makes it possible to determine the stress at any point \((A)\) lying on the outline of the beam under consideration.

Part II. Bending of Curved Rods of Figured Outline

3. Application of the hypothesis of broken sections to certain particular cases of non-straight beams

The hypothesis of broken sections can be applied to certain cases of non-straight beams.

Let us consider beams that have a cross-section lying in a plane passing through both centers of curvature of the upper and lower parts of its contour. With the aid of the hypothesis of broken sections it becomes possible to determine rather simply the stresses in the section coinciding with the indicated plane.

As a first example, let us consider the beam shown in Fig. 15.

The beam is a plate of constant thickness \(b\), having a projection in its upper part and a concavity in its lower part. The plane \(nn\) is the transverse plane of symmetry of the beam.

Let the beam be subjected to pure bending by a certain moment \(M\). The problem will consist in determining the stresses in the section \(AB\).

Let the line \(tt\) represent the neutral layer of the beam. Let us draw the coordinate axes \(X\) and \(Y\), choosing point \(D\) as the origin of coordinates. Then, in general form, the upper contour of the beam may be represented by the equation \(y_1=f_1(x_1)\), and the lower contour by the equation \(y_2=f_2(x_2)\); moreover, we shall assume that, for the equation of the upper contour, the positive ordinate axis is directed upward, and for the equation of the lower contour the positive ordinate axis is directed downward. This will be convenient later in determining the signs before \(\rho_1\) and \(\rho_2\).

Fig. 15

In accordance with the adopted notation, the first and second derivatives for the upper contour will be

\[ \frac{dy_1}{dx_1} \quad \text{and} \quad \frac{d^2y_1}{dx_1^2}, \]

and for the lower contour

\[ \frac{dy_2}{dx_2} \quad \text{and} \quad \frac{d^2y_2}{dx_2^2}. \]

For a beam of the type under consideration we shall have:

\[ \left. \begin{array}{ll} \text{for point } A, & \dfrac{dy_1}{dx_1}=0;\quad \dfrac{d^2y_1}{dx_1^2}=\dfrac{1}{\pm \rho_1};\quad y_1=y_a=AD, \\[1.2em] \text{for point } B, & \dfrac{dy_2}{dx_2}=0;\quad \dfrac{d^2y_2}{dx_2^2}=\dfrac{1}{\pm \rho_2};\quad y_2=y_b=DB \end{array} \right\} \tag{15} \]

where \(\rho_1\) and \(\rho_2\) are the corresponding radii of curvature of the contours at points \(A\) and \(B\). Before these radii one must take the plus sign if the contour at the given point has a concavity, and the minus sign if the contour at the given point has a convexity. For the beam shown in Fig. 15, the plus sign should be taken before both radii.

Let us take a point \(A_1\), at a distance \(dx\) from point \(A\), and draw through it the broken section \(A_1D_1B_1\).

Since the elementarily small segment \(DD_1\) of the neutral layer may be regarded as coinciding with the axis of abscissas, it becomes possible, for determining the stresses in the section \(AB\), to apply formula (3).

For the type of beam under consideration, taking into account conditions (15), we shall have, according to expressions (5):

for the upper part of the beam

\[ \left. \begin{gathered} m=m_1=1+\frac{y_a}{\pm \rho_1},\\ q=q_1=1;\quad y=y_a;\quad u=u_1 \end{gathered} \right\}; \tag{16} \]

for the lower part of the beam

\[ \left. \begin{gathered} m=m_2=1+\frac{y_b}{\pm \rho_2},\\ q=q_2=1;\quad y=y_b;\quad u=u_2 \end{gathered} \right\}. \tag{17} \]

Then the expression for the stresses for the upper part of the beam, i.e., for the points of the segment \(AD\) of the section, from formula (3) will be

\[ \sigma_1= \frac{d\varphi E m_1 u_1} {d x_0\left[m_1+(1-m_1)\dfrac{u_1}{y_a}\right]}, \tag{18} \]

or, substituting \(m_1\) and carrying out elementary transformations, we obtain

\[ \sigma_1= \frac{d\varphi E u_1} {d x_0\left(1-\dfrac{u_1}{\pm \rho_1+y_a}\right)}. \tag{19} \]

For the lower part of the beam, i.e., for the points of the segment \(BD\), in the corresponding way we find

\[ \sigma_2= \frac{d\varphi E m_2 u_2} {d x_0\left[m_2+(1-m_2)\dfrac{u_2}{y_b}\right]}, \tag{20} \]

or, after substituting \(m_2\) and some elementary transformations, we obtain

\[ \sigma_2= \frac{d\varphi E u_2} {d x_0\left(1-\dfrac{u_2}{\pm \rho_2+y_b}\right)}. \tag{21} \]

From the condition of equilibrium, the sum of the projections of the forces acting on the left or right part of the beam must be equal to zero. Taking into account that the beam is subjected to pure bending, the indicated condition of equilibrium may be represented in the following form:

\[ \int_0^{y_a}\sigma_1 b\,du_1 = \int_0^{y_b}\sigma_2 b\,du_2. \tag{22} \]

Substituting \(\sigma_1\) and \(\sigma_2\) from (19) and (21) and carrying out the integration, after cancellation by \(E \cdot b \cdot \dfrac{d\varphi}{dx_0}\) and changing the signs to the opposite ones, we obtain

\[ (\pm \rho_1 + y_a)^2 \left[ \frac{y_a}{\pm \rho_1 + y_a} + \ln \left( \frac{\pm \rho_1}{\pm \rho_1 + y_a} \right) \right] = (\pm \rho_2 + y_b)^2 \left[ \frac{y_b}{\pm \rho_2 + y_b} + \ln \left( \frac{\pm \rho_2}{\pm \rho_2 + y_b} \right) \right]. \tag{23} \]

The signs before \(\rho_1\) and \(\rho_2\) must be taken in accordance with the concavity or convexity of the contours, as was indicated above. In addition, one may write

\[ y_a + y_b = h. \tag{24} \]

By jointly solving the last two equations with respect to \(y_a\) and \(y_b\), one can determine the position of the neutral layer, i.e., point \(D\).

Let us now find the stresses at points \(A\) and \(B\). For this purpose we write the equation of equilibrium of the moments of the external forces and the elastic forces with respect to point \(D\) in the following form:

\[ M = \int_{0}^{y_a} \sigma_1 b u_1 \, du_1 + \int_{0}^{y_b} \sigma_2 b u_2 \, du_2 . \]

Substituting \(\sigma_1\) and \(\sigma_2\) from expressions (18) and (20), we obtain

\[ M = E \frac{d\varphi}{dx_0} \left[ \int_{0}^{y_a} \frac{m_1 b u_1^2 \, du_1} {m_1 + (1-m_1)\dfrac{u_1}{y_a}} + \int_{0}^{y_b} \frac{m_2 b u_2^2 \, du_2} {m_2 + (1-m_2)\dfrac{u_2}{y_b}} \right]. \tag{25} \]

Denoting the first integral standing in the brackets by the letter \(U_a\), and the second by the letter \(U_b\), we obtain

\[ E \frac{d\varphi}{dx_0} = \frac{M}{U_a + U_b}. \]

Substituting the value found for \(E \dfrac{d\varphi}{dx_0}\) into formula (18) and setting \(u_1 = y_a\), we find the stress at point \(A\):

\[ \sigma_a = \frac{M}{U_a + U_b} \cdot m_1 y_a . \tag{26} \]

Similarly, the stress at point \(B\) is found from formula (20):

\[ \sigma_b = \frac{M}{U_a + U_b} \cdot m_2 y_b . \tag{27} \]

The solutions of the integrals \(U_a\) and \(U_b\) will be as follows:

\[ U_a = \frac{b m_1 y_a^3}{(1-m_1)^3} \left[ \frac{1}{2} - 2m_1 + m_1^2(1.5 - \ln m_1) \right], \]

\[ U_b = \frac{b m_2 y_b^3}{(1-m_2)^3} \left[ \frac{1}{2} - 2m_2 + m_2^2(1.5 - \ln m_2) \right]. \]

Let us note that in these expressions

\[ \frac{y_a^3}{(1-m_1)^3}=-\rho_1^3 \]

and

\[ \frac{y_b^3}{(1-m_2)^3}=-\rho_2^3, \]

which may simplify the calculations.

In the case when \(m_1=1\) or \(m_2=1\), which will occur for a rectilinear contour, the integrals \(U_a\) and \(U_b\), respectively, will have the solutions

\[ U_a=\frac{b y_a^3}{3}, \]

\[ U_b=\frac{b y_b^3}{3}. \]

Let us note that the stresses at points \(A\) and \(B\) of the rod under consideration (Fig. 15) can also be calculated by means of the graph in Fig. 10. In this case the stresses \(\sigma_a\) and \(\sigma_b\) must be represented by the following expressions:

\[ \sigma_a=\frac{2MB_1B_2m_1y_a}{(B_1y_b^3m_2+B_2y_a^3m_1)b}; \tag{28} \]

\[ \sigma_b=\frac{2MB_1B_2m_2y_b}{(B_1y_b^3m_2+B_2y_a^3m_1)b}. \tag{29} \]

In these expressions \(B_1\) and \(B_2\) should be determined from the graph in Fig. 10: when determining \(B_1\), proceeding from the parameters of the upper contour curve for point \(A\), i.e. \(\dfrac{y_a}{\rho_1}\) and \(\alpha_1\), equal to zero, and when determining \(B_2\), proceeding from the parameters of the lower contour curve for point \(B\), i.e. \(-\dfrac{y_b}{\rho_2}\) and \(\alpha_2\), equal to zero.

Fig. 16

As a second example, let us consider the beam shown in Fig. 16.

This example is of special interest because, for a beam of this form, experimental data are available in the literature, and it appears possible to compare the theoretical results with the experimental data.

In the book by Prof. S. P. Timoshenko [1] are given the results of an experimental determination of stresses in section \(ADB\) of the beam shown in Fig. 16, having the form of a plate of constant thickness \(b\) and subjected to pure bending. The stress diagram obtained from the experiment is shown in Fig. 16 by the heavy line. The principal dimensions of the tested plate were as follows:

\[ H = 28.6\ \mathrm{mm}, \]

\[ \rho_1 = 1.6\ \mathrm{mm}, \]

\[ h = 19.1\ \mathrm{mm}, \]

\[ \frac{\rho_1}{h} = 0.0831 \cong \frac{1}{12}. \]

Let us set the problem of constructing the stress diagram for section \(ADB\), proceeding from theoretical calculations. The beam (plate) under consideration is a special case of the beam of the preceding example (Fig. 15).

The peculiarity of the present case is determined by the fact that the lower contour of the beam is rectilinear. Thus, for the beam under consideration we shall have:

\[ \begin{aligned} &\text{for point } A \qquad &&m_1 = 1 + \frac{y_a}{\rho_1},\\ &&&q_1 = 1;\\ &\text{for point } B \qquad &&m_2 = 1,\\ &&&q_2 = 1. \end{aligned} \]

Then the expression for the stress at points of the segment \(AD\) may be found from formula (18), which is written as:

\[ \sigma_1 = \frac{d\varphi E u_1} {dx_0\left(1-\dfrac{u_1}{\rho_1+y_a}\right)}, \tag{30} \]

and the expression for the stress at points of the segment \(DB\) may be found from formula (20), which takes the form

\[ \sigma_2 = \frac{d\varphi E u_2}{dx_0}. \tag{31} \]

Let us proceed to determine the position of the neutral layer, i.e., point \(D\). From equation (22), for the case under consideration we obtain

\[ (\rho_1+y_a)^2 \left[ -\frac{y_a}{\rho_1+y_a} -\ln\left(\frac{\rho_1}{\rho_1+y_a}\right) \right] = \frac{y_b^2}{2}. \]

In addition, we have:

\[ y_a + y_b = h. \]

Solving the last two equations simultaneously by trial, we obtain

\[ y_a = 0.380h = AD, \]

\[ y_b = 0.620h = DB. \]

According to the experimental data,

\[ y_a = 0.425h = AD_1, \]

\[ y_b = 0.575h = D_1B. \]

Thus, the discrepancy in the position of the neutral layer, theoretical and experimental, is obtained in the form of the distance

\[ DD_1 = 0.425h - 0.380h = 0.045h = 0.86\ \mathrm{mm}. \]

Let us proceed to a comparison of the stresses found theoretically and experimentally. Using formulas (30) and (31), one can find the ratio of the theoretical stresses at points \(A\) and \(B\). For this we set, in formulas (30) and (31),

\[ u_1 = y_a \quad \text{and} \quad u_2 = y_b, \]

after which we divide \(\sigma_1\) by \(\sigma_2\). Then we obtain

\[ \frac{\sigma_a}{\sigma_b} = \frac{\sigma_1}{\sigma_2} = \frac{y_a}{\left(\dfrac{\rho_1}{\rho_1 + y_a}\right)y_b}. \]

Since the theoretical values \(y_a\) and \(y_b\) were obtained as

\[ y_a = 0.380\,h \]

and

\[ y_b = 0.620\,h, \]

and it is also known that

\[ \rho_1 = 0.0834\,h, \]

we obtain

\[ \frac{\sigma_a}{\sigma_b} = 3.38. \]

From the experimental data, the ratio of the stresses at points \(A\) and \(B\) is equal to

\[ \frac{\sigma_a}{\sigma_b} = 2.97, \]

i.e., the discrepancy is \(14\%\).

From comparison of the results obtained it follows that at point \(A\) the theoretical stress will be somewhat greater than the experimental one (by approximately \(7\%\)), while at point \(B\) the theoretical stress will be somewhat less than the experimental one (by approximately \(7\%\)). The curve of distribution of the theoretical stresses in the section \(ADB\) is plotted in Fig. 16 with a thin line.

Of all the questions considered in the present example, the greatest interest is presented by the magnitude of the stress at point \(A\), as the maximum stress in the given structure.

It turned out that the theoretical value of this stress, found on the basis of the hypothesis of broken sections, differs from the value found experimentally by approximately \(7\%\). There is reason to believe that a significant part of this discrepancy is due to the inaccuracy of the experimental data, since according to the experimental data the area of the positive-stress diagram is not quite exactly equal to the area of the negative-stress diagram (see Fig. 16), which contradicts the condition of equilibrium.

In conclusion to this example, we point out that the absolute values of the stresses at points \(A\) and \(B\) can be found from formulas (28) and (29) with the corresponding determination of the quantities entering into these formulas.

Fig. 17

As a third example, let us consider the curved bar shown in Fig. 17, subjected to bending by couples of forces applied at its ends.

The sections of the bar will be rectangular. Let both contours have a common center of curvature at the point \(O\).

In this particular case, the sections drawn through the point \(A\) and a nearby point \(A_1\) are not broken, but plane.

As a consequence, the problem reduces to the ordinary problem of a bar with a curvilinear axis, solved on the basis of the hypothesis of plane sections.

Let us also note that the formulas derived from the hypothesis of broken sections, in this particular case, reduce to the usual formulas of the hypothesis of plane sections.

9. Extension of the hypothesis of broken sections to the case when the curved-axis bar has non-rectangular sections

We shall restrict ourselves to considering the sections shown in Fig. 18. The lines \(nn\) in Fig. 18 represent the neutral layer.

In the case of a double-tee section [Fig. 18(a)] each of the integrals \(U_a\) and \(U_b\) entering into formulas (26) and (27) must be divided into two, namely

\[ U_a= \int_{0}^{y_1} \frac{m_1 b_2 u_1^2\,du_1}{m_1+(1-m_1)\dfrac{u_1}{y_a}} + \int_{y_1}^{y_a} \frac{m_1 b_1 u_1^2\,du_1}{m_1+(1-m_1)\dfrac{u_1}{y_a}}, \tag{32} \]

\[ U_b= \int_{0}^{y_2} \frac{m_2 b_2 u_2^2\,du_2}{m_2+(1-m_2)\dfrac{u_2}{y_b}} + \int_{y_2}^{y_b} \frac{m_2 b_3 u_2^2\,du_2}{m_2+(1-m_2)\dfrac{u_2}{y_b}}. \tag{33} \]

Let us recall that in these expressions \(u_1\) and \(u_2\) are the distances of any points of the upper and lower parts of the section from the neutral layer.

Fig. 18

To determine the position of the neutral layer, equation (22) is rewritten as follows:

\[ \int_{0}^{y_1}\sigma_1 b_2\,du_1 + \int_{y_1}^{y_a}\sigma_1 b_1\,du_1 = \int_{0}^{y_2}\sigma_2 b_2\,du_2 + \int_{y_2}^{y_b}\sigma_2 b_3\,du_2. \tag{34} \]

In the case of the sections shown in Fig. 18(b) and 18(c), in the written expressions one should take \(b_3\) equal to \(b_2\).

Part III. Application of the hypothesis of broken sections to the determination of normal stresses in plates of shaped outline under simultaneous tension and bending

10. Limits of applicability of the hypothesis of broken sections under tension of plates of shaped outline

In the first and second parts of the present work, formulas were derived for determining normal stresses in the bending of beams of shaped outline.

In view of the fact that many machine parts having a shaped outline are subjected not only to bending but also to tension or compression, it is very useful to extend the hypothesis of broken sections to the case of a combination of the indicated deformations.

For this purpose, let us first consider the case of axial tension of a plate of shaped outline and establish the limits of applicability of the hypothesis of broken sections for this case.

Let the plate of shaped outline have the form shown in Fig. 19. The thickness of the plate is constant and equal to \(b\).

The plate is subjected to tension under the action of forces \(P\) directed along its axis.

Through two infinitely close points \(A\) and \(A_1\), taken on the contour of the plate, draw the broken sections \(ADB\) and \(A_1D_1B_1\). The section \(ADB\) in the present

Fig. 19

Fig. 20

case will prove to be plane, because point \(A\) is a point of inflection of the contour, and the normal at this point will be directed perpendicular to the axis of the plate. Under the influence of the tensile forces \(P\), deformation will occur, and the section \(ADB\) will move relative to the section \(A_1D_1B_1\) into a certain position \(A_2D_2B_2\).

On the basis of the hypothesis of broken sections it is assumed that the sections do not curve. Then the fibers located between the sections \(A_1D_1B_1\) and \(ADB\) will receive equal absolute elongations, equal to \(DD_2\).

Consider the fiber \(KF\), located between the broken sections \(ADB\) and \(A_1D_1B_1\) and taken at a distance \(u\) from the axis of the rod.

The relative elongation of the fiber under consideration is:

\[ \varepsilon=\frac{FN}{KF}=\frac{FN(\rho+y_a)}{DD_1(\rho+y_a-u)}. \]

The stress in the same fiber will be

\[ \sigma=\varepsilon E=\frac{FN(\rho+y_a)E}{DD_1(\rho+y_a-u)}. \tag{35} \]

It follows from expression (35) that the law of stress distribution will be hyperbolic (Fig. 20).

The asymptotes of the hyperbolas will be lines passing through the centers of curvature \(O_1\) and \(O_2\) parallel to the axis of the plate.

From the equilibrium condition one can write

\[ P=2\int_{0}^{y_a}\sigma\cdot b\,du \]

or

\[ P=2\int_{0}^{y_a}\frac{FN\cdot(\rho+y_a)\cdot E\cdot b\cdot du}{DD_1\cdot(\rho+y_a-u)}. \]

Integrating, we obtain

\[ P=-\frac{2E\cdot FN\cdot(\rho+y_a)\cdot b\cdot \ln\left(\frac{\rho}{\rho+y_a}\right)}{DD_1} \]

Whence we find

\[ \frac{E\cdot FN}{DD_1}=\frac{P}{2(\rho+y_a)\cdot b\cdot \ln\left(\frac{\rho+y_a}{\rho}\right)}. \]

Substituting into expression (35), we obtain

\[ \sigma=\frac{P}{2(\rho+y_a-u)\cdot b\cdot \ln\left(\frac{\rho+y_a}{\rho}\right)}. \tag{36} \]

The greatest stress will be at point \(A\) when \(u=y_a\). This stress will be

\[ \sigma=\frac{P}{2\rho b\ln\left(\frac{\rho+y_a}{\rho}\right)}. \tag{37} \]

The stress-concentration coefficient may be found as follows:

\[ K=\frac{\sigma}{\sigma_0}, \]

where \(\sigma_0\) is the stress in the prismatic part of the plate, equal to

\[ \sigma_0=\frac{P}{b\cdot h}=\frac{P}{2by_a}. \tag{38} \]

On the basis of expressions (37) and (38),

\[ K=\frac{y_a}{\rho\ln\left(\frac{\rho+y_a}{\rho}\right)}, \]

or

\[ K=\frac{y_a}{\rho\ln\left(1+\frac{y_a}{\rho}\right)}, \tag{39} \]

or

\[ K=\frac{h}{2\rho\ln\left(1+\frac{h}{2\rho}\right)}. \tag{40} \]

We shall call the obtained quantity \(K\) the theoretical stress-concentration coefficient for the case of tension.

In the graph of Fig. 21, the thin line III is plotted for the theoretical coefficient \(K\) as a function of the ratio \(\frac{\rho}{h}\).

On the same graph, the curves (I and II) for experimental values of the stress-concentration coefficient are plotted with heavy lines.

Curve I is plotted for the condition \(\dfrac{H}{h}=2.5\) by means of the table given in Prof. S. P. Timoshenko’s book Course of Strength of Materials, 1931 ed., p. 578.

Fig. 21

This curve may be regarded as limiting, since a further increase of the ratio \(\dfrac{H}{h}\) has practically no effect on the stress-concentration coefficient.

Curve II is taken from Prof. N. M. Belyaev’s book Strength of Materials, 1939 ed., p. 604. This curve is plotted under the condition

\[ H = h + 2\rho . \]

In tension, the stress-concentration coefficient is affected not only by the ratio \(\dfrac{\rho}{h}\), but to a considerable degree also by the ratio \(\dfrac{H}{h}\), as follows from the experimental data. The latter circumstance limits the application of the theoretical formula (37), as well as formulas (39) and (40), since they do not take into account the influence of the magnitude of the ratio \(\dfrac{H}{h}\).

From consideration of the graph in Fig. 21 it follows that the theoretical curve (thin line III) differs little from the experimental curves (I and II) for values of the ratio \(\dfrac{\rho}{h}\) greater than 0.35.

Moreover, the theoretical curve agrees fairly well throughout its entire length with the upper limiting experimental curve I.

Thus, it may be considered that the hypothesis of broken sections and formulas (37, 39, and 40) obtained from it are practically applicable for determining normal stresses in tension if the shape of the plate satisfies at least one of the two conditions:

\[ \left. \begin{array}{l} \dfrac{\rho}{h} \ge 0.35 \\[6pt] \dfrac{H}{h} \ge 2.5 \end{array} \right\} \tag{41} \]

In this case, as follows from the graph, the error will not exceed 3–4%.

We shall keep the indicated circumstances in mind below when considering questions on the applicability of the hypothesis of broken sections for determining normal stresses in tension of parts of one configuration or another.

For comparison, in the graph of Fig. 21, the curve of the stress-concentration coefficient computed with the aid of the hypothesis of cylindrical sections is plotted as a dashed line, according to the formula, which we give without derivation:

\[ K=\frac{1}{\frac{n}{\sqrt{2n+1}}\ln\left(\frac{\sqrt{2n+1}+1}{\sqrt{2n+1}-1}\right)}, \]

where

\[ n=\frac{\rho}{y}=\frac{2\rho}{h}. \]

In Table 2 we give a comparison of the stress-concentration coefficients obtained by various investigators for the case of tension of a flat specimen with semicircular notches.

Table 2¹)

	\(\dfrac{\rho}{h}\)	0.2	0.25	0.35	0.4	0.5
According to Fischer		1.9	—	—	1.6	—
According to Wahl		—	1.9	—	—	1.55
According to Preiss and Berg		—	—	1.65	—	—
According to Frocht		2.03	1.92	—	—	1.52
According to Kuhn		1.7	1.68	1.65	1.62	1.60
According to Afanas’ev’s calculations		2.08	1.97	1.77	1.67	1.50
According to the hypothesis of broken sections		2.00	1.85	1.60	1.55	1.45

Let us note that, by means of the hypotheses under consideration, it is possible to determine the stress at an arbitrary point lying on the contour of the notch (see Fig. 20). Thus, according to the hypothesis of broken sections, the stress at point \(A_1\) will be

\[ \sigma=\frac{P\cos\alpha}{2b\rho\ln\left(1+\frac{y}{\rho\cos\alpha}\right)\left(\cos^2\alpha+\frac{G}{E}\sin^2\alpha\right)}, \]

where \(y\) is the ordinate of point \(A_1\),

¹) In compiling the table, the experimental data were borrowed from the article by N. N. Afanas’ev [4].

\(\alpha\)—the angle of inclination of the tangent at the point \(A_1\),
\(G\)—the modulus of elasticity in shear,
\(E\)—the modulus of elasticity in tension.

11. Example of determining normal stresses on the basis of the hypothesis of broken sections in a straight-axis rod subjected to tension and bending

Let us consider the plane straight-axis rod shown in Fig. 22, having a constant thickness \(b\). Thus, the sections of the rod are rectangular.

We shall assume that the rod satisfies at least one of the conditions (41).

The rod is stretched by forces \(P\) applied along its axis, and is bent by a moment \(M\) acting in the plane of the rod. Obviously, under the given conditions the greatest stress will be at the point \(A\). The problem will consist in determining this greatest stress.

The stress from bending at the point \(A\) can be found from formula (12):

\[ \sigma=\frac{M\cdot B}{b\cdot y^2}, \]

where \(M\)—the bending moment,
\(b\)—the thickness of the rod,
\(y\)—the ordinate of the point \(A\), equal to \(AD\),
\(B\)—a coefficient determined from the graph in Fig. 10.

Fig. 22

The stress from tension at the point \(A\) can be found from formula (37):

\[ \sigma_1=\frac{P}{2\rho b\ln\left(\frac{\rho+y_a}{\rho}\right)}, \]

since in the conditions of the problem it was assumed that the rod satisfies at least one of the conditions (41).

The resultant stress at the point \(A\) will be

\[ \sigma_a=\sigma+\sigma_1. \]

Thus, with the aid of the hypothesis of broken sections, with sufficiently high accuracy and little expenditure of time, it is possible to solve a complex problem connected with stress concentration under conditions of the simultaneous action of bending and tension.

12. Example of determining normal stresses on the basis of the hypothesis of broken sections for a rod with a curvilinear axis, subjected to bending and tension (Fig. 23).

The sections of the rod are rectangular. The width of the sections \(b\) is constant. We shall assume that, for the given rod, \(\frac{\rho_1}{R} > 0.35\), and, consequently, the rod satisfies one of the conditions (41), which is sufficient for the applicability

hypothesis of broken sections. The rod is subjected to the action of forces \(P\) applied to its ends.

The problem will consist in determining the stress at point \(A\). It is evident that in the section \(ADB\) the rod will experience tension and bending. The stress at point \(A\) due to bending is determined on the basis of the data set forth in item 8 of the present work.

From expressions (23) and (24), \(y_a\) and \(y_b\) may be found, and thereby the position of the neutral layer (point \(D\)) is determined, for the time being without taking account of the tension of the rod.

Fig. 23

The stress at point \(A\) due to bending must be determined by formula (28) or (26)

\[ \sigma_a=\frac{2MB_1B_2m_1y_a}{(B_1)y_b^3m_2+(B_2)y_a^3m_1)b}. \tag{42} \]

To determine the stress \(\sigma_a\), it is necessary to know the magnitude of the bending moment \(M\). The magnitude of this moment will be determined subsequently.

Let us take some point \(C\) in the section \(ADB\) and apply to it two directly opposite forces equal to \(P\) (see Fig. 23).

The force \(P\) applied at point \(C\) will stretch the rod in the section \(ADB\). The position of \(C\) must be chosen so that the tensile force \(P\) does not produce rotation of the section \(ADB\), but only produces its translational displacement relative to the adjacent broken section \(A_1D_1B_1\). Thus, under the influence of the force \(P\) applied at point \(C\), the section \(ADB\) will assume, relative to the section \(A_1D_1B_1\), some position \(A_2D_2B_2\), parallel to \(ADB\). Then all the fibers lying between the sections \(ADB\) and \(A_1D_1B_1\) will receive identical absolute elongations, equal to \(DD_2\).

In accordance with what has been said, Fig. 23 (b) shows the diagram of normal stresses due to the tensile action of the force \(P\).

It is evident that, from the condition of absence of rotation of the section \(ADB\), the position of point \(C\), as the point of application of the tensile force, must be determined by the position of the center of gravity of the area of the stress diagram. Point \(C\) must be the projection of the indicated center of gravity onto the line \(AB\).

To determine the stress diagram, one may write the following relations, taking into account that in the present example \(\rho_2\) is negative:

\[ \frac{\sigma_1}{\sigma_0} = \frac{DD_1}{AA_1} = \frac{\rho_1+y_a}{\rho_1}; \tag{43} \]

\[ \frac{\sigma_2}{\sigma_0} = \frac{DD_1}{BB_1} = \frac{\rho_2+y_b}{\rho_2}; \tag{44} \]

\[ P=\int_0^{y_a}\sigma\cdot b\cdot du+\int_0^{y_b}\sigma\cdot b\cdot du. \tag{45} \]

The latter expression, by means of formula (37), can be transformed into the following:

\[ P=\sigma_1\rho_1 b\cdot \ln\left(\frac{\rho_1+y_a}{\rho_1}\right) +\sigma_2\rho_2 b\cdot \ln\left(\frac{\rho_2+y_b}{\rho_2}\right). \tag{45a} \]

Solving equations (43), (44), and (45a) jointly, we obtain

\[ \sigma_1= \frac{P}{ \rho_1 b\cdot \ln\left(\frac{\rho_1+y_a}{\rho_1}\right) + \frac{\rho_1 b(\rho_2+y_b)\ln\left(\frac{\rho_2+y_b}{\rho_2}\right)} {\rho_1+y_a} }, \tag{46} \]

\[ \sigma_2= \frac{P}{ \frac{\rho_2 b(\rho_1+y_a)\ln\left(\frac{\rho_1+y_a}{\rho_1}\right)} {\rho_2+y_b} + \rho_2 b\cdot \ln\left(\frac{\rho_2+y_b}{\rho_2}\right) }; \tag{47} \]

\[ \sigma_0= \frac{P}{ b(\rho_1+y_a)\ln\left(\frac{\rho_1+y_a}{\rho_1}\right) + b(\rho_2+y_b)\ln\left(\frac{\rho_2+y_b}{\rho_2}\right) }. \tag{48} \]

The quantities found determine the stress diagram shown in Fig. 23(в). Having constructed the stress diagram, one can determine the center of gravity of its area.

Let us note that, in determining the position of the center of gravity of the area of the stress diagram, one may set the force \(P\) in expressions (46), (47), and (48) equal to unity, whereby the distance \(l_0\) of the center of gravity of the diagram from point \(A\), measured vertically, will not change, since \(l_0\) can be expressed exclusively through the geometrical parameters of the rod under consideration.

It can be proved that point \(C\) coincides with point \(D\), and thus \(l_0=AD=y_a\). Indeed, the integrals entering expression (22), after substituting into them \(\sigma_1\) and \(\sigma_2\) from expressions (19) and (21), may be regarded as quantities proportional to the static moments of the corresponding parts of the diagram of tensile stresses shown in Fig. 23(в).

The equality of the indicated integrals shows that point \(D\) is the projection of the center of gravity of the diagram of tensile stresses onto the line \(AB\). Thus, point \(C\), mentioned above, coincides with point \(D\), and consequently \(l_0\) is equal to \(y_a\) and is determined from equations (23) and (24).

Having found \(l_0\), it becomes possible to determine the magnitude of the bending moment

\[ M=P(l_1+l_0), \]

where \(l_1\) is the distance between the applied force \(P\) and point \(A\) (see Fig. 23).

Having determined the magnitude of the bending moment, we find the stress at point \(A\) due to bending by formula (42). The resultant stress at point \(A\) will be \(\sigma_{a1}=\sigma_a+\sigma_1\).

Thus, the problem posed has been solved.

Let us note that it would be incorrect to solve this problem by means of the hypothesis of plane sections, since the centers of curvature of the rod axis, \(O_1\) and \(O_2\), do not coincide. In solving this problem by means of the hypothesis of plane sections, the tacit assumption is made that the center of curv—

curvature \(O_1\) coincides with the center of curvature \(O_2\), i.e., a distortion of the initial conditions of the problem occurs. This latter circumstance can be tolerated only in the case when the distance between the centers of curvature is insignificant. The hypothesis of broken sections is in no way connected with such a restriction. The centers of curvature may even lie on both sides of the rod. However, we have been able to solve the problem by means of the hypothesis of broken sections only for a rod with a curvilinear axis that has a transverse plane passing through both centers of curvature.

The question of solving the problem for a rod of more complex shape with a curvilinear axis by means of the hypothesis of broken sections remains open for the present.

13. Extension of the Method for Solving the Previous Problem to the Case in Which the Cross Sections of a Rod with a Curvilinear Axis Are Not Rectangular

The solution of the problem of determining the stresses in a rod subjected to tension and bending, under the condition that the rod has a cross section of one of the types indicated in Fig. 18, presents no fundamental difficulties. The determination of the stress due to bending and the finding of the position of the neutral layer can be carried out by the method indicated in Section 9. In determining the stresses due to tension, the integrals entering into expression (45) must be divided into parts in accordance with the different width of the cross section.

In order to determine the position of point \(C\), the diagram of tensile stresses must be transformed in a known manner, namely, the stress at each point must be multiplied by the corresponding width of the section. Then the centroid of the area of the transformed diagram is determined, and thereby the position of point \(C\) is found. Point \(C\), as before, coincides with point \(D\).

14. Some Experiments Concerning the Hypothesis of Broken Sections

In the course of carrying out the investigations set forth in the present work, the author constantly tried to compare the theoretical results obtained from the hypothesis of broken sections with experimental data available in the literature.

In most cases, the experimental data confirmed the sufficient accuracy of the hypothesis of broken sections. Especially good agreement between the theoretical results obtained from the hypothesis of broken sections and the experimental data occurred in the sections of the work dealing with the bending of beams and plates of shaped outline (see, for example, Figs. 6, 7, and 16).

Less satisfactory agreement was obtained in the study of the tension of shaped plates, as a result of which, in this part, it was necessary to introduce certain restrictions on the applicability of the hypothesis of broken sections.

In view of the fact that the verification, carried out in the present work, of the theoretical results obtained from the hypothesis of broken sections by comparing them with experimental data was nevertheless indirect in character, it seemed desirable to carry out a direct observation of the behavior of a broken section during deformation of a specimen. For this purpose, test specimens were made of rubber. The specimens had the form of shaped plates of constant thickness. On the lateral surfaces of the specimens, broken sections were applied in ink according to the rules established in the theory of the given question [see Fig. 1(6)]. After this the specimens were subjected to deformation.

Experiments showed that, in the general case, when specimens undergo bending with noticeable curvatures, broken sections do not undergo distortions, whereas the control plane sections drawn on the specimen undergo significant distortions (see the photograph on p. 46).

A slight curvature of the broken section was observed in the case when the contour of the specimen profile had, at the point through which the broken section was drawn, a discontinuous change in the radius of curvature. This latter circumstance agrees with the remarks made in item 4 concerning Fig. 8. Slight curvatures of the broken section were also observed for comparatively sharp angles of inclination of the sides of the broken section to the neutral layer or, in other words, for large angles \(\alpha\) of inclination of the tangent to the outline of the specimen. This circumstance explains a certain discrepancy between the theoretical and experimental curves shown in Fig. 6, the discrepancy becoming noticeable for contour points corresponding to angles \(\alpha\) exceeding \(20^\circ\).

In the case of axial tension of shaped rubber specimens, more noticeable distortions of the broken sections were observed, these distortions consisting both in a change of the angle between the sides of the broken section and in curvature of the sides themselves. The indicated distortion of the broken section occurred even in the case when the specimen satisfied conditions (41) and the stress calculated by the theoretical formula (37) coincided with the value of the stress found by the optical method. This circumstance seemed very strange, since the theoretical formula (37) was obtained on the basis of the hypothesis of broken sections, i.e., a hypothesis assuming that, during deformation, broken sections are not distorted. This kind of contradiction has the following explanation.

In Fig. 24(a) the rubber specimen under test is shown before deformation. On its surface are drawn the broken section \(ADB\) and the section \(A_0D_0B_0\).

Fig. 24                Fig. 25

In Fig. 24(b) the same specimen is shown in the stretched state. The broken section \(ADB\), drawn on the surface of the specimen before its deformation, has acquired the form indicated by the lines \(A_1D_1B_1\) (in the figure the distortion of the broken section is shown exaggerated).

In the same Fig. 24(b), the original configuration of the broken section \(A_1DB_1\) is shown by a dashed line for comparison.

Let us consider the fibers enclosed between the sections \(A_0D_0B_0\) and \(A_1D_1B_1\). The theoretical deformation of these fibers will be determined by the position of the undistorted broken section \(A_1DB_1\). The actual deformation of the same fibers will be determined by the distorted section \(A_1D_1B_1\).

The theoretical and actual deformations will be the same for the fibers whose position is determined by the points of intersection of the undistorted

...and distorted sections, i.e., with the points \(A_1;\ K;\ N\) and \(B\). It is evident that the theoretical and actual stress in the indicated fibers will also be the same. In accordance with what has been said, Fig. 25 shows the stress diagrams. The solid line represents the diagram of the actual stresses, and the dashed line—the theoretical stresses.

Thus, despite the actual curvature of the broken section, the maximum actual stress may turn out to be the same as the theoretical one, as was the case for the specimen under consideration. This explains the contradiction noted above. The inaccuracies of the theoretical formulas associated with the distortion of broken sections affect chiefly the character of the stress distribution over the section, and not the magnitude of the maximum stress. But, owing to the insignificance of the distortion of the broken sections, these inaccuracies are extremely small if the shape of the specimen satisfies conditions (41).

Part IV. Tension, bending, and torsion of rods having the shape of bodies of revolution

15. Determination of the stress under tension in a rod having the shape of a body of revolution

Consider the rod shown in Fig. 26. The rod has a groove encircling it. The transverse sections of the rod are circles. The rod is subjected to tension by forces \(P\).

The problem will consist in determining the stresses at points lying at the bottom of the groove. An analogous problem for a rod having the shape of a plate was solved in the third part of the present work by means of the hypothesis of broken sections.

For a rod having the shape of a body of revolution, the hypothesis of broken sections passes into the hypothesis of conical sections, which assumes that, during deformation of the rod, the conical sections are not distorted and the generators of the cones are not curved.

Fig. 26

In the rod shown in Fig. 26, a conical section \(ADB\) has been drawn through the point \(A\), lying at the bottom of the groove. This conical section, in the present special case, has become plane, since the generators of the hypothetical cone are perpendicular to the contour of the rod. A second conical section \(A_1D_1B_1\) has been drawn through the nearest point \(A_1\). The vertex of the second conical section will be the point \(D_1\), lying on the axis of the rod.

Consider the fiber \(KF\), enclosed between the conical sections and taken at a distance \(r\) from the axis of the rod. Under tension of the rod, the section \(ADB\) will move into the position \(A_2D_2B_2\) (see Fig. 26). In this case the fiber \(KF\) under consideration will receive the elongation \(FN\).

The stress in the fiber under consideration may be represented by means of expression (35), which, with the notation given in Fig. 26, takes the following form:

\[ \sigma=\varepsilon E=\frac{FN\cdot(\rho+y_a)E}{DD_1(\rho+y_a-r)}. \tag{49} \]

From the equilibrium condition one may write

\[ P=\int_F \sigma\, dF . \tag{50} \]

In the last expression, \(dF\) represents an elementary area of the section \(ADB\).

From consideration of Fig. 26 (b) it follows that the elementary area \(dF\) can be expressed as follows:

\[ dF=r\,d\psi\,dr . \tag{51} \]

Substituting into (50), we obtain

\[ P=\int_0^{2\pi}\int_0^{y_a}\sigma\,r\,dr\,d\psi, \]

or

\[ P=\int_0^{2\pi}\int_0^{y_a} \frac{FN(\rho+y_a)E\,r\,dr\,d\psi} {DD_1(\rho+y_a-r)} . \tag{52} \]

Integrating, we find

\[ P=\frac{FN(\rho+y_a)E\,2\pi \left[(\rho+y_a)\ln\left(\frac{\rho+y_a}{\rho}\right)-y_a\right]} {DD_1}. \]

Determining

\[ \frac{FN(\rho+y_a)E}{DD_1} = \frac{P} {2\pi\left[(\rho+y_a)\ln\left(\frac{\rho+y_a}{\rho}\right)-y_a\right]} \]

and substituting into expression (49), we obtain

\[ \sigma= \frac{P} {2\pi(\rho+y_a-r)\left[(\rho+y_a)\ln\left(\frac{\rho+y_a}{\rho}\right)-y_a\right]} . \tag{53} \]

Taking \(r=y_a\) in the last expression, we obtain the stress at point \(A\), and also at all points lying at the bottom of the cut-out,

\[ \sigma_a= \frac{P} {2\pi\rho\left[(\rho+y_a)\ln\left(\frac{\rho+y_a}{\rho}\right)-y_a\right]}, \tag{54} \]

or

\[ \sigma_a= \frac{P} {2\pi n y_a^2\left[(n+1)\ln\left(\frac{n+1}{n}\right)-1\right]}, \tag{55} \]

where

\[ n=\frac{\rho}{y_a}. \]

The expression obtained for the stress at point \(A\) is approximate, like expression (37), found earlier for a plate of arbitrary shape by means of the hypothesis of broken sections.

Expressions (54) and (55) will have practically sufficient accuracy provided at least one of the conditions (41) is satisfied, which for the present case take the following form:

\[ \left. \begin{aligned} \frac{\rho}{d} &> 0.35,\\ \frac{D}{d} &> 2.5. \end{aligned} \right\} \tag{56} \]

If none of these conditions is fulfilled, then the stress calculated by formulas (54) or (55) will be greater than the actual stress, similarly to what occurred in tension of a shaped plate (see the graph in Fig. 21).

From formula (55) an expression may be obtained for the stress-concentration coefficient for points lying at the bottom of the groove of the rod shown in Fig. 26. The stress-concentration coefficient will be

\[ K=\frac{\sigma_a}{\sigma_0}, \tag{57} \]

where

\[ \sigma_0=\frac{P}{\pi y_a^2}. \tag{58} \]

Taking expression (55) into account, we obtain

\[ K=\frac{1}{2n\left[(n+1)\ln\left(\frac{n+1}{n}\right)-1\right]}, \tag{59} \]

where, as before,

\[ n=\frac{\rho}{y_a}. \]

The graph in Fig. 27 presents curves showing the values of the stress-concentration coefficient as a function of the ratio \(\rho:d\).

The solid line represents the curve of values of the stress-concentration coefficient calculated by formula (59). The dashed line is drawn for comparison as the curve of values of the stress-concentration coefficient obtained from the hypothesis of spherical sections.

Fig. 27

The hypothesis ...

spherical sections is obtained from the hypothesis of cylindrical sections, as applied to rods having the form of bodies of revolution).

According to the hypothesis of spherical sections, in the case of the groove shown in Fig. 26, the stress-concentration factor will have the expression, which we give without derivation:

\[ K=\frac{1}{2n\ln\left(\frac{2n+1}{2n}\right)}, \tag{60} \]

where

\[ n=\frac{\rho}{y_a}. \]

16. Determination of the stress in bending in a rod having the form of a body of revolution

Let us consider the rod shown in Fig. 28. The rod has a groove encircling it. The transverse sections of the rod are circles. The rod under consideration is subjected to pure bending by moments \(M\).

The problem will consist in determining the stress at point \(A\). We shall solve the problem posed on the basis of the hypothesis of conical sections, which was already applied in the preceding section. In Fig. 28, just as was done in Fig. 26, conical sections \(ADB\) and \(A_1D_1B_1\) are drawn through the nearby points \(A\) and \(A_1\). As in Fig. 26, the conical section \(ADB\) in this particular case will be plane.

Fig. 28

Let us consider a fiber of the material \(KF\), taken between the indicated conical sections at a distance \(r\) from the axis of the rod and at a distance \(u\) from the neutral layer.

When the rod is bent, the section \(ADB\) will rotate relative to the conical section \(A_1D_1B_1\) through a certain angle \(\Delta\varphi\) and will occupy the position \(A_2DB_2\).

The fiber \(KF\) will receive an elongation equal to

\[ FN=u\cdot\Delta\varphi . \]

From Fig. 28 (b) it follows:

\[ u=r\cdot\sin\psi . \]

Then

\[ FN=r\cdot\sin\psi\cdot\Delta\varphi . \]

The stress in the fiber under consideration may be obtained from formula (49) in the following form:

\[ \sigma=\varepsilon E= \frac{r\cdot\sin\psi\cdot\Delta\varphi\cdot(\rho+y_a)E} {DD_1(\rho+y_a-r)}. \tag{61} \]

Further, from the condition of equilibrium one may write

\[ M=2\int_F \sigma\,dF\cdot r\cdot \sin\psi, \]

or

\[ M=2\int_0^\pi \int_0^{y_a} \frac{r^2\sin^2\psi\,\Delta\varphi(\rho+y_a)E\,dr\,d\psi} {DD_1(\rho+y_a-r)}. \]

Integrating, we obtain

\[ M=\frac{2\Delta\varphi(\rho+y_a)E}{DD_1} \left[ (n+1)^3\ln\left(\frac{n+1}{n}\right) -(n+1)^2-\frac{1}{2}(n+1)-\frac{1}{3} \right]\frac{\pi}{2}y_a^3, \]

where

\[ n=\frac{\rho}{y_a}. \]

Determining from this expression the quantity

\[ \frac{\Delta\varphi(\rho+y_a)E}{DD_1} \]

and substituting it into formula (61), we obtain

\[ \sigma= \frac{M\cdot r\cdot \sin\psi} {2\left[ (n+1)^3\ln\left(\frac{n+1}{n}\right) -(n+1)^2-\frac{1}{2}(n+1)-\frac{1}{3} \right](\rho+y_a-r)\frac{\pi}{2}y_a^3}. \tag{62} \]

Taking in the last expression

\[ r=y_a \]

and

\[ \psi=\frac{\pi}{2}, \]

we find the stress at point \(A\):

\[ \sigma_a= \frac{M} {\pi\cdot n\cdot y_a^3 \left[ (n+1)^3\ln\left(\frac{n+1}{n}\right) -(n+1)^2-\frac{1}{2}(n+1)-\frac{1}{3} \right]}, \tag{63} \]

where, as before,

\[ n=\frac{\rho}{y_a}. \]

The result obtained is approximate, just as the results obtained for the bending of shaped plates, based on the hypothesis of broken sections, were approximate (see, for example, the graph in Fig. 7).

The stress-concentration coefficient in bending of a rod having the form shown in Fig. 28 may be found as follows:

\[ K=\frac{\sigma_a}{\sigma_0}, \]

where

\[ \sigma_0=\frac{M}{\dfrac{\pi y_a^3}{4}}. \]

Then, taking expression (63) into account, we obtain

\[ K=\frac{1}{4n\left[(n+1)^3\ln\left(\frac{n+1}{n}\right)-(n+1)^2-\frac{1}{2}(n+1)-\frac{1}{3}\right]}. \tag{64} \]

The graph in Fig. 29 presents curves showing the values of the stress-concentration coefficient as a function of the ratio \(\rho:d\) for the case of bending of the rod shown in Fig. 29.

The solid line gives the curve of the values of the stress-concentration coefficient calculated by formula (64).

For comparison, the dashed line gives the curve of values of the stress-concentration coefficient obtained according to the hypothesis of spherical sections.

According to the hypothesis of spherical sections, in the case of the undercut shown in Fig. 28, the stress-concentration coefficient will have the expression, which we give without derivation:

\[ K=\frac{1}{4n\left[(2n+1)\ln\left(\frac{2n+1}{2n}\right)-1\right]}, \tag{65} \]

where

\[ n=\frac{\rho}{y_a}. \]

The theoretical results found cannot be compared with experimental data obtained by the optical method, since the optical method is suitable only for flat rods and cannot be used for bodies of revolution¹.

In some handbooks the results obtained by the optical method for flat rods are artificially transferred to bodies of revolution, for example, when determining the stress concentration in a shaft fillet. However, such an artificial transfer cannot be regarded as sufficiently justified.

From a comparison of the graphs shown in Figs. (7), (21), (27), and (29), it follows that the stress-concentration coefficient found on the basis of the hypotheses considered in the present work is obtained, for bodies of revolution, lower than for flat rods.

Fig. 29

17. Determination of shear stresses in torsion of rods having the shape of bodies of revolution

The hypotheses of conical and spherical sections can be used to determine shear stresses in torsion of rods having the shape of bodies of revolution.

Let the rod shown in Fig. 30 be twisted by moments \(M\) applied to its ends. The problem will consist in determining the shear stresses at points lying at the bottom of the groove. We shall solve the posed problem by means of the hypothesis of conical sections.

Consider the material element \(KFK_1F_1\), bounded by the conical sections \(ADB\) and \(A_1D_1B_1\) and located at a distance \(r\) from the axis of the rod (see Fig. 30).

Fig. 30

The shear stress on the lateral faces of the element under consideration may be expressed as follows:

\[ \tau=\gamma \cdot G, \]

where \(\gamma\) is the relative shear, and \(G\) is the modulus of elasticity in shear.

In our case

\[ \tau= \frac{\Delta \theta \, r(\rho+y_a)G} {DD_1(\rho+y_a-r)}, \tag{66} \]

where \(\Delta \theta\) is the angle of rotation of the section \(ADB\) relative to the conical section \(A_1D_1B_1\). We note that expression (66) is, to a certain extent, analogous to expression (49).

Further, from the equilibrium condition, taking Fig. 30(b) into account, we may write

\[ M=\int_F \tau \cdot r^2 \cdot d\psi \cdot dr. \tag{67} \]

Solving equations (67) and (66) simultaneously and taking \(r\) equal to \(y_a\), we obtain the expression for the stress at points lying at the bottom of the groove in the following form:

\[ \tau_a= \frac{M} {2\pi n y_a^3\left[(n+1)^3\ln\left(\frac{n+1}{n}\right)-(n+1)^2-\frac{1}{2}(n+1)-\frac{1}{3}\right]}, \tag{68} \]

where

\[ n=\frac{\rho}{y_a}. \]

The stress concentration coefficient may be obtained in the following manner:

\[ K=\frac{\tau_a}{\tau_0}, \tag{69} \]

where

\[ \tau_0=\frac{M}{\dfrac{\pi y_a^3}{2}}. \tag{70} \]

We note that, after substitution, the stress concentration coefficient will obtain an expression identical with expression (64), and consequently the graph constructed in Fig. 29 for the case of bending of the rod may be extended to the case of torsion considered here.

According to the hypothesis of spherical sections, the following expression for the shear stresses at points lying on two grooves can be obtained in an analogous manner:

\[ \tau_a= \frac{M}{ 2\pi n y_a^3\left[(2n+1)\ln\left(\frac{2n+1}{2n}\right)-1\right] }, \tag{71} \]

where

\[ n=\frac{\rho}{y_a}. \]

The stress-concentration coefficient can be determined from expression (69).

Let us note that the stress-concentration coefficient will receive an expression identical with expression (65), and consequently the dashed curve plotted in the graph of Fig. (29) may be referred to the case considered here. It should be noted that, according to the hypothesis of conical sections, the stress-concentration coefficient is obtained higher than according to the hypothesis of spherical sections.

In the book by S. V. Serensen [6] data are given for the torsion of a round specimen with a groove, obtained by measurements on an electrical model (Thum and Bautz). Comparing these data with the graph of Fig. 29, one may note that satisfactory coincidence of the curves is obtained only for \(D:d=1.2\). For the remaining values of \(D:d\), larger discrepancies occur. From comparison of the graph of Fig. 29 with the indicated literature data, one must conclude that formulae 68 and 71 may be used only when one of two conditions is present:

\[ \frac{D}{d}\geqslant 1.2 \]

or

\[ \frac{\rho}{d}\geqslant 1. \]

Fig. 31

Let us note that the hypothesis of spherical sections gives an exact solution of the problem of torsion of a cone, since in the theory of elasticity it is proved that, in torsion of a cone, spherical sections are not distorted.

We shall proceed to determine the shear stress at any point of the contour of the shaft. In Fig. 31 a longitudinal section of a shaft of variable diameter is shown.

Take on the fillet in Fig. 31 an arbitrary point \(A\) and draw through it the conical section \(ADB\). The other conical section \(A_1D_1B_1\) is drawn through the nearest point \(A_1\). Between the indicated conical sections consider an elementary ring, whose section will be \(KFK_1F_1\) and whose radius is \(r\).

The shear stress in the ring under consideration can be expressed, by analogy with formula (3) (see Part 1 of the present work), as follows:

\[ \tau=\gamma G=\frac{d\theta\cdot G\cdot m\cdot q\cdot u\cdot \cos\alpha} {dx_0\left[m+\left(q^2-m\right)\frac{u}{yq}\right]} . \tag{72} \]

In the expression written,

\[ \begin{aligned} \gamma&\text{ is the relative shear,}\\ G&\text{ is the modulus of elasticity in shear,}\\ d\theta&\text{ is the angle of rotation of the conical section }ADB\text{ relative to the conical}\\ &\quad\text{section }A_1D_1B_1,\\ m\text{ and }q&\text{ are determined by expressions (5), while the remaining quantities are given}\\ &\quad\text{in Fig. 31.} \end{aligned} \]

Further, from the condition of equilibrium one may write

\[ M=\int_{0}^{yq}\tau\cdot 2\pi\cdot u^2\cdot \cos^2\alpha\cdot du . \tag{73} \]

Solving (72) and (73) together and then taking

\[ u=yq, \]

we obtain the expression for the stress at point \(A\) in the following form:

\[ \tau= \frac{M} {2\pi\frac{a}{n}\,y_a^3 \left[ \frac{1}{3a}-\frac{1}{2n}+\frac{a}{n^2} -\frac{a^2}{n^3}\ln\left(\frac{a+n}{a}\right) \right](a+n)^2}, \tag{74} \]

where

\[ n=\pm \frac{\dfrac{y_a}{\rho}}{\cos^2\alpha}; \qquad a=\frac{\cos\alpha\pm\dfrac{y_a}{\rho}}{\cos^2\alpha}. \tag{75} \]

In the last expressions the upper sign should be taken in the case of concavity of the contour, and the lower sign in the case of convexity. \(y_a\) is the ordinate of the point \(A\) under consideration. In the case of a fillet,

\[ y_a=\frac{d}{2}+\rho(1-\cos\alpha). \]

According to data in the literature, from experiments carried out by the principle of electrical analogy [1], the greatest shear stress in the fillet is obtained at point \(A\) (see Fig. 31), lying at an angle \(\alpha\) approximately equal to twenty degrees. Then the stress-concentration factor in the case of a fillet may be found from the formula:

\[ K=\frac{\tau_a}{\tau_0}; \]

where \(\tau_a\) must be calculated by formula (74) for the angle \(\alpha\) equal to twenty degrees, while \(\tau_0\) must be determined as the stress in the cylindrical part of the shaft having the smaller diameter:

\[ \tau_0=\frac{M}{\dfrac{\pi d^3}{16}}. \]

In an analogous way, the shearing stress at point \(A\) of the fillet may be found on the basis of the hypothesis of spherical sections, namely:

\[ \tau_a= \frac{M\sin^4\alpha} {2\pi y_a^3 \left[ (1-c^2)\ln\left(\frac{c-\cos\alpha}{c-1}\right) +c(1-\cos\alpha)+\frac{1}{4}(1-\cos 2\alpha) \right](c-\cos\alpha)} , \tag{76} \]

where

\[ c=\cos\alpha-\frac{\rho}{y_a}\sin^2\alpha . \tag{77} \]

In comparison with experimental data (S. V. Serensen [6], Fig. 175), formulas (74) and (76) give considerably overestimated values of the stresses. Thus, for example, at \(\rho:d=0.1\), the stress-concentration factor obtained by means of formula (76) is \(1.58\), whereas according to experimental data it has the value \(1.44\), even under the condition that \(D:d=2\). It should be noted that the formulas considered here do not take into account the influence of the ratio \(D:d\), whereas, according to experimental data, this ratio strongly affects the magnitude of the stress under torsional deformation. For the reasons indicated, formulas (74) and (76) can give satisfactory results only when the ratio \(\rho:d>1\), since in this case the influence of the ratio \(D:d\) dies out.

18. Determination of shearing stresses in the case of shear in rods of shaped outline

Let us consider the rod shown in Fig. 32. The rod has grooves on its sides. Shear occurs under the action of forces \(P\).

The formula for determining the greatest shearing stress at points \(A\) and \(B\), lying at the bottom of the groove, may be written by analogy with formula (37) and will have the form

Fig. 32                Fig. 33

\[ \tau_a=\frac{P}{2\rho b\ln\left(\frac{\rho+y_a}{\rho}\right)} . \tag{78} \]

The stress-concentration factor may be determined from formulas (39) or (40).

For the case of the rod shown in Fig. 33, having an external circular groove, we shall have a formula analogous to formula (54)

\[ \tau_a= \frac{P} {2\pi\rho\left[ (\rho+y_a)\ln\left(\frac{\rho+y_a}{\rho}\right)-y_a \right]} . \tag{79} \]

The stress-concentration factor may be determined from formulas (59) or (60).

Conclusion

1. The application of the hypotheses described to the calculation of rods of variable cross-section should be regarded as an artificial, approximate method for determining stresses in the rods indicated.

2. The hypothesis of broken sections, which for bodies of revolution becomes the hypothesis of conical sections, gives results that closely agree with experimental data in the case of tension and bending; consequently, from a practical point of view, this hypothesis may be regarded as sufficiently accurate for the indicated cases of deformation and within certain limits of applicability.

3. The hypothesis of cylindrical sections, touched upon in part in the present work, which for bodies of revolution becomes the hypothesis of spherical sections, gives satisfactory results for certain ratios of the parameters of the part in the case of torsion of a rod of variable cross-section. In addition, this hypothesis gives results coinciding with the exact solution in the case of bending and tension of a wedge and torsion of a cone.

References

1. S. P. Timoshenko.—Theory of Elasticity, translation. ONTI, 1934.

2. E. Coker and L. Filon.—A Treatise on Photo Elasticity, Cambridge, 1931. Russian translation: Optical Method for the Investigation of Stresses, ONTI, 1936.

3. H. Neuber.—Kerbspannungslehre, Grundlagen für genaue Spannungsrechnung, Berlin, 1937.

4. N. N. Afanasiev.—Approximate calculation of the stress concentration coefficient. Journal of Technical Physics, 1935, vol. VI, no. 7.

5. I. V. Podzolov.—Allowable stresses for steels. Oborongizdat, Moscow, 1940.

6. S. V. Serensen.—Dynamic strength in mechanical engineering. Moscow, 1945.

Appendix

Photographs of a rubber specimen in the free and deformed states. Before deformation, the broken section ADB was marked on the specimen in ink. As is evident from the photographs, the broken section was not distorted during deformation of the specimen, which testifies to the sufficiently high accuracy of the hypothesis of broken sections under bending deformations.

1. The stress-concentration coefficients calculated by the Neuber method and by the Afanas'ev method for round specimens are somewhat higher than those given in the graphs of Figs. 27 and 29 [3, 4]. ↩