S. P. Gomelya.

Mining engineer-technologist, lecturer at the Tomsk Technological Institute
of EMPEROR NICHOLAS II.

ON THE QUESTION

OF THE RATIONAL CALCULATION OF A HOOK.

A theoretical investigation of the simplest rational forms of a hook and data for its calculation and design.

With 9 drawings in the text.

Tomsk.
Steam Typolithography of V. I. Orlova.
1913.

On the Problem of Rational Calculation of a Hook.

Engineer-Technologist S. P. Gomelli.

1. The hook is among the most responsible parts used in mechanical engineering. It has wide application both in hoisting machines and in the rolling stock of railways.

In both of these fields, the failure of this comparatively simple part always threatens grave consequences. In hoisting machines the hook, being the part most exposed to accidental failure, can, more than any other part, “destroy human lives and, in general, cause damage and losses” (Glück “Cranes and Machinery”). The role of the hook in the rolling stock of railways is no less significant, and failures of the draw hook have more than once been the cause of very serious railway catastrophes.

It is obvious that these circumstances, as well as the comparatively high cost of the material and manufacture of the hook, compel one to treat the calculation and construction of this part with special attention, and therefore a more detailed elucidation of the question of hook design will not be superfluous.

It must be noted that certain irregularities are admitted in the methods ordinarily used for calculating and designing a hook. Thus, when calculating a hook for hoisting machines, the arrangement of the forces acting upon it is taken to be more favorable with respect to the stresses arising in it than may be the case in reality. Then, usually, in calculating the diameters of the holes of the hook and of the suspension ropes, the influence of their inclination is not taken into account, and the lower section of the hook is checked only for shear by a plane passing through its axis; as a result, at normal values of the permissible stresses, this section may turn out to be…

insufficiently strong. In exactly the same way, the usually accepted calculation of a hook by the Grashof formula, the magnitudes of the ratio of the parallel sides of the trapezoid obtained in the dangerous section of it, are rather arbitrary and unsatisfactory in the sense of the most advantageous use of the hook material. Finally, it should be pointed out that, when calculating a hook by the Grashof formula, the ordinary practice is to give it a shape that does not correspond to the calculation itself: the calculation is always carried out on the assumption that the center of curvature of the calculated section of the hook coincides with the center of its opening; when drawing the hook, however, this circumstance is usually not taken into account, as a result of which, in determining the stresses, an error is obtained, which in some designs is not negligible.

Since, in determining the breaking dimensions of a hook, formulas for the eccentric tension of short bars are very often used, below the calculations of a hook both for eccentric tension and by the Grashof formula are considered,—with, of course, in both cases, as far as possible, all the circumstances just noted being taken into account.

1. The dimensions of a hook depend not only on the maximum load \(Q\) for which it is to be calculated, but also on its form, in which the radius \(a\) of the hook opening is of essential importance: the larger this radius, the larger must be the magnitude of the moment bending the hook, together with which the weight of the hook itself also increases. Therefore the diameter of the hook opening should be given minimum dimensions, sufficient only for suspending the maximum load from the hook by means of ropes or chains.

When the hook is hooked onto the loops of ropes or chains, the forces \(S\) acting in them depend on the angle of inclination of the ropes \(z\) to the vertical and are expressed by the formula \(S = \dfrac{Q}{4}\dfrac{1}{\operatorname{Cos} z}\), from which it is seen that, when \(z\) varies from \(0^\circ\) to \(90^\circ\), \(S\) increases from \(\dfrac{1}{4}Q\) to infinity. Obviously, for the calculation of the ropes and the hook it is necessary to set some limiting value of \(S\).

Let us take the limiting value \(S\) to be equal to \( \frac{1}{2}\varphi\), which corresponds, when the rope pair of each loop is parallel (Fig. 1), to the angles of their inclination to the vertical \(z=60^\circ\). Or, in general, to the inclination of the two resultants of the tensions of each loop at the same angles of \(60^\circ\) to the vertical.

Fig. 1.

Let us denote the diameter of the hemp rope for suspending to the hook a load through \(d\), the allowable stress in it being taken as \(100\ \mathrm{kg}/\mathrm{cm}^2\); the diameter of wrought iron we shall denote by \(d'\), and the allowable stress in it, in view of the impossibility of bends at the corners of the object being lifted, for example as for a calibrated chain, we shall take equal to \(320\ \mathrm{kg}/\mathrm{cm}^2\). Under these conditions we obtain, for the rope,

\[ d=0.079\sqrt{Q_{\mathrm{cm}}} \]

and for the chain

\[ d'=0.045\sqrt{Q_{\mathrm{cm}}}. \]

But the diameter of the hook opening, according to Fig. 2, is

\[ 2a=2\left(\frac{d}{2}+\frac{d}{2\sin\alpha}\right); \]

therefore, substituting into this formula the values of \(d\) and rounding the numbers, we obtain, for the rope:

\[ 2a=0.17\sqrt{Q_{\mathrm{cm}}}. \]

and for the chain:

\[ 2a=0.1\sqrt{Q_{\mathrm{cm}}}. \tag{1} \]

The latter formula (1), as giving minimum dimensions for the hook opening, should be adopted for determining the value \(a\). The other formula may be applied only in the design of hooks of small lifting capacity, in which the maximum load is suspended on hemp ropes.

1. The calculation of the hook must be carried out under the condition of the most unfavorable arrangement of the forces acting on it. If we assume, as is usually done, that the maximum load on the hook \(Q\) coincides with the vertical passing through the center of its opening, we would obtain one dangerous section \(I—O\) (Fig. 2). If we suppose that the loops of the rope are directed at an angle to the vertical, so that the resultant of their tensions makes with it an angle of \(60^\circ\),—

Fig. 2.

then one would obtain a section more disadvantageous for the strength of the hook; moreover, besides section I—O lying in the horizontal plane, the dangerous section III—O, identical in conditions with the first, will also be found. In addition, it is possible at the same time to calculate for bending and tension the section IV—O, also lying in a vertical plane, which in the case of the first supposition, i.e. when the force \(Q\) acting on the hook is directed vertically, would, according to calculation in compression, prove weak in comparison with the dimensions usually assigned to the hook in this place in practice.

Since the hook, by its form, represents a curvilinear bar with a radius of curvature, determined by geometrical means, of the centers of gravity of the transverse sections that is rather small in comparison with the height of the dangerous section, its calculation by the formulas for eccentric tension of a rectilinear bar will be the less accurate the relatively smaller the aforementioned radius of curvature. However, the calculation of a hook by the simple formulas for eccentric tension, often used in practice, can, with a corresponding choice of the allowable stress, give dimensions satisfactory with respect to the strength of the hook.

At the same time, when the material of the hook is sufficiently ductile, the load applied to it, increasing beyond the elastic limit, diminishes its curvature, owing to which the hook, in its form, approaches more and more the calculated one, acquiring thereby an ever greater resistance.*)

*) K. Bach, “Machine Details,” p. 557.

Experimental investigations of bars in rupture have shown that the actual force that destroys the bar is even somewhat greater than the calculated one. Such results may be explained by the fact that the distribution of normal stresses in any section of a bent bar, beyond the limits of elasticity of the material, no longer follows the law of a straight line; moreover, the normal stresses over the height of the section are distributed more favorably with respect to the strength of the bar, following the stress diagram obtained at rupture of the bar.*)

Nevertheless, up to the limit of elastic deformation, which is always taken as the basis for calculations of the strength of machine parts, calculation by the formula for eccentric tension is only approximate.

1. The calculation of a hook must be based on the condition of its greatest strength with the least expenditure of material; at the same time, of course, the form of the hook sections must be sufficiently simple so as not to increase the cost of its manufacture. Thus, the transverse sections of a hook may be given either the form of a trapezoid (in particular, a triangle, rectangle, or square), or an ellipse (in a particular case, a circle). Sections of the latter kind are less advantageous in the sense of utilizing the resistance of the hook material, which works chiefly in bending; therefore we shall consider only hooks with a trapezoidal section.

The dangerous sections of the hook (Fig. 2), I--O and III--O, for \(\alpha = 60^\circ\), are subjected to bending and tension from the force \(Q\). Let the parallel sides of the trapezoid obtained in the dangerous section I--O be denoted by \(b_1\) and \(b_2\), its height by \(a\), the distances of the center of gravity of the trapezoid \(C\) from the parallel sides \(a b_1\) and \(b_2\) respectively by \(c_1\) and \(c_2\), and the area of the trapezoid itself and its moment of inertia by \(f\) and \(J\).

The maximum normal tensile stress \(\varepsilon_1\) and compressive stress \(\varepsilon_2\) are expressed by the formulas:

\[ \varepsilon_1 = \frac{Q}{f} + \frac{Q(a+c_1)c_1}{J} \quad \text{and} \quad -\varepsilon_2 = \frac{Q}{f} - \frac{Q(a+c_1)c_2}{J}. \]

) Berlov, Parts of Hoisting Machines*, p. 580.

In Fig. 2 the magnitudes of these stresses are represented graphically, namely the segments \(1—5\) and \(2—4\), with point 7 corresponding to the center of gravity \(C\) of the dangerous section; segment \(8—5\) corresponds to the normal stress \(\dfrac{Q}{f}\) in the dangerous section from tension alone, while the segments \(1—8\) and \(2—6\) correspond to the maximum stresses from bending; in the tensile fibers the segment \(8—1\) is

\[ \frac{M c_1}{J}=\frac{Q(a+c_1)c_1}{J}, \]

and in the compressed fibers the segment \(2—6\) represents

\[ \frac{M c_2}{J}=-\,\frac{Q(a+c_1)c_2}{J}. \]

Substituting \(m\) in place of \(\dfrac{b_1}{b_2}\) and \(k\) in place of \(\dfrac{h}{a}\), and also taking into account that for a trapezoid

\[ J=\frac{h^2(b_1^2+4b_1b_2+b_2^2)}{36(b_1+b_2)} =\frac{h^2 f}{18}\,\frac{m^2+4m+1}{(m+1)^2}, \]

\[ \frac{c_1}{h}=\frac{2+m}{3(m+1)} \quad\text{and}\quad \frac{c_2}{h}=\frac{2m+1}{3(m+1)}, \]

we obtain:

\[ \xi_1=\frac{Q}{f}\left\{1+\left|\frac{1}{k}+\frac{2+m}{3(m+1)}\right| \frac{2+m}{3(m+1)}\, \frac{m^2+4m+1}{18(m+1)_2}\right\}, \]

whence one may then determine

\[ \xi_1=\frac{Q}{f}\left\{ \frac{2[3(m+1)+k(2+m)](2+m)} {(m^2+4m+1)k} +1 \right\} \]

and, in an analogous manner,

\[ \xi_2=\frac{Q}{f}\left\{ \frac{2[3(m+1)+k(2+m)](2m+1)} {(m^2+4m+1)k} -1 \right\}. \]

If the maximum allowable stress in the dangerous section of the hook is denoted by \(\xi\), then for calculating the dangerous section we obtain the following expressions:

\[ f_1=\frac{Q}{\xi}\left\{ \frac{2[3(m+1)+k(2+m)](2+m)} {(m^2+4m+1)k} +1 \right\}. \tag{2} \]

\[ f_2=\frac{Q}{\xi}\left\{ \frac{2[3(m+1)+k(2+m)](2m+1)} {(m^2+3m+1)k} -1 \right\}. \tag{3} \]

1. Of the two values obtained for the area of the dangerous section of the hook, \(f_1\) and \(f_2\), one must of course choose the larger, so that the stress of the material of the opposite sign should not exceed the permissible value. As regards the magnitude \(k\), which is equal to \(\frac{h}{a}\), then, as is not difficult to notice, with its increase the area of the dangerous section of the hook decreases. Thus, from this point of view, it would be advantageous to take the value of \(k\) as large as possible, if there did not exist for it a certain limit, determined by practical considerations (convenience of using the hook, stability against deformations in the plane of the easiest bending, etc.). Usually the practical limit of the magnitude \(k\) is taken to be equal to 2.5.

As regards the values of \(m\), by which the form of the dangerous section is determined (when \(m\) is equal to zero and infinity we obtain, in section, a triangle; when \(m\) is equal to one, a rectangle; and for intermediate values of \(m\), a trapezoid), it is usually chosen so that the area \(f\) of the dangerous section, for a given permissible stress of the material, should be as small as possible.

Thus, in order to determine the most advantageous value of \(m\), for which the area of the calculated section is the least, it is necessary to determine the minima of both expressions (2) and (3) for \(f_1\) and \(f_2\); for this purpose, taking their derivatives with respect to \(m\) and equating them to zero, we obtain two independent equations:

\[ (f_1)'_m = \frac{m^2 - 2(k+1)m - (5+4k)} {(m^2+4m+1)^2} = 0 \tag{4} \]

\[ (f_2)'_m = \frac{(5+k)m^2 + 2m - (1+k)} {(m^2+4m+1)^2} = 0 \tag{5} \]

From the first equation (4) we obtain

\[ m = (k+1) \pm \sqrt{k^2+6k+6}. \]

The second derivative of \(f_1\) with respect to \(m\), represented in the form of the expression

\[ (f_1)''_m = 6k\, \frac{[2m - 2(k+1)](m^2+4m+1)} {(m^2+4m+1)^4} - \frac{2[m^2 - 2(k+1)m - (5+4k)](m^2+4m+1)(2m+4)} {(m^2+4m+1)^4}, \]

in the given case has the same sign as the expression \(2m - 2(k+1)\), so that, by substituting the roots of equation (4), we obtain:

\[ 2m - 2(k+1) = \pm 2\sqrt{k^2+6k+6}, \]

and, consequently, for \(m = k + 1 + \sqrt{k^2 + 6k + 6}\), \(f_1\) will have a minimum, while for \(m = k + 1 - \sqrt{k^2 + 6k + 6}\) we obtain a maximum of \(f_1\).

From the second equation (5), in an analogous manner, we obtain:

\[ m = \frac{-1 \pm \sqrt{k^2 + 6k + 6}}{5 + k}, \]

whereby the second derivative of \(f_2\) with respect to \(m\) has the same sign as the expression \(2(k + 5)m + 2\), which, for

\[ m = \frac{-1 + \sqrt{k^2 + 6k + 6}}{5 + k}, \]

being greater than zero, determines the minimum of \(f_2\); and for the other root—the maximum of \(f_2\).

Thus, the minimum for the areas of the dangerous sections of the hook corresponds to the roots of equations (4) and (5) with positive radicals, and their maximum to the roots of the same equations with negative radicals.

For an example let us consider the variations of \(f_1\) and \(f_2\) for \(k = 2\). Substituting this value of \(k\) into equations (4) and (5) and solving them, we find: for the areas \(f_1\), the minimum occurs at \(m = 7.7\), the maximum of \(f_1\)—at \(m = -3.7\); for the areas \(f_2\), the minimum will occur at \(m_1 = 0.53\), and the maximum of \(f_2\)—at \(m = -0.81\).

Fig. 3.

In Fig. 3 are shown curves indicating the variations of the values of \(f_1\) and \(f_2\) for \(k = 2\) as functions of \(m\), the values of which are laid off on the axis of abscissas. The curve for \(f_1\), LHDAC, has its minimum at point C, and the curve corresponding to \(f_2\) has its minimum at point B. The maxima of \(f_1\) and \(f_2\),

found in the negative parts of the curves represented in the drawings by dotted lines, have no practical significance in calculating the hook, because for the trapezoidal section of a hook \(m\) must always be positive.

1. Since, in determining the value of \(m\) corresponding to the smallest value of the area of the dangerous section, one has to deal with both curves \(f_1\) and \(f_2\), it is also necessary to find the point of their mutual intersection, for which \(f_1=f_2\), or, what is the same, \(\xi_1=\xi_2\).

Equating the values \(f_1\) and \(f_2\), (2) and (3), we obtain:

\[ \frac{2[3(m+1)+k(2+m)](2+m)}{(m^2+4m+1)k}+1 = \frac{2[3(m+1)+k(2+m)](2m+1)}{(m^2+4m+1)k}-1, \]

whence we find:

\[ m=k+1 \; ^*) \tag{6} \]

Substituting \(m+1\) in place of \(m\) in expressions (4) and (5), we obtain:

\[ (f_1)'_{m=k+1}=-\frac{Q}{k^2} \quad \text{and} \quad (f_2)'_{m=k+1}=\frac{Q}{k^2}(k+1); \]

thus, since \(k\) is always greater than zero, \((f_1)'_{m=k+1}\) is always positive, while \((f_2)'_{m=k+1}\) is negative.

Thus, the curve \(f_1\) at the point \(A\) of its intersection with the other curve \(f_2\) decreases down to its minimum, while the curve \(f_2\) at the same point increases from its minimum; i.e., the abscissa of the point of mutual intersection of both curves

\[ m=k+1 \]

lies between the abscissas of their minima:

\[ m=k+1+\sqrt{k^2+6k+6} \quad \text{and} \quad m=\frac{-1+\sqrt{k^2+6k+6}}{5+k}. \]

Consequently, the factual, i.e. acceptable for calculating the hook, minimum of the area of the dangerous section corresponds to the point \(A\) of intersection of both curves.

If the abscissa of the point of intersection of these curves were not situated between the abscissas of their minima, then the factual minimum of the area of the dangerous section would not correspond to the condition of equality of the permissible stresses \(\xi_1=\xi_2\), and would not coincide with the indicated point of their intersection, but with that of the minima which is larger—a case which does not occur in calculating the hook by Grashof’s formula and which will be considered below.

\[ \text{\(^*)\ Besides this there exist negative roots: } m=-1,\; m=-0.27 \text{ and } m=-3.73, \]
which likewise have no significance for calculating the hook.

Thus, in calculating a hook for eccentric tension, the most advantageous form of its cross sections—in the sense of the least area—is obtained for \(k = n + 1\). This formula is almost always used when determining the most advantageous values of \(m\) (and quite often also when calculating a hook by Grashof’s formula). However, the grounds from which this formula is ordinarily derived, \(\varepsilon_1=\varepsilon_2\), constitute only a special case, which does not always apply, and, consequently, they are not entirely correct.

The area of the dangerous section, under the condition stated above, is expressed by the formula

\[ f=\frac{Q}{\delta}\,\frac{3(k+2)}{k}. \tag{7} \]

Assigning to \(k\) the values \(1, 2, 3\), and \(\infty\), we find for \(f\) the following corresponding five quantities: \(9\frac{Q}{\delta},\ 6\frac{Q}{\delta},\ 5\frac{Q}{\delta}\), and \(3\frac{Q}{\delta}\); from this one may conclude that the benefit from increasing \(k\) very quickly begins to decrease with the increase of the value of \(k\) itself. Thus this circumstance is also one of the reasons why one should not pass beyond the values of \(k\) established by practice.

Knowing \(f\), it is not difficult to determine \(b_1\) and \(b_2\). Indeed,

\[ 2f=b_2(m+1)b_1,\quad \text{whence one obtains:}\quad b_2=\frac{Q}{\delta}\,\frac{6}{k^2 a}; \tag{8} \]

7. The external outline of the hook must be drawn in such a way that its outer contour passes through the points I and III of the dangerous sections I—O and III—O (Fig. 4) determined earlier. If the external outline of the hook over a large part of its contour can be described by a single radius, then the center of the arc is located on the line II O, making an angle with the dangerous sections I—O and III—O in halves. The magnitude of this radius is determined if a third section is calculated, for example the vertical section IV—O (Fig. 4).

The section IV—O likewise must be given a trapezoidal form with the most advantageous ratio \(m\) of the lengths of the parallel sides of the trapezoid; moreover, in order to simplify the calculation, it may be assumed that the center of gravity of the section IV—O is at the same distance from the center of the hook opening as in the sections I—O and III—O. In fact, the center of gravity of this section approaches the center somewhat ...

of the opening \(O\) and will take position \(I\), sufficiently close to the position of the center \(n\) adopted by us; with a certain allowance, there will be obtained some excess in the margin of strength. Then it will no longer be difficult to calculate the section \(IV—O\) for bending and tension from the force \(Q \operatorname{Sn}\alpha = 0.866\,Q\), situated at a lever arm distance \(a\) from the center of gravity of this section, i.e. in exactly the same way as the force \(Q\) is situated with respect to the dangerous sections \(I—O\) and \(O—III\).

Fig. 4.

If one sets as the value of the larger parallel side of the trapezoid obtained in the section \(IV—O\), taking it equal to \(\gamma b_1\) (where \(\gamma\) is a value determined from the conditions of constructiveness and close to unity), then, on the basis of equation (8), the height of the trapezoid \(h'\) may be determined.

Denoting the ratio \(\dfrac{h'}{a}\) by \(k'\), we obtain the equation:

\[ \frac{\gamma b_1}{k' + 1} = \frac{6Q \operatorname{Sn}\alpha}{\xi k'^2 a}. \]

Since for the dangerous sections \(I—O\) and \(III—O\) the ratio \(\dfrac{h}{a} = k\), then

\[ b_1 = \frac{6Q(k+1)}{\xi k^2 a}; \]

substituting the value of \(b_1\) into the preceding equation, we obtain an equation giving the dependence between \(\gamma\), \(k\), and \(k'\):

\[ \gamma (k+1) k'^2 - \operatorname{Sn}\alpha\, k^2 k' - \operatorname{Sn}\alpha\, k^2 = 0, \]

from which we determine \(k'\):

\[ k' = \frac{ \operatorname{Sn}\alpha\, k^2 \pm \sqrt{(\operatorname{Sn}\alpha\, k^2)^2 + 4\gamma (k+1) k^2 \operatorname{Sn}\alpha} }{ 2\gamma (k+1) }. \]

For \(k = 2\), \(\alpha = 60^\circ\), and \(\gamma = 1\), we obtain \(k' = 1.8\) and \(m' = 2.8\); laying off downward along the vertical \(OIV'\), from the center of the hole, of the hook eye \(O\), a segment equal to \(2.8a\), we find the point \(IV'\) and thereby determine both the position of the center and the magnitude of the radius \(R\) of the arc that coincides with the outer outline of the hook passing through the three points \(I\), \(III\), and \(IV'\).

The section of the hook \(V—O\), making an angle of \(30^\circ\) with the vertical angle, can also be calculated for eccentric tensile force \(Q \sin \alpha\), where \(\alpha = 30^\circ\). Having specified \(\gamma\), by formula (8) we determine \(k''\), the value of which is equal to the ratio:

\[ \frac{h''}{a} \]

(where \(h''\) is the height of the trapezium in the section \(V—O\)); and then we find \(m''\) from the formula: \(m'' = k'' + 1\).

For \(k = 2\) and \(\gamma = 0.8\), we obtain, for the section \(V—O\), \(k'' = 1.4\); consequently, \(h'' = 1.4a\) and \(m'' = 2.4\).

The sections \(VI—O\) and \(VII—O\) are under the same conditions as the sections \(IV—O\) and \(V—O\), so that they have the same design dimensions as these.

In Fig. 4 the outline of a hook is shown, made for \(k = 2\) according to the calculation given; in this case the centers of the circles of the outer contour of the hook are located at the intersection of the straight line \(HO\) with the vertical that divides the radius \(AO\) in half.

§. A more accurate calculation of the hook can be carried out by Grashof’s formula, with respect to which the formula we have used for eccentric tension is only a special case.

Grashof’s formula determines the magnitude of the normal stress in the fibers of a curvilinear bar, if the following are known: \(M\)—the moment of the external forces applied to the bar with respect to the center of gravity \(C\) of the section under consideration—positive when, under its influence, the curvature of the axis of the bar increases; \(N\)—the normal force at the center of gravity of the section \(C\)—positive if it stretches the fibers; \(r\)—the radius of curvature before deformation of the axis of the bar, coinciding with the geometric locus of the centers of gravity of its transverse sections; \(x\)—the distance of the point at which the normal stress of the fiber is being determined from the principal axis of inertia of the section, perpendicular to the plane of curvature at the same point \(C\); and \(f\)—the area of the section.

With the indicated notation, the magnitude of the normal stress in the section under consideration, \(\zeta\), is expressed by the formula

\[ \zeta=\frac{N}{f}+\frac{M}{fr}+\frac{M}{\eta f}\cdot\frac{x}{r+x}, \tag{10} \]

in which

\[ \eta=-\frac{1}{f}\int \frac{x}{r+x}\,df, \tag{11} \]

where the values of \(x\) are negative for fibers situated on the side of the curvature axis of the bar toward the center of gravity \(C\).

Let us, as before, denote the maximum tensile stress of the fibers in the dangerous section of the hook by \(\zeta_1\), and their maximum compressive stress by \(\zeta_2\).

The center of curvature \(T\) of the dangerous section \(AI\) (Fig. 5) does not always coincide with the center of the hook opening \(O\), which may be given not only the form of a circle, but also that of an ellipse or of some other curve.

If the ratio of \(AD\) to \(AO\) is equal to \(n\), then, adopting the former notation for the elements of the dangerous section, i.e. denoting the parallel sides of the trapezoid integral by \(b_1\) and \(b_2\), the distances of its center of gravity from them by \(c_1\) and \(c_2\), the height of the trapezoid by \(h\), and the ratio \(\frac{h}{a}\) by \(k\), we obtain:

Fig. 5.

\[ \zeta_1=\frac{Q}{f}\left(1-\frac{a+c_1}{na+c_1} +\frac{a+c_1}{na+c_1}\cdot\frac{c_1}{na}\cdot\frac{1}{\eta}\right), \]

\[ -\zeta_2=\frac{Q}{f}\left(1-\frac{a+c_1}{na+c_1} -\frac{a+c_1}{na+c_1}\cdot\frac{c_2}{na+h}\cdot\frac{1}{\eta}\right). \]

Taking the common factor

\[ \frac{1}{\,n+\frac{c_1}{a}\,} \]

outside the brackets, then

\[ \zeta_1=\frac{Q}{f}\, \frac{1}{\,n+\frac{c_1}{a}\,} \left[n-1+\left(1+\frac{c_1}{a}\right)\frac{c_1}{na}\cdot\frac{1}{\eta}\right]. \]

\[ -\xi_2=\frac{Q}{f}\,\frac{1}{\,n+\frac{c_1}{a}\,} \left|\,n-1-\left(1+\frac{c_1}{a}\right)\frac{c_2}{na+h}+\frac{1}{\gamma}\,\right|. \]

Substituting

\[ \frac{c_1}{h}=\frac{m+2}{3(m+1)},\qquad \frac{c_2}{h}=\frac{2m+1}{3(m+1)},\qquad \frac{h}{a}=k, \]

we obtain expressions for the areas of the critical sections \(f_1\) and \(f_2\), as functions of the maximum permissible stress \(\xi\) in the stretched or compressed fibers of the section:

\[ f_1=\frac{Q}{\xi} \left\{ \frac{[3(m+1)+k(m+2)](m+2)k} {3n(m+1)[3n(m+1)+k(m+2)]}\,\frac{1}{\gamma} + \frac{3n(m+1)-3(m+1)} {3n(m+1)+3(m+2)} \right\} \tag{12} \]

\[ f_2=\frac{Q}{\xi} \left\{ \frac{[3(m+1)+k(m+2)](2m+1)k} {3n(m+1)[3n(m+1)+k(m+2)(k+n)]}\,\frac{1}{\gamma} - \frac{3n(m+1)-3(m+1)} {3n(m+1)+3(m+2)} \right\} \tag{13} \]

1. In the expressions given, it is necessary to replace the quantity \(\gamma\) by its analytical expression, determined from formula (11). Let us find the value of \(\gamma\) for a trapezoidal section. To this end we shall consider the trapezoid (Fig. 6) as a figure consisting of two geometrical elements: the parallelogram \(KLRP\) and the figure \(RNTMP\), bounded by two parallel straight lines \(RN\) and \(MP\), and two straight lines \(PR\) and \(MN\). We place the origin of coordinates, \(OX\) and \(OY\), at the center of gravity of the trapezoid \(C\). Denoting by \(ST\) the length of the segment parallel to the bases of the trapezoid and passing through its center of gravity, and by \(b_0\) [[unclear: continuation]], we find:

Fig. 6.

\[ \gamma=\frac{1}{f} \left| b_0\int_{-c_1}^{c_2}\frac{x\,dx}{r+x} - \frac{b_1-b_2}{h} \int_{-c_1}^{c_2}\frac{x^2\,dx}{r+x} \right|. \]

Since

\[ b_0=b_2+\frac{(b_1-b_2)c_2}{h}, \]

we obtain:

\[ \eta=-\frac{1}{f}\left\{\left|b_2+\frac{(b_1-b_2)c_2}{h}\right| \left|c_1+c_2-r\ln\frac{r+c_2}{r-c_1}\right] -\frac{b_1-b_2}{h}\left[\frac{c_2^2-c_1^2}{h}-r(c_1+c_2)+r^2\ln\frac{r+c_2}{r-c_1}\right]\right\}, \]

or

\[ \eta=\frac{r}{f}\left[\left\{b_2+\frac{r+c_2}{h}(b_1-b_2)\right\} \ln\frac{r+c_2}{r-c_1}-(b_1-b_2)\right]-1 . \tag{14} \]

We make the substitution:

\[ r=na+c,\qquad f=\frac{(b_1+b_2)h}{2},\qquad \frac{b_1}{b_2}=m,\qquad \frac{h}{a}=k, \]

and determine \(\eta\):

\[ \eta= \frac{2\{3n(m+1)+k(m+2)\}}{3k(m+1)^2} \left\{\left[1+\frac{n+k}{k}(m-1)\right]\ln\frac{n+k}{n}-(m-1)\right\}-1 . \tag{15} \]

For \(n=1\), the expression obtained takes the following form:

\[ \eta_{n=1}= \frac{2\{3(m+1)+k(m+2)\}}{3k(m+1)^2} \left\{\left[m+\frac{m-1}{k}\right]\ln(1+k)-(m-1)\right\}-1 . \tag{15 bis} \]

1. If, substituting the found value of \(\eta\) into expressions (12) and (13), we obtain general formulae for determining the area of the cross-section of the hook as a function of the greatest permissible stress \(\xi\), the maximum load \(Q\), and the relative quantities \(m\), \(n\), and \(k\).

Fig. 7.

Denoting, as before, the areas computed on the basis of the stresses in the tensile and compressed parts respectively by \(f_1\) and \(f_2\), we find the following expressions for them:

\[ f_1=\frac{Q}{\varepsilon}\, \frac{ k^2[3(n+1)+k(m+2)](m+1)(m+2) }{ 2[3n(n+1)+k(n+2)]\left\{[(n+1)+k(m-n)]_n^{\,n+k}(m-1)-\frac{3}{2}k^2(m+1)^2\right\} +\frac{Q\,3n(m+1)-3(m+1)}{\varepsilon\,3n(m+1)-k(m+2)} } \tag{16} \]

\[ f_2=\frac{Q}{\varepsilon}\, \frac{ k^3[(n+1)+k(m+2)](m+1)(2m+1) }{ 2(n+k)^3[(3n+1)+k(m+2)] \left\{[3n+1+k(m+2)][(n+1)+k(m-1)]-\frac{3}{2}k^2(m+1)^2\right\} -\frac{Q\,3n(m+1)}{\varepsilon\,3n(m+1)-k(m+2)} } \tag{17} \]

For \(n=4\), i.e. when the center coincides with the center \(O\) of the hook opening, the expressions for the fibers nearest to the rod branch have the following form:

\[ f_{1,n-1}= \frac{Q}{\varepsilon}\, \frac{ k^3(m+1)(m+2) }{ 2[3(n+1)+k(m-1)]\{[1-n(k+1)]-\frac{3}{2}k^2(m+1)^2\} } \tag{18} \]

\[ f_{2,n-1}= \frac{Q}{\varepsilon}\, \frac{ k^3(n+1)(2m+1) }{ 2(k+1)^3[3(n+1)+k(m-1)]\{[1-n(k+1)]-\frac{3}{2}k^2(m+1)^2\} } \tag{19} \]

If, in the calculation of an elliptical hook, it is required to refer all quantities to the extension fibers, then expressions (16) and (18) are used for the fibers drawn in equations (2) and (3). To obtain the case when we must take formulas referring to the peripheral point \(n\), then \(\ln\left(1+\frac{k}{n}\right)\) may be expanded in the following form:

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

Substituting this expression into formula (16), we obtain:

\[ f_1=\frac{Q}{\varepsilon} = \frac{ k^2[3(n+1)+k(n+2)](n+1)+2 }{ 2n^2[3(n+1)+\frac{k}{n}(n+1)^2] } +\frac{k}{n}(m+1)^2 -\frac{ \frac{m+2}{2\cdot 3}\left(\frac{k}{n}\right)^2 +\frac{m+3}{3\cdot 4}\left(\frac{k}{n}\right)^3 +\frac{m+4}{4\cdot 5}\left(\frac{k}{n}\right)^4+\cdots }{ 3(m+1)+\frac{k}{n}(m+2) } +\frac{Q}{\varepsilon} \frac{ 3(m+1)-3\frac{1}{n}(m+1) }{ 3(m+1)+\frac{k}{n}(m+2) }. \]

For \(n=\infty\), this expression takes the following form:

\[ f_{1,n=\infty} = \frac{Q}{\varepsilon} \left\{ \frac{ k^3\{3(m+1)+k(m+2)\}(m+2)+2 }{ 18n^3(m+1)^2+4m+4\left(\frac{k}{n}\right)+1 } \right\}, \]

or, finally, a single formula:

\[ f_{1,n} = \frac{Q}{\varepsilon} \left\{ \frac{ 2[3(m+1)+2]+1 }{ (m^2+4m+1)k } \right\}, \]

which is identical with formula (2).

In exactly the same way, expression (17), for \(n=\infty\), may be reduced to formula (3) for the eccentric tension of a straight bar.

Thus, having obtained by simple calculation the formula for eccentric tension, just as for compression, when the section of a curved beam is rectangular, we are able (as will be seen below) to calculate quite simply the eccentric tension of any beam whose cross-section is rectangular, both by the formula of Grashof and by the formula of Grashof.

1. Formulas (18) and (19) may be represented in the following form:

\[ f_{1,\ n=1} = \frac{Q}{\varepsilon}\, \frac{k^3}{2}\, \frac{(m+1)(m+2)}{\alpha m^2+\beta m+\gamma}, \tag{20} \]

\[ f_{2,\ n=1} = \frac{Q}{\varepsilon}\, \frac{k^3}{2(k+1)}\, \frac{(m+1)(2m+1)}{\alpha m^2+\beta m+\gamma}, \tag{21} \]

with

\[ \left. \begin{aligned} \alpha &= (k+3)(k+1)\ln(k+1)-2.5\,k^2-3k,\\ \beta &= 2k(k+2)\ln(k+1)-4k^2,\\ \gamma &= 0.5\,k^2+3k-(2k+3)\ln(k+1). \end{aligned} \right\} \tag{22} \]

From the expressions obtained for \(f_1\) and \(f_2\) it is clear that the area of the dangerous section depends on \(m\) and \(k\), and, while \(m\) may vary from \(0\) to \(\infty\), the quantity \(k\), as has already been indicated, has a certain practical limit \(k = 2.5\).

Assigning the value of \(k\), we can, if \(n=1\), determine the value of \(m\) for which the area of the section of the hook under consideration, for a given allowable stress \(\varepsilon\), will be the least.

Since the magnitude of the sought area is determined by two functions of \(m\), (18) and (19), in calculating it one must take, of the two values of the functions \(f_{1,\ n=1}\) and \(f_{2,\ n=1}\) for the same \(n\) and the same \(m\), the value which is larger, so that the maximum stress of the material of the opposite sign shall not prove to be higher than the allowable \(\varepsilon\); under this condition one can find the actual minimum of the required area, which coincides either with the minimum of \(f_{1,\ n=1}\), or with the minimum of \(f_{2,\ n=1}\), or else with the point of mutual intersection of the curves determined by these functions.

The point of intersection of the curves is in fact a minimum only in the case when at it one function is increasing and the other decreasing; in the opposite case it coincides with one of the minima of the indicated functions.

Thus, in order to determine the values of \(m\) for which the area of the dangerous section of the hook, when it is calculated by Grashof’s formula, is obtained as the smallest for a given value of the allowable stress, it is first necessary to find those values of \(m\) at which

curves \(f_{1,n-1}=f_{2,n=1}\). For this it is necessary to equate to one another the right-hand sides of expressions (20) and (21), whereby we obtain the equation:

\[ \frac{(m+1)(m+2)}{\alpha m^2+\beta m+\gamma} = \frac{(m+1)(2m+1)}{(\alpha m^2+\beta m+\gamma)(k+1)}, \]

from which the desired values of \(m\) are determined:

\[ m=-1;\quad m=-\frac{2k+1}{k-1}\quad \text{and}\quad m=\frac{1}{2\alpha}\left(-\beta \pm \sqrt{\beta^2-3\alpha\gamma}\right)^{*} \]

For \(k>0\) the values \(\alpha,\beta\), and \(\gamma\) are positive, so that only negative values of \(m\) correspond to the found points of intersection of the two curves under consideration; and since, for the calculation of the hook, only positive \(m\) has meaning, then, if \(k>1\), it is sufficient to investigate the curves only for \(m\) greater than zero.

Let us take the derivatives with respect to \(m\) of expressions (20) and (21).

\[ (f_{1,n-1})'_m = \frac{Q}{\varepsilon}\,\frac{k^3}{2}\, \frac{(\beta-3\alpha)m^2-2(2\alpha-\gamma)m-(2\beta-3\gamma)} {(\alpha m^2+\beta m+\gamma)^2} \tag{23} \]

\[ (f_{2,n-1})'_m = \frac{Q}{\varepsilon}\,\frac{k^3}{2(k+1)}\, \frac{(2\beta-3\alpha)m^2-2(\alpha-2\gamma)m-(\beta-3\gamma)} {(\alpha m^2+\beta m+\gamma)^2} \tag{24} \]

Equating these derivatives to zero, we obtain equations from which the values of \(m\) corresponding to the maxima and minima of the functions under investigation are determined; here, from the first equation we obtain:

\[ m_1= \frac{(2\alpha-\gamma)\pm \sqrt{(2\alpha-\gamma)^2+(\beta-3\alpha)(2\beta-3\gamma)}} {\beta-3\alpha} \tag{26} \]

and from the second:

\[ m_2= \frac{(\alpha-2\gamma)\pm \sqrt{(\alpha-2\gamma)^2+(2\beta-3\alpha)(\beta-3\gamma)}} {2\beta-3\alpha}, \tag{27} \]

Thus, from each equation two values of \(m\) are determined; in order to find out which of them correspond to maxima and which to minima, we also find the second derivatives of the functions under consideration (20) and (21):

*) In the last case \(f_{1,n-1}\) is equal to \(f_{2,n=1}\), and both are equal to \(\pm\infty\); hence, at these points the function undergoes a discontinuity.

\[ (f_{1,\,n-1})''_{m}=\frac{Q}{\xi}\,k^{3}\left\{ \frac{(\alpha m^{2}+\beta m+\gamma)^{2}\,[(\beta-3\alpha)m-(2\alpha-\gamma)]}{(\alpha m^{2}+\beta m+\gamma)^{4}} - \frac{[(\beta-3\alpha)m^{2}-2(2\alpha-\gamma)m-(2\beta-3\gamma)]\,[(\alpha m^{2}+\beta m+\gamma)(2\alpha+\beta)]}{(\alpha m^{2}+\beta m+\gamma)^{4}} \right\}. \]

\[ (f_{2,\,n-1})''_{m}=\frac{Q}{\xi}\,\frac{k}{k+1}\left\{ \frac{(\alpha m^{2}+\beta m+\gamma)^{2}\,[(2\beta-3\alpha)m-(\alpha-2\gamma)]}{(\alpha m^{2}+\beta m+\gamma)^{4}} - \frac{[(2\beta-3\alpha)m^{2}-2(\alpha-2\gamma)m-(\beta-3\gamma)]\,(\alpha m^{2}+\beta m+\gamma)(2\alpha+\beta)}{(\alpha m^{2}+\beta m+\gamma)^{4}} \right\}. \]

Since \((\alpha m^{2}+\beta m+\gamma)^{2}\) cannot be a negative quantity, and since the second terms in these derivatives, for the values \(m_{1}\) and \(m_{2}\) that make the first derivatives vanish, also vanish, the signs of the second derivatives are the same as the signs of the expressions:

\[ (f_{1,\,n-1})''_{m}\quad\text{has the sign of:}\quad (\beta-3\alpha)m_{1}-(2\alpha-\gamma) \]

and

\[ (f_{2,\,n-1})''_{m}\quad\text{has the sign of:}\quad (2\beta-3\alpha)m_{2}-(\alpha-2\gamma). \]

Substituting into these expressions the values \(m_{1}\) and \(m_{2}\), the roots of equations (26) and (27), we find that the functions \((f_{1,\,n-1})''_{m}\) and \((f_{2,\,n-1})''_{m}\) have the same signs as the signs of the radicals:

\[ \pm\sqrt{(2\alpha-\gamma)^{2}+(\beta-3\alpha)(2\beta-3\gamma)} \]

and

\[ \pm\sqrt{(\alpha-2\gamma)^{2}+(2\beta-3\alpha)(\beta-3\gamma)}, \]

i.e. each function will have one minimum, corresponding to the positive radicals, and one maximum—the negative ones; moreover, one of these minima is the actual minimum, for which the ratio \(m\) corresponds to the minimal area of the dangerous section at the given stress of the material \(\xi\).

It must be noted that when \(m\) is positive and when \(k>1\), the ratio

\[ f_{1,\,n-1}:f_{2,\,n-1} \]

is equal to

\[ \frac{(m+2)(k+1)}{2m+1}, \]

and is always greater than unity; thus, for every \(k\) greater than unity, the actual minimum must coincide with the minimum of \(f_{1,\,n-1}\). But since it is not advantageous to give the quantity \(k\) values less than unity, because this increases the area of the calculated section of the hook, it may be assumed that \(k\) for the calculated sections of the hook is always greater than \(1\), and,

therefore, the largest value of \(m\) found in the sense indicated above can in such cases be determined by the formula:

\[ m_1= \frac{(2\alpha-\gamma)+ \sqrt{(2\alpha-\gamma)^2+(\beta-3\alpha)(2\beta-3\gamma)}} {\beta-3\alpha}, \tag{28} \]

in which the values of the quantities \(\alpha\), \(\beta\), and \(\gamma\) are given by formulas (22).

If \(k\) were less than 1, then, in order to find the largest value of \(m\), it would be necessary to take into account both formulas (26) and (27), and the question would have to be solved by a complete investigation of both curves determined by functions (18) and (19), and intersecting at

\[ m=\frac{1-2k}{1-k}. \]

1. Having found the most advantageous value of \(m\), we substitute its value in formula (20) and calculate the area of the design section \(f_{1,n-1}\). Knowing this area, it is then not difficult to find also the lengths \(b_1\) and \(b_2\) of the parallel sides of the trapezoid obtained in the section under consideration, from the equations:

\[ 2f_{1,n-1}=b_2(m+1)ka \quad \text{and} \quad b_1=mb_2. \]

For an example let us determine the values of the elements of a trapezoidal section of a hook for \(k=2\).

From formulas (22) we find: \(\alpha=0.4790\); \(\beta=1.5776\) and \(\gamma=0.3098\).

Substituting these quantities in formula (28), we obtain \(m_1=10.69\)—the most advantageous, for \(k=2\), value of the ratio between the parallel sides of the trapezoid. After substituting the values of \(m\) and \(k\) in equation (20), the area of the design section is determined as

\[ f_{1,n-1}=8.25\,\frac{Q}{\xi}, \]

where \(Q\) is the maximum load lifted by the hook, and \(\xi\) is the allowable stress in the stretched fibers of the hook. The length of the smaller parallel side of the trapezoid, \(b_2\), on the basis of the preceding, will be equal to

\[ \frac{16.50\,Q}{23.38\,\xi a}, \]

or, since \(a=0.05\sqrt{Q}\), then

\[ b_2=\frac{282.3}{\xi}\,a, \]

whereas the length of the larger side is

\[ b_1=\frac{3018}{\xi}\,a. \]

In Fig. 7 (p. 17) are presented, for the particular case when \(k=2\), both curves determined by functions (18) and (19), with the values of the areas \(f_{1,n-1}\) and \(f_{2,n-1}\) plotted along the ordinate axis, and along the abscissa axis—

m. The minimum of the first curve coincides with point C, and the minimum of the second curve with point B. Both these curves have common points of intersection (which do not, however, lie between the abscissae of their minima) in the parts of the curves corresponding to negative values of m, which are shown in the drawing by dotted lines.

In addition, below is given Table I of the values*) for various \(k\) of the most advantageous (in the sense of the magnitude of the area) ratios \(m_1\) of the parallel sides \(b_1\) and \(b_2\) of trapezoidal sections in calculating a hook by Grashof’s formula, as well as the values of the coefficient

\[ H=\frac{k^3}{2}\,\frac{(m+1)(m+2)}{\alpha m^2+\beta m+\gamma} \]

for determining the corresponding areas of the trapezoids by equation (20). Alongside these are also given the values of the coefficients \(m_0=k+1\) and \(H_0=\frac{3(k+2)}{k}\) from formulae (6) and (7), for determining analogous quantities in calculating the hook as a rectilinear bar in eccentric tension. For comparison of the results of the one and the other calculation, the table also gives the values of the coefficients \(H_0'\), determined by Grashof’s formula, for \(m=m_0\), and in the last column of the table the ratios \(\frac{H_0'}{H_0}\) are indicated.

TABLE I.

\(k\)	\(m_1\)	\(H\)	\(H_0'\)	\(H_0\)	\(m_0\)	\(\dfrac{H_0'}{H_0}\)
0,75	5,2	12,5	13,5	11,0	1,75	1,22
1,00	6,0	10,8	11,3	9,0	2,00	1,25
1,25	7,0	9,7	10,2	7,8	2,25	1,31
1,50	8,1	9,0	9,4	7,0	2,50	1,34
1,75	9,4	8,6	8,8	6,4	2,75	1,38
2,00	10,7	8,3	8,6	6,0	3,00	1,43
2,25	12,2	8,0	8,3	5,7	3,25	1,46
2,50	14,2	7,8	8,1	5,4	3,50	1,50
3,00	19,9	7,6	7,8	5,0	4,00	1,57
4,00	49,3	7,4	7,6	4,5	5,00	1,69
5,00	\(\infty\)	7,3	7,6	4,2	6,00	1,80

*) In the computations, natural logarithms were taken from five-place tables.

1. According to the calculation given, from formulas (20) and (28), the hook must be so designed that the locus of the centers of gravity of its radial sections, \(CB\), is situated on a circle whose center coincides with the center of the hook opening \(O\) (Fig. 8).

In the case when the hook opening is drawn along an arc of a circle, the line \(HO\), coinciding with the direction of the force \(Q\), with the configuration we have adopted as the most unfavorable for the stresses in the hook, is the axis of symmetry with respect to the sections \(I—O\) and \(III—O\), \(IV—O\) and \(VI—O\), \(V—O\) and \(VII—O\), which are situated pairwise under identical conditions with respect to the maximum moments and forces acting on them; in this case the simplest outline for the external contour of the hook must be a circle whose center is located on the line \(HO\).

Fig. 8.

If the height of some third section of the hook were also found, in addition to the two dangerous sections \(I—O\) and \(III—O\), for example, of section \(IV—O\), then by this very fact both the magnitude of the radius \(R\) for this circle and the position of its center \(D\), which must lie on the line \(HO\), would be determined.

Let us find the dimensions of the trapezoidal section \(IV—O\), and of the completely identical section \(VI—O\), having first specified one of the dimensions of the trapezoid, for example, the length of its larger base \(b'_1\). In these sections the hook is subjected to bending under the influence of the moment of the force \(Q\) acting at a lever arm \(OB \sin 60^\circ\), and to tension from the force \(Q \sin 60^\circ\), whose combined action is equivalent to the action of—

...the action of the force \(Q\,\mathrm{Sn}\,60^\circ\), applied at the center of the hook opening and directed normally to the plane of section IV—O.

Thus, in determining the area of the section or the stresses of the fibers, we may in the present case make use of the formulas derived earlier, in which, instead of the force \(Q\), the force \(Q\mathrm{Sn}60^\circ\) must be taken. For the elements of section IV—O we shall use the former designations, but with a prime, so that (Fig. 8)

\[ EB=c'_1=c_1;\qquad EM'=h'=k'a. \]

The ratio of the parallel sides of the trapezoid \(m'=\dfrac{b'_1}{b'_2}\) is determined as a function of the quantity \(k'\) from the condition that the center of gravity of the computed section remains on the circular axis of the hook. Indeed, if the inner opening of the hook is likewise described by an arc of a circle, then

\[ \frac{c'_1}{h'}=\frac{m'+2}{3(m'+1)}=\frac{c_1}{h'},\quad \text{where } c_1=\frac{(m+2)ka}{3(m+1)}; \]

whence we obtain:

\[ m'=\frac{c_1-\dfrac{2}{3}k'a}{\dfrac{k'a}{3}-c_1} = \frac{k(m+2)-2k'(m+1)}{k'(m+1)-k(m+2)}. \tag{29} \]

This formula indicates that, if the height of the trapezoid \(h'\) is divided into three equal parts, then the point \(B\), corresponding to its center of gravity, divides the middle portion in the ratio of the lengths of which is equal to \(\dfrac{1}{m'}\). This property may subsequently be used for the graphical determination of the ratio \(m'\), if the position of the center of gravity of the trapezoid and its height \(h'\) are known.

If \(h'\), and therefore \(k'\), are not known, then, for determining the elements of the trapezoid, one may compose another equation, expressing the condition of strength of the hook in the section under consideration on the basis of equations (10) and (14):

\[ f'_{1,n=1} = \frac{Q\mathrm{Sn}60^\circ}{\sigma}\frac{c_1}{a} = \frac{t'}{ r\left[m'(k'+1)\ln(k'+1)-\ln(k'+1)-m'+1\right]-\dfrac{1}{f'} }. \]

Substituting into this expression the value of \(m'\) from equation (29), we obtain for determining \(k'\) a transcendental equation, which can...

it would have to be solved either by successive approximation or by a graphical method.

However, the latter method is rather complicated; therefore, instead of specifying \(b'_1\), it is more convenient to specify the radius \(R-a\), and consequently also the center \(D\), of the outer contour of the hook. Then, analytically or directly from the drawing, the value

\[ k'=\frac{R'}{a'} \]

is determined, and, on the basis of equation (29), the value \(m'\) is obtained by the graphical method indicated above.

The values of these quantities, \(k'\) and \(m'\), must then be substituted into an expression analogous to (20), which has the following form:

\[ f'_1=\frac{Q\sin 60^\circ}{\delta}\, \frac{k'^3(m'-1)(m'-2)} {\alpha' m'^2+\beta' m'+\gamma'}, \tag{30} \]

in which the values \(\alpha'\), \(\beta'\), \(\gamma'\) are determined by formulas (22) for \(k=k'\). From this expression the area of the trapezoid \(f'_1\) is determined (if \(k'>1\)); and then from the equation

\[ 2f'_1=a'k'b'_1\,\frac{m'+1}{m'} \]

we obtain:

\[ b'_1=b_1\, \frac{f'_1\,m'(m+1)k} {f_1\,m(m'+1)k'}. \tag{31} \]

In exactly the same way, the magnitudes of the ratios \(k''\) and \(m''\) can be found, and the lengths of the parallel sides of the trapezoids \(b''_1\) and \(b''_2\) can be determined in sections \(V-O\) and \(VII-O\), for which in the preceding formula (30) the acting force must be taken equal to \(Q\sin 30^\circ\).

Section \(II-O\), with respect to the forces acting upon it, is in the same conditions as section \(IV-O\). Its height \(h_0\) and the value of the ratio of the parallel sides of the trapezoid \(m_0\) can be determined graphically; as for the magnitude of the larger base of the trapezoid \(b'_0\), for constructive reasons it may be taken equal to \(b_1\), as in the adjacent dangerous sections; from this the strength of the hook in section \(II-O\) will be somewhat increased. This surplus of strength at this place will not be superfluous, since in cases where the point \(E\) of application of the force \(Q\) is displaced, owing to friction of the suspended rope toward the end of the hook, the dangerous section approaches section \(II-O\).

Finally, it should be pointed out that section \(V-O\) must be checked for shear by the force \(Q\), and the shank of the hook—for tension.

If \(k=2\), the position of the center of the arc for the outer outline of the hook may be given by the coordinates: \(x=-\dfrac{a}{2}\) and \(y=-\dfrac{a}{2}\tg 30^\circ\) (Fig. 8). The radius of the arc \(R\) and the height \(h'\) of sections \(IV—0\) and \(VI—0\) in this case are determined from the triangles \(OID\) and \(OIVD\), whence we obtain:

\[ R^2=6\frac{1}{3}a^2,\quad \text{or}\quad R=2{,}51a; \]

to determine \(h'\) we compose the equation:

\[ 6\frac{1}{3}a^2=(a+h')^2+\frac{1}{3}a^2-(a+h')\frac{a}{\sqrt{3}}, \]

from which we find:

\[ h'=1{,}75a^*), \quad \text{thus } k'=4{,}75. \]

Taking into account that the most advantageous ratio \(m\) of the parallel sides of the trapezia, corresponding to the minimum values of the areas of the calculated sections of the hook, for \(k=2\) is, according to the data given above, \(10{,}69\), we find on the basis of formula (29) the value of the ratio \(m'=3{,}15\).

Substituting the obtained values \(k'\) and \(m'\) into equation (30), we obtain:

\[ f'=7{,}63\,\frac{0}{6}; \]

and then from equation (31) we find:

\[ b_1'=0{,}89\,b_1 \quad \text{and}\quad b_2'=0{,}28\,b_1. \]

For sections \(V—0\) and \(VII—0\), on the basis of equation (30), we obtain:

\[ k''=1{,}45,\quad m''=1{,}0 \quad \text{and}\quad b_1''=b_2=0{,}5\,b_1. \]

The height of section \(II—0\), \(h_0\), is determined as the difference \(HO-a\), or

\[ h_0=2{,}51a+\frac{a}{\sqrt{3}}-a=2{,}09a. \]

In order that the center of gravity of this section be removed from the center of the hook opening \(O\) by the same distance as the centers of gravity of the other calculated sections, it is necessary that the ratio of the parallel sides of the trapezium \(m_0\) satisfy the equation:

\[ \frac{(m_0+2)\,2{,}09}{3(m_0+1)} = \frac{(m+2)\,2}{3(m+1)}, \]

from which we obtain \(m_0=2{,}48\).

1. In the calculation of the hook presented above, the ratios of the parallel sides of the trapezoidal sections were determined under the assumption that the axis of these sections represents an arc of a circle, with its center coinciding with the center of the hook opening; respectively, the adopted

*) The other root gives a value of \(h'\) for a non-existent section.

for calculating it, his principal formulas (18) and (19), which are a special case of the general formulas (16) and (17), for \(n=1\). However, very often Grashof’s formula, which relates to the case when \(n=1\), is incorrectly applied even when \(n\) is in fact not equal to unity.

Indeed, in the most commonly used method of constructing a hook, its outer contour is drawn as an arc of a circle from a center lying on a horizontal line passing through the center of the hook opening \(O\), and the distance between these centers is seldom taken equal to from \(\frac{1}{4}a\) to \(\frac{3}{4}a\) or more. With such a construction of the hook contour, its axis, coinciding with the geometric locus of the centers of gravity of the cross sections, unless special measures are taken, will have variable curvature, and the center of curvature of the dangerous section \(1—0\) will not coincide with the center of the hook opening, i.e. \(n\) will not be equal to 1. Consequently, in this case the formula \(\varepsilon = -\frac{Q}{r_0 f}\frac{x}{r+x}\) cannot be applied.

To illustrate the magnitude of the inaccuracy that results, let us determine the stress \(\varepsilon\) in the dangerous section of the hook for the case when \(m=3\) and \(k=2\). If the center of curvature of the dangerous section of the hook and the center of its opening coincide, then the magnitude of this stress \(\varepsilon\) is obtained from formula (18):

\[ \varepsilon = \frac{Q}{f}\,8.56 . \]

If, however, the center of curvature of the axis is distant from the center of the hook opening by \(\frac{1}{4}a\), then, taking in formula (16) \(n=\frac{3}{4}\), we obtain:

\[ \varepsilon = \frac{Q}{f}\,9.23 . \]

Thus, from the use of a formula that does not correspond to the case, significant errors are obtained, the magnitudes of which, in percent, for different \(n\) and for \(k=2\) and \(m=3\), are given in the following Table I:

TABLE II.

m	1	3/4	2/3	1/2	1/3	1/4
\( \dfrac{f_z}{Q} \)	8.56	9.23	9.56	10.49	12.16	13.69
%	0	8	11	11	42	60

1. Although in the literature on the calculation of hooks it is generally held that the most advantageous forms of hook sections, from the standpoint of the most rational use of the material, correspond to the minimum values of their areas, this view is not entirely correct, just as the usually adopted ratios of the sides of the trapezoids corresponding to the minima of the calculated areas of hook sections are also incorrect, in accordance with what has been set forth above.

Indeed, for a curvilinear form of the hook, the volume element \(d\omega\), enclosed between two transverse sections whose angle between them is \(dz\), is expressed, on the basis of Guldin’s theorem, by the product:

\[ d\omega = f_z r_z\, dz, \]

where \(f_z\) is the area of the transverse section of the hook at the angle \(z\) to the origin of coordinates, and \(r_z\) is the radius of curvature of the hook axis in the section under consideration.

In order to obtain the volume of the entire hook \(\omega\), it is necessary to integrate the indicated expression over the limits of the arc corresponding to the hook axis, so that

\[ \omega = \int f_z r_z\, dz. \tag{32} \]

If the axis of the calculated sections of the hook represents an arc of a circle whose center coincides with the center of the hook opening, then the quantity \(r\) within the limits of the calculated sections is constant, and then the volume will be equal to

\[ \omega = r \int f_z\, dz. \]

Thus, the volume of the hook—and therefore also the expenditure of material on it—increases not only together with the areas of the transverse sections of the hook, but also at the same time with the magnitude, in these sections, of the radii of curvature of the axis, which, other conditions being equal, are the smaller, the greater the ratio of the parallel sides of the trapezoids \(m\).

Consequently, the value of the product \(fr\) for a given section of the hook characterizes the degree of suitability of the form of this section from the point of view of the most advantageous utilization of the material of the hook.

Let us express the value of the product \(fr\) for the case when the axis of the hook is an arc of a circle whose center coincides with the center of the opening of the hook, in terms of the variables \(m\) and \(k\), which have their previous meanings.

Taking into account expressions (18) and (19), and also that

\[ r=a+c_1=a\,\frac{3(m+1)+k(m+2)}{3(m+1)}, \]

we obtain:

\[ f_{1}r=\frac{Q}{\sigma}\,6ak^3 \frac{[3(m+1)+k(m+2)](m+2)} {\alpha m^2+\beta m+\gamma}, \tag{33} \]

and

\[ f_{2}r=\frac{Q}{\sigma}\,\frac{ak^3}{6(k+1)} \frac{[3(m+1)+k(m+2)](2m+1)} {\alpha m^2+\beta m+\gamma}, \tag{34} \]

where \(\alpha,\ \beta\), and \(\gamma\) have their previous meanings:

\[ \begin{aligned} \alpha&=(k+3)(k+1)\ln(k+1)-2.5k^2-3k,\\ \beta&=2k(k+2)\ln(k+1)-4k^2,\\ \gamma&=0.5k^2-3k-(2k+3)\ln(k+1). \end{aligned} \tag{35} \]

1. The points of intersection of the curves determined by the functions (33) and (34) may be obtained from the equation:

\[ \frac{[3(m+1)+k(m+2)](m+2)} {\alpha m^2+\beta m+\gamma} = \frac{[3(m+1)+k(m+2)](2m+1)} {(\alpha m^2+\beta m+\gamma)(k+1)}, \tag{36} \]

from which the corresponding values of its roots are found:

\[ m=-\frac{3+2k}{3+k},\quad m=-\frac{2k+1}{k-1} \quad\text{and}\quad m=\frac{1}{2\alpha}\left(-\beta\pm\sqrt{\beta^2-4\alpha\gamma}\right). \]

If \(k>1\), then all these roots are negative and, consequently, for this the two curves do not intersect for any positive value of \(m\).

For \(k\) greater than 1, the values of \(f_1r\) are always greater than the values of \(f_2r\); therefore, in this case the minimum of expression (33) corresponds to the actual minimum of the product \(fr\), and hence to the condition of the most advantageous use of the hook material, and in that case the required value of \(m\) is determined as the root of the equation \((f_1r)'_m=0\).

Find the first derivative with respect to \(m\) of \((f_1 r)\):

\[ (f_1 r)'_m = \frac{Q}{\varepsilon}\,\frac{ak^3}{6}\, \frac{A_1m^2+2B_1m+C_1}{(\alpha m^2+\beta m+\gamma)^2}, \tag{37} \]

where

\[ \begin{aligned} A_1&=\beta(3+k)-\alpha(9+4k),\\ B_1&=\gamma(3+k)-\alpha(6+4k),\\ C_1&=\gamma(9+4k)-\beta(6+4k) \end{aligned} \tag{38} \]

Equating the trinomial \(A_1m^2+2B_1m+C_1\) to zero, we determine the roots of the resulting quadratic equation as

\[ m=\frac{-B_1\pm\sqrt{B_1^2-A_1C_1}}{A_1}. \tag{39} \]

In order to determine which value corresponds to the required minimum, we take the second derivative with respect to \(m\): \((f_1r)''_m\):

\[ (f_1r)''_m = \frac{Q}{\varepsilon}\,\frac{ak^3}{6}\, \frac{(\alpha m^2+\beta m+\gamma)^2(A_1m+B_1m)}{(\alpha m^2+\beta m+\gamma)^4} - \]

\[ -\frac{Q}{\varepsilon}\,\frac{ak^3}{6}\, \frac{(A_1m^2+2B_1m+C_1)(\alpha m^2+\beta m+\gamma)(2\alpha+\beta)} {(\alpha m^2+\beta m+\gamma)^4}. \]

For the values of \(m\) that make \((f_1r)'_m\) vanish, this second derivative takes the same sign as the sign of the radical in expression (39); thus, the required minimum corresponds to the following value of \(m\):

\[ m=\frac{-B_1+\sqrt{B_1^2-A_1C_1}}{A_1}, \tag{40} \]

where \(A_1\), \(B_1\), and \(C_1\) are determined from expressions (38) and (35).

In the case when \(k\) is equal to or less than \(1\), the question of the most advantageous value of \(m\), in the sense of the most rational use of the material of the hook, is complicated by the necessity of investigating both curves determined by expressions (33) and (34), which then necessarily intersect at the point corresponding to the positive

\[ m=\frac{1+2k}{1-k}. \]

As for the minimum of the function \(f_2r\), it corresponds to the value of \(m\) determined as the root of the equation \((f_2r)''_m=0\) with a positive radical:

\[ m=\frac{-B_2+\sqrt{B_2^2-A_2C_2}}{A_2}, \tag{41} \]

where

\[ \left. \begin{aligned} A_2 &= \beta(6+2k)-\alpha(9+5k)\\ B_2 &= \gamma(6+2k)-\alpha(3+2k)\\ C_2 &= \gamma(9+5k)-\beta(3+2k) \end{aligned} \right\} \tag{42} \]

In those cases where, in calculation by formulas (40) or (41), negative values are obtained for \(m\), while the functions \(f_{1r}\) and \(f_{2r}\) decrease continuously within the limits of positive \(m\), the most advantageous trapezoidal form of the hook with respect to material consumption is obtained at \(m=\infty\), which corresponds to triangular sections of the hook.

1. We present Table III of the values of \(m_1\), corresponding for different values of \(k\) to the condition of the most advantageous utilization of the hook material, and also the values of the coefficient

\[ H_1=\frac{k^3(m+1)(m+2)}{2}\cdot\frac{1}{\alpha m^2+\beta m+\gamma} \]

for determining the cross-sectional areas of the hook obtained in this case by formula (20), as well as the coefficient \(l=\dfrac{b_1\delta}{a}\), by which the form of the section itself is characterized.

TABLE III.

\(k\)	\(m_1\)	\(H_1\)	\(\dfrac{b_1\delta}{a}\)
0,75	9,0	12,6	12096
1,00	13,5	10,9	8112
1,25	24,8	9,9	6629
1,50	75,1	9,2	4550
1,75	\(\infty\)	8,7	3977
2,00	\(\infty\)	8,4	3360
2,25	\(\infty\)	8,1	2880
2,50	\(\infty\)	7,9	2528
3,00	\(\infty\)	7,6	2027
4,00	\(\infty\)	7,4	1480
5,00	\(\infty\)	7,3	1168

On the basis of the data in the last column of Table III it may be seen that the width of the design section \(b_1\) is the smaller, the larger \(k\) is and the greater the value of the allowable stress of the material \(\delta\). Consequently, in order that the form of the hook section should not prove too ...

expanded, it is necessary, for smaller values of $\varepsilon$, to take larger values of $k$.

The comparative advantages of calculating the hook by Grashof’s formula at the most advantageous value of $m_1$ from the standpoint of utilization of the material, and also of calculating at values of $m_0$ corresponding to the minima of the calculated areas of the hook sections, as compared with the calculation usually adopted by the same formula, but with $m = k + 1$, are shown in Table IV, which gives, for various values of $k$, the values of the products $f_1 r_1$, $f_0 r_0$, and $fr$ for all three indicated cases.

TABLE IV.

$k$	$f_1 r_1$	$f_0 r_0$	$fr$
0,75	16,0	16,1	18,1
1,00	14,8	14,9	16,3
1,25	14,2	14,3	15,7
1,50	13,9	14,0	15,4
1,75	13,8	14,0	15,4
2,00	13,9	14,2	15,7
2,25	14,2	14,5	15,9
2,50	14,5	14,8	16,3
3,00	15,4	15,5	17,3
4,00	17,2	17,4	19,4
5,00	19,6	19,6	22,0

From a comparison of the results in the table given, it may be concluded that the most advantageous values for $k$, corresponding to the smallest values of the products $f_1 r_1$, $f_0 r_0$, and $fr$, by which the consumption of material for the hook is characterized, are obtained for values of $k$ lying approximately within the limits from 1,25 to 2,25.

For values of $k$ above or below the indicated limits, these products increase rapidly: thus, for example, in passing from $k = 2$ to $k = 5$ their values increase by more than 40%.

In exactly the same way, the consumption of material increases by more than 10% when the value of $m$ is determined by the formula $m = k + 1$, and not by the formulas for determining the most advantageous value of $m$.

At the same time, the difference between the results following from calculating the hook by the formulas for determining the minimum section and

according to the formulas for the most advantageous section, corresponding to the minima of the products \(f_1 r_1\), is insignificant and, in general, no more than \(2\%\). However, from a purely theoretical point of view, the latter method is more correct and, moreover, simpler, since for values of \(k\) above 1.75, as is seen from Table III, it leads to a simple triangular form of the design section.

As regards the further calculation of the hook, in the sense of determining the elements necessary for drawing it, in the present case the same path is applicable as was indicated earlier.

1. Since the sharp edges formed by the trapezoidal sections of the hook, upon contact with the links of a chain or a rope, may damage them and themselves be damaged,—in order to avoid this, the larger parallel side of the trapezium is replaced by an arc smoothly joined to its inclined sides.

With such a disturbance of the form of the trapezium, its area, the magnitude \(\eta\), and other elements influencing the calculation of the hook are also changed. It is desirable, of course, that with such a modification of the hook section the deviations from the calculation performed earlier should be as small as possible.

Since, according to the fundamental formulas of Grashof,

\[ \delta=\frac{N}{f}+\frac{M}{fr}+\frac{M}{\eta fr}\frac{x}{r+x}, \]

where

\[ \eta=-\frac{1}{f}\int \frac{x}{r+x}\,df, \]

then, in view of the above, it is first of all necessary that \(r\) remain unchanged, i.e. that the position of the center of gravity \(C\) of the section not change (Fig. 9). This can be achieved by making the rounding in the wide part of the trapezium and correspondingly narrowing or shortening its narrow part; if at the same time the arc of the rounding is tangent to the larger parallel side of the trapezium which it replaces, then the value of \(x\) for the most remote fibers is also not changed; nevertheless the values of \(\eta\) and \(f\) do change.

For the rounded section of the hook, the values of the corresponding coefficients \(\eta_0\) and \(f_0\) may be determined either graphically or analytically. If a planimeter for measuring areas is available, it is convenient to use Banjulin’s graphical method for determining \(\eta_0\). For this purpose, from the center of curvature of the hook axis \(O\), lying in the plane

Fig. 9.

of the transverse section under consideration, draw the straight line \(Oy\) to the points of the contour of the section, and then from its center of gravity \(C\) draw lines \(Cy'\) parallel to them until they intersect the perpendiculars from the points \(y\) on the axis of symmetry of the section at the points \(y'\) of the desired curve. The ratio of the area \(\varphi_0\), bounded by this curve and consisting of a positive and a negative part, to the area of the figure itself \(f_0\) will give the required value \(\eta_0\) for the rounded section of the hook, i.e.

\[ \eta_0=\frac{\varphi_0}{f_0} \tag{43} \]

It is not difficult to notice from the very method of construction and from the formulas (15) given earlier that \(\eta_0\) does not change if the ordinates of the section are proportionally increased or decreased.

Therefore, if some value of \(\eta_0\) has been found for a definite section of the hook, then, knowing its area \(f_0\), it is easy to determine the stress in the hook \(\mathcal{E}\) by formula (10) and, conversely,—having specified the permissible stress \(\mathcal{E}\), one can determine the required area of the section \(F_0\) by the formula:

\[ F_0=\frac{Q}{\mathcal{E}}+\frac{M}{\mathcal{E}r}+\frac{M}{\eta_0\mathcal{E}r}\,\frac{c_1}{r-c_1}, \tag{44} \]

*) The indicated construction graphically reproduces the formula \(\displaystyle \int \frac{x y\,dx}{r+x}\), where \(x\) and \(y\) are the coordinates of the hook section under consideration, and \(r\) is the distance of its center of gravity \(C\) to the center of the opening \(O\).

which, for the case when \(n=1\), takes the form:

\[ F_0=\frac{Q}{6}\frac{1}{\eta_0}\frac{c_1}{a} \tag{44 bis} \]

Then it is already easy to determine the magnitude of the ratio \(\psi\) by which it is necessary to increase the ordinates of the rounded section:

\[ \psi=\frac{F_0}{f_0} \tag{45} \]

where \(F_0\) is determined by formula (44), and \(f_0\) is the area of the rounded section for which \(\eta_0\) was computed.

19. In order to avoid laborious graphical constructions and the use of a planimeter, which is not always at hand, one may with great convenience make use of the following method for determining the coefficient \(\eta_0=\dfrac{\varphi_0}{f_0}\).

Having determined the principal dimensions of the trapezoidal section, we round it off by inscribing an arc tangent to the larger of the parallel sides of the trapezoid, from some center lying on its axis; at the same time we also make the junctions between the larger sides of the trapezoid and the constructed arc.

In order that the center of gravity, after this change in the section of the hook, remain in its place, we cut off its narrower part by a line perpendicular to the axis of the section. The position of this line can be determined analytically, but since this gives an equation of the third degree, it is more convenient to determine it experimentally. For this it is necessary to cut the rounded section of the hook out of paper and freely suspend it at one point together with a thread plumb line; the intersection of the axis of the section with the axis of the thread determines the center of gravity of the figure. By gradually cutting off, with lines perpendicular to the axis of the section, the narrow part of the trapezoid*), one can bring the plumb line—and, therefore, the center of gravity of the figure—into coincidence with the former center of gravity of the trapezoid.

As for determining the magnitude \(\eta_0\) for the rounded and shortened section, this can be found quite simply and, moreover, very accurately by analytic means. For this purpose we replace the rounded part of the section by a broken line, so that the whole fig—

*) One may also cut off the inclined sides of the trapezoid—but such a method is more complicated.

the figure consisted of trapezoids having a common axis (see Fig. 9). The area of this figure \(f_0\) will be equal to the sum of the areas of the trapezoids composing it and, consequently, is determined without difficulty. The quantity \(\eta_0\) can be determined with the aid of formula (11).

Denoting the elements of each constituent trapezoid by a special subscript \(i\), we may write:

\[ \eta_0=-\frac{1}{f_0}\sum \int \frac{x_i\,df_i}{r+x_i}. \]

But since

\[ -\int \frac{x_i\,df_i}{r+x_i}=\eta_i f_i, \]

we obtain:

\[ \eta_0=\frac{1}{f_0}\sum \eta_i f_i \tag{46} \]

The quantity \(\eta_i\) must be determined for each separate trapezoid by formula (14).

If we denote the elements of the trapezoids composing the section of the hook, in accordance with their successive order, by the subscripts \(3,4,5,\ldots\) (the subscripts 1 and 2 were used for certain elements of the principal trapezoid), then, retaining the previous letter designations, we obtain the following formula for determining \(\eta_0\):

\[ \eta_0=\frac{r}{f_0}\left\{ \left[b_3+(r-c_3)\frac{b_4-b_3}{h_3}\right]\ln\frac{r-c_3}{r-c_4}+b_3-b_4+\right. \]

\[ +\left[b_4+(r-c_4)\frac{b_5-b_4}{h_4}\right]\ln\frac{r-c_4}{r-c_5}+b_4-b_5+ \]

\[ +\left[b_5+(r-c_5)\frac{b_6-b_5}{h_5}\right]\ln\frac{r-c_5}{r-c_6}+b_5-b_6+ \]

\[ \cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot \]

\[ \left.+\left[b_n+(r-c_n)\frac{b_{n+1}-b_n}{h_n}\right]\ln\frac{r-c_n}{r-c_{n+1}}+b_n-b_{n+1}\right\}-1. \]

Since the quantities \((r-c_0)\), \((r-c_1)\), etc., represent the distances of the parallel sides of the trapezoids from the center of curvature \(O\) of the hook in various

in the section under consideration, then, denoting them by \(l\) with the corresponding indices and making certain transformations, we obtain:

\[ \eta_0=\frac{r}{f_0}\left\{ \left[b_3+l_3\frac{b_4-b_3}{h_3}\right]\ln\frac{l_3}{l_4} +\right. \]

\[ +\left[b_4+l_4\frac{b_5-b_4}{h_4}\right]\ln\frac{l_4}{l_5} + \]

\[ +\left[b_5+l_5\frac{b_6-b_5}{h_5}\right]\ln\frac{l_5}{l_6} + \]

\[ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot \]

\[ \left. +\left[b_n+l_n\frac{b_{n+1}-b_n}{h_n}\right]\ln\frac{l_n}{l_{n+1}} +b_3-b)^{n+1} \right\}-1 \tag{47} \]

Here

\[ f_0=\frac{1}{2}\left\{(b_3+b_4)h_3+(b_4+b_5)h_4+(b_5+b_6)h_5+\cdot\ \cdot\ \cdot\right. \]

\[ \left.\cdot\ \cdot\ \cdot +(b_n+b_{n+1})h_n\right\}. \tag{48} \]

Knowing \(\eta_0\) and \(f_0\), it is easy to determine, from formulas (44) and (45), the coefficient \(\psi\) by which the ordinates of the rounded section (or, what is almost the same, the parallel sides of the trapezoids composing it, \(b_3, b_4\ldots\)) must be multiplied in order to obtain a section in which the most highly stressed fibers, under the action of the external forces applied to the hook, undergo a tension whose magnitude is equal to the prescribed allowable stress \(\sigma\).

1. On the basis of the foregoing we arrive at the following conclusions.

a) The calculation of the hook of lifting machines for a single vertical force equal to the maximum load is insufficient, since it leads to sections in the lower part of the hook that are too weak. A more correct calculation is obtained if it is assumed that the hook is subjected to the action of two forces, symmetric with respect to the vertical axis and equal in magnitude and direction to the resultants of the tensions of the independent strands of the hoisting rope.

Taking the limiting angles of inclination of the indicated resultants to the vertical to be \(60^\circ\), one obtains, as a result of the calculation, satisfactory dimensions of the parts of the hook.

b) Calculation of the hook for eccentric tension leads to an excessively low estimate of the stresses arising in its calculated sections; the difference between the stresses obtained in this case and the

computed by the more exact Grashof formula, the greater the value of the ratio \(k\) (the height of the section to the radius of the opening of the hook). Thus, when using the formula for eccentric tension to calculate a hook, the stresses allowable for the material must be reduced as \(k\) increases, in accordance with the data of Table III.

c) The condition of equality, in the design section of the hook, of the maximum stresses from tension and compression is not, in general, an indication that the area of this section itself will be the smallest. To determine the minimum design section it is necessary to investigate in greater detail the analytical expressions for the magnitude of its area, (14) and (15), taking into account not only the points of intersection of the curves corresponding to these functions, but also their minima. In the particular case when the calculation of the hook is carried out for eccentric tension, the minimum of the design area corresponds to the point of intersection of both curves, or, in other words, to the aforementioned condition of equality of the maximum stresses in the tensile and compressive fibers.

d) The section of the hook most advantageous in the sense of utilization of the material corresponds not to the minimum of the design area, but to the minimum of the product of the area of the design section by the radius of curvature of the hook axis in this section; in addition, of course, both products obtained from the conditions of the maximum allowable stress in the compressed and in the stretched parts of the design sections of the hook must likewise be taken into account.

On the basis of the data of Tables IV and III it may be seen that the smallest value of the mentioned product corresponds to the triangular form of hook at a value of \(k\) equal to 1.75. However, in order not to obtain excessively wide, non-constructive hook sections, the question of choosing the value of \(k\) must always be coordinated with the magnitude of the allowable stress \(\sigma\), being guided in this by the data of the tables given earlier.

e) For the calculation of a rounded hook section, instead of the graphical-construction methods of Bantlin or Tolle associated with them, one may use the fairly simple and accurate analytical formula (47), provided only that the rounding arc of the hook is replaced by a broken line symmetric with respect to the axis of the section.

.................

ERRATA.

Pages	Lines	Printed	Should be
6	6 from top	III—0	III—0 at \(30^\circ\) to the vertical
9	4, 8 and 9 from bottom	before fractional expressions	omit the coefficient \(\dfrac{Q}{\delta k}\)
9	4 from bottom	\((5+4)\)	\((5+4k)\)
10	7 from top	\(k^2+6k+5\)	\(k^2+6k+6\)
11	13 from top	\(m+1\)	\(k+1\)
11	14 from top	\(-\dfrac{Q}{k\delta}\)	\(-\dfrac{Q}{k\delta}\,\dfrac{1}{k^2+6k+6}\)
11	14 from top	\(\dfrac{Q}{\delta k}\,k+1\)	\(\dfrac{Q}{k\delta}\,\dfrac{k+1}{k^2+6k+6}\)
11	15 from top	\((f_1)'\)	\((f_2)'\)
11	15 from top	\((f_2)'\)	\((f_1)'\)
15	16 from top	AD	AT
16	7 and 9 from top	\(+\,3(m+2)\)	\(+\,k(m+2)\)
17	3 from top	\(\dfrac{c_2^2-c_1^2}{h}\)	\(\dfrac{c_2^2-c_1^2}{2}\)
17	5 from top	\(-\,f\)	\(-\,\dfrac{f}{r}\)
17	7 from top	\(na+c\)	\(na+c_1\)
18	1 from top	\((m+1^2)\)	\((m+1)^2\)
18	2 and 4 from top	\(-\,k(m+2)\)	\(+\,k(m+2)\)
18	9 from top	(18)	(17)
20	3 from top	(11)	(21)
21	12 from top	\(3d\)	\(3a\)
22	3 from top	\((f_{1,\,n=-1})''_m=-\dfrac{Q}{\delta}\,\dfrac{k}{k+1}\)	\((f_{2,\,n=-1})''_m=\dfrac{Q}{\delta}\,\dfrac{k^2}{k+1}\)
30	14 from top	\(d\omega\)	\(da\)
31	9 from bottom	\(\dfrac{1}{2\alpha}\) and \((-\beta\pm\sqrt{\beta^2-4\alpha\gamma}\)	\(\dfrac{1}{2\alpha}(-\beta\pm\sqrt{\beta^2-4\alpha\gamma})\)
36	4 from top	at the end of the line	omit the sign *)
36	4 from bottom	\(C_1\)	\(c_1\)
37	15 from top	large	lateral
40	5 from top	III	I and III

Literature.

M. N. Berlov, “Details of Hoisting Machines,” 1909.

L. G. Kiefer, “Load-Lifting Machines,” 1910.

L. Z. Ratnovsky, “Hoisting Cranes,” issue 2, 1910.

A. M. Samus, “Course on Hoisting Machines,” 1896.

A. Ernst, “Die Hebezeuge,” 1903.

W. Pickersgill, “Lasthebemaschinen,” 1905.

H. Bethmann, “Die Hebezeuge,” 1908.

A. Bottcher, “Cranes,” 1908.

L. Rousselet, “Les ponts roulants actuells,” 1908.

R. Drozdov, “Report on Experiments with a Reinforced Coupling,” Bulletin of the Society of Technologists, 1905.

A. Pedersen, “A diagram for designing hoisting hooks,” American Machinist, February 15, 1908.

V. Marmor, “Calcul exact d’un crochet,” Revue de Mécanique, 1905.