I. I. Bobarykov.

Professor of the Tomsk Technological Institute of EMPEROR NICHOLAS II.

ON THE QUESTION

OF THE RESISTANCE OF WOOD

TO SHEAR.

Experimental study.

Tomsk.

Siberian Partnership of the Printing Trade, corner of Dvoryanskaya and Yamskoy Lane, own building.

1915.

I. I. Bobarykov.

On the Question of the Resistance of Wood to Shear

Wood is one of the important building materials, and therefore the study of its mechanical properties is of essential importance in construction. Experiments on the resistance of wood to external forces have been carried out since long ago,¹) and at the present time much material can be found for elucidating the question from various sides.

By its nature wood has a very heterogeneous structure, and its resistance to the acting forces depends not only on the kind of deformation, but also on the direction in which it is produced. A whole series of factors sometimes has a very noticeable influence on the numerical values that determine one or another mechanical quality of wood. In addition to the conditions of growth and the time of felling, the position of the specimen being tested in the trunk, the specific gravity, and especially the degree of moisture are of importance.

In accordance with the conditions under which wood is usually subjected to loads in structures, the majority of investigators, in determining the strength of wood, test it in compression, bending, less often in tension and shear, and only in exceptional cases in torsion. It is therefore quite natural that the resistance of wood to compression has been studied best of all for various species, chiefly coniferous ones. It is enough to mention such works as those of Bauschinger,²) Tetmajer,³) Wijkander,⁴) Schwappach,⁵) Jonson,⁶) Janka,⁷) and others.

¹) For example, Chevandier et Wertheim, Sur les propriétés mécaniques du bois, 1846. Barlow, A treatise on the strength of materials, 1867. These works contain some information of a historical character.
²) Untersuchungen über die Elasticität und Festigkeit von Fichten—und Kiefern Bauhölzer, 1883, 1889.
³) Methoden und Resultate der Prüfung der schweiz. Bauhölzer, 1896.
⁴) Untersuchung d. Festigkeits-Eigenschaften schwedischen Holzarten, 1897.
⁵) Untersuchungen über Raumgewicht und Druckfestigkeit d. Holzes wichtiger Waldbäume, 1897.
⁶) Mechanical tests made at Washington University, 1983.
⁷) Untersuchungen über die Elasticität und Festigkeit der österreichischen Bauhölzer, 1900, 1904, 1913.

In structures, wood is very often subjected, in addition to compressive stresses, also to shearing (tangential) stresses; this occurs in bending, in notches, keys, in torsion, in the fastening of disk pins in wooden shafts, etc. The resistance of wood to cutting forces is by no means the same: it is rather considerable if the latter are directed perpendicular to the fibers and, generally speaking, small when the shear is directed parallel to them. This case is of special importance, and therefore most investigators have had it in view in their experiments. Usually, in order to carry out a shear test, a specimen is cut in the form of a prism, which is then subjected to shearing, the plane in which this shearing takes place being directed either radially with respect to the annual rings, or tangentially to them, or obliquely. For example, Tetmayer (see footnote No. 3) used square prisms 4.5–5.5 cm at the base and 10 cm high. The shear planes were directed along 1–1, 2–2, 3–3, 4–4 (Fig. 1); Wijkander took prisms of small height, 20 × 20 cm with a height of 10 cm. Bauschinger (footnote No. 2) divided prisms into four quadrants, from which he obtained eight prisms with triangular and trapezoidal bases (Fig. 2), and the shearing took place along planes I—I and II—II, i.e. in the radial direction. Rudeloff⁸) cut plates 3 × 4 and 4 × 4 cm, according to Fig. 3; thus the shear planes were either radial or tangential. In Julius’s⁹) experiments they ran in the tangential direction. A special arrangement occurred in

Fig. 1.  Fig. 2.  Fig. 3.

⁸) Untersuchungen an Fichtenholz, 1912.
⁹) The physical characteristics of the hardwoods of Western Australia, 1908.

In Jonson’s experiments (footnote No. 6), shearing here occurred along the planes \(ab\), \(cd\) (Fig. 5). Julius also used the same method, in parallel with Warren’s apparatus. The shearing planes had dimensions of \(1 \times 2\) sq. inches.

Fig. 4.

Fig. 5.

Of the Russian investigations, the most thorough test was carried out by Zhuravsky\(^{10}\); the arrangement of the experiment is shown in Fig. 4.

Either the tenons or the middle part along the planes \(mn\) were subjected to shearing (depending on the dimensions). In the first case \(mn = 8''\), with length \(\delta = 0.2''\), \(b = 0.2''\text{–}0.8''\), width \(a = 0.8''\). In the second arrangement \(mn = 1.5''\text{–}2.6''\), \(a = 0.4\).

In all the experiments indicated, the stress \(k_2\) at failure was determined from the well-known relation:

\[ k_2 = P : F, \]

where \(P\) and \(F\) respectively denote the shearing force and the area of failure. Thus, the law of stress distribution is assumed in advance to be known, namely, a uniform distribution is accepted. This assumption is undoubtedly incorrect, since with the aid of the theory of elasticity it can be shown that the distribution of stresses in shearing follows a more complex law\(^{11}\). The same conclusion as to the impossibility of uniform distributions in shearing must be reached on the basis of logical reasoning. This, among other things, is indicated in the article by V. Mramornov\(^{12}\), who

\(^{10}\) Results of the investigation of the Howe system applied to the bridges of the St. Petersburg–Moscow Railway, 1855.

\(^{11}\) Filon. Philosoph. Transactions of Royl. Society, Series A, V.

\(^{12}\) On stresses in the shearing of wood. 1913. Journal of the Society of Siberian Engineers.

makes an attempt to obtain the desired law and, on the basis of his experiments on shearing, comes to the conclusion that the tangential stresses are distributed according to a hyperbolic law.

Zhuravsky, in the work cited above, says that, according to his observations, the resistance to cutting increases in proportion to the increase of the shearing area only until the length \(mn\) (Fig. 4) becomes equal to 9 times the depth of the notch. With a further increase of this length the resistance does not increase.

In Table I some figures from these experiments are given.

TABLE I.

Strong coincides with fine-grained. If we take the products of columns 1 and 4, we obtain

\[ \begin{aligned} \text{rows } 1,\,3,\,5\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ 44,\,50 \text{ and } 51.2 \\ \text{” } 2,\,4,\,6\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ 30,\,36.2 \text{ and } 33.8. \end{aligned} \]

Thus, already in Zhuravsky’s experiments one may see indirect confirmation of the non-uniformity of the distribution of stresses.

Of course, the circumstance indicated does not at all devalue the investigations carried out. They give average figures and, in some cases, relations that will hold under any law of stress distribution. For example, in Fig. 6 (see p. 5) a curve is shown depicting the change in the mean temporary resistance to shear of wood with change in the degree of moisture; the diagram is taken from the above-mentioned investigation by Jonson. On the verticals is plotted the degree of moisture in \(\%/\%\), and on the horizontals the stress in pounds per square inch. In general, here the same picture is observed as in compression tests. The resistance increases with ש

with decreasing moisture content; at the same time the influence of the degree of moisture is the more significant, the smaller it is.

Fig. 6.

In the article “Shearing tests”¹ Izod gives figures relating to the resistance of wood to shearing, this being carried out partly in a direction perpendicular to the fibers, simultaneously in two parallel planes, and partly along the fibers. Assuming, as usual, a uniform distribution of stresses, Izod obtained the figures given in Table II, from which we note an enormous difference in the resistance of wood along and across the fibers.

TABLE II.

From the point of view of calculation in structures, knowledge of the average resistance is insufficient, which makes the choice of a factor of safety entirely arbitrary. In the work by V. Mramornov cited above, the author establishes a law of distribution, taking as a basis the assumption that this law does not depend on the length of the sheared portion. In that case, having obtained two points of the required hyperbola, one can find the greatest stress at failure. Without stopping here to consider this method, we deem it necessary to note that in all tests in shearing the phenomenon of bending always occurs, and therefore the results obtained cannot yield absolute figures; the latter must inevitably include an error, which is not so easy to determine.

Shear phenomena are usually studied in torsion, in which it may be assumed with a sufficient degree of accuracy that, in the plane of the cross-section, only shearing stresses develop. According to the law of duality of tangential stresses, the latter must also appear in radial planes. If it is assumed that the law of the straight line according to which the stresses are distributed, reaching a maximum at the periphery, may be admitted also beyond the limits of elastic phenomena, then from torsion tests one can determine the resistance of wood to shear along the fibers; in this case failure will, evidently, occur at the weakest point. Tests on the resistance of wood to torsion have been carried out, but in no considerable number. Thus Bevan, in his article “Experiment of the modulus of torsion”², gives data for the modulus of elasticity of the second kind \(G\) for several kinds of wood.

TABLE III.

Data on the torsion of wood were obtained by Thurston³. In Table IV are given figures for the temporary resistance to shear for various species.

TABLE IV.

TABLE V.

One may also point to the article by Bouicseau⁴, in which are presented the results of torsion tests and the determination both of the modulus of elasticity \(G\) and of the temporary resistance to shear. Table V. The figures presented do not give a sufficiently clear picture of the influence of various factors, such as, for example, the position of the specimen in the tree, the influence of moisture, etc. Therefore there is felt to be a need to set up torsion tests in order to obtain the necessary data on the resistance of wood to shear.

With the intention of proceeding to determine the mechanical properties of construction timber from Tomsk province, chiefly pine and larch, I considered it necessary to include in these tests also the study of resistance to shear. And therefore I decided to set up preliminary torsion tests in order to clarify the circumstances to which attention must be paid. It was proposed to conduct, in parallel, shear tests in order to establish the relation between the stresses in torsion and in shear. Owing to a shortage of material, the task could not be fully carried out.

For carrying out the tests, in the spring of 1912 two trees were acquired at the warehouse of the Administration of State Properties of Tomsk province. According to information received from the Administration, the pines were felled

in the winter of 1911/12 in the Turgai forest along the Chudym River (the right tributary of the Ob River), in the Yuksinskaya allotment; they were delivered on barges in logs 6 arshins long. Each tree yielded two logs. The trees were cut at a height of about one meter above ground level and had a butt diameter of about 12 vershoks, and at the top cut about 9 vershoks. The forest was entirely healthy, with very few knots. In outward appearance the trees differed only somewhat in the average thickness of the annual rings; the tree with the finer layers was 256 years old, the second, with coarser layers, 259 years old. We shall denote them as tree I and tree II.

The cutting for the extraction of specimens was carried out in the same way: the tree was sawn into five boards approximately 60 mm thick; then it was cut perpendicular to the axis into pieces of different heights, 22 pieces in each tree. The numbering of the pieces ran from the top toward the butt. Pieces I, III, VI, IX, XII, XIV, XVII, XX, and XXII, each 426 mm long, served for torsion specimens; pieces V, VIII, XVI, and XIX, each 120 mm long, for compression specimens, and, finally, pieces II, IV, VII, X, XIII, XV, XVIII, and XXI, each 450 mm long, were used for cleavage, bending, etc. From the pieces, by sawing the boards lengthwise, prismatic bars of square cross-section approximately \(60 \times 60\) mm were obtained; moreover, the marking of the pieces for selecting these bars was done with the definite intention of obtaining the greatest possible number of specimens; Fig. 7 gives a diagram of the distribution of specimens in piece III of tree II. The hatched parts were discarded partly as slab pieces, and in the middle of the tree because the principal crack passed there. Specimens 36 and 42 proved, when mounted on the machine, to have small internal cracks, for which reason they were rejected. Thus, only part of the material was necessarily used, though a fairly considerable part; thus, for tree I, of the possible torsion specimens, 18% were unfit, and for tree II—16%.

Fig. 7.

The torsion specimens had the form and mean dimensions indicated in Drawing 8. The tests were carried out on an Amsler machine, which was checked before and after the investigation. The mean diameter of the specimens

Drawing 8.

was equal to \(d = 50\) mm. On the cylindrical part of the bar, two marks were made at a distance \(l = 150\) mm from one another; mirrors were fastened at these sections, by means of which the elastic angles \(\alpha\) of twist were read, and with change of the twisting moment by 2 kg·m. From these data the value of \(G\) was determined from the formula

\[ G = \frac{32}{\pi}\frac{Ml}{\alpha d^4}. \]

Strict proportionality between stress and deformation was not observed in all bars, as may be seen, for example, from Drawing 9, which shows diagrams obtained for specimens Nos. 43, 119, 142, 169, and 189 of wood No. I (Table A). Whereas for specimen No. 119 this relation is represented, although with very large radii of curvature. As an example we give two records.

Owing to the circumstance indicated, it must be acknowledged that the modulus of elasticity \(G\) varies, decreasing with the load; but since this decrease is insignificant, it is possible to compute \(G\) as an average quantity; the stresses were computed by the formula

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

The elastic limit was not determined; assuming that it lies not far from the proportional limit, the stresses \(k_1\) corresponding to this latter limit were computed. It is self-evident that the value of these figures is approximate, on the basis of what has been said above.

The quantities \(k_1\) and \(k_2\)—the stress at failure—are placed in Tables A and B (see the appendix).

After failure, which was determined by the drop of the load on the machine and the appearance of a longitudinal crack, a cylinder was immediately cut from the middle of the specimen, with a diameter corresponding to the diameter of the specimen and with a height of about 30 mm (on the average). This cylinder was weighed and then subjected to drying at a temperature of about \(98^\circ\) C.\(^{17}\) In this way the degree of moisture \(\varphi\) at the moment of carrying out the experiment was found. In addition, knowing the weight of the dried cylinder, the specific gravity of the specimen was determined. The volume was obtained by direct measurement; this method was preferred to the method of immersion in liquid, because the former is simpler; the degree of error in both methods is approximately the same. The average thickness of the annual rings was obtained by dividing the diameter of the cylinder by the number of rings counted in the given cylinder.

In Tables A and B all these data are presented.

As we see, the moisture content of the wood (regarding the specimens dried in the oven as absolutely dry) varies within the limits of \(6\)—\(13.5\%\); the humidity in the laboratory room varied, depending on the state of ventilation, from 40 to 50% on the average.

With such a comparatively small water content, it may be assumed that the specific gravity varies proportionally to the change in moisture. Let the specific gravity be denoted by \(\Delta\); then we shall have

\[ \frac{\Delta-\Delta_o}{\varphi-\varphi_o}=Const.=A, \]

where \(\Delta_o\) and \(\varphi_o\) refer to absolutely dry wood, but \(\varphi_o=0\). As is seen from column 6 of Tables A and B, the indicated ratio varies very little.

\(^{17}\) In a cabinet with double walls, between which distilled water boiled.

Below are given the mean values for all sections of the trees tested:

On the basis of (4) we may write

\[ \Delta = A \varphi + \Delta_o \]

\[ \Delta_1 = A \varphi_1 + \Delta_o \]

or

\[ \Delta_1 = \Delta - A(\varphi - \varphi_1) \]

Let, for example,

\[ \varphi_1 = 10\%, \]

then

\[ \Delta_1 = \Delta A(\varphi - 10) \]

For tree II, \(A = 0.257\); in such a case we obtain

\[ \Delta_{10} = \Delta + 2.57(1 - .01\varphi). \tag{19} \]

Thus we have the possibility of referring the data obtained to a moisture content of \(10\%\) and then comparing them with one another. Let us see what the error will be if the numerical values of the specific gravity are left without reduction. For tree II we have

\[ \varphi_{max.} = 12.7\% \;(\text{No. }70), \quad \varphi_{min.} = 6.8\% \;(\text{No. }125), \quad \Delta_{min.} = 36.3 \]

and according to (6) we have

\[ \Delta'_{10} = 36.3 + 2.57(1 - 0.1 \cdot 12.7) = 35.6 \]

\[ \Delta''_{10} = 36.3 + 2.57(1 - 0.1 \cdot 6.8) = 37.2 \]

and

\[ \frac{\Delta''_{10} - \Delta'_{10}}{\Delta}\,100 = 4.2\%. \]

Consequently, the greatest deviation does not exceed \(4.2\%\), but since such a simultaneous coincidence of unfavorable circumstances does not

observed, the actual error due to neglecting the influence of moisture will not exceed 2%. Therefore no reduction of the specific gravity to any definite degree of moisture was made.

It may be supposed that a change in moisture within the comparatively small limits of 6–7% should not be strongly reflected in the magnitude of the temporary resistance. Turning to the diagram indicating the dependence between \(k_2\) and \(\varphi\) according to Jonson’у (Fig. 6), we note that the temporary resistance falls sharply with increasing moisture, and this is especially noticeable at small values of \(\varphi\). A similar phenomenon was also observed by Janka in the curves he constructed for the case of compression. In order to form an idea of the actual influence of a change in \(\varphi\) under torsion, I made special observations. Namely, from logs 4, 15, and 21—wood I—and 7, 11, and 18—wood II—small beams were selected. From each small beam two specimens for torsion were prepared in the form indicated in Fig. 8, but of considerably shorter length. This was done with the aim of having paired specimens, situated as close as possible to one another along the length and, consequently, possessing approximately the same specific gravity and moisture.

After fabrication, one small beam was set in the machine, while the other was at the same time placed in a drying cabinet to determine the degree of moisture. After drying was completed, the second small beam, which had been kept in a desiccator, was subjected to torsion. In this way, for each small beam from the log, two figures were obtained—the temporary resistance at a certain degree of moisture and in an absolutely dry state. The results of these tests are collected in Tables \(C\) and \(D\), corresponding to woods I and II. Looking through the figures found, we note, first, that the resistance of wood to shear by no means always decreases with increasing moisture; for the case under consideration the opposite relation may occur. The greater part of the specimens showed a lower stress in shear in the completely dry state⁶. Secondly, the fluctuations in the change of the magnitude, together with drying, are rather considerable and do not reveal any definite law of the phenomenon. The result obtained is, to a certain extent, in contradiction with the accepted views on the influence of moisture. Of course, I cannot draw an exact conclusion on the basis of the available material, but I believe that there is no reason to consider impossible a decrease in resistance at very small degrees of moisture; it is necessary

one must keep in mind that under these conditions the wood becomes little elastic and, to a considerable degree, brittle. The curves shown in drawing 9 were obtained precisely in the last experiments and give diagrams for wet (a) and dry (b) specimens from wood I. Returning to Tables C and D, let us determine the mean values of the ratio

\[ m=\frac{k_2-k'_2}{\varphi}, \]

where \(k_2\) and \(k'_2\) are the temporary resistance for wet and dry specimens. We shall have

Wood I.

Cut 4—\(m=0.31\) kg/cm.\(^2\)

” 15—\(m=0.70\) ”

” 21—\(m=0.93\) ”

Wood II.

Cut 7—\(m=0.94\) kg/cm.\(^2\)

” 11—\(m=1.03\) ”

” 18—\(m=1.40\) ”

It is easy to see that the decrease in resistance per 1% of moisture increases, approximately, from the top toward the butt. Plotting, on a chosen scale, distances along the axis of abscissas, and the values of \(m\) vertically, we obtain two straight lines I and II, i.e. it may be assumed that \(m\) varies inversely proportionally to the distance from the soil level.

In such a case the greatest decrease per 1% of moisture will be 1.6 kg/cm. for cut 22 of wood II.

If we take as the normal moisture content \(\varphi = 10\%\), then specimen No. 252 will have the greatest deviation from this mean value,

Fig. 10.

whose moisture content was \(\varphi = 7\%\). The stress, reduced to the normal moisture content for this specimen, would be

\[ k_{10} = k - m(10 - \varphi) = 87.5 - 3 \cdot 1.6 = 82.7 \ \text{kg}/\text{cm}^2, \]

which corresponds to a deviation of \(5.5\%\).

Not having in view, in the present work, the obtaining of exact dependences, I did not reduce the obtained stresses to a single moisture content of \(10\%\). But it is self-evident that, in a further investigation of the phenomenon of shearing of wood, it will be necessary to keep the indicated circumstance in mind.

As for clarifying the question of the influence of the degree of moisture content on the modulus of elasticity, special experiments in this direction were not carried out, because it was difficult to dry long specimens; moreover, over the length of one meter one cannot speak of constant moisture content and specific gravity. To determine the change in the magnitude \(G\) together with \(\varphi\), it is necessary to arrange the experiments in a somewhat modified form.

If, using the data of Tables A and B, one compiles the mean values for each section, then Tables VI (wood I) and VII (wood II) are obtained.

TABLE VI.

TABLE VII.

For clarity, the changes in these quantities together with the change in the distance of the section from the butt are presented in the form of curves in Fig. 11 (tree I) and Fig. 12 (tree II). Here the general law is confirmed that the specific weight of the tree increases as one approaches the butt, with the increase being greatest in the last section. Here, incidentally, the greatest resin content also appears. Approximately parallel to the curve of specific weight run the curves of temporary resistance and of the modulus of elasticity. This circumstance suggests that between the quantities \(G\), \(k\), and \(\Delta\) there exists a certain constant relation. Indeed, from Tables VI and VII we find.

TABLE No. VIII.

Thus, one may take approximately, for both trees,

\[ k_2 \approx 1.7\,k_1 \quad \text{and} \quad k_2 \approx 190\,\Delta. \]

As was indicated above, the resistance of wood to compression has been studied sufficiently well for many of its varieties. It is therefore of undoubted interest to determine what relation exists between the resistance of pine to shear and to crushing. When cutting up the trees, these tests were provided for, and for them four sections were set aside. From these, bars 12 cm long were sawn out, with a square base \(6 \times 6\ \mathrm{cm}^2\), i.e. the ratio of height to the side of the base was taken equal to 2. This height was chosen deliberately, so as to make it possible to set up a delicate apparatus for finding the modulus of elasticity \(E\) of the first kind. Crushing was carried out in Amsler’s press, of 60-ton capacity. The deformations were read by means of Bach’s sector instrument. The results obtained are collected in Tables E and F, respectively for Trees I and II. The stresses, as well as the modulus of elasticity, increase with approach to the butt.

If the mean data by sections are calculated, Table IX is obtained.

Fig. 11.

Visible graph labels: \(G\), \(K_2\), \(K_1\), \(\Delta\); ordinate markings include \(8000\), \(7000\), \(6000\) with units \(k/c^2\), and \(100\), \(80\), \(60\), \(40\), \(20\) with units \(k/c^2\); additional markings \(0.5\), \(0.4\); abscissa markings \(0, 1, 2, 3, 4, 5, 6, 7\), \(m\).

Fig. 12.

(Graph with curves labeled \(\xi\), \(K_2\), \(\Delta\), and \(\delta\); horizontal axis \(m\). Visible scale labels include \(9000\ \mathrm{K}/\mathrm{c}^2\), \(6000\), \(100\ \mathrm{K}/\mathrm{c}^2\), \(80\), \(60\), \(40\), \(20\), \(0\), as well as \(0.4\), \(0.3\), \(1^\circ\), and \(0^\circ 9\).)

Fig. 13.

TABLE IX.

As was to be expected, the resistance of the wood increases from the top log section toward the butt. The same is also observed with the modulus of elasticity. In drawing 13 the changes in the quantities \(E\) and \(k_2\) are shown with the change in position relative to the butt. The dotted lines refer to wood I, the solid lines to wood II. For clarity, on the same drawing are plotted lines representing the changes in the modulus of elasticity of the second kind \(G\), the temporary resistance to shear \(k_s\), and the thickness \(\delta\) of the end rings. These quantities were taken for the log sections 6, 9, 17, and 20, nearest to 5, 8, 16, and 19. The direction of the lines indicates the relation among the quantities represented. The stress \(k_1\), corresponding to the limit of proportionality, is calculated approximately, in view of the circumstances that were mentioned in connection with the torsion tests. It is not without interest to note that the ratio between the temporary resistance in compression and in torsion is approximately constant and is equal to the ratio between the stresses at the limit of proportionality. This can be seen from the following Table X, which gives a comparison of the indicated ratios. To make comparison possible, the mean data have been taken for specimens coinciding in direction in two neighboring log sections, in compression and torsion.

TABLE X.

i.e., on average it comes out to about seven.

Finally, comparing in the same way the moduli of the first and second kind with one another, we observe that the ratio between \(E\) and \(G\) fluctuates comparatively little, increasing from the cut end toward the butt.

Drawing 14.

TABLE XI.

and on average it may be taken as about 11.

Wishing to make use of the results of the torsion tests in phenomena of shear, it is necessary, for comparison, to have figures for resistance to shear obtained from a direct test upon it. It was indicated above that it is impossible to put such observations in pure form, since shear is inevitably accompanied by bending. Moreover, owing to the uncertainty of the law of distribution of stresses, one can obtain only a mean value for the quantity \(k_2\). It was therefore decided nevertheless to make several tests in order to obtain such figures. First the specimen was given the form shown in Drawing 14. A bar of square section (approximately \(6 \times 6\)) had thinner parts \(A\) and \(B\), which served for gripping in the machine. Perpendicular to the fibers, on opposite sides, saw-cuts \(aa\) were made, with the calculation that, under the action of the forces \(P, P\), sliding would occur along the plane \(AB\). The length of splitting was 100 m/m. At the points \(c, c, c, c\), at a distance of 2 cm from one another, mirrors were installed. The latter were fastened in a support having the form indicat-

to that in Fig. 15, and, in order to reduce its weight, made of aluminum; the collar \(b\) could rotate on a pin forming one whole with the base \(a\); into the latter were set needles \(d, d\), representing feet; by sticking the latter into the wood, they fixed the position of the mirror. The distance between the feet was 3 mm. The mirrors were set so that the shearing line passed between the needles. By means of tubes with scales, placed at a distance of 2 meters from the mirrors, readings were observed which were the result of the combined action of shear and skewing (the work was carried out on a 2-ton Amsler machine). The loading began at 50 kg and, increasing at each observation by 100 kg, after the reading was reduced to 50 kg. The greatest load was \(P = 550\) kg. From the results obtained, the arithmetic means were taken, i.e., the readings at 100 kg. In order to eliminate the influence of skewing, after the end of the test in the form described, at a load of 50 kg, the mirrors were removed and again set at the same points \(c\text{—}c\), but with the base turned through \(90^\circ\), i.e., in the new position the needles were arranged along the line \(AB\) between the saw cuts (Fig. 14). Then loading and unloading proceeded in the same way as before; average figures were computed, which were, of course, smaller than the first ones. The differences obtained by subtracting the corresponding figures of the second series from the figures of the first were accepted as quantities proportional to the shear deformations, since the phenomenon took place within the limits of elasticity. In order to be certain that an analogous phenomenon took place on the other side of the bar, a mirror was set at point e-5, corresponding to c-1. The conduct of the tests required great effort because of the painstaking and difficult installation of four mirrors and tubes in such a small space; all observations were repeated. The results of tests on 6 specimens showed that the largest readings are given by the end mirrors, as is seen from the following Table XII.

Fig. 15.

TABLE XII.

If these quantities are laid off, on the chosen scale, assigning them to points 1–9, we obtain the curves shown in Fig. 16. Since these curves do not make it possible to derive a definite law of stress distribution, further experiments in this direction were discontinued, in order to single them out as an independent problem. It is thought, nevertheless, that the result obtained may be regarded as sufficient to establish the presence of a maximum of stresses at \(a\) and \(b\) and a minimum in the middle.

Fig. 16.

The experiments just described confirmed the presence of skewing of the beams, and to a fairly considerable degree.

Wishing to weaken the influence of this circumstance on the resistance in shear, the following observations were made on beams of a somewhat different shape—namely, prepared according to the type shown in Fig. 17. Here splitting must occur simultaneously along the planes \(bc\)—\(bc\), over a length likewise of 100 mm. Comparing the forms of Figs. 14 and 17 with one another, it is easy to see that, all other conditions being equal, the bending moment in the latter case will be approximately 60% smaller and, of course, cannot be eliminated altogether. The number of specimens for splitting could not be made large, since here it was necessary to locate the plane of splitting in a perfectly definite manner, so as to guarantee the phenomenon in a more or less proper form.

Fig. 17.

A whole series of beams had to be rejected, because during failure extraneous phenomena appeared, such as, for example, the tearing of some—

rupture of fibers, shear in an oblique direction, waviness of fibers, etc. All these circumstances artificially increased the resistance of the wood.

TABLE XIII.

Table XIII gives the results obtained for 58 specimens of wood I, taken, as is seen, from different cuts, i.e. at different heights. For each specimen in tension (cleavage) a specimen from an adjacent cut for torsion, lying in the same direction, was selected. It may be observed that the ratio $\gamma$ between the temporary resistances for torsion and for tearing out (the mean value), although not constant, fluctuates only slightly. This circumstance confirms the correctness of the assumption that the strength of wood with respect to shearing can be determined from torsion tests. Unfortunately, the figure for the mean temporary resistance obtained from cleavage tests cannot characterize the magnitude of the resistance, since it depends on the length of the cleaved portion (up to a known limit). Therefore the ratio found in Table XIII is also correct for a cleavage length of 100 m/m, but

taking into account that for each tree one can find that limiting length of shearing beyond which the magnitude of the mean resistance already remains unchanged, it is evidently possible to obtain such a value of \(\eta\) as will characterize the given kind of timber.

The theory of plane bending indicates that, under this kind of deformation, tangential stresses develop in the material of the bar. If one selects a form of cross-section such that failure occurs as a result of the development of large shearing forces, then it would be possible to calculate the magnitude of the temporary resistance by yet another method. Wishing to make use of this circumstance, it was decided to prepare specimens for bending tests from bars 2 and 21 of tree II. For this purpose three specimens were cut from each bar; the cross-section had the form of an I-beam (Fig. 18), with a small web height and with thickened flanges; the dimensions \(a = 14—15\) mm, \(h = 25\) mm, \(b \approx 5\) mm, \(c = 6—7\) mm were chosen so that failure would occur as a result of shearing along the neutral layer.

Fig. 18.

The bending of the specimen was carried out in such a way that it was placed on two supports with a span of 140 mm. The force \(P\) was applied at the middle of the bar; the load was gradually increased until failure. The stresses were calculated according to the known relation

\[ k = \frac{3P\left(ah^2 - b^2(a-c)\right)} {4\left(ah^3 - (a-c)b^3\right)c}. \]

The results of these determinations are given in Table \(G\). Each stress figure is the mean of three parallel tests on material from one and the same bar. Sections 2 and 21 are situated between sections 1 and 3, and 20 and 22, the bars from which served as material for the torsion tests. If one takes the mean values of the stresses in the specimens of sections 1, 3, 20, and 22 corresponding to the specimens of Table \(G\), and compares them with the mean values of the latter, the following table is obtained.

TABLE XIV.

For cuts located at opposite ends of the tree, we obtain one and the same ratio between the stresses found from bending tests and from torsion tests. The former turns out to be 20% higher than the latter stress. A difference of this kind is very simply explained by the presence of friction, which must be overcome in order to destroy the beam by bending. For compression we do not know the coefficient of friction and cannot introduce the corresponding correction. An approximate consideration permits us to think that the friction force will be approximately about 20% of the magnitude \(k\).

If the law of stress distribution is taken to be a hyperbola, then from the shearing experiment one can derive the value of the greatest temporary resistance. At my request V. Mramornov carried out several experiments on small boards taken from different places in tree II. The shearing surface was marked out as in the beam shown in Fig. 14, but of small thickness. The line \(AB\) had to run along one and the same layer.

Each figure is the average of 2–3 observations. The number of experiments is not large, since there was no longer enough material, and chiefly because it is very difficult to select strictly straight-grained beams, as a result of which many of them were rejected. In order that the figures obtained in this way could be compared with the data found from torsion tests, corresponding pieces for torsion were sought for each beam; at the same time the condition was imposed that, as far as possible, the beams being compared should lie in one direction along the length of the tree. Table XV gives the results of the indicated experiments.

TABLE XV.

It is self-evident that one can no longer expect complete agreement, because wood is far too inhomogeneous a material. But the difference established from Table XV is not great, and in any case gives grounds to suppose that experiments in this direction are desirable.^21)

There may also arise the question of the change in the magnitude of this temporary resistance in the cross section with change in distance from the center. In this respect it should be noted that it is impossible to establish an entirely exact regularity. In Fig. 19 diagrams are given showing the course of the values \(k_2\) (solid lines), specific gravity \(\Delta\) (small dash), and \(G\) (large dash) for sections 1, 3, 6, 9, 12, 14, and 17 of tree I, and 1, 3, 6, 12, and 17 of tree II. The direction is taken from the center of the section toward the south. In Fig. 20 analogous diagrams are given for sections 1, 12, and 22 of tree I, and 3, 12, and 17 of tree II; the direction is from the center toward the east.

In general the following peculiarity is observed: the temporary resistance does not follow the change in specific gravity, but rather the reverse; meanwhile, when examining the changes along the length of the trunk, we noted the circumstance that the temporary resistance increases together with the increase in specific gravity (Fig. 11). Further, the smallest value of the temporary resistance along the radius is obtained between the heartwood and the sapwood. Approximately the same is observed also in the diagrams of Fig. 20.

What has been set forth above may lead to conclusions of the following kind. The torsion test gives the magnitude of the least resistance to shear; the temporary resistance \(k_2\) changes with position in the trunk; between the temporary resistance to torsion and to compression there may exist a simple dependence, just as between the moduli \(E\) and \(G\) for one and the same tree. In experiments for determining the mechanical properties of wood it is desirable to conduct parallel torsion experiments, and in this connection it is necessary to clarify the influence of the degree of moisture on the various factors that determine these mechanical properties.

I consider it my duty to express my gratitude to my assistant in the mechanical laboratory of the Tomsk Technological Institute, mechanical engineer A. P. Malyshev, who gave me great assistance in carrying out the experiments.

April 1914
Tomsk.

^21) Such a study is also being planned in the Mechanical Laboratory of the Tomsk Technological Institute in the near future for oak, larch, and birch.

Wood I

• Experiment No. 1
• Experiment No. 5
• Experiment No. 6

• Experiment No. 9

• Experiment No. 8

• Experiment No. 14
• Experiment No. 7

• Experiment No. 1

• Experiment No. 3

• Experiment No. 9
• Experiment No. 12
• Experiment No. 17

Axis markings visible on the diagrams include: \(0.2,\ 0.4,\ 0.6\); \(20,\ 40,\ 100,\ 160\); \(5000,\ 6000,\ 7000,\ 8000,\ 9000\); and \(kg/cm^2\).

Wood II.

Fig. 11.

Table A.

Table B.

Table C,

Table D.

Table E.

Table F.