PROCEEDINGS
of the Tomsk Technological Institute
of Emperor Nicholas II.

Vol. 8. 1907. No. 5.

A. M. Krylov.

On the Strength Coefficient of Wood under Compression and Bending.

1–35.

On the Strength Coefficient of Wood under Compression and Bending.

Toward the end of my stay abroad (the winter and part of the summer semester of 1903) I came to Stuttgart. While continuing there, at factories located in the vicinity of the city, my practical acquaintance with mechanical engineering in general and with steam boilers in particular, I had, thanks to the kind assistance of Prof. Bach, the opportunity to devote part of my time to purely laboratory work. The material chosen for this work was wood; the aim was to investigate the influence of the height of the specimen on compressive strength and, if possible, to give a theoretically grounded explanation of the splintering or shearing of the fibers of the material that is observed in this process, almost always at one and the same angle of \(45^\circ\). For this purpose I carried out about 280 tests on specimens of oak, beech, spruce, fir, and pine, having at the base a square with a side of 7 cm and various heights from 7 cm to 35 cm. In becoming acquainted with the literature of the question, I unexpectedly came upon two new questions which could serve as the subject of a number of independent investigations. For one of them I collected a small amount of material, which required further treatment and, chiefly, experimental verification—this was the question of the ratio of the strength in compression and in bending. On the other question I had the opportunity there to carry out separately a small series of experiments, which yielded quite a satisfactory result. This is the question of a complete characterization of the physical and mechanical properties of wood by its specific weight in an absolutely dry state. The immediate occasion for thinking about this was acquaintance with a small brochure by Prof. R. Hartig, Holzuntersuchungen, Altes und Neues, in which his observations are collected concerning the influence of various factors on the growth and increment of trees of different species. Considering wood as a living organism, I came to the conclusion that in its specific weight in an absolutely dry state, as in a mirror, there should be reflected all the diverse conditions under which it has to grow. In this case the properties of the wood should reflect the relative situa-

its orientation to the sun. I ordered specimens cut from one and the same disk of a trunk of spruce, fir, and pine, crosswise, as indicated in Fig. 1. When they were tested, the results shown in Table I were obtained.

Table.

These data pertain to specimens of spruce, which gave the most evident confirmation of the indicated dependence. The difference between the greatest specific gravity (0,54 for the specimen marked 2) and the smallest (0,46 with mark 3) amounts to 15%; the difference in the coefficients of strength of the same specimens likewise gives 15%. As for the specimens of pine and fir, they also indicated different specific gravities and coefficients of strength (for pine 9, 25%, for fir about 9%) depending on the position in the trunk of the tree, but I did not observe such regularity in their distribution over the section as in spruce. Assuming upon my arrival in Tomsk to devote part of my time to the investigation of the two questions noted, I discontinued further experiments in this

direction*). Unfortunately, being engaged in responsible work in an entirely different branch of the applied sciences, I cannot for the time being bring my intention to completion, and in the present article I venture to give a systematic account of what I had prepared as early as 1903, with only a very small addition of material from the new investigations of G. Janka.

In studying the mechanical properties of wood and determining its strength under the action of one or another load, the principal attention must be directed to determining its strength in compression, bending, and shearing; precisely these three kinds of loading are most often encountered in construction practice for the wooden parts of structures. A torsional load is encountered comparatively rarely (in the shafts of large water wheels), and tensile loading still more rarely. But apart from these practical considerations, testing wood in tension has to be abandoned for the following reasons as well. Since the tensile strength of wood is very great, while it resists shearing very weakly, the conduct of such an experiment is associated with considerable difficulties. In order to prevent premature shearing of the material in the head for gripping, the specimen must have considerable dimensions, and this makes it difficult to mount it properly in the machine; sometimes, in order to increase the force of friction of the fibers against one another and their resistance to shearing, special glued-on heads are used, but this does little to simplify matters. The specimen itself must also be selected with particular care from the trunk or log and must be well prepared; the direction of the fibers must be exactly parallel to the line of action of the forces; on the length of the specimen over which measurements of deformation are to be made there must be no interrupted fibers, otherwise what will occur is not a pure tensile stress, but one combined with shearing forces. The only respect in which tensile tests are interesting and important is the determination of the modulus of elasticity and of the elastic limit of the material; but, unfortunately, carrying out such an experiment and measuring it through to the end with reliable results is almost impossible. Chevandier and Wertheim**) already observed that in well-dried wood the elastic limit is very close to, or almost coincides with, the breaking stress; the same was later observed by Bauschinger in most cases, especially in testing coniferous species.

* I do not give here data relating to the question of the influence of the height of the specimen on compressive strength, since I do not consider them sufficient for deciding this question. I shall only point out that oak alone gives a fairly uniform decrease in strength, on the average by \(11\%\), when the height of the specimen is increased each time by 7 cm.

** In strength, if it is taken relative to specific weight, wood is in no way inferior to iron; for example, for pine, with a safety factor of 10, the safe tensile stress is given as \(1.2\ \mathrm{kg}/\mathrm{mm}^2\); since its specific weight is about \(0.6\), then \(\frac{1.2}{0.6}=2\); for iron this ratio, with a much smaller safety factor, is equal to or somewhat greater than 1.

In view of this, it is impossible to make reliable observations up to the elastic limit; moreover, not long before rupture—which in wood is not accompanied by the “flow” period of the material characteristic of iron, but appears suddenly, when the load has reached the greatest possible magnitude—splitting of the specimen begins; the rupturing fibers disturb the proper position of the measuring instruments, and the reading of their indications made under such conditions leads to inaccurate conclusions. The general character of the phenomena described was the same in the tests of Nördlinger, Tetmayer, and others; in view of this, as the elastic limit of wood in tension it will be quite permissible to take its stress at rupture. Consequently, determinations of the modulus of elasticity are also inaccurate, since, according to Bauschinger’s practice, it is customary to determine it by observing deformations up to some arbitrarily chosen limit (i.e., up to a load equal to \(1/3\) of the breaking load). The international commission that met in Munich in 1884 to develop uniform methods for testing materials concluded that, for characterizing the mechanical properties of wood, testing it in compression and bending is quite sufficient.

In addition to the reasons indicated above, the basis for such a decision should also be considered to include the circumstance that a tensile test can be performed with a specimen of small cross-section; and since the qualities of wood vary considerably depending on the position of the specimen in the trunk, the tensile strength is subject to considerably greater fluctuations than its strength in compression or bending. As a measure of the working capacity of wood, of its mechanical properties, the commission recognized the work of bending of a beam, calculated from the bending diagram up to the fracture of the specimen. In his later investigations on the elasticity and strength of fir and pine, Bauschinger came to the conclusion that this work, adopted as a measure of the mechanical qualities of wood, characterizes them less than the bending strength itself; finally, still later, the same investigator came to the conclusion that only compressive strength is the actual and true indicator in judging the strength of wood as a building material. In contrast to what has been said regarding the tensile test, a compression test parallel to the fibers is simpler, more reliable, and easier to perform. As has been found by parallel experiments with specimens of different preparation, the inaccuracy in the preparation of the compressed surfaces has a very slight effect on the final result of the tests; from the data of Johnson’s experiments it may be concluded that, for the final result, it remains immaterial whether these surfaces are planed or left in the condition in which they are obtained with a good saw cut. Wijkander’s experiments showed that, so long as the height of the specimen does not exceed \(3/2\) times the side of the base,

remain without influence, and the knots present in the specimen; the latter, however, must not be very large, must be well connected with the surrounding fibers, and distributed more or less uniformly over the specimen. As in the first case, here too it is important to measure the deformations of the specimen; specimens intended for observing deformations, contrary to what has been said, must be carefully worked, especially on the clamping surfaces. But even in this case the determination of the modulus of elasticity and of the limit of compression is difficult, since the height of the specimen cannot be large; otherwise it will give a different coefficient of strength than that obtained on an ordinary cubical specimen, or on a prism with a height of \(1\frac{1}{2}\)--\(2\) times the side of the base. A wood specimen tested in compression does not give such a sharp picture of the destruction of the material as in tension or bending. When the load reaches its greatest value, the specimen, so to speak, “gives way”; in a test on Amsler–Laffon’s press, in which the magnitude of the load is given by a mercury column, the moment when the resistance of the specimen has reached its maximum is marked by a slight slowing in the rise of the mercury column, a brief stopping of it at one level, and the beginning of its subsequent fall. At this moment the specimen still has no signs of destruction on its surfaces; if the experiment is stopped at this moment and then repeated again, the mercury manometer again shows almost the same load at the limit of destructive stress; as compression continues, the appearance of fracture lines on the sides of the specimen becomes noticeable; these are formed by fibers displaced over one another. The manometer at this point shows a smaller load; lowered to a certain height, the mercury column remains again almost at the same level; this indicates that the already strongly deformed specimen, with fibers already partly destroyed, can still withstand a fairly large load, approximately 18--22 percent less than the compressive strength. Continuing the experiment, one observes that after the level of the mercury column has stood for quite a long time at an unchanged height, it again begins to rise, i.e. the resistance of the specimen increases. The compression of the wood in this process increases. Observations of the deformations give the following result: proportionality between stresses and shortenings of the specimen is maintained approximately up to 50% of the load corresponding to the strength of the material; the deformations remaining after larger loads are very small in comparison with the elastic, disappearing ones; the limit of the latter must be considered very close to, or coincident with, the coefficient of strength of the material. This will correspond to what was said above concerning the phenomena as they are observed in compression, namely, upon reaching the greatest height, the level of the mercury column at first rises very slowly for a short time and then remains for some time at a constant height; at this moment the deformations of the bar increase considerably faster than the stresses—

phenomena, the elastic limit has been exceeded; the period of insignificant rise of the column must correspond to that state of stress when the deformations, while still remaining elastic to a significant degree, nevertheless diminish in comparison with the strongly increasing permanent changes in the height of the prism. Unfortunately, as in tension, here also an exact and true determination of the elastic limit and of the modulus is associated with great difficulties. The data of Bauschinger’s and Tetmayer’s experiments give grounds for supposing that the modulus of elasticity has one and the same value both in tension and in compression; if this is not true for all species, then, at least for the most commonly used ones, such an assumption will be very close to reality*).

It was already indicated above that, after a whole series of investigations on the question of the resistance of wood to forces acting upon it, Bauschinger came to the conclusion that its compression strength may serve as the best and very reliable indicator of its quality. This is partly understandable: in all species the tensile strength of wood exceeds that in compression by a round factor of 2; the strength of wood in bending is also, for all species, greater than the compressive strength, by approximately \(1 \frac{1}{2}\) times. It is evident that, for one and the same species, under identical testing conditions, the qualities of the material must be better in that batch of timber in which the compressive strength is greater. The strengths in compression, tension, and bending appear to be in some kind of dependence on one another**). This is also indicated by the numerical values, given in Table II, of the strength coefficients for various species.

Table II.

*) For the purposes of the present article this is not of great importance; instead of the stress distribution in the cross-section of a bent beam assumed below, it would be necessary to consider it as following the law \(E=\alpha\sigma^m\), recommended by Bach for materials with different moduli of elasticity.

**) In 1860 Nördlinger indicated that the strengths in tension and bending are in a regular dependence: with an increase in the first, the second also increases. Below we shall see that tensile strength plays a guiding role in determining bending strength. But since all tests of wood have, since the eighties, been conducted chiefly to determine compressive strength, we shall also consider the question of the relation between precisely it and bending strength.

Table II contains the average values of the strength in compression and bending, as they were obtained by Tetmayer in the investigation of the mechanical properties of construction timber in Switzerland. Arranging the species in the order of increasing compressive strength, we see that they also proceed in the same order with respect to bending strength. We see that a deviation is found only for beech, which, with a lower compressive strength than oak, gives a greater bending strength; but under constant strength in tension the latter plays almost always the decisive role; and thus there is no exception in this case, since the tensile strength of beech is considerably higher than that of oak (1340 kg/cm² versus 964 kg/cm²). But Tetmayer’s data do not constitute an exception; in Table III, in the same

Table III.

Table IV.

In this order are given the mean values obtained in testing Swedish structural species (Wijkander), and in Table IV—the same data for fir from various localities in Austria (G. Janka)*. The only exception in the series in Table III is elm (Ulme). Finally, in Table V are given the coefficients of strength in compression and bending according to Johnson’s experiments.

Table V.

With regard to the latter, we note that the moisture content of the specimens varied within considerable limits; for example, in White Pine from 87.2% to 10.2%, and, notwithstanding this, the same dependence was clearly preserved. The moisture content of wood, as is known, is one of the factors most strongly affecting its strength. With a decrease in it, the strength of wood in compression increases; one of the real causes of this must be the well-known property of wood to swell and shrink. A specimen tested with a large percentage content of moisture must, in essence, sustain the same load as in the case of a lower degree of moisture; but since all its dimensions are thereby increased, then, referring it to this changed area, we obtain a smaller strength.

Table VI.

*) “The greater the strength in compression given by the trunk, the greater was also its strength in bending, although the relation between their coefficients was not constant.”

As the strength in compression increases with a decrease in its moisture content, the strength in bending also increases; Table VI shows the results of Bauschinger’s experiments with pine. But the dependence between the strengths in compression and in bending can be illustrated not only by the numerical material of their coefficients of strength, but also by the almost complete coincidence of the moduli of elasticity under the two different methods of loading. In Fig. 2 lines are plotted showing the changes in the modulus of elasticity under compression and bending for individual trunks tested by G. Janka. The figures of the horizontal row give the numbers of the trunks, and on the verticals the corresponding moduli are laid off. We see that both lines for the most part run parallel to one another; if we remember that the modulus of elasticity in bending is always obtained as somewhat smaller, then the proximity of one line to the other, or their coincidence, will be very probable.

Everything stated above leads involuntarily to the conclusion that there is some dependence between the strength in compression and in bending. It is clear that if it were possible to clarify it more or less precisely and determine it, the task of testing the mechanical properties of wood would be considerably simplified; moreover, this would make it possible for the practitioner to be guided in his calculations by more reliable and dependable data than at present, since the actual testing of a batch of timber supplied would become considerably cheaper and would undoubtedly pay for itself by the technical-economic calculation that would take the place of the now widely practiced empirical data.

Turning now to the consideration of the phenomena in bending and to the clarification of the indicated dependence, it should be pointed out that in the present case we are dealing with a phenomenon analogous to that observed in cast iron. The strength of the latter in bending exceeds that in tension by as much as 2 times; this contradiction to the data of the theory of bending, as is known, led to the staging of a number of interesting experiments by Bach concerning the clarification of the dependence of the first strength on the second. But there is an essential difference between these two materials; whereas in cast iron the modulus of elasticity is a variable quantity, changing as a function of the stresses that are produced in one or another fiber, and different under tensile and compressive stresses, in wood their difference is so small that the modulus \(E\) may be regarded as exactly the same for both types of loading; moreover, in cast iron there is no limit of proportionality between stresses and deformations, while in wood it is clearly expressed and extends to very large loads. Whereas in cast iron the indicated dependence between tension and bending can be quite satisfactorily explained by establishing the law \(E = \alpha\sigma^m = \dfrac{\lambda}{l}\), for wood it is possible to explain it using only Hooke’s law on the proportionality of deformations to stresses.

The theory of bending gives formulas on the basis of considering the deformations and stresses in a beam supported at both ends and loaded at the middle of its length by a vertically directed force, the stresses of the fibers being assumed to lie within the limits of elasticity of the material. In this, the following four assumptions are made: 1) the system of external forces acts in one plane passing through one of the principal axes of the bar, and gives, for each cross-section, only one couple of forces; 2) plane sections perpendicular to the axis of the bar remain such even after bending; 3) the modulus of elasticity for all fibers, independently of the stress in them, is the same and equal, both in compression and in tension; 4) the fibers exert no influence upon one another and are as if independent of one another.

Experience shows that, under the influence of forces, a bar bends: one side of it becomes convex, the other concave; the extreme fibers of the first prove to be the longest, and likewise those of the second the shortest.

The former are elongated and therefore must be regarded as stressed by tensile forces; the latter are under the action of compressive forces; since we are dealing with an entire bar, it must be inferred, as a consequence of what has been said, that there are fibers whose stress is zero and whose length does not change. The axis of the bar coinciding with such fibers is called the neutral axis. The theory of bending, on the basis of the assumptions made, indicates to us, from purely geometrical relations, that the stress of the fibers is proportional to their distance from the neutral axis and that the latter passes through the center of gravity of the section. So long, therefore, as the stresses remain within the limits of elasticity, the mathematical conclusions agree fairly well with the phenomena of reality; but as soon as the latter limit is passed, the result is a discrepancy between the one and the other. The greatest contradictions are found in those materials which either have no limit of proportionality at all, or whose moduli of elasticity in compression and tension are not equal to one another. Such a result is a consequence of the assumptions indicated above; in reality, besides the moment of the couple of forces, there is always present a shearing force and the shear stresses due to it; there is always a mutual influence of fibers upon one another, increasing their strength both by the force of friction and by cohesion between them, and thereby forming an obstacle to the free deformations that they exert upon one another. Only one thing may be considered permissible even beyond the limit of elasticity, namely, that the originally plane sections of the bar remain such up to fracture and are perpendicular to the neutral axis of the bar. In conformity with this state of things, laboratory practice has adopted as a rule the determination of the strength of a material in bending by a direct bending test, and especially for those materials whose—

in bending for which it differs most strongly from that determined by direct experiment for the strength in tension or compression. But this does not exclude the possibility of simplifying the question of testing materials, if it were possible, for one material or another, to find a simple and uncomplicated dependence between the bending strength and the compressive or tensile strength.

Leaving aside the influence of the shearing force, the theory of bending, in solving the questions subject to it, makes use of two conditions of equilibrium: 1) the sum of the compressive stresses must be equal to the sum of the tensile stresses in each cross-section of the beam; 2) the sum of the moments of the internal forces of compression and tension with respect to any point of the section must be equal to the sum of the moments of the external forces with respect to the same point. The stress of an arbitrarily chosen fibre situated at a distance $\eta$ from the neutral axis may be expressed as follows:

\[ \sigma = \pm E \frac{\eta}{\rho}, \]

where $E$ is the modulus of elasticity and $\rho$ is the radius of curvature of the bent neutral layer; the sign $+$ or $-$ depends on whether the fibre in question is on the side of the stretched or the compressed fibres. The first condition of equilibrium leads to the conclusion that the neutral axis passes through the centre of gravity of the beam section. The second condition relates the moment of the external forces and the internal stresses in the form of the dependence

\[ M = \frac{\sigma J}{\rho}, \]

where $J$ is the moment of inertia of the section with respect to the neutral axis. So long as the internal stresses, up to the outermost fibre, remain within the limits of elasticity, the distribution of stresses, following the law of linear variation, may be represented as in Fig. 3. We assume that the modulus of elasticity in tension and in compression is one and the same; fibres equidistant from the neutral axis will be equally stressed in compression and in tension. Let us suppose that the indicated distribution of internal forces over the section corresponds to such a load that any further increase of it already entails the appearance of residual deflections, i.e. the beam has already been brought to the elastic limit. We said above that the elastic limit both in compression and in tension lies near the strength coefficient, and indicated that the compressive strength of wood is considerably less than its tensile strength. It is natural to suppose that the elastic limit in bending will be reached precisely at the moment when the stress of the outermost compressed fibre is equal to the stress of simple compression at failure. Fig. 2 showed us the variation of the moduli of elasticity in compression and

from bending for different trunks, and we saw that both lines run, for the most part, parallel to one another, but do not coincide. But the latter circumstance can easily be explained by the fact that, in calculating the modulus of elasticity, the following circumstances were not taken into account.

First, the action of shearing forces was not taken into account. The deflection of a beam resting with its two ends on supports and loaded at the middle of its length is given by the formula

\[ f_1=\frac{Pl^3}{48EI}, \]

where \(P\) is the load and \(l\) is the span between the supports. The shearing forces give an additional deflection, the magnitude of which can be expressed by the formula

\[ f_2=0.3\frac{Pl}{E_1F}, \]

where \(E_1\) is the modulus of elasticity under shearing stress and \(F\) is the area of the cross-section of the beam; the observed deflection \(f\) is the sum of these two, independent of one another, and was equal to

\[ f=\frac{Pl^3}{48EI}+0.3\frac{Pl}{E_1F}. \]

It is clear that if \(E\) were calculated by the formula

\[ E=\frac{Pl^3}{48I\left(f-\dfrac{0.3Pl}{E_1F}\right)}, \]

then larger values would be obtained for it. It is known that, with incorrectly chosen ratios of the height of the beam to its length, the error in calculating the modulus may reach 30%, if the deflection due to shearing forces is disregarded. In G. Janka’s experiments the length \(l\) was equal to \(150\ \mathrm{cm}\), and the beam had the form of a square in cross-section with a side of about \(10\ \mathrm{cm}\). From the ratio \(\dfrac{l}{h}=15\) it is seen that the beam’s span was sufficient so that fracture would not occur precisely from these forces; but, taking into account the slight resistance of wood to shearing forces, and therefore the small value of the modulus \(E_1\), the dimensions of the beam were such that it was necessary to take into account, in calculating \(E\), also the deflection \(f_2\).

Judging from the data of Chevandier and Wertheim, \(E_1=\dfrac{1}{5}E\), which can easily be explained by the layered structure of wood. Substituting this value of \(E_1\) into the preceding formula, we obtain

\[ f=\frac{Pl}{EF}\left[4\frac{l^2}{h^2}+1.5\right]. \]

The ratio \(f_2 : f_1\) is:

\[ \frac{f_2}{f_1} = \frac{0.3Pl.4Ebh^3}{Pl^3E_1bh} = 6\left(\frac{h}{l}\right)^2. \]

Beam 10b had, for example, a height of 10.885 cm; the deflections differed from one another as \(0.03156 : 1\); the error in calculating the modulus \(E\) was, consequently, about 3%.

Secondly, the bending moment, owing to the friction of the beam and the support, which is significant at large loads (at the elastic limit it amounted to 50% of the breaking load (1400 kgr.)), is in reality less than that introduced into the calculation.

The magnitude of the corrected moment will be:

\[ M=\frac{Pl}{4}\left(1-2\mu\frac{h}{2l}\right) = \frac{Pl}{4}\left(1-\mu\frac{h}{l}\right) \]

\(\mu\)—the coefficient of friction. The friction of oak on cast iron is already significant in itself, but here, owing to the position of the beam, along the line of support, depending on the slight hardness of pine, an indentation must have formed, which further intensified the influence of this circumstance. Taking \(\mu\), as for dry oak on cast iron, equal to 0.5, we find that for the same beam the value \(\mu \frac{h}{l}\) reaches 0.0363, i.e. the error will be about another 3.6%. In general, even with such an inaccurate calculation, owing to the absence of data on the coefficient of friction of pine, we see that the modulus \(E\) was calculated as 6.78% less than the true one. The modulus of elasticity for specimen 10 was determined as:

\[ E \text{ in compression} = 111 \text{ tons per } \square \text{ cm.}, \]

\[ E \text{ in bending} = 102.4 \text{ tons per } \square \text{ cm.}, \]

i.e. \(E\) in bending was determined to be

\[ \frac{111-102.4}{111}\cdot 100 = 7.8\% \]

less than in compression. Introducing the above correction, we obtain for \(E\) in bending the value

\[ E=1.0678\cdot 102.4=109.34\ \mathrm{tn/cm^2}, \]

and, taking into account what has been said concerning the coefficient \(\mu\), we may suppose that the two moduli coincide. This result is of importance for us in that the diagram of the stress distribution in the section has been constructed by us correctly.

It may be regarded as an established rule that the stress of a bent beam should not exceed the elastic limit of the same material under simple tension or compression. Experiment, indeed, shows that for many materials the beginning of noticeable residual deflections coincides with the moment when the outermost fibers receive stresses up to the elastic limit. Let us suppose that there is

material, exactly and strictly elastic up to a certain limit, and that its elastic limit in compression and in tension is one and the same; let us further assume that beyond the elastic limit the material is so plastic that even the slightest increase in load already entails the appearance of significant residual deformations. Good soft iron will in part resemble such a material. We know that beyond the elastic limit, both in compression and in tension, the deformations of iron increase considerably more rapidly, and moreover so rapidly at first that on the diagram expressing the dependence between them we obtain an almost horizontal line (Fig. 4). We load a bar of such a material in bending; let us trace approximately what must occur with the fibers of such a material, and what the distribution of stresses over the section must be. The elastic limit; the outermost stretched and compressed fibers have been deformed by the full amount of the elastic deformations possible for them. The load has increased somewhat. The outermost fibers must now give a more considerable elongation or shortening than was the case before; their elastic resistance has been exceeded. In the case of simple tension, when all the fibers simultaneously receive such an elongation in the direction of the acting force, after a certain time equilibrium is established between it and the internal stresses; the same occurs in compression. In bending, however, owing to the influence of the neighboring fibers, the free assumption by the outermost fibers of the deformation corresponding to the stress is impossible; their deformation, as parts of the bent bar, can be only that which is determined by the geometrical relations, i.e., must correspond to a small increase in connection with the decrease of the radius of curvature, according to the expression \(i=\frac{\eta}{\rho}\). The influence of the adjacent fibers will be manifested, consequently, in the fact that the outermost one will not balance the entire tensile force that falls to its share; part of it will be taken up by those lying nearest to it below; with the latter, depending on the magnitude of the load, either the same thing will occur as with the outermost ones, or they will now be stressed to the elastic limit. Let us represent the phenomenon as follows: there is a bar connected by a hinge with a support (moving without friction) (Fig. 5); it is under the action of a force \(P\), and in the indicated position is held by a row of wires fastened in an immovable object. The wires are made of the elastic material we have assumed, and of a material very plastic beyond the elastic limit. It is easy to see that the stresses in the wires decrease in proportion to the decrease of their distance from the horizontal axis passing through the center of the hinge; the one nearest to it will be stressed least of all. A force \(P_1<P\) caused in the upper wire a limiting—elastic stress; a force \(P\) is applied; the upper wire must receive a stronger tensile strain than it can sustain in the

in reality, corresponding to the deflection of the beam into the position indicated by the dotted line. Let the share of the upper wire bear the force \(qP\), where \(q\) is a coefficient depending on \(\eta\), the distance of this wire from the axis. Owing to the non-free deformation, it cannot balance this force: it will balance only a part of it, of magnitude close to that force which produces the limiting—elastic stress; the excess is taken up by the second wire; let us suppose that the same thing occurs with this and also with the third wire, and that the force \(P\) will come into equilibrium with the internal stresses only when the fourth wire from the top is loaded up to the elastic limit. Let \(\varepsilon\) be the deformation corresponding to a distance 1 from the axis. At a distance \(\eta\) of a wire loaded to the elastic limit it will be \(\varepsilon\eta=e\); the stress in it will be \(\sigma=Ee\). The sum of the internal forces of all the wires whose stresses are within the limits of elasticity will be \(\sum\sigma=\sum E\varepsilon\). It will be less than \(P\), as is not difficult to verify if we construct the diagram of the change of stress according to the linear law in these two positions of the bar (Fig. 6). The difference \(P-\sum E\varepsilon\) will be balanced by the three upper wires, and the stresses in them will differ so little from one another that they may be taken as equal among themselves, as is indicated by the straight line parallel to the vertical plane. The same will occur in the bent beam on the side of the compressed fibers. Accordingly, the distribution of stresses in the bent beam may be represented by us as shown in Fig. 7. The fibers lying nearer to the axis passing through the center of gravity of the section will be stressed within the elastic limits; the outermost and some part of those adjacent to them will be stressed somewhat more highly. As the load increases, the number of fibers stressed within the elastic limits will diminish. If such a beam is unloaded, internal stresses will remain in it; the law of their distribution may be indicated, approximately, as in Fig. 7 by the dotted line. The elastically stressed fibers, striving to give the beam its original form, will increase the elastic deflection, producing, on the side of the fibers previously stretched, compressive stresses, and conversely. The sum of these residual internal stresses must, of course, be equal to zero, and this is the only condition of equilibrium.

If the material possesses different strengths in compression and in tension (with one and the same modulus of elasticity), then the distribution of stresses in the section will be somewhat different. The elastic limit in bending will be reached at the moment when the outermost tensile or compressive fibers, depending on which strength is smaller, are stressed up to the elastic limit corresponding to this type of loading. We know, for example, that cast iron is approximately much stronger in compression than in tension; the outermost fibers on the convex side of such a beam will more quickly attain the limiting-elastic stress-

—than that of compression. Assuming that the modulus of elasticity is one and the same, the distribution of stresses in the section may be represented as in Fig. 8. We conclude from this that the stress of the extreme fiber on the compressed side is no longer equal to the stress of the corresponding stretched fiber, as it was up to the elastic limit, but is greater than it.

Since the condition of equilibrium requires equality of the compressive and tensile forces, it may be inferred from it that the neutral axis must move toward the compressed fibers; it will occupy such a position that the area of the triangle representing the law of distribution of stresses on the compressed side will be equal to that of the figure representing the change of tensile stresses. In a material such as wood, residual deformations must begin first of all on the side of the compressed fibers; beyond the elastic limit, the neutral axis must shift toward the side of the stretched fibers. We already see what practical significance may attach to confirmation by suitable experiments of the following conclusions. Any part of a structure, correctly calculated and designed, must not carry a stress greater than that corresponding to the elastic limit of the material. In bending of wooden beams, elastic deflections must be related to the strength of the wood in compression. Consequently, whatever the breaking load in bending may be, the stress for such a beam must be taken, so to speak, within the limits of the elastic resistance of wood in compression. From this stress we must choose the allowable one, guided by one or another degree of reliability or strength, depending on the character of the structure and on the responsibility of the part being calculated within it.

Fracture of a bent wooden beam, as is known, occurs in the great majority of cases from rupture of the lower fibers of the beam, the fibers in tension. Only in rare cases, when on the side of the compressed fibers near the dangerous section of the beam there happens to be some similar place (an unsound place, a poorly ingrown knot, interrupted fibers, etc.), does the fracture have such a character that the strength of these fibers plays the decisive role. As in tension, so also in bending, fracture occurs more or less suddenly, sometimes accompanied by a slight crackling of the fibers shortly before failure, resulting from splitting of the fibers and rupture of part of them; on the side of the compressed fibers, only after the elastic limit has been passed does the same characteristic phenomenon of fiber displacement become noticeable as takes place when a specimen is tested in compression. We have already indicated above that such a specimen, clearly bearing traces of fiber failure, is still capable of withstanding a considerable load, approximately 0.75–0.8 of the greatest load, continuing to deform in the direction of the acting forces. If we had at hand diagrams that corresponded well and accurately to the law of change of deformations—

Proceeding from the stresses in compression and tension, it would not be difficult for us to trace the course of the distribution of stresses in the section under bending, when the elastic limit of the material is exceeded. One need only turn to iron, and then it will at once become clear what is meant here. In Fig. 9 such diagrams are shown for compression and tension, constructed in rectangular coordinate axes in such a way that the tensile stresses and elongations are directed toward the positive quantities, and the compressive stresses and shortenings toward the negative quantities. From such a combined diagram one can trace how the stresses should be distributed in the section of a bent beam. Up to the elastic limit they will obey Hooke’s law of proportionality; further, as the load increases, and therefore also the stresses of the outermost fibers, the deformations of the latter should increase faster than the stresses, and still further the flow of the material should begin, when it becomes completely plastic. Let us arrange these separate diagrams for tension and compression in the same way as we usually construct the diagram of stress distribution in the section of a bent beam; that is, let the deformations be laid off on the vertical axis, and the corresponding stresses on the horizontal axis. We shall obtain the diagram of Fig. 10; such a diagram will correspond to a certain state of the bent beam beyond its elastic limit.

A small part of the central fibers, stressed within the elastic limit, will obey Hooke’s law; in the rest, the dependence of the deformations on the stresses will be expressed by curves similar to those obtained in simple tension and compression when the elastic limit has been passed.

Let us draw, approximately, diagrams expressing the dependence of deformations and stresses in compression and tension of wood, taking into account what was said at the beginning about the general character of the phenomena observed in this. The diagram in compression must have the form of Fig. 11, and in tension that of Fig. 12; let us now construct the diagram, as in the preceding case, from these two for bending, placing the diagram for tension below (Fig. 13). It will indicate to us approximately that distribution of stresses in the section of a bent beam which should occur for some state of equilibrium of it beyond the elastic limit. Suppose that we are dealing with a material whose strength coefficients in compression and tension are quite the same as those for wood, but which also possesses the property that, upon reaching the elastic limit in tension, the material becomes very plastic; a small increase in the stress of the stretched fibers should give, beyond the elastic limit, a large extension.

Up to the elastic limit, the distribution of stresses in the section must obey Hooke’s law of proportionality (with \(E\)—the modulus of elasticity, the areas of the triangles being equal). Let us denote by \(L_1\) the elastic lim—

elasticity in compression and through \(L_2\)—in tension. Until such time as the stress of the extreme fibers on the compressed side is below \(L_1\), the beam will undergo only elastic deflections. The onset of residual deflections must correspond to that load \(P_1\), which is computed from the formula

\[ \frac{P_1 l}{4}=\frac{b h^2}{6} L_1 \quad \text{or} \quad P_1=\frac{2}{3}\frac{b h^2}{l} L_1 . \]

(\(l\) is the length of the beam between the supports, \(b\) the width in section, and \(h\) the height.) At all loads greater than \(P_1\), residual deflections must occur; but they will be very small, since on the other side of the beam, on the side of the stretched fibers, only elastic deformations take place; consequently, after the force ceases to act, under the influence of the corresponding elastic internal forces, the beam will tend to assume its initial position; in full measure, however, this will be impossible for it, since on the side of the compressed fibers there is already elasticity lost in its full initial measure; in such fibers there will now be produced a small tensile stress (Fig. 14). The magnitude of the residual deflection corresponding to these stresses, arising after the beam is unloaded, will disappear, increasing the elastic deflection. We increase the load and bring it to the value \(P_2\). The number of fibers of the compressed side stressed somewhat above the limit of elasticity will increase, the neutral axis will move farther downward, and the stress of the extreme stretched fiber will keep increasing. Let \(P_2\) be that load at which a significant inelastic deflection is obtained. Let us see in what relation it must stand to \(P_1\). This moment of the onset of more noticeable inelastic deflections must coincide with the time when the stress of the extreme stretched fiber reaches the value \(L_2\). Let the distribution of stresses in the section shown in Fig. 14 correspond precisely to this moment.

Let us call the ratio

\[ \frac{L_2}{L_1}=m>1; \]

obviously, on a certain scale \(ay=L_2\), \(cy^1=L_1\); denote \(oy=i_2\), \(oy^1=i_1\), and \(ob^1=i'\); from the similarity of the triangles \(obb^1\) and \(oay\) we have: \(L_2:L_1=i_2:i'\), or \(i_2=mi'\).

The first condition of equilibrium of the beam gives us:

the sum of the compressive stresses must be equal to the sum of the tensile stresses

\[ b(i_1-i')L_1+b\frac{L_1 i'}{2}=b\frac{L_2 i_2}{2}. \]

\[ L_1 i'=2L_1 i_1-L_2 i_2. \]

Since \(L_2=mL_1\) and \(i_2=mi'\), then

\[ L_1 i'=2L_1 i_1-m^2 i' L_1; \]

hence

\[ i'=\frac{2i_1}{1+m^2} \quad \text{and} \quad i_2=\frac{2mi_1}{1+m^2}. \]

We see that the inequality of stresses at the elastic limit in tension and compression entails a displacement of the neutral axis toward the more elastic fibers, i.e., when \(m>1\), \(i_2\) constitutes part of \(i_1\).

The second condition: the sum of the moments of the internal forces of compression and tension, with respect to any point of the section of the beam, is equal to the moment of the external forces with respect to the same point.

Let the center of moments be point \(o\).

We have the equality

\[ \frac{bL_2 i_2}{2}\cdot \frac{2}{3} i_2 + \frac{bL_1 i'}{2}\cdot \frac{2}{3} i' + \frac{bL_1(i_1+i')}{2}(i_1-i') = M. \]

Replacing \(L_2\) by the equal quantity \(mL_1\) and \(i_2\) by \(mi'\), we find:

\[ 2m^3 i'^2 + 3 i_1^2 - i'^2 = \frac{6M}{bL_1}. \]

Since

\[ i'=\frac{2i_1}{1+m^2}, \]

then

\[ 4(2m^3-1)+3(1+m^2)^2 = \frac{6(1+m^2)^2}{bL_1 i_1^2}M, \]

\[ (m+1)(3m^3+3m^2+m^2+m+m^2-1)=\text{eodem} \]

\[ (m+1)^2(2m^2+2m-m^2-1)=\text{eodem} \]

\[ (m+1)^3(3m-1)=\text{eodem} \]

Besides this, we have

\[ i_2=\frac{2mi_1}{1+m^2} \]

and \(i_2+i_1=h\); whence

\[ i_1=\frac{(1+m^2)h}{(1+m)^2}. \]

Carrying out the substitution, we obtain

\[ M=\frac{3m-1}{1+m}\cdot \frac{bh^2L_1}{6}. \]

But

\[ M=\frac{P_2l}{4}, \]

and

\[ \frac{2}{3}\cdot \frac{bh^2}{l}L_1=P_1; \]

therefore,

\[ P_2=\frac{3m-1}{1+m}P_1. \]

This is the desired relation between the loads \(P_1\) and \(P_2\). Within the limits of the loads \(P_1\) and \(P_2\), the material we have assumed will have small residual deflections; up to \(P_1\) they will be entirely elastic, after \(P_2\) they will begin to increase considerably, while the elastic deflections will greatly diminish and, conversely, the residual ones will increase considerably. Turning to wood, we must draw the following conclusion. Since its elastic limit lies very close to the ultimate strength in tension, the load \(P_2\), mentioned above, must either be the one destroying the bent beam, or be very close to it; the stress developing in the extreme stretched fibers ...

under its influence must be equal to the coefficient of strength of the wood in tension, or be greater, but differ from it very little. As for the stress \(L_1\) on the side of the compressed fibers at the very moment when the bent beam begins to acquire residual deflection, it, obviously, must in wood be very close to its coefficient of strength in compression. Let us now see what experience gives us. The data given below pertain to Südtirol fir, whose mechanical properties were investigated in detail by G. Janka and A. Hadek. We note here that even Tetmajer, in testing Swiss larch, came to the quite definite conclusion that, when beams of one and the same wood are bent, significantly different coefficients of strength are sometimes obtained, depending on whether the pith is located below or above. Pine, fir, spruce, larch, and oak all yielded a higher coefficient of strength in bending for those beams in which the pith was located on the side of the compressed fibers; it is known that the central part of the trunk has a lower coefficient of strength both in compression and in tension. In accordance with the consideration set forth above concerning the relation between the loads of the beam \(P_1\) and \(P_2\), it follows that the decisive factor in the fracture of a bent beam must be the strength of the wood in tension; this is indeed confirmed by the data for all five species (see Table VII).

Table VII.

In Table VIII are given the results of experiments on Südtirol fir. The stresses are calculated in \(\mathrm{kg}/\mathrm{cm}^2\); in one column are entered those stresses observed in the compression test at which a markedly increasing reduction in the length of the specimen was observed;

They are called Fliessgrenze; that is, it seems, the stresses at which the deformations begin to grow faster than the corresponding stresses. In the second case—the stresses in bending corresponding to the elastic limit.

Table VIII.

The arithmetic mean of the former, for all 12 tested logs, is 258.5 kg/cm²; of the latter, 268 kg/cm²; the difference between them is 3.54%.

Table IX.

Table IX gives the results relating to beams in which the pith was located on the side of the compressed fibers; in one column are the coefficients of strength in compression, and in the other the limits of elasticity in bending; the difference between them is 1.46%. Table X contains the data relating to beams whose pith was turned toward the side of the stretched fibers. In one column are the stresses bordering on the beginning of increasing deformations in compression, and in the other the limits of elasticity in bending. The difference between them is 3.08%.

Table X.

Table XI gives the mean values indicated above, and also includes data concerning fir from Nordtirol, Wienerwald, and Erzgebirge, which was tested by G. Janka (the results of the latter experiments were published in 1904).

Table XI.

From this table we see that the proportional limit in bending of fir beams from Südtirol and Erzgebirge lies very close to the “flow” limit (Fliessgrenze), and for fir from the other two localities to the strength coefficient in compression. The difference in the first case is 2.48%, in the second 1.53%; in both cases it is so small that, in practice, the stresses at the proportional limit in bending may be considered to coincide with those with which we have just compared them. Here, incidentally, it should be recalled how difficult it sometimes is, in a compression test, to mark exactly one or another moment in the change of deformations as a function of the stresses. The results of the experimental data, therefore, we find to agree in fact quite well with the conclusions drawn above.

According to Tetmajer’s experiments, fir has a strength coefficient of \(602^{\mathrm{kg}}/\mathrm{cm}^2\), and likewise for compression \(276^{\mathrm{kg}}/\mathrm{cm}^2\). The ratio is \(2,18\). The load \(P_2\), which must produce fracture of the bent beam, will be

\[ P_2=\frac{3\cdot 2,18-1}{1+2,18}\cdot P_1=1,74P_1. \]

We have already said above that the load \(P_1\) corresponds to that load on the beam at which, on the side of the compressed fibers, the extreme fiber is stressed up to the proportional limit, and have indicated that the stress in them must be close to the strength coefficient in compression or coincide with it. Calculating the stresses by the usual bending formula, derived on the assumption that the neutral axis at the moment of fracture passes through...

and the center of gravity of the section, we shall obtain for them under loads \(P_2\) and \(P_1\) the same ratio as that in which \(P_1\) and \(P_2\) themselves stand. Summing up the general result of his experiments with fir from southern Tyrol, G. Janka found that the strength in bending is equal to \(1.72\) times that in compression. This multiplier is smaller than the one we computed from Tetmayer’s data by only \(0.02\). We see how negligible this difference is, and at the same time we must note a certain generality of the conclusions we have drawn: the coefficient \(m\) was taken by us from data for Swiss fir, while the computations showed its full suitability also for fir from southern Tyrol.

Thus, the investigation carried out above as to what the stress distribution in the section of a bent beam should be, and what the dependence should be between the coefficients of strength in compression and in bending, finds quite satisfactory confirmation also in the experimental data. But in the consideration made above, when drawing the diagram of the stress distribution in the section, we took these stresses to be mutually equal for all those fibers which are stressed beyond the elastic limit. In reality, they must differ somewhat from one another; the law of their variation must be expressed by some curve, as is seen from the combined diagram for tension and compression of iron when applied to bending. The general form of the equation of such a curve was first proposed by Hodkinson; a more convenient structure for such an equation was proposed by Saint-Venant; and we shall apply it in this form to the particular case under consideration. Before proceeding further to the conclusions, we must agree as to what law of variation of the stresses in the fibers we shall take as the basis of our consideration. Experience indicates that it will not be a large and arbitrary assumption that, beyond the elastic limit, the plane sections of a beam remain plane and perpendicular to the curve of bending. If this is so, then the stresses in the fibers will vary in proportion to their distance from the neutral axis. This follows directly from the geometrical relations (Fig. 15). We have:

\[ \frac{ef}{00_1}=\frac{\rho+\eta}{\rho} \]

and hence

\[ \frac{ef-00_1}{00_1}=\frac{\eta}{\rho}=i=\text{relative elongation of the fiber } ef. \]

The stress of this fiber will be

\[ \sigma=Ei=E\frac{\eta}{\rho}, \]

or, replacing \(\frac{1}{\rho}\) by the equal quantity \(\frac{M}{EJ}\), we obtain

\[ \sigma=\frac{M}{J}\cdot\eta . \]

(We assume, as before, that the modulus of elasticity \(E\) is the same in tension and compression and does not depend on the stresses.)

Let there be a beam of rectangular section with height \(h\) and width \(b\) (Fig. 16); the distance between the supports is \(l\); the position of the neutral axis at the moment of fracture is determined by the line \(00_1\); \(e_1\) and \(e_2\) are the distances

from the neutral axis of the extreme stretched and compressed fibers. We further assume that the stresses of the stretched fibers up to rupture vary proportionally to their distance from the neutral axis. Let \(Z\) be the greatest stress of the extreme stretched fiber at the moment of fracture.

The stress of any intermediate fiber, at a distance \(\eta_1\) from the axis, will be \(\sigma_1\)

\[ \sigma_1 = Z \frac{\eta_1}{e_1}. \]

For \(\eta_1 = e_1\), \(\sigma_1 = Z\); for \(\eta_1 = 0\), \(\sigma = 0\).

The compressive stresses vary according to a certain curve (Fig. 17). Let \(D\) be the stress of the extreme compressed fiber; the stress \(\sigma_2\) of an arbitrarily chosen fiber between the axis and the extreme compressed fiber must be represented as a function of \(D\) and \(\eta_2\); it is quite satisfactory for the circumstances of fracture to take the following dependence among these three quantities:

\[ \sigma_2 = D \left[ 1 - \left(1 - \frac{\eta_2}{e_2}\right)^n \right]. \]

For \(\eta_2 = e_2\), \(\sigma_2 = D\); for \(\eta_2 = 0\), \(\sigma_2 = 0\); for all intermediate values of \(\eta_2\) it will be less than \(D\), gradually approaching zero.

For the curve given by this equation, we impose the following condition: at the point of intersection through which the neutral axis passes, this curve must have a tangent coinciding with the straight line expressing the dependence of the tensile stresses on the distance \(\eta_1\). For this it is necessary that the tangents of their angles be equal, i.e.

\[ \frac{\sigma_1}{\eta_1} = \frac{d\sigma_2}{d\eta_2} \quad \text{for } \eta_2 = 0. \]

We have:

\[ \frac{Z}{e_1} = \frac{nD}{e_2} \left(1 - \frac{\eta_2}{e_2}\right)^{n-1} \quad \text{for } \eta_2 = 0 = \frac{nD}{e_2}. \tag{1} \]

We set up two equations, using the two conditions of equilibrium of the beam; the scale in which Fig. 17 is drawn is known.

The sum of the tensile stresses

\[ b \sum \sigma_1 = \frac{Z e_1}{2}\, b \]

The sum of the compressive stresses

\[ b \sum \sigma_2 = b \int_0^{e_2} \sigma_2\, d\eta_2 \]

\[ b \int_0^{e_2} D \left[ 1 - \left(1 - \frac{\eta_2}{e_2}\right)^n \right] d\eta_2 = b D e_2 - \frac{b D e_2}{n+1} = \frac{b n D e_2}{n+1}. \]

The first condition of equilibrium gives us:

\[ \frac{Z e_1}{2} = \frac{n D e_2}{n+1}. \tag{2} \]

From (1) we have \(Z=\dfrac{nDe_1}{e_2}\); after substitution we obtain

\[ \frac{e_1^2}{2}=\frac{e_2^2}{n+1}. \tag{3} \]

Since \(e_1+e_2=h\), solving (3) jointly with this, we find

\[ e_1=\frac{h\sqrt{2}}{\sqrt{2}+\sqrt{n+1}} \tag{4} \]

\[ e_2=\frac{h\sqrt{n+1}}{\sqrt{2}+\sqrt{n+1}}. \tag{5} \]

The second condition of equilibrium requires the equality of the moment of the external forces, \(M\), to the sum of the moments \(m_1\) and \(m_2\) of the internal forces in the section with respect to some point. The center of moments is the point of intersection of the plane of the section and the neutral axis.

\[ M=m_1+m_2. \]

\[ m_1=\text{mom. of the internal tensile forces} =b\int_0^{e_1}\sigma_1\eta_1\,d\eta_1. \]

\[ m_2=\text{mom. ” ” compressive} =b\int_0^{e_2}\sigma_2\eta_2\,d\eta_2. \]

\[ m_1=\frac{bZ}{e_1}\int_0^{e_1}\eta_1^2d\eta_1 =\frac{bZe_1^2}{3}. \]

\[ m_2=b\int_0^{e_2}D\left|\,1-\left(1-\frac{\eta_2}{e_2}\right)^n\,\right|\eta_2d\eta_2 \]

\[ m_2=bD\int_0^{e_2}\eta_2d\eta_2 -bD\int_0^{e_2}\left(1-\frac{\eta_2}{e_2}\right)^n\eta_2d\eta_2 \]

The first integral for \(m_2\) gives \(\dfrac{bDe_2^2}{2}\) (with the \(+\) sign).

Setting \(\eta_2=u\) and \(\left(1-\dfrac{\eta_2}{e_2}\right)^n d\eta_2=dv\), we integrate further by the formula

\[ \int udv=uv-\int vdu. \]

\[ bD\int_0^{e_2}\eta_2\left(1-\frac{\eta_2}{e_2}\right)^n d\eta_2 =bD\left|\begin{matrix} e_2\\ 0 \end{matrix} -\eta_2\frac{e_2}{n+1}\left(1-\frac{\eta_2}{e_2}\right)^{n+1} -\right. \]

\[ -bD\int_0^{e_2}-\frac{e_2}{n+1}\left(1-\frac{\eta_2}{e_2}\right)^{n+1}d\eta_2. \]

Substitution of the limits turns the first term of the right-hand side into zero; discarding it, we integrate the second term.

\[ -bD\left|\begin{matrix} e_2\\ 0 \end{matrix} \frac{e_2^2}{(n+1)(n+2)} \left(1-\frac{\eta_2}{e_2}\right)^{n+2} =\frac{bDe_2^2}{(n+1)(n+2)}. \]

\[ M=m_1+m_2=\frac{bZe_1^2}{3}+\frac{bDe_2^2}{2}-\frac{bDe_2^2}{(n+1)(n+2)}, \]
whence

\[ M=\frac{bZe_1^2}{3}+\frac{n(n+3)bDe_2^2}{2(n+1)(n+2)}. \tag{6} \]

From (4) and (5) we substitute \(e_1^2\) and \(e_2^2\)

\[ M=\frac{bZ}{3}\cdot\frac{2h^2}{(\sqrt{2}+\sqrt{n+1})^2} +\frac{n(n+3)bD}{2(n+1)(n+2)}\cdot \frac{(n+1)h^2}{(\sqrt{2}+\sqrt{n+1})^2}. \]

From (1) we replace \(Z\), and at the same time from (4) and (5) \(\frac{e_1}{e_2}\):

\[ M=\frac{bnD}{3}\cdot \frac{2h^2\sqrt{2}}{\sqrt{n+1}(\sqrt{\,}+2\sqrt{n+1})^2} +\frac{bnDh^2(n+3)}{2(n+2)(\sqrt{2}+\sqrt{n+1})^2}. \]

\[ M=\frac{bnDh^2}{(\sqrt{2}+\sqrt{n+1})^2} \left[ \frac{2\sqrt{2}}{3\sqrt{n+1}}+ \frac{n+3}{2(n+2)} \right]. \]

\[ M=\frac{bh^2}{6}\cdot \frac{nD}{(\sqrt{2}+\sqrt{n+1})^2} \left[ 4\sqrt{\frac{2}{n+1}}+ 3\frac{n+3}{n+2} \right]. \]

\[ M=\frac{bh^2}{6}\cdot D\cdot \frac{ n\left[ 3\frac{n+3}{n+2}+4\sqrt{\frac{2}{n+1}} \right] }{ (\sqrt{2}+\sqrt{n+1})^2 }. \tag{7} \]

The right-hand side of equality (7) consists of three factors; the first of them is the section modulus for a beam of rectangular cross-section; the second is the coefficient of crushing strength. If the usual formulas for bending were applicable beyond the elastic limit as well, then the product of these two factors would be equal to the breaking moment; but, by virtue of the relations indicated above beyond the elastic limit, a dependence between the internal forces that arise and the moment of the external forces is established by equality (7). Here the whole factor containing the exponent \(n\) is that coefficient by which the calculated dangerous moment must be multiplied in order to obtain the actual breaking moment. For \(n=3.57\) this multiplier will be equal to \(1.74\).

For this value of \(n\), from (4) and (5) we determine

\[ e_1=0.4h \quad \text{and} \quad e_2=0.6h. \]

We see that, under the assumed variation of compressive stresses along the curve expressed by the equation for \(\sigma_2\), the position of the neutral axis at the moment of fracture deviates considerably from the axis passing through the center of gravity of the section. The area over which the compressive forces develop is one and a half times greater than that over which the tensile forces are distributed. And immediately one is struck by the wastefulness in the use of this material, which is so widely practiced in [[unclear: word cut off at page bottom beginning with “iz-”]]

preparation of floor ceiling wooden beams with a rectangular cross-section, for here too, as with cast iron and as Bach’s experiments have shown, material situated near the center of gravity of the section is used much less effectively than follows directly from the theory of bending.

From (1) we find that, at the bending moment, the stress of the extreme tensile fibers must be

\[ Z=\frac{e_1}{e_2} nD=\frac{2}{3}\,3.57D=2.38D. \]

The exponent \(n\) must have different values for individual species; the indicated value, \(3.57\), is suitable for fir. Let us compare some data from experiment with the preceding conclusions.

Fir from Switzerland (Tetmajer) \(D=276;\ Z=2.38D=656\).

    ”    Sweden (Wijkander) \(D=369;\ Z=2.38D=878\).

The mean values for \(Z\) according to Tetmajer are ... 602, Wijkander’s ... 814; in the first case the value computed by us is greater than that obtained by experiment by \(6\%\), in the second by \(7.8\%\). Both figures are below \(10\%\), i.e. they do not exceed the limits which, for coniferous species, are recognized as quite admissible, in view of the great variability of the strength of wood, by American investigators (Johnson).

Above we have considered two possible cases of variation of stresses in the section of a bent beam; in the first, the compressive stresses vary according to a straight-line law only for fibers lying close to the neutral axis, while for all the others, lying above the fiber stressed to the elastic limit, they are equal to one another; in the second, they vary according to a certain curve for all fibers of the compressed side. The first is simpler and, perhaps, more nearly corresponds to the actual relations, if one takes into account that in a beam broken in bending there is always noticeable a more or less deeply extending line, over its height, of fibers pushed over one another. The position of the neutral axis in this case is determined by the values \(e_1 = {}^{3}/_{7}h\) or \(e_2 = {}^{4}/_{7}h\). The general form of the diagram of stress distribution in the section is indicated in Fig. 18; but both that and the other conclusion may be considered quite satisfactorily to resolve the question posed concerning the dependence of bending strength on compressive strength. A series of experiments that would make it possible to establish more definitely the values of \(m\) and \(n\) would be both necessary and desirable in order to verify both the correctness of the assumptions made in considering the question and the correctness of the indicated conclusions. That they are close to reality is, incidentally, confirmed by the following information, found by me in the literature. As early as 1837, Prof. Barlow, on the basis of his experiments with timber, indicated the position of the neutral axis of a bent beam at the moment of bending, at a distance of \(5/8\) of the height from the extreme compressed fiber;

then, in 1898, Prof. Neely (North America), from a comparison of numerous data obtained in testing the mechanical properties of constructional species, derived the following formula for the dependence of the bending moment on the strength in compression:

\[ M = D_a (r_1 + r_2)b; \]

\(D_a\) is the sum of the compressive stresses, \(r_1\) and \(r_2\) are the distances of the centers of gravity of the areas on which the tensile and compressive forces act from the neutral axis, and \(b\) is the width of the beam. The sum \(r_1\) and \(r_2\) was found to be sufficiently constant and equal to \(^{3}/_{5}h\).

It is not difficult to see that almost the same value of the sum \(r_1 + r_2\) is given by the values found above for \(e_1\) and \(e_2\).

The distance of the center of gravity of the area of the tensile forces

\[ r_1 = {}^{2}/_{3}\cdot 0.4h \]

and of the compressive forces

\[ r_2 = \frac{n(n+3)bDe_2}{2(n+1)(n+2)} : \frac{bnDe_2}{n+1} = \frac{e_2(n+3)}{2(n+2)} = 0.3\frac{n+3}{n+2}h \]

\[ r_1 = 0.266h \quad \text{and} \quad r_2 = 0.353h \]

\[ r_1 + r_2 = 0.619h \infty {}^{3}/_{5}h. \]

The strength of wood in compression may, in this way, be regarded as sufficient to characterize for us its qualities for two of the most common loading cases for it; as regards the strength of wood in shearing along and across the fibers, experiments concerning this kind of resistance have until recently been almost nonexistent, and therefore the question remains open as to the possibility of there existing a known relation between it and the compressive strength, and also between the moduli of elasticity under these two kinds of stress. If it were possible to establish the values of those multipliers for different species by which, on multiplying, we would obtain from the compressive strength also the strength of wood in bending, then this would greatly facilitate the testing of this material with respect to its mechanical properties. But it is known that the coefficient of strength of wood in compression depends on the degree of moisture of the specimen at the moment of testing and on the specific weight of the given specimen in the absolutely dry state. Even for one and the same trunk it does not represent a constant quantity; experience shows that a specimen taken from the upper part of the trunk resists the action of compressive forces less than specimens from the lower-lying parts of the trunk; likewise, specimens taken from different sides of one and the same trunk have different coefficients of strength.

The technical-mechanical properties of wood grown in one locality are not similar to those for wood of the same species but grown in another locality and under other conditions. In view of this, the laboratory practice of recent years has developed a system of detailed tables, which should be filled in with data relating to the test-

species under study. In them there should be indicated the soil of growth, the geographical position of the locality, the age of the tree, its position in the forest (indication of the nearest edge or clearing of the forest with respect to the cardinal points), the width of the annual rings and the ratio between their spring and summer parts, and the specific gravity of the wood and its moisture content. But of all these factors, whose influence is intended to be established, only the last two always prove to stand out more sharply from among the rest. All investigations lead to one and the same conclusion: that the compressive strength of wood increases uniformly with an increase in its specific gravity in the absolutely dry state, and the dependence between them in this state is expressed by a straight line; likewise, an increasing moisture content of the wood entails a decrease in the strength of the wood; the change in strength in this case occurs differently, depending on the state of water saturation in which the specimen is found. The influence of moisture is generally noticeable more sharply than the influence of the specific gravity of the wood. As for the remaining factors, no more or less regular dependence of the compressive strength of wood on them has been observed; in one and the same forest there are trees with greater and lesser specific gravity, with greater and lesser compressive strength. At the same time, both for trees of one forest and for trees of one and the same species from different localities, there apparently remains in force one and the same law of variation of compressive strength with specific gravity along a straight line, the equation of which may be written in the form \(D=\alpha \delta_c-\beta\): the test must be carried out at one and the same degree of moisture, or the data must be reduced to that degree; without indication of the percentage moisture content, the results of the tests have no significance. In the equation of the straight line, \(D\) is the compressive strength, and \(\alpha\) and \(\beta\) are constants; \(\delta_c\) is the specific gravity at a definite moisture content \(c\) (usually, according to Baumhauer’s proposal, \(c=15\%\)). Experimental data do indeed indicate the existence of such a direct dependence; in Table XII are set down the values for \(D\) and \(\delta\), as they were determined for fir from 10 different localities by Schwappach and Janka.

Table XII.

Schwappach characterizes the condition of his specimens by indicating that they were air-dry, i.e., with the same percentage of moisture as the air-dry state of wood when stored in a closed dry room; Janka’s data are reduced to a 15% moisture content. From the table it is evident that, as the specific gravity increases, the compressive strength of wood also increases, although not uniformly over all intervals. If these data are plotted in rectangular coordinates, laying off the specific gravity on one axis and the coefficients of compressive strength on the other, we obtain a series of points that are situated quite symmetrically about a straight line passing in part through some of them (Fig. 19). Obviously, this straight line will express the mean and more general variation of the strength of wood of a given species with specific gravity at a definite percentage moisture content. In the same figure lines are plotted expressing the law of variation of compressive strength with specific gravity in the normal—air-dry—condition of the specimens (14–16%) for the firs Nordtirol, Wienerwald, and Erzgebierge. Each of the lines is very close to a straight line; however, the constants \(\alpha\) and \(\beta\) in this case must be different from those above.

A direct dependence of the strength of wood on its specific gravity was already observed in the eighteenth century by Buffon, specifically for oak, whose strength was found to be directly proportional to its specific gravity. Such a dependence does not seem strange and, on the contrary, is necessary and natural for such a material, layered and porous, as wood. Its mass consists of a series of fibers, between which are situated ducts and vessels; through them, during the life of the tree, nutritive sap moves. Its anatomical structure is different for deciduous and coniferous species, and this explains above all the different hardness, strength, brittleness, weight, etc., of these species. The less these nutrient passages, vessels, and voids of the cells themselves, of which the fibers are composed, are filled with sap, water, or air-

...only the walls of such vessels, the fibers of which make up the woody mass, will resist the external forces. The more of them there are per unit area, the greater their resistance will be; hence there follows a direct dependence of the strength of wood on its specific gravity; and, indeed, in an absolutely dry state this strength should have the greatest value and should increase continuously and uniformly with an increase in the specific gravity in this state (Fig. 19). Experience indicates, however, that even in this state, for one and the same species grown in different places, straight lines are obtained that lie very close to one another and are almost parallel to one another. It remains unexplained whether this depends on the peculiarities of the locality where the tree grew, or on other causes. In his experiments Sehald, after heating wood in a stream of hot air at \(100^\circ\) C., observed a certain increase in specific gravity. In his opinion, this occurs from incipient oxidation in the constituent parts of the specimen, from the formation of acetic and formic acids; he considers it possible that such oxidation begins from the moment the wood is dried; finally, he observed that at \(100^\circ\) C. not all the water is removed from the specimen. In the presence of such phenomena, the observed lack of coincidence in the strength of wood in the absolutely dry state at a given specific gravity for one locality with that at the same gravity for another locality may be due precisely to the different capacity of the wood for the above-mentioned processes. This is all the more probable because, in all his investigations of wood, Hartig finds that specific gravity is the best indicator of its quality, and that in all the attempts of many other investigators it has not been possible to note and trace such a direct dependence between the compressive strength of wood and the above-mentioned factors as soil, climate, etc. If, however, one takes into consideration the circumstance that the specific gravity of wood is almost always a characteristic of the most or least favorable and favorable conditions for the growth of a given species in a certain locality, it will become obvious that no such direct dependence between them can be found, for the influence of all these conditions in their totality has already found its reflection in the specific gravity. Reflected in it are the quality of the soil, the abundance or deficiency of heat and sunlight, moisture, the structure of the wood, the quantity and composition of the substances filling the passages and pores of the tree, the climatic conditions under which the life of the tree proceeded, the mechanical action on the tree of accidental forces (prolonged one-sided winds, etc.). By virtue of this, it would appear entirely superfluous, in testing wood, to continue attempts to establish a dependence between compressive strength and those factors whose influence on the specific gravity of wood may be considered fully proven. On the contrary, it seems necessary and expedient, in work on test...

to the testing of wood, introduce investigations on the influence of the anatomical structure of the fibers on the strength of wood, as well as study more precisely the influence of moisture upon it. It has already been stated above that with an increase in moisture the strength of wood decreases. This occurs for two reasons: moist fibers are less able to resist the acting forces than dry ones; secondly, with increasing moisture in a specimen there occurs a change in its dimensions, connected with the property of wood to swell when it is impregnated with water. Within small limits of change in moisture, this change in the dimensions of the area of the specimen is not large; but what is characteristic is precisely the commonality of the course of changes in dimensions and in the compressive strength of wood. The latter, having its greatest value in the absolutely dry state, begins to decrease when the specimen is wetted; the most rapid decrease in strength is observed between moisture-content limits of 7% to 15%; more slowly it occurs from 0% to 7% and from 15% to 25%; within the limits of 25%–50%, and most often at 33%–40%, there is observed an almost sudden decrease in the strength of the specimen, which, with a further increase in moisture, decreases so slowly that it may be regarded as constant. The dimensions of the area of the specimen’s cross-section have their smallest value in the absolutely dry state; with an increase in moisture up to 25% there is a rapid and almost proportional to the quantity of absorbed water increase in the dimensions of the area of the section; between 25% and 40% it proceeds more slowly, and beyond this limit up to the greatest saturation so slowly and uniformly that it may be regarded as constant. In explanation of this one may say that a noticeable decrease in the area of the specimen’s section, when we dry it, occurs from those pores as soon as the walls of the fibers and cells begin to dry; the loss of water contained in the vessels and passages of the wood, however, does not have a great influence on this phenomenon. Together with this, changes in the dimensions of the area depend, like strength, also on the specific gravity of the wood; in heavier wood the change in dimensions is also greater (Fig. 19). Thus, the change in the compressive strength of wood and the change in the dimensions of the area of the specimen’s section are evidently the consequence of one and the same cause, and this cause must be recognized as the difference in the anatomical structure of the timber. For both, a magnitude of change corresponding to one percent of moisture can be found; this is especially important for the change in the compressive strength of wood, for in this way it would be easy to recalculate the strength in the absolutely dry state into that which should correspond to a given percentage of moisture. The direct dependence of compressive strength on the specific gravity in the absolutely dry state, on the other hand, makes it possible for each species to find that number by which the product of the specific gravity would give the compressive strength, and from it we would at once find also the strength in bending.

Tomsk, September, 1906.
A. Krylov.

List of Literary Sources

1) Dr. H. Nordlinger. Die technische Eigenschaften der Hölzer. Stuttgart, 1860.

2) ” ” ” Die gewerblichen Eigenschaften der Hölzer. Stuttgart, 1890.

3) Dr. A. Schwappach. Untersuchungen über Raumgewicht und Druckfestigkeit des Holzes wichtigen Waldbäume. I. Die Kiefer, Berlin, 1897; II Fichte, Weisstanne, Weymouthskiefer und Rothbuche. Berlin, 1898.

4) Dr. R. Hartig. Holzuntersuchungen. Altes und Neues. Berlin, 1901.

5) J. Bauschinger. Mitteilungen aus dem mechanisch-technischen Laboratorium der Königl. techn. Hochschule in München; Heft IX, 1883; H. XIV, 1886; H. XVI, 1887.

6) L. Tetmayer. Methoden und Resultate der Prüfung der Schweizerischen Bauhölzer. 1884.

7) Wijkander. Untersuchung der Festigkeitseigenschaften Schwedischer Holzarten. Göteborg, 1897.

8) M. Rudeloff. Mitteilungen a. d. Kgl-techn. Versuchsanstalten zu Berlin, 1886, 1889, 1895.

9) A. Hadek und Janka. Untersuchungen über Elasticität und Festigkeit der Oesterreichischen Bauhölzer. I. Fichte Südtirols, Wien, 1900; II. Fichte von Nordtirol, vom Wienerwalde und Erzgebirge. Wien, 1904.

10) Timber Physics, P. I, II, Washington., 1892.
” ” Bulletin № 22, The white Pine. Washington, 1899.

Fig. 1.

Fig. 4.

Fig. 6.

Fig. 3.

Fig. 5.

Fig. 2.

Fig. 7.

Fig. 8.

Fig. 9.

Fig. 10.

Fig. 11.

Fig. 12.

Fig. 13.

Fig. 14.

Fig. 15.

Fig. 17.

Fig. 16.

Fig. 18.

![Graph: dependence of wood strength on specific gravity and moisture content.]

Specific gravity in the absolutely dry state

Specific gravity in the air-dry state

Specific gravity in the absolutely dry state

Specific gravity in the air-dry state

Visible curve labels include: pine, spruce, birch, oak, larch [[unclear: several additional handwritten species names and point labels]].

Marginal note:

The numerator gives the resistance in compression, reduced to 1% moisture content; the denominator gives the resistance in bending, reduced to the absolutely dry state.

\[ \frac{870,\; 910,\; 1030\ \text{kg/cm}^2}{420,\; 500,\; 720\ \text{kg/cm}^2} \]

[[unclear: remaining small handwritten annotations and several curve labels in the graph]]