or the H-girder or rail is designed to resist bending in one particular direction, but if, as in a tall pillar, it be necessary to resist bending in all directions alike, it is obvious that the tubular or cylindrical construction best meets the case; for it is plain that this hollow tubular pillar is but the
-girder turned round every way, in a “solid of revolution,” so that on any two opposite sides compression and tension are equally met and resisted, and there is now no need for any substance at all in the way of web or “filling” within the hollow core of the tube. And it is not only in the supporting pillar that such a construction is useful; it is appropriate in every case where stiffness is required, where bending has to be resisted. The long bone of a bird’s wing has little or no weight to carry, but it has to withstand powerful bending moments; and in the arm-bone of a long-winged bird, such as an albatross, we see the tubular construction manifested in its perfection, the bony substance being reduced to a thin, perfectly cylindrical, and almost empty shell. The quill of the bird’s feather, the hollow shaft of a reed, the thin tube of the wheat-straw bearing its heavy burden in the ear, are all illustrations which Galileo used in his account of this mechanical principle[619].
Two points, both of considerable importance, present themselves here, and we may deal with them before we go further. In the first place, it is not difficult to see that, in our bending beam, the strain is greatest at its middle; if we press our walking-stick hard against the ground, it will tend to snap midway. Hence, if our cylindrical column be exposed to strong bending stresses, it will be prudent and economical to make its walls thickest in the middle and thinning off gradually towards the ends; and if we look at a longitudinal section of a thigh-bone, we shall see that this is just what nature has done. The thickness of the walls is nothing less than a diagram, or “graph,” of the “bending-moments” from one point to another along the length of the bone.
Fig. 332.
The second point requires a little more explanation. If we {678} imagine our loaded beam to be supported at one end only (for instance, by being built into a wall), so as to form what is called a “bracket” or “cantilever,” then we can see, without much difficulty, that the lines of stress in the beam run somewhat as in the accompanying diagram. Immediately under the load, the “compression-lines” tend to run vertically downward; but where the bracket is fastened to the wall, there is pressure directed horizontally against the wall in the lower part of the surface of attachment; and the vertical beginning and the horizontal end of these pressure-lines must be continued into one another in the form of some even mathematical curve—which, as it happens, is part of a parabola. The tension-lines are identical in form with the compression-lines, of which they constitute the “mirror-image”; and where the two systems intercross, they do so at right angles, or “orthogonally” to one another. Such systems of stress-lines as these we shall deal with again; but let us take note here of the important, though well-nigh obvious fact, that while in the beam they both unite to carry the load, yet it is always possible to weaken one set of lines at the expense of the other, and in some cases to do altogether away with one set or the other. For example, when we replace our end-supported beam by a curved bracket, bent upwards or downwards as the case may be, we have evidently cut away in the one case the greater part of the tension-lines, and in the other the greater part of the compression-lines. And if instead of bridging a stream with our beam of wood we bridge it with a rope, it is evident that this new construction contains all the tension-lines, but none of the compression-lines of the old. The biological interest connected with this principle lies chiefly in the mechanical construction of the rush or the straw, or any other typically cylindrical stem. The material of which the stalk is constructed is very weak to withstand compression, but parts of it have a very great tensile strength. Schwendener, who was both botanist and engineer, has elaborately investigated the factor of strength in the cylindrical stem, which Galileo was the first to call attention to. {679} Schwendener[620] shewed that the strength was concentrated in the little bundles of “bast-tissue” but that these bast-fibres had a tensile strength per square mm. of section, up to the limit of elasticity, not less than that of steel-wire of such quality as was in use in his day.
For instance, we see in the following table the load which various fibres, and various wires, were found capable of sustaining, not up to the breaking-point, but up to the “elastic limit,” or point beyond which complete recovery to the original length took place no longer after release of the load.
|
Stress, or load in gms. per sq. mm., at Limit of Elasticity |
Strain, or amount of stretching, per mille | |
|---|---|---|
| Secale cereale | 15–20 | 4·4 |
| Lilium auratum | 19 | 7·6 |
| Phormium tenax | 20 | 13·0 |
| Papyrus antiquorum | 20 | 15·2 |
| Molinia coerulea | 22 | 11·0 |
| Pincenectia recurvata | 25 | 14·5 |
| Copper wire | 12·1 | 1·0 |
| Brass wire | 13·3 | 1·35 |
| Iron wire | 21·9 | 1·0 |
| Steel wire | 24·6 | 1·2 |
In other respects, it is true, the plant-fibres were inferior to the wires; for the former broke asunder very soon after the limit of elasticity was passed, while the iron-wire could stand, before snapping, three times the load which was measured by its limit of elasticity: in the language of a modern engineer, the bast-fibres had a low “yield-point,” little above the elastic limit. But nevertheless, within certain limits, plant-fibre and wire were just as good and strong one as the other. And then Schwendener proceeds to shew, in many beautiful diagrams, the various ways in which these strands of strong tensile tissue are arranged in various cases: sometimes, in the simpler cases, forming numerous small bundles arranged in a peripheral ring, not quite at the periphery, for a certain amount of space has to be left for living and active tissue; sometimes in a sparser ring of larger and {680} stronger bundles; sometimes with these bundles further strengthened by radial balks or ridges; sometimes with all the fibres set