been very different if we had built up our pillar of cards or slates lying obliquely to the lines of pressure, for then at once there would have been a tendency for the elements of the pile to slip and slide asunder, and to produce what the geologists call “a fault” in the structure.
Somewhat more generally, if AB be a bar, or pillar, of cross-section a under a direct load P, giving a stress per unit area = p, then the whole pressure P = pa. Let CD be an oblique section, inclined at an angle θ to the cross-section; the pressure on CD will evidently be = pa cos θ. But at any point O in CD, the pressure P may be resolved into the force Q acting along CD, and N perpendicular to it: where N = P cos θ, and Q = P sin θ = pa sin θ. The whole force Q upon CD = q · area of CD, which is = q · a ⁄ (cos θ). {686} Therefore qa ⁄ (cos θ) = pa sin θ, therefore q = p sin θ cos θ, = ½ p sin 2θ. Therefore when sin 2θ = 1, that is, when θ = 45°, q is a maximum, and = p ⁄ 2; and when sin 2θ = 0, that is when θ = 0° or 90°, then q vanishes altogether.
Fig. 338.
This is as much as to say, that a shearing stress vanishes altogether along the lines of maximum compression or tension; it has a definite value in all other positions, and a maximum value when it is inclined at 45° to either, or half-way between the two. This may be further illustrated in various simple ways. When we submit a cubical block of iron to compression in the testing machine, it does not tend to give way by crumbling all to pieces; but as a rule it disrupts by shearing, and along some plane approximately at 45° to the axis of compression. Again, in the beam which we have already considered under a bending moment, we know that if we substitute for it a pack of cards, they will be strongly sheared on one another; and the shearing stress is greatest in the “neutral zone,” where neither tension nor compression is manifested: that is to say in the line which cuts at equal angles of 45° the orthogonally intersecting lines of pressure and tension.
In short we see that, while shearing stresses can by no means be got rid of, the danger of rupture or breaking-down under shearing stress is completely got rid of when we arrange the materials of our construction wholly along the pressure-lines and tension-lines of the system; for along these lines there is no shear.
To apply these principles to the growth and development of our bone, we have only to imagine a little trabecula (or group of trabeculae) being secreted and laid down fortuitously in any direction within the substance of the bone. If it lie in the direction of one of the pressure-lines, for instance, it will be in a position of comparative equilibrium, or minimal disturbance; but if it be inclined obliquely to the pressure-lines, the shearing force will at once tend to act upon it and move it away. This is neither more nor less than what happens when we comb our {687} hair, or card a lock of wool: filaments lying in the direction of the comb’s path remain where they were; but the others, under the influence of an oblique component of pressure, are sheared out of their places till they too come into coincidence with the lines of force. So straws show how the wind blows—or rather how it has been blowing. For every straw that lies askew to the wind’s path tends to be sheared into it; but as soon as it has come to lie the way of the wind it tends to be disturbed no more, save (of course) by a violence such as to hurl it bodily away.
In the biological aspect of the case, we must always remember that our bone is not only a living, but a highly plastic structure; the little trabeculae are constantly being formed and deformed, demolished and formed anew. Here, for once, it is safe to say that “heredity” need not and cannot be invoked to account for the configuration and arrangement of the trabeculae: for we can see them, at any time of life, in the making, under the direct action and control of the forces to which the system is exposed. If a bone be broken and so repaired that its parts lie somewhat out of their former place, so that the pressure-and tension-lines have now a new distribution, before many weeks are over the trabecular system will be found to have been entirely remodelled, so as to fall into line with the new system of forces. And as Wolff pointed out, this process of reconstruction extends a long way off from the seat of injury, and so cannot be looked upon as a mere accident of the physiological process of healing and repair; for instance, it may happen that, after a fracture of the shaft of a long bone, the trabecular meshwork is wholly altered and reconstructed within the distant extremities of the bone. Moreover, in cases of transplantation of bone, for example when a diseased metacarpal is repaired by means of a portion taken from the lower end of the ulna, with astonishing quickness the plastic capabilities of the bony tissue are so manifested that neither in outward form nor inward structure can the old portion be distinguished from the new.
Herein then lies, so far as we can discern it, a great part at least of the physical causation of what at first sight strikes us as a purely functional adaptation: as a phenomenon, in other words, {688} whose physical cause is as obscure as its final cause or end is, apparently, manifest.