flexural

and axes.

715. Hence in a rod of isotropic substance the principal Principal axes of flexure (§ 599) coincide with the principal axes of inertia rigidities of the area of the normal section; and the corresponding flexural rigidities [§ 596] are the moments of inertia of this area round these axes multiplied by Young's modulus.

716. The interpretation of the results [§ 712 (2), (3)] to which the analytical investigation has led us is simply that if we imagine the whole rod divided, parallel to its length, into infinitesimal filaments (prisms when the rod is straight), each of these shrinks or swells laterally with sensibly the same freedom as if it were separated from the rest of the substance, and becomes elongated or shortened in a straight line to the same extent as it is really elongated or shortened in the circular arc which it becomes in the bent rod. The distortion of the cross section by which these changes of lateral dimensions are necessarily accompanied is illustrated in the annexed diagram,

and conical

produced in the four sides of a

prism by

principal plane.

Anticlastic in which either the whole normal section of a rectangular beam, curvatures or a rectangular area in the normal section of a beam of any figure, is represented in its strained and unstrained figures, rectangular with the central point 0 common to the two. The flexure flexure in a is in planes perpendicular to YOY,, and concave upwards (or towards X); G the centre of curvature, being in the direction indicated, but too far to be included in the diagram. The straight sides AC, BD, and all straight lines parallel to them, of the unstrained rectangular area become concentric arcs of circles concave in the opposite direction, their centre of curvature, H, being for rods of gelatinous substance, or of glass or metal, from 2 to 4 times as far from O on one side as G is on the other. Thus the originally plane sides AC, BD of a rectangular bar become anticlastic surfaces, of curvatures 1/p and - σ/p, in the two principal sections. A flat rectangular, or a square, rod of India rubber [for which σ amounts (§ 684) to very nearly, and which is susceptible of very great amounts of strain without utter loss of corresponding elastic action], exhibits this phenomenon remarkably well.

Experi

mental

717. The conditional limitation (§ 588), that the curvature is illustration. to be very small in comparison with that of a circle of radius equal to the greatest diameter of the normal section (not obviously necessary, and indeed not generally known to be necessary, we believe, when the greatest dianieter is perpendicular to the plane of curvature), now receives its full explanation. For unless the breadth, AC, of the bar (or diameter perpendicular to the plane of flexure) be very small in comparison with the mean proportional between the radius, OH, and the thickness, AB, the distances from OY to the corners A', C' Uncalcu- would fall short of the half thickness, OE, and the distances of ordinary to B, D' would exceed it by differences comparable with its own amount. This would give rise to sensibly less and greater shortenings and stretchings in the filaments towards the corners than those expressed in our formulæ [§ 712 (2)], and so vitiate the solution. Unhappily mathematicians have not hitherto succeeded in solving, possibly not even tried to solve, the beautiful problem thus presented by the flexure of a broad very thin band (such as a watch spring) into a circle of radius

lated effects

bendings of a thin flat spring.

comparable with a third proportional to its thickness and its breadth. See § 657.

necessity

limitation,

curvature

when a thin

is bent in a

718. But, provided the radius of curvature of the flexure Hence is not only a large multiple of the greatest diameter, but also for stricter of a third proportional to the diameters in and perpendicular § 628, of to the plane of flexure; then however great may be the ratio than $ 588 of the greatest diameter to the least, the preceding solution is flat spring applicable; and it is remarkable that the necessary distortion plane perof the normal section (illustrated in the diagram of § 716) to its does not sensibly impede the free lateral contractions and expansions in the filaments, even in the case of a broad thin lamina (whether of precisely rectangular section, or of unequal thicknesses in different parts).

pendicular

breadth.

to flexure

plate: by a

ing stress;

taneous

stresses in

at right

one another.

719. Considering now a uniform thin broad lamina bent Transition in the manner supposed in the preceding solution, we have of a plate. precisely the case of a plate under the influence of a simple bending stress (§ 638). If the breadth be a, and the thickness b, the moment of inertia of the cross section is b. ab, and Flexure of a therefore the flexural rigidity is Mab3, or Mb3 if the breadth single bendbe unity. Hence a couple K (§ 637) would bend it to the curva- by simulture 12K/Mb3 length-wise (or across its length), and (§ 716) bending, would produce the curvature 12oK/Mb breadth-wise (or two planes across the breadth), but with concavity turned in the contrary angles to direction. Precisely the same solution applies to the effect of a bending stress, consisting of balancing couples applied to the two edges, to bend it across the dimension which hitherto we have been calling its breadth. And by the principle of superposition we may simultaneously apply a pair of balancing couples to each pair of parallel sides of a rectangular plate, without altering by either balancing system the effect of the other; so that the whole effect will be the geometrical resultant of the two effects calculated separately. Thus, a square plate of thickness b, and with each side of length unity, being given, let pairs of balancing couples K on one pair of opposite sides, and A on the other pair, be applied, each tending to produce concavity in the same direction when positive. If and

A denote the whole curvatures produced in the planes of these couples, we shall have

cylindrical

Stress in 720. To find what the couples must be to produce simply curvature: cylindrical curvature, «, let λ = 0. We have

=

in spherical Or to produce spherical curvature, let λ. This gives

curvature:

.(3).

clastic

in anti- Or lastly, to produce anticlastic curvature, equal in the two curvature. directions, let x=-λ. This gives

Hence, comparing with § 641 (10) and § 642 (16), we have, for A the cylindrical rigidity, and for h and k the synclastic and anticlastic rigidities of a uniform plate of isotropic material,

The coefficient A which appears in the equation of equilibrium of a plate urged by any forces [§ 644 (6) and §§ 649...652], and c, which appears in its boundary conditions, are [§ 642 (16)] given in terms of h and k thus simply :

for anti

ure of a plate

also by

trac- transition com- rectangular

from simple torsion of

721. It is interesting and instructive to investigate the Same result anticlastic flexure of a plate by viewing it as an extreme case clastic flexof torsion. Consider first a flat bar of rectangular section arrived at uniformly twisted by the proper application of tangential tions [§ 706 (10)] on its ends. Let now its breadth be parable with its length; equal, for instance, to its length. We prism." thus have a square plate twisted by opposing couples applied in the planes of two opposite edges, and so distributed over these areas as to cause uniform action in all sections parallel to them when the other two edges are left quite free. If, lastly, we suppose the thickness, b, infinitely small in comparison with the breadth, a, in (46) of § 707, we have

The twist per unit of length gives at in the length a, which [§ 640 (4)] is equivalent to an anticlastic curvature (according to the notation of § 639), equal to T. And the balancing couple N applied in only one pair of opposite sides of the square is, as we see by § 656, equivalent to an anticlastic stress (according to the notation of § 637) II = N/a. Hence, for the anticlastic rigidity, according to § 642 (13), we have

which agrees with the value (6) otherwise found in § 720, by the composition of flexures.

traction in

section of

rectangular

It is most important to remark-(1) That one-half of the Analysis of part nrab3 in the value of N given by the formula (46) of normal § 707, is derived from a and ẞ as given by (8) of § 706, and the twisted term тху of y by (45);—and (2) That if we denote by y' prism. the transcendental series completing the expression (45) for y, it is the term nƒƒx dxdy of § 706 (17), that makes up the other half of the part of N in question, and that it does so as follows, according to the process of integrating by parts, in which it is to be remembered that to change the sign of either x or y, simply changes the sign of y' :

dy dy

Gdx (10),