Since the strengths of beams attached at one end or supported on props, the other dimensions being the same, vary as the squares of tho vertical depths, it follows that the most advantageous position. when the areas of the transverse sections are equal, is that in which the broadest surface is in a vertical position. In this manner girders and joists in edifices are invariably placed.
When a beam or bar is attached at one end to a wall, or when it turns upon its middle poiut like the great lever of a steam-engine, if it be required that the beam should be equally strong in its whole length, it should be made to taper towards its extremities. When the depth of the beam is constant, the breadth at any point should ho propor tional to its distance from the extremity. the breadth is to be constant, the vertical face of the beam shenld have tho form of the common parabola ; and when both breadth and depth vary, a longitu dinal section of the beam should have the figure of that which is called a cubical parabola.
If a weight bo applied at any point in the length of a beam which is supported on two props, the strain produced by it will be the greatest when it is placed in the middle ; and the strain varies as the product of the distance of the weight from the points of support. For let AB (fig. 3) represent a beam eupported at A and 8, and let o be any point in it. Imagine a weight w to be applied at any point r; then, by the nature of the lever, ne:rn::w: the pressure exerted by w on the point A, ; and this term expresses the re-action of AB the prop at A in consequence of the weight at P. Then also A AB is equal to the strain at c produced by this re-action. Again, i111400 a weight w' to be applied at re; then we shall have, as before, : r' n : : w' : L'' r' and this last term expresses the reaction of the AB prop at a in consequence of the weight at P' : also . P' o ie equal • An to the strain at o produced by this reaction. The sum of these strains is equal to the whole strain at o produced by the two weights. But when P and P' coincide with c, wo have r' A=A sr, p B=BC; and the sum of the two strains 0+ W'. a B or, putting w" for w n AB ve, we have the strain at c, in consequence of the weight w" placed there, equal to C. n and if w„ be a constant weight applied An at any point in AB, tho strain will vary as n o, B C. This rectangle, and consequently the strain, is a maximum when c is in the middle of the lino.
If a weight be diffused over a beam which is fixed at one end to a wall, it may be considered as acting at its centre of gravity, which, if the diffusion be uniform, will Ito in the middle of the length of the beam. The momentum of the strain will consequently be equal to
half of that which would result from an equal weight attached to the opposite end.
When a body is compressed in a direction perpendicular to the length of the fibres, the points of support being very near and on opposite sides of the place at which the force is applied, the strain to which the body is then subject has been called by Dr. Young the force of dctrusion. This species of strain sometimes occurs in the construction of machinery ; but few experiments have yet been made to determine the strength by which materials resist it. From these however it appears that the strength is proportional to the area of the transverse section, and that it varies from four•thirds twice the strength by which the same material would resist a strain in the direction of its length.
Such machines as capstans and windlasses, and axles which revolve with their wheels, are, when in action, subject to be twisted, so that their fibre, tend to become curved in oblique directions ; and the strain thus produced is called that of torsion. The most natural way of investigating the strength of materials to resist this kind of strain is probably that which was adopted by Dr. Robison : this mechanician imagined the cylindrical body to be composed of an infinite number of concentric hollow cylinders inserted in each other ; and, supposing the , whole to be cut by a plane perpendicular to the axis, he conceived that two particles in the circumference of any one of the concentric circles would resist the effort to separate them, by a force proportional to their distance from the common axis. Hence, if the radius of the whole cylinder be r, and that of any one of the internal cylinders be x; also, if s' represent the force of cohesion between any two particles in r' the outer circumference, we have r : r' : : x : x. The last term ex presses the like force in the circumference of the cylinder, whose radius is x, and the momentum of cohesion is T. But as all the particles in that circumference exert the same power, and the number of particles is proportional to x, it follows that 7 x3 will represent the sum of all the forces in the latter circumference, and 7 x3 dx will represent the sum in a hollow cylinder whose thickness, in the direction of the radius, is infinitely small and equal to dx. Then, by the rules of integration, we have for tho strength of the whole cylinder the r' expression ;. which, between x = 0 and x = r, becomes r and hence the whole strength of any cylinder varies as the cube of the radius.