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CHAPTER III.
OF THE FORM AND STRENGTH OF MATERIALS.
Modes of Estimation, Stress and Strain, Resistance, Extension, Compression, Lateral Strain, Stiffness, Tubes, Strength, Place of Strain, Incipient Fracture, Shape of Timber, Torsion, Limit of Bulk, Practical Remarks.

When materials are employed for mechanical purposes, the power or strength with which they resist external force, depends not merely upon the nature of the material, but upon its shape, its bearings, and upon the manner in which force is applied to it. It is, therefore, important, to consider, not only the qualities of individual substances, but likewise the laws, which are common to different materials, by which they act in resisting mechanical change, from forces applied to them.

Modes of estimation.--Two methods are employed in estimating the strength of materials, in different forms and situations; one by mathematical computation, the other by actual experiment. Neither method is to be looked upon as precisely accurate in its results; yet these results furnish approximations to truth, which, in many cases, it is very useful to understand.

Stress and strain.--Professor Robison, and some other writers on the strength of materials, have enumerated four modes, commonly called strains, by which any force or stress acting upon a solid body may operate to overcome the cohesion of its particles. These are, 1. By extension, producing a tendency to rupture; as in the case of ropes, tie-beams, king-posts, &c. 2. By compression, tending to shorten or crush the material; as in columns, walls, and foundations. 3. By transverse strain, tending to break or bend the material; as in beams, rails, and oars. 4. By torsion or twisting; as in screws, rudders, and axles fixed to wheels. To these, Dr. Young has added another, viz., detrusion or pushing aside, as in the case of a pin or thread operated on by the blades of scissors. The changes called flexure, or bending; fracture, or breaking; and alteration, or permanent change of form without separation, are effects of force exerted on materials.

Resistance.--To these disturbing influences, bodies oppose certain qualities, which depend, in part, upon the nature of the material, and in part on its form, condition, and connection. These are, their strength, by which they resist all permanent changes resulting from mechanical force, but more particularly fracture. Their hardness, by which they resist impressions, or superficial changes. Their stiffness, by which they resist flexure or bending. Their elasticity, by which they regain their original size and form, after any force producing mechanical change in them has ceased to operate. Their tenacity, by which they undergo permanent alteration without fracture. This quality is called ductility, when exposed to extension, and malleability, when exposed to compression. Some authors add the term resilience, to express the quality by which a body resists impulse, like that of a blow, in contradistinction from strength, by which it resists pressure.

Extension---When a bar of any material is drawn in the direction of its length, its resistance, or strength, will be proportionate to its size at the weakest point; i.e., to the area of its cross section at that point. The tie-beam of a roof, the posts of a printing press, and the shaft or piston rod of a pump, are exposed to this kind of strain; and their weakest point is commonly found at the place where they are perforated, or mortised, to connect them with the other parts. Various experiments have been made to determine the comparative strength with which different substances resist extension. Although they do not fully agree in their results, they nevertheless, when taken collectively, afford approximations of some use for practical purposes. An idea of the relative strength of the metals, when extended, may be obtained from Mr. Rennie's experiments, detailed in the Philosophical Transactions for 1818. His experiments were made with bars, six inches long, and a quarter of an inch square. The average number of pounds avoirdupois, which they supported respectively, is in round numbers as follows. Steel, about 8000 pounds. Hammered iron, about 4000. Gun metal and wrought copper, 2000. Cast copper and brass, 1000. Tin, 300. Lead, 100. Experiments have been made on the longitudinal strength of the wood of different European trees; and similar experiments, sufficiently varied, on the trees of this continent, might be a valuable addition to our knowledge.

Compression.--When a bar or beam is compressed in the direction of its length, it resists more powerfully than in any other way. If the beam be long, and its strength be overpowered by pressure, it bends, and then breaks; but if its thickness be as much as a seventh part of its length, it commonly swells in the middle, splits, and is crushed. When a stone block or pillar is crushed, the parts nearest to the force break away, and slide off diagonally at the sides, leaving a pyramidal base. The lower stories of buildings, the piers and piles of bridges, the spokes of carriage wheels, and the legs of furniture, are subjects of this force. According to Mr. Tredgold, a cubic inch of malleable iron will support, without alteration, a weight of about 17,000 pounds; cast iron, 15,000; brass, 7000; oak and mahogany, nearly 4000; tin, 3000; lead, 1500. Granite is crushed by 11,000 pounds to the square inch; white marble, by 6000; Portland stone, by 4000.

When a force acts on a homogeneous straight column, in the direction of its axis, it can only extend or compress it equally through its whole substance. But if the direction of the force is not in the axis, but parallel to it, the extension or compression will then be partial. In a rectangular column or block, when the compressing force is applied to a point more distant from the axis than one sixth of the depth, the remoter surface will be no longer compressed, but extended. In this case, the distance from the axis of the neutral point, or that which is neither compressed nor extended, will be inversely as that of the point to which the force is applied. For example, a weight or compressing force being applied on one side of the block or column CDEF, Fig. 1, and acting in a direction parallel to its axis, the compression will extend only to the line AB, the parts beyond this being extended.

[Illustration: Fig 1]

Lateral strain.--When a beam is acted on transversely, or by a force applied to its side, the effect produced is the joint result of extension and compression. For if it be moved or bent by such a force, from its original direction, the part which becomes convex is extended, while the part rendered concave is compressed. The properties by which a beam resists lateral pressure, are, its stiffness and its strength.

Stiffness.--The stiffness of any substance is measured by the force required to cause it to bend or recede through a given small space in the direction of the force. It appears to be governed by different laws from those of the strength which resists fracture. When a force is applied to a beam transversely, its stiffness is directly as the breadth, and the cube of the depth of the beam, and inversely as the cube of its length.[A] Thus, if we have a beam which is twice as long as another, we must make it, in order to obtain an equal stiffness, either twice as deep, or eight times as broad. When a beam is supported at both ends, its stiffness is twice as great as that of a beam of half the length inserted in a wall, or otherwise firmly fixed, at one end. If both ends are firmly fixed, the stiffness is quadrupled.[A]

[A] Gregory's Mathematics for Practical Men, 389; also Young's Nat. Philosophy, i. 139, and Tredgold's Elements of Carpentry, 31.

Tubes.--A tube or hollow beam is much stiffer than the same quantity, or weight, of matter in a solid form. The stiffness is increased nearly in proportion to the square of the diameter; since the cohesion and repulsion are equally exerted, with a smaller curvature, and act also on a longer lever. We see this principle applied in nature to the stems of reeds, and the bones and quills of animals.

Strength.--The strength of beams of the same kind, and fixed in the same manner, in resisting a transverse force which tends to break them, is simply as their breadth, as the square of their depth, and inversely as their length. Thus if a beam be twice as broad as another, it will also be twice as strong; but if it be twice as deep, it will be four times as strong; for the increase of depth not only doubles the number of the resisting particles, but also gives each of them a double power, by increasing the length of the levers on which they act. The increase of the length of a beam must obviously weaken it, by giving a mechanical advantage to the power which tends to break it; and some experiments appear to show, that the strength is diminished in a proportion greater than that in which the length is increased.

The strength of a beam supported at both ends, like its stiffness, is twice as great as that of a single beam of half the length, which is fixed at one end; and if both the ends are firmly fixed, the strength of the whole beam is again doubled.

[A] The quantity of timber being the same, a beam will be stronger in proportion as the depth is greater; but there is a certain proportion between the depth and breadth, which, if it be exceeded, the beam will be liable to overturn and break sidewise. To avoid this, the breadth should never be less than that given by the following rule, unless the beam be held in its position by some other means.

Divide the length in feet, by the square root of the depth in inches, and the quotient multiplied by the decimal O.6 will give the least breadth that should be given to the beam.--Tredgold's Carpentry, p. 32.

Place of strain.--If a weight or other stress be placed on any given point of a horizontal bar which is supported at both ends, the strain on that point will be proportional to the rectangle of the two segments into which the point divides the bar. Hence, the place where the strain would be greatest is in the middle of the bar, and a given weight would be most likely to break it in that place.

Incipient fracture.--An incipient or partial fracture, at the place of strain, weakens a beam more, than if the whole side of the beam were cut away to the same depth as the fracture. This is because the sound, or stronger parts of the beam tend to straighten themselves, and thus increase the curvature at the point which is weakened. The same cause occasions the breaking of glass in the direction of a cut made by a diamond, or of a crack which has commenced. It also explains the ease with which a bent twig may be cut off, if we begin on the convex or strained side. Mr. Emerson asserts that a triangular beam, which is so strained that the greatest extension takes place at one of its angles, is rendered stronger, rather than weaker, by cutting away this angle to a small depth, so as to convert the beam into a four-sided figure; thus producing the seeming paradox of a part being stronger than the whole. A sharp angle is indefinitely weak, and fracture is more likely to begin in an angle than in a broad surface.

Shape of timber.--It may be inferred from the consideration of the nature of the different kinds of resistance, that if we have a cylindrical tree a foot in diameter, which is to be formed into a prismatic beam by flattening its sides, we shall gain the greatest stiffness by making the breadth or thickness six inches, and the depth ten and a half; the greatest strength by making the breadth seven inches, and the depth nine and three quarters.

Torsion.--The kind of strain called torsion or twisting, consists in the lateral displacement or detrusion of the opposite parts of a solid, in opposite directions; the central particles only remaining in their natural state. The strength, or rather stiffness, with which the shaft of a wheel, or crank resists torsion, increases in a rapid ratio to its diameter. Professor Robison has calculated, that the power of resisting torsion is as the cube of the diameter; and the more recent estimates of M. Duleau make it as the fourth power of the diameter. If the length vary, the resistance to the force of torsion will be inversely as the length, for obvious reasons. It is advantageous in machinery to increase the diameter of shafts which are exposed to this strain, the amount of material remaining the same. For this purpose, they are sometimes made hollow, and sometimes winged with lateral projections.

Limit of bulk.--It is important to recollect that when the bulk of a substance employed becomes very considerable, its own weight may bear so great a proportion to its strength, as to add materially to the load to be supported. In most cases, the weight of bodies increases more rapidly than their strength, and thus causes a practical limitation of the magnitude of our machines and edifices. Thus a roof, or a bridge, may be very strong, when of small, or moderate size; but if the size be extended beyond a certain limit, although the materials and proportion of parts remain the same, yet the structure will not support its own weight. We see also a similar limit in Nature; for if trees and animals were made many times larger than we now find them, and of the same kinds of substance, they would not sustain their own weight. Small animals endure greater comparative violence, and perform greater feats of strength in proportion to their size, than large owes. It has been observed that whales are larger than any land animals, because their weight 15 more equally supported by the pressure of the medium in which they swim.

Practical Remarks.--In frames of houses, and for various other purposes, beams are used of a prismatic form, having straight, parallel sides. But such beams, when exposed to a lateral strain, are not of equal, or duly proportioned strength throughout; and therefore a part of them is superfluous. This consideration is not of much importance in ordinary practical cases. But in cases where economy of the material is important, as in cast-iron rail-roads, also in machinery where it is desirable that the moving parts should be as light as possible, consistently with the requisite strength, it becomes of consequence to ascertain the best form for resisting a force with the smallest amount of material. Mathematicians have calculated the forms of different beams, which are suited to give them, at all points, a strength proportionate to the pressure they sustain, supposing the material to be of uniform texture. But the outline which answers merely to mathematical truth, is, in many cases, too scanty for actual employment; so that in order to obtain sufficient length for a secure connexion of the beam with its bearings, it is necessary to include the mathematical figure in a somewhat similar one, of larger dimensions. The following rules are, most of them, given in substance by Mr. Tredgold.

If a beam be supported at both ends, and a load applied at some one point between the supports, and always pressing downwards, the best plan appears to be, to make the under side, or that opposite the load, perfectly straight; and to make the breadth equal throughout the whole. The upper side should be shaped as in Fig. 2, being highest where the load rests.

[Illustration: Fig. 2]

The same form is proper for a beam supported in the middle, as the beam of a balance. If the beam be strained, sometimes from one side and sometimes from the other, as in the beam of a steam-engine, then both sides should be of the same form, and EA and FB should each be equal to half CD, as in Fig. 3.

[Illustration: Fig. 3.]

If a beam be of equal thickness, and a weight, or force, be applied to its flat side, the shape may be such as is represented in Fig. 4.

[Illustration: Fig. 4.]

If a beam be intended to support a weight uniformly-distributed throughout its length, or a load rolling over it, the line bounding the compressed side should be a half-ellipse, the other side being straight, as in Fig. 5.

[Illustration: Fig. 5.]

Where it is necessary that the upper side should be straight, the above form may be inverted, and the ends adapted to the bearings.

Beams which are fixed at one end only, and support weights, should decrease as they recede from the wall, or point of fixture. If the weight be at the extremity, the outline, in a beam cut from a vertical plank, should be parabolic; but if equally distributed throughout, it may be straight.

[Illustration: Fig. 6.] [Illustration: Fig. 7.]

If a beam be firmly fixed at both ends, and supports a weight in the middle, it should be largest at the ends and in the middle, as in Fig. 8.

[Illustration: Fig. 8.]

For resisting a cross strain, it is advantageous that the edges of a beam should be made thicker than the rest of its substance, so that a section of the beam would be nearly such as is seen in the subjoined figure.[A]

[Illustration: Fig. 9.]

When it is designed that a shaft should be stiff in all directions, it should be tubular, or else ribbed on all sides.

It must be recollected that the foregoing rules prescribe only a general form, the proportions of which must vary with the nature of the material, and the degree of resistance, or load to be supported.

Works which treat of the strength of materials.--ROBISON'S Mechanical Philosophy, 4 vols. 8vo. 1822, vol. i. p. 369, &c.;--BARLOW on Timber, 8vo. 1823;--YOUNG'S Natural Philosophy, 2 vols. 4to. 1807; vol. i. p. 135, &c; vol. ii. art. 333, &c.;--RENNIE, in the Philosophical Transactions, 1818;--DULEAU, Annales de Chimiť, tom. xii.;--TREDGOLD'S Elementary Principles of Carpentry, 4to. 1820;--THIDGOLE'S Essay on Cast Iron, 8vo. 1824;--EMERSON'S Mechanics;--GREGORY'S Mechanics, 3 vols. 8vo. edit. 1826;--GREGORY'S Mathematics for Practical Men, 8vo. 1825.

[A] For the form best suited to resist longitudinal pressure, see the article Column in Chap. VII.


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