IV. The weight of a floor of timber has been calculated at 35 to 40 lbs. per square foot; 20 lbs. is usually allowed. A single joisted floor without counter floor, from 1260 to 2000 lbs. per square. A framed floor with counter flooring, from 2500 to = 4000 lbs. per square. V. Partitions, or any other additional weights brought upon the floor, must also be taken into consideration. This is calculated at from 1480 to 2000 lbs. per square. VI. The weight of the load to be carried must always include that of the girder itself. STRAINS ON BEAMS AND GIRDERS. 1628c. These we shall consider under the heads I. TRANSVERSE STRAIN (1628g.), which consists partly of the action of Tension as well as of Compression, each of them being dependent upon the Cohesion of the material. Under II. TENSION (1630c.), will be considered the neutral axis (1630c.), deflection of beams (1630e.), with the modulus of elasticity (1630i), impact or collision (16300.), and the tensile strength (1630p.). Under III. COMPRESSION (1630w.) is considered Deflection of pillars, and Detrusion (1631n.). The subject IV. TORSION (1631x.) closes this section. 162 d. Timber is permanently injured if more than even of the breaking weight is placed on it. Buffon allowed, which is now the custom, for the safe load. Fairbairn states that for bridges and warehouses, cast iron girders should not be loaded with more thanor of the breaking weight in the middle. For ordinary purposes, for cast iron is allowed for the permanent load (Barlow). A little more than can be allowed for wrought iron beams, as that material, from its extensile capability, does not suddenly give way (Warr); but they should never be loaded with more than 4th (Fairbairn). Girders, especially those of cast iron, which are liable to be less strong than intended from irregu larity in casting and cooling, should be proved before use to a little more than the extent of the safe load; this proof, however, should never exceed the half of the breaking weight, as the metal would be thoroughly weakened. Tredgold observes that a load of of the breaking weight causes deflection to increase with time, and finally to produce a permanent set. The Board of Trade limits the working strain to 5 tons, or 11,200 lbs. per square inch, on any part of a wrought iron structure. 1628e. Of all the circumstances tending to invalidate theoretical calculations, the sun is about the worst. Mr. Clark writes, about the Britannia tubular bridge: "Although the tubes offer so effectual a resistance to deflection by heavy weights and gales of wind, they are nevertheless extremely sensitive to changes of temperature, so much so that half an hour's sunshine has a much greater effect than is produced by the heaviest trains or the most violent storm. They are, in fact, in a state of perpetual motion, and after three months' close observation, during which their motions were recorded by a self-registering instrument, they were observed never to be at rest for an hour. The same may almost he said of the large bridges over the dock passages. The sun heats the top flange, whilst the wind, after sweeping along the water, impinges on the bottom flange, keeping it cool and causing it to contract, whilst the top flange is being expanded by the sun, so putting a camber on the bridge much exceeding the deflection caused by the heaviest working load. At the Mersey Docks the top flanges of the bridges are painted white, to assist in meeting the difficulty." TRANSVERSE STRAIN. 1628g. The strength of beams in general is directly as the breadth, directly as the breadth x depth square of the depth, and inversely as the length; thus But a certain length supposed quantity must, however, be added to express the specific strength of any material, a quantity only obtained by experiments on that material. This supposed quantity is rebreadth x depth x S presented by S. We then obtain -breaking weight. Therefore, in length experiments, a simple transposition of the quantities evolves the value of S; thus length x breaking weight S, which S then becomes a constant. As regards the usual form breadth x depth? of a cast-iron girder, using C as a constant for a signification in a girder, similar to that of S in a beam, the formula area of section x dept.2x-breaking weight. The values of S and C are only applicable to a beam or girder of a similar sectional form to that from length which the value was derived, since this constant expresses the specific strength of that form of section. 1628h. Another formula for estimating the strength of beams rests on the knowledge of the resistance (or r) offered by any material to fracture by a tensile or crushing force, and the depth of the neutral axis (or n) of this area in the beam; the latter, of course, x breadth x depth cannot be calculated. except from experiment. The rule is nx length weight. See RESISTANCE, in Glossary. == breaking 16281. TABLE OF THE TRANSVERSE STRENGTH OF TIMBER: 1 Inch Square, 1 Foot Long. 1 x BW 1628k. The results of Barlow, Nelson, Moore, Denison, and some others, are collected in the above table, which gives a mean of the whole (Warr); Barlow's values are also noted separately, being those usually supplied in the Handbooks; and obtained by Barlow's for=S, from experiments on a projecting beam or arm; or from the formula =S, when a beam supported at the ends is under trial. A measurable set is produced by a straining force very much less than that to which the material will be likely to be exposed in practice. Without having this principle in mind, the differences between the actual breaking weight and the permanent set weight of some writers will be misunderstood. The practical man, however, will use one third or some other proportion of these values, as noticed in par. 1628d. (See another Table, par. 1630s). mula, 16281. TABLE OF THE TRANSVERSE STRENGTH OF METALS: 1 Inch Square, 1 Foot Long. 1628m. Fairbairn's experiments on cast irons obtained from the principal iron-works, and made into bars 1 inch square and 5 feet long, proved that the longer beams are weaker than the shorter in a greater proportion than their respective lengths; that the strength does not increase quite so rapidly as the square of the depth; that the deflection of a beam is proportional to the force or load; and that a set occurs with a small portion of the breaking weight. In 59 experiments, the strongest; Ponkey} 7-122 No. 3, cold blast Spec. Grav. Break. Wt. 578 lbs. Ult. deflect. 1.74, hard. '884 tons. For In 59, experiments, the weakest; Plaskynas-6.916 ton, No. 2, hot blast Mean value 440 lbs., affording for the specific strength, S = 1980 lbs., or the rule including n, a comparison of two specimens gave n=2.63. 1628n. Morries Stirling has considerably strengthened cast iron by adding a portion of malleable cast iron. Four experiments, by Hodgkinson, gave the following results :No. 2 quality (20 per cent. scrap), bars 9 ft. long, 2 ins. square S=2248 S=1682 S=2803 S=1996 Compressive power, No. 2 54.62 tons. No. 3 16280. Hodgkinson also found the average breaking weight in pounds of a bar of cast iron, 1 inch square and 4 feet 6 inches long between the supports, Average of 21 samples of hot blast iron Average of 21 samples of cold blast iron to be as follows:- The superior transverse strength of cold blast iron equals nearly 2 SHAPES OF BEAMS AND GIRDERS. 1628p. "Calculation affords the following shapes for iron beams, as being enabled to do the most work with the least expenditure of substance. Beams supported at one end: L If the load be terminal and the depth constant, the form of the beam in breadth should be wedge-shaped, the breadth increasing as the length of the beam (the latter measured from the loaded end). II. If the breadth be constant, the square of the depth must vary as the length, or the vertical section will be a parabola. III. When both breadth and depth vary, the section should present a cubical parabola. IV. When the beam supports only its own weight, it should be a double parabola, that is, the upper as well as the lower surV. When face should be of a parabolic form, the depth being as the square of the length. a beam is loaded evenly along its surface, the upper surface being horizontal, the lower one should be a straight line meeting the upper surface at the outer end, and forming a triangular vertical section; the depth at the point of support being determined by the length of the beam and the load to be sustained. VI. If an additional terminal load be added to such a beam, the under surface should be of a hyperbolic curvature. VII. And in a flanged beam, the lower flange should describe a parabolic curve (as in example IV.). 1628q. "Beims supported at both ends. I. A beam loaded at any one point, as scale beams and the like, should have a parabolic vertical section each way from the loaded point, Fig. 6136. A. fig. 613b. II. In flanged beams, the lines may be nearly straight, and approach the straight lines more as the flanges are thinner. III. A beam loaded uniformly along the whole of its length, should have an elliptic outline for the upper surface, the lower one being straight. This form applies to girders for bridges and other purposes where the load may be spread. IV. With thin flanges, a beam so circumstanced should be of a parabolic figure. V. If a flanged beam have its upper and lower sides level, and be loaded uniformly from end to end, the sides of the lower flange should have a parabolic curvature." (Gregory.) VI. In the case of example III., Fairbairn observes that the greatest strength will be attained, while the breadth and depth is allowed to be diminished A towards the ends. This diminution should take place in curved lines which are strictly parabolic. The most convenient way of doing this is by preserving a horizontal level in the bottom flange, diminishing its width, as well as the height of the girder, as fig. 613c. Thus the spaces bb should be square on plan for the bearings on the wall, &c., and equal to the width of the bottom flange at the centre; the intermediate length / to be curved to the form prescribed. The width of the bottom flange is to be reduced near the ends to one half of its size in the middle, and the total depth of the girder reduced at the ends in the same proportion. At the middle of the bearing, a flange may be cast on to connect the upper and lower flanges, and this will give additional stiffness to the girder. 1628r. Gregory further remarks on this subject: when the depth of the beam is uniform, and (VII.) the whole load is collected in one point (as A, fig. 613d.) the sides of the beam B E Fig. 613d. PLANS. -E- Fig. 613e. should be straight lines, the breadth at the ends, B, being half that where D the load applied. VIII. When the load is uniformly distributed (fig. 613e.) the sides should be portions of a circle, the radius of which should equal the square of the length of the beam divided by twice its breadth. When the breadth of the beam is uniform and (IX.) the load is collected in one point, the extended (under) side should be straight, the depth at the point where the load is applied twice that at the ends, and the lines connecting them straight (fig. 613b.) See example I. When the load is uniformly distributed, X. the extended (under) side should be straight, and the compressed (upper) side a portion of a circle whose radius equals the square of half the length of the beam divided by its depth. See examples III. and VI. When the transverse section of a beam is a similar figure throughout its whole length; XI. if the load be collected at one point, the depth at this point should be to the depth at its extremities as 3:2: the sides of the beam being all straight lines. XII. When the load is uniformly distributed, the depth in the centre should be to the depth at the end as 3: 1, the sides of the beam being all straight lines. VARIOUS LAWS AFFECTING BEAMS AND GIRDERS. 16288. The principles on which the rules subjoined are founded may be seen in Gregory, Mechanics, &c. and Barlow, Strength of Materials, but divested, certainly, of the refine ment of Dr. T. Young's Modulus of Elasticity, and some other matters, which we cannot help thinking unnecessary in a subject where, after exhausting all the niceties of the ques tion, a large proportion of weight is considered too much for the constant load. 1628t. The transverse strength is that power, in the case of a beam, exerted in opposing a force acting in a direction perpendicular to its length. The following formula and rules apply to the various positions in which a bean or girder is placed. = W เ I. If a beam be loose (or supported) at both ends, and the II. If a beam be loose at both ends, and the weight be ap- III. If a beam be loose at both ends, and the weight be applied at an intermediate point; the spaces m with n=1IV. If a beam be fixed at both ends, and the weight be applied in the middle, it will bear one half more than if both ends be loose (I.) V. If a beam be fixed at both ends, and the weight be applied uniformly along the same length, it will bear three times more than the load in the middle of No. 1, than if both ends be loose VI. If a beam be fixed at both ends, and the weight be ap- = W 8 m n VIII. If a beam be fixed at one end only, and the weight be applied in the middle, it will bear half as much again as at the end. IX. If a beam be fixed at one end, and the weight be applied uniformly along its length, it will bear double the load at the end. X. If a beam be fixed at one end only, it is as strong as one of equal breadth and depth, and twice the length which is fixed at both ends. XI. If a beam be supported in the middle and loaded at each end, it will bear the same weight as when loose at both ends and loaded in the middle (as I.) XII. If a beam be continued over three or four points and the load be uniformly distributed, it will suffice to take the part between any two points of support as a beam fixed at both ends. XIII. If some of the parts have a greater load than the others, it will be near enough in practice to take the parts so loaded as supported at the ends only. XIV. If a beam be inclined and supported at both ends, it has its breaking weight equal to that of the same beam when horizontal, multiplied by the length of the inclined beam and divided by the horizontal distance. NOTE. In calculating for the strength of a beam or of a girder, it is usual to reckon on the ends being loose, from the difficulty of fixing the ends in a sufficient manner to warrant the rule in that case being followed: and when the ends are solidly embedded, they should penetrate the wall for a distance equal to at least three times the depth of the beam or girder (par. 1630m.); but this precaution is seldom carried out in practice. 1628u. For the effect of running loads over bars, we must refer to Professor Willis's experiments at Cambridge, given at the end of Barlow's Strength of Materials, &c., 1851. 1628v. Two geometrical methods of finding the best proportion of a beam to be cut out of a given cylinder have been propounded. The stiffest beam, says Tredgold, that can be cut out of a round tree, is that of which the depth is to the breadth as 3 to 1, or nearly as 1.7320508 to 1; this is in general a good proportion for beams that have to sustain a considerable load. The required propor tions are obtained by dividing a diameter as ab in fig. 613f, into two equal parts, ac and cb, then drawing with a and b as centres two arcs through e to cut the circle ine and f; the points aebf being joined, the figure is that of the stiffest beam that can be cut out of a cylinder, to resist a perpendicular strain. It is also observed by Tredgold that the strongest beam which can be cut out of a round tree is that of which the depth Fig. C15/. Fig. 613g. Fig. 615h |