Method of using the above Table for Horizontal Timbers.

1625. To find the strength of a beam of fir 23 98 ft. long and 5.330 by 9.594 in. Against these dimensions in the Table VI. we find 10378 as the breaking weight. In the Table VII. we find the primitive horizontal strength of oak is to that of fir as 1000 to 918. Hence 1000 918::10378 to a fourth term which=9527; which expresses the greatest strength of such a beam of fir, or that which would break it. Cutting off the last figure on the right hand, that is, taking one tenth, we have 952 for the greatest weight with which such a beam should be loaded.

1625a. If the beam be of chesnut, whose primitive strength is 957, the proportion becomes 1000 957::10378: 9931=the greatest strength of such a piece, and 31 the greatest weight with which it should be loaded.

Method of Application for the vertical bearing Strength.

10

1626. To find the vertical strength of an oak post 9.594 in. square, and 9.594 ft. high, we shall find in Table VII. for the primitive vertical strength, 807 for 19.188 lines English superficial. But as this strength diminishes as the relative height of the post increases, which in this cases is 12 times, we must (1601.), take only of 807, according to the progression there given, that is, 672.

13254-756

1626a. The post being 9.594 in. square, its area will be 9.594 x 122=13254756, and = 692·34, and 692·34 × 672·5=465000, which divided by 10=46500; is the weight with which without risk the post may be loaded.

19.188

16266. If the post be of fir, whose primitive vertical strength to that of oak is as 851 to 807, we have only to use the proportion 807: 465000::851: 490980, which divided by 10=49098; the greatest weight with which it should be loaded.

Method for obtaining the absolute or cohesive Strength.

1627. In respect of this species of strength, which is that with which timber resists being drawn asunder in the direction of its fibres by weights acting at its end, it is only necessary to multiply the area of the section of the piece reduced to lines by the tabular number 1821, if it be oak, and divide the product by 19.188, and the quotient will show the greatest effort it can bear.

13254-75X1821
19.188

1627a. Thus for a piece of oak 9.594 in. square, we have = 1260700 (in round numbers), which divided by 10 gives the greatest weight that should be suspended to the piece.

16276. From Table VII. it will be seen that in the direction of the absolute strength, becch is the strongest wood, and that strength will be 13254-75×2480 (the tabular number) 1363850, which will give 136385 for the greatest weight to be attached.

Of the Strength of Timbers in an inclined Position.

A

19:188

D

=

1628. If we suppose the vertical piece AB to become inclined to the base, experiment proves that its strength to resist (fig. 613a) a vertical effort diminishes as its inclination increases; so that, if from the upper part in D a vertical Dƒ be let fall, and from the points of the base the horizontal line BC be drawn, the strength of the piece diminishes as Bf increases whence, I. The strength of a vertical piece is to that of an inclined piece of the same length and scantling as the length AB is to Bf, or as the radius is to the sine of the inclination of the piece. II. Vertical pieces have the greatest strength to resist a weight, and the weakest are pieces which lie horizontally.

B

Fig. 6132.

1628a. The first of these results furnishes an easy method of finding, by the aid of the last table, the strength of a piece of timber whose length and inclination are known. Thus, suppose a piece of oak inclined 4 692 ft. and 9.594 ft. long; its size being 8528 by 9.594 ins., whose area, therefore, is 11781 74 lines. This must be divided by the tabular number 19-188, and the quotient will be 614. In table VII.,807 is the primitive vertical strength of oak for 19-188 lines superficial of section; but as the piece is more than 12 times the width of its base, we are, as before observed, to take only of 807, or 672-5, which is to be multiplied by 614, and the product is 412915. Then the proportion 9.594: 4692::412915: 843400 is the strength, which, divided by 10= 84340, is the greatest load to which the inclined piece ought to be subjected.

1628b. In Practical or Constructive CARPENTRY, Chap. III. Sect. iv., tables of scantlings for timbers are given more immediately useful to the practical architect than those deducible from the above rule. These tables and rules, however, have been approved by others,

especially by Cresy in his Encyclopædia of Civil Engineering; but in consequence of the very large amount of information obtained, since the first edition of this work, resulting from the investigations of scientific and practical experimentalists, the following condensed summary of the new ideas on the strength of BEAMS, GIRDERS, and PILLARS, both in timber and iron, must necessarily be submitted for the consideration of the reader and student.

1628c. The term beam is applied herein to large rectangular sections; that of girder, to large irregular shapes; and those of bar and irons, to small rectangular and irregular

forms.

1628d. Beams and girders are calculated for the following classes of buildings:1. Light workshops and factories, public halls, churches, and other buildings in which people only accumulate, with warehouses for light goods. For all these an allow ance of 1 cwt., or 168 lbs. per square foot of floor surface, will include the weight of the joisting, the flooring, and the load upon it.

II. Storehouses for heavy goods, or factories in which heavy machinery and goods are placed. For these an allowance of 24 cwt. or 280 lbs. per square foot of floor surface will include the same weights.

III. Ordinary dwelling houses. For these an allowance of 1

cwt, or 140 lbs. per square

foot, will include the same weights. JV. Tredgold calculates 40 lbs. per square foot for the weight of a ceiling, counter floor, and iron girders, with 120 lbs. per foot more, supposing the floor to be covered with people at any time, making together a weight equal to 160 lbs. as the least stress that ought to be taken. Partitions, or any other additional weights brought upon the floor, must also be taken into consideration. (See further s. v. Weight, in GLOSSARY, Addendum.)

V. The weight of the load to be carried must always include that of the girder itself.

STRAINS ON BEAMS AND GIRDERS.

1628e. 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 aris (1630c.), deflection of beams (1630e.), with the modulus of Elasticity (1630i.), impact or collision (16309.), 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.

1628f. Timber is permanently injured if more than even of the breaking weight is placed on it. Fairbairn states that for bridges and warehouses, cast iron girders should not be loaded with more that or 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 th. (Fairbairn). Girders, especially those of cast iron, which are liable to be less strong than intended from irregularity 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.

TRANSVERSE STRAIN.

1628g. The strength of beams in general is, directly as the breadth, directly as the breadth x depth2 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 re breadth 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 breaking weight S, which S then becomes a constant.

breadth x depth

As regards the usual form 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 breaking weight. The values of S and C are only applicable to a beam or girder of a similar sectional form to that from

area of section x depth2x C

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,

cannot be calculated, except from experiment. The rule is breadth x depth2 = breaking

weight. See RESISTANCE, in Glossary.

nx length

1628i. TABLE OF THE TRANSVERSE STRENGTH OF TIMBER: 1 Inch Square, 1 Foot Long. Barlow. Tredgold Hurst. Moles

Warr.

Value of

worth.

Gravity.

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

1 x B W
4 a d

mula ix BW

=S, when a beam supported at the ends is under trial. A measureable 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. 1628f.

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.

Spec. Grav.

In 59 experiments, the strongest; Ponkey]

No. 3,

In 59, experiments, the weakest; Plaskynaston, No. 2, hot blast

7.122

6.916

Break. Wt.

Ult. deflect.

'884 tons. For

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

No. 3 No. 2 No. 3

Compressive power, No. 2

445-5714 pounds 456 9090 pounds

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, to be as follows:Average of 21 samples of hot blast iron Average of 21 samples of cold blast iron The superior transverse strength of cold blast iron equals nearly 24 per cent. son experimented, in 1846 and 1847, on bars of different kinds of cast iron, 1 inch square and 3 feet bearing. The results are given in the Civil Engineer, 1850, pp. 194-9.

SHAPES OF BEAMS AND GIRDERS.

R. Stephen

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: I.

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 surface should be of a parabolic form, the depth being as the square of the length. V. When 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. "Beams supported at both ends. I. A beam loaded at any one point, as scale bcams and the like, should have a parabolic vertical section each way from the loaded point,

A

Fig. G136.

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

towards the ends. This
diminution should take place
in curved lines which are
strictly parabolic. The most 1
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 I 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 should be straight lines, the breadth at the ends, B, being half that where D the load is applied.

B

E

Fig. 613d.

PLANS.

E

Fig. 613e.

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