EF

supports CC (fig. 613w.). Weights were placed at each end at equal distances from the supports, and the weights being gradually increased, the bar broke simultaneously through at EE. On another trial, a bar broke only in one point F, being a little nearer to the middle. This was considered a sufficient proof that a portion of the metal might be removed from the middle of the bar without diminishing its lateral strength, and that by adding this metal about the points EE, the lateral strength would be increased.

W

Various Problems.

Fig. 613w.

1630 I. When a beam (as sections A and B), with the ends supported, is to be calculated to support a permanent weight in the middle, the formula for obtaining the breadth and depth are feet W lbs-b, and IW =ď2. W weight to be supported, S safe weight, or of the ultimate strength of an inch bar; b and d in inches.

S÷3d2

S÷3 b

II. When a similar beam for obtaining the breadth and depth had the ends fixed, the

III. When a similar beam projects from a wall, and is loaded at the ends, the formula

S÷36-25=d2.

IV. When a similar beam has to support a load placed at some distance from the end (fig. 613u), the effective length must first be obtained by the rule, par. 1629y. Then the formula for the depth is

To find the diagonal of a uniform square cast iron beam, to support a given strain in the direction of that diagonal, when the strain does not exceed the elastic force of the material (Tredgold):—

V. When the beam is supported at the ends, and loaded in the middle, the formula is

VI. When a similar beam has not the strain in the middle of the length

= diagonal in inches. Here A and B refer to fig. 613u.

1630b. To obtain dimensions, &c. of beams and girders :

I. To find the depth of a beam, the length, breadth, and weight being given. For feet W lbs. -d inches. If no breadth or depth be given, let n = any

II. To find the breadth of a beam, the depth, length, and weight being given. For

A and B,

The proportion between the breadth and depth which will afford the best result
is 6 10 depth, in timber.
:
The formula for the least breadth a beam for a
I feet

III. To find the length, the weight, depth, and breadth being given :

IV. To find the constant S, the length, depth, breadth and breaking weight per foot in

length, inch square, being given, For A and B,

V. To find the area of the bottom flange, the length, load, and depth being given. I feet permanent load in tons distributed

beam, or tension not more than 5 tons per square inch.

For E, tension = 2 load +

VI. To find the multiple of depth and area of the bottom flange, the length and load being

given. For D girder, 1 P=d x a.

VII. To find the area of the top flange. For girder, bottom +1.

VIII. To find the area of the side plates. For E girder, area of bottom.

TENSION, ETC.

1630c. The neutral axis. A timber beam supported at the ends and pressed down in the middle by a weight, will have its lower fibres extended, while the upper fibres are

FF

pushed together. Since there are these two strains, there will be some line or point in the depth which is labouring under neither the one nor the other; this is the neutral axis. The further the fibres are from the neutral line, the more they will resist deflection from the load. It might be inferred that the material should be placed so far above and below the neutral line as other circumstances will allow, in order that they may be in a position to exercise the greatest power. The most simple application of these views is shown in Laves's girder (described in CARPENTRY). "As cast iron resists fracture about six times more powerfully under compression than under tension, it is useless to give as much area of material in the upper or compressed, as in the lower or extended, flange of a cast iron beam." Hodgkinson (Experimental Researches, 1846, p. 484-94) states that the position of the neutral axis in cast iron rectangular beams, at the time of fracture, is situated at about of the whole depth of the beam below its upper surface. The sectional area of the top flange of a cast iron girder must be rather more than of the bottom flange, to keep the position of the neutral axis at of the depth. In sudden fractures it was from to of the depth.

1630d. Tredgold, Iron, 1st edit. 1822, p. 53, considered the line of neutral axis in this section to be in the middle of the depth. He notices the curious fact put forth by Du Hamel, who cut beams one-third, one-half, and two-thirds through, and found the weights to be borne-by the uncut beam 45 lbs. ; and by those cut 51 lbs., 48 lbs. and 42 lbs. respectively, which would indicate that less than half the fibres were engaged in resisting extension, although it does not prove that two-thirds of the thickness contributed nothing to the strength, as Robison imagines. Barlow found that in a rectangular beam of fir, the neutral axis was about five-eighths of the depth, as shown by the section of tracture. Warr gives for cast iron, the value of n or neutral axis 2.63; n=6 when the line may come in the middle. Attention should be given to the highly valuabie paper by the Astronomer Royal (Prof. Airy), On the Strains in the Interior of Beams and Tubular Bridges, read in 1862 before the British Association at Cambridge. It is given in the Athenæum for October 11; and its further elucidation in the last edition (1864) of Fairbairn's Application of Cast Iron, &c.

1630e. Deflection. The deflection of a beam supported at the ends and loaded in the middle, is directly as the cube of the length, inversely as the cube of the depth, and inload x length3 versely as the breadth; therefore, deflection. Beams have been said to breadth x depth 3 bear considerable deflection without any injury to the elasticity of the material. Buffor and Tredgold considered the elasticity to remain perfect until one-third of the breaking weight is laid on. Hodgkinson was perhaps the first who practically showed that in a cast iron beam, and part of the breaking weight caused a visible set after that weight was removed; while another beam took a visible set with th part of its breaking weight. He found the permanent set in cast iron beams to be as the square of the load applied. He also found that cast iron beams bore two-thirds, and even more, of their breaking weight for long periods, without any indication of failing. Gregory (Mechanics for Practical Me". 4th edit. 1862) considers that, though the above rule may be correct for beams about 5 feet in length, it does not apply when they are much longer. Thomas Cubitt found by his experiments that, when the length became about 20 feet, the set was only as the weight; and that with larger beams the set was still less. Fairbairn found the impropriety of adopting any rule founded on elastic limits, since it was evident that, while the elasticity of a bar is injured as soon as a weight was applied, the particles or fibres take up fresh positions until the antagonistic forces in the beam are brought nearly to equality, wher one-third or two-thirds of the breaking weight will affect the subsequent deflection of the beam. 1630f. For a rectangular beam of cast iron supported at both ends and loaded in the middl to the extent of its elastic force, = deflection. For similar beams, loaded uni formly, multiply by 025 in place of 02. (Tredgold). It has been stated that the ultimate strength of a girder of the usual proportions may be approximately ascertaine from its deflection under proof, on the assumption that a load equal to half the breaking weight will cause a deflection of a of its length (Dobson). The proportion of th greatest depth of a beam to the span is so regulated, that the proportion of the greates deflection to the span shall not exceed a limit which experience has shown to be consisten with convenience. That proportion, from various examples, appears to be for the workin load, = 2 from to; for the proof load, 600 from to go (Rankine). 1630g. Mr. Dines, when superintending upwards of two hundred experiments for M Cubitt, on cast-iron girders (as section D) whose dimensions are limited, found that whe the load in the centre is taken as ths of the breaking weight, the following formulæ ma be used, (d depth in centre; length in feet) :-I. When the top and bottom flanges an

defl.

12 feet 02
d inches

defl.

=

equal, and the girder parallel, or equal depth throughout, deflection. II. When the flanges are not equal, and the girder is not parallel,

12

12

35 d

deflection. III. When the beam

has no top flange, and the depth varies, = deflection (Gregory).

leet3 Wcwts. C

4 binches dinches

30d=

1630h. The formulæ given by Hurst, Handbook, &c. for finding deflection, which occur under Stiffness of beims, are, I. When supported at the ends and loaded in the middle, -deflection inches. II. For cylinders, 24 d or diam. inches = deflection inches. HI. If the beam be fixed at one end and loaded at the other, the deflection = 16 times the product. IV. If fixed at one end and uniformly loaded, 6 times. V. If supported at both ends and uniformly load d, gths. VI. If fixed at both ends and loaded in the middle, ¡th. VII. If fixed at both ends and uniformly loaded, ths. He gives the following:TABLE OF THE RELATIVE STRENGTH OF BODIES TO RESIST DEFLECTION = C.

:

VIII. The deflection of a rectangular beam is to a cylindrical one, as 1 to 17. IX. When the deflection is taken as th of an inch per foot in length (which is considered to be safe under a proof of § of the breaking weight) then for a beam supported at both ends and = b; // 12 WC = d; =W; b=1; a=C; but XI. For cylinders, 4/17/2WC = diameter.

12 W

C

bd3 C
W
XI. For an uniform load

loaded in the middle, for or 2C for take ths, as before. 16301. The modulus of elasticity, or resistance of materials to stretching, is the term given to the ratio of the force of restitution to the force of compression. It is the measure of the elastic force of any substance. By means of it, the comparative stiffness of bodies may be ascertained. Thus from the following table it will be perceived that a piece of cast tron is 10-7 times as stiff as a piece of oak of equal dimensions and bearing. Resilience, or toughness of bodies, is strength and flexibility combined; hence any material or body which bears the greatest load, and bends the most at the time of fracture, is the toughest. The modulus is estimated by supposing the material to present a square unit of surface, and by any weight or force to be extended to double, or compressed into one-half the original length; such a weight will represent the modulus.

TABLE OF THE MODULUS OF ELASTICITY; WITH THE PORTION OF IT (LIMITING THE COHESION OF THE MATERIAL, OR) WHICH WOULD TEAR THEM ASUNDER LENGTHwise.

•8225 x

1

120 x 120

1,596,300
1,700,000

-S 1,200,000 R
1,714,500

Brass wire

Copper wire, (2)

Iron wire

17,000,000 R

25,300,000 R

1,530,000 15,800,000

Stones, &c.

Portland stone
Slate, Welsh, average
Window glass

White marble

8,580,000 8,000,000 R 2,520,000

R, from Rankine, Civil Engineering.

1630k. Hence, the modulus of elasticity being known for any substance, the weight may be determined which a given bar, nearly straight, is capable of supporting. For instance, in fir, supposing its height 10,000,000, a bar one inch square and 10 feet long may begin to bend with the weight of a bar of the same thickness, equal in length to × 10,000,000 feet 571 feet, that is, with a weight of about 120 lbs; neglecting the effect of the weight of the bar itself. If we know the force required to crush a bar or column, we may calculate what must be the proportion of its length to its depth in order that it may begin to bend rather than be crushed. (Gregory, p. 382.) 16301. For a rectangular beam supported at both ends and the weight applied in the middle, Gregory, p. 388, gives the formula deflection in inches in the middle Here M modulus of elasticity in pounds; 7 length in feet; W weight in pounds; b breadth and d depth both in inches. Fenwick, Mechanics of Construction, gives the formula Mbas or deflection. Here I length is in inches; and I moment of inertia of the section which for a rectangle, = b d3.

W 13

48 MI

432 13 W
Mbd3

=

4 W 3

1630m. As it may often be necessary to calculate the deflection for an arm from that of beam, or vice versa, we notice the statement made by Barlow, edit. 1837, that "the deflection of a beam fixed at one end in a wall and loaded at the other, is double that of a beam o twice the length, supported at both ends, and loaded in the middle with a double weight. But by his editor in 1851, the word double was altered to equal. Certain experiments mad by us on both the beam and the arm, tended to prove that the former was correct (Builder 1866, p. 124); but scientific investigations show that mathematically the latter is correct

but as they mainly depend upon the perfect manner in which the tail of the arm is secured, the former, or double deflection, is recommended to be anticipated in practice.

1630. There is no such thing as permanent elasticity in any rigid material, and the only possible way of constructing a beam which will return to its original form after the load is removed, is a compound or trussed beam, put together in such a way that the permanent alteration of one material counterbalances that of the other. All beams, without exception, will settle in the course of time, even with the lightest load. Not only the load, but the changes of temperature afford a permanent cause of this settling. Facts on this point are difficult to obtain, as the experiments require to be extended over years, and on the same piece of material. Iron rods, one inch square, which may carry 60,000 lbs. before they are torn, stretch permanently by a load of less than 20,000 lbs. The best wrought iron cannot bear more than one-sixth of its load, without being permanently altered. These data apply only where the material is permanently at rest; if motion or accidental increase of burden happens, the above rules and numbers are considerably modified. As elasticity in material varies as much as its strength, and does not follow the same rules as cohesion, it is advisable to make experiments in each particular case where important structures are to depend upon the smallest quantity of material. (Overman).

16300. Impact or Collision. A second force, after direct pressure, is that of impact, says Fairbairn, involving a proposition on which mathematicians are not agreed. For practical purposes, we may suppose a heavy case equal to 2,240 lbs. or one ton, falling from a height of 6 feet upon the floor. According to the laws of gravity, a body falling from a state of rest obtains an increase of velocity in a second of time equal to 32 feet and during that period falls through a space of 16 feet. This accelerated velocity is as the square roots of the distances; and a falling body having acquired a velocity of 8.05 feet in the first foot of its descent, and 6 feet being the height from which a weight of one ton is supposed to fall, we have √6×805=2·449 × 8·0519-714 for the velocity in a descent of 6 feet. Then 19714 × 2240=44,159 lbs. or nearly 20 tons, as the momentum with which the body impinges on the floor. In the present state of our knowledge, this momentum may probably be taken as the measure of the force of impact.—“ On the effects of impact, the deflections produced by the striking body on wrought iron are nearly as the velocity of impact, and those on cast iron greater in proportion to the velocity. The experiments and investigations made for the ? Commissioners on Railway Structures are extremely valuable. Their results showed that "the power of resisting impact increases with the permanent load upon the beam; the greater the weight at rest upon the beam, the greater must be the momentum of a striking body in order to break it. This is satisfactory, as it diminishes the risk from falling weights in warehouses: the more nearly the weight upon the floors approaches the point at which danger begins, the greater is their power of resisting sudden impacts. Comparing the mean results of the experiments on bars not loaded, "we find that the transverse is to the impactive strength as 2685 3744, or as 1 : 1.39. Similarly, when the bar is loaded with 28 lbs. in the centre, the transverse is to the impactive strength as 2685: 4546, or as 1:1.69; and when 391 lbs. is spread uniformly over the bar, the transverse is to the impactive strength as 2685: 5699, or as 12.12."—(Fairbairn, p. 228).

1630p. Tensile strength is that power of resistance which bodies oppose to a separation of their parts when force is applied to tear asunder, in the direction of their lengths, the fibres or particles of which they are composed. Tredgold's assertions of the principles have been combated by Gregory, to whose work we must refer the student for the reasons he gives. If a piece of No. 10 iron wire bears a tension of 2,000 lbs. before it breaks, ten wires will bear ten times 2,000 lbs. If the sections of 50 wires of this number, form the contents of one square inch, then it will bear a stress of 50 x 2000 lbs. before it is torn asunder, provided the wires are so arranged that each will carry its full weight. But it does not follow that a bar of wrought iron of one square inch will carry an equal weight, not even if the iron be of the same quality. If a solid iron rod of one square inch will carry 50,000 lbs., it does not follow that a rod of 10 square inches in section will carry ten times as much. When welded together, the capacity for resistance appears to be weakened. This observation applies to almost every kind of material, and varies only in degree. The tables of cohesion are generally computed to the tearing of the material, but our calculations should never go beyond the excess of elasticity, for fear of injuring the material. (Overman.)

1630q. If the strain upon a rod or strut be greatest on any one side, that side must sustain the whole force or break. This consideration is of great practical moment in estimating the value of all kinds of ties, as king and queen posts, &c.—(Tredgold).

1630r. The formula for the strength of tie-rods, suspension bars, &c. is C tons x area of section in inches = W tons-a quarter to be taken for safe weight-or C lbs. x area of section in inches = W lbs. . C being obtained from one of the columns in Tables I and II.