Page:Encyclopædia Britannica, Ninth Edition, v. 22.djvu/627

 STRENGTH OF MATERIALS G03 worked at a blue heat not only run a much more serious risk of fracture in the process than when worked either cold or red-hot, but become deteriorated so that brittleness may afterwards show itself when the metal is cold. 1 52. The following table gives a few representative data regarding the strength of the more important materials used in engineering (the figures are gathered from the writings of Barlow, Hodgkinson, Kirkaldy, Thurston, Rankine, Unwin, Clark, and others) :

Graphic repre- sentation of distri- buted stress. Ultimate Strength. Tons per Square Inch. Elasticity. Tons, per sq. in. fount's lodulus E. Mod. of Rigidity C. ft / /. ; )JtolO$ 14 15 to 20 27 to 29 24 across 2G along 24 across 22 along 19 acres. 27 to 29 25 19 to 24 25 to 50 35 26 to 32 average about 30 30 to 45 40 to 65 about 28 80 72 25 to G5 42 36 to 58 GO to 75 >Iong the the fibre

,, 20 9 to 13 11 fibre 1 IS to 22, or about tor/, 5000 - to 6000 12,000 - to 13,000 1300 to 2500 American (ordnance) ... ,, ,, strengthened by successive fusions. Vrought-iron Finest Lowmoor and York-/ -5000 Soft Swedish ,, ,, average about Steel Mild steel plates (Siemens or ) about "|of/< 12,000 > to 13,000 1 13,000 13,000 7000 5000 to 5200 J Whitworth's fluid-compressed 40 48 to 6S 70 194 150 10 to 14 15 to 16 28 8 to 13 22 22 11 to 23 1 5 to 26 35 10 to 14 ,, (hard) Steel wire, ordinary, about Tempered steel rope wire rolled 5 8000 5500 6400 2SOO 1500 2200 /4500t 1 6500 6000 2400 33 to 70 28 to 32 2 to 3 7 to 10 2 0-9 3 to 7 1 to 35 " 4 21 to 5J 4 to 7 4 to 6 4 to 7 4 to 7 5500 Tin 3 4 n 1 1000 800 BOO 950 Timber Oak Riga- fir. Ash 2 to 4 4 4 3 2i to 5 li"to2J 2 to 2i 1J to 3 i 750 950 650 7000 Teak Stone Slate Brick JtoJ 53. Space admits of no more than a account of some of the more simple straini machines and engineering structures. The stress which acts on any plane surfs an imaginary cross-section of a strained piece, may be represented by a figure formed by setting up ordinates A, Bb, f &c., from points on the surface, the w length of these being made proportional to the intensity of stress at each point. This gives an ideal solid, which may be called the stress figure, whose height shows the distribution of stress over the surface which forms its base. A line * drawn from g, the centre of gravity of the stress figure, parallel to the ordinates short and elementary ug actions that occur in ice AB (fig. 25), such as V_ ^-"^ 3 jfa f c Fig. 25. Centre of A, &c. , determines the point C, which is called the centre of stress, stress. and is the point through which the resultant of the distributed stress acts. In the case of a uniformly distributed stress, ab is a plane surface parallel to AB, and C is the centre of gravity of the surface AB. "When a bar is subjected to simple pull applied axially that 1 Stromeyer, " The Injurious Effect of a Blue Heat on Steel and Iron," Min. Proc. Inst. "C.E., vol. Ixxxiv., 1880. s to say, so that the resultant stress passes through the centre of ravity of every cross-section, the stress maybe taken as (sensibly) uniformly distributed over any section not near a place where the "orm of the cross-section changes, provided the bar is initially in a state of ease and the stress is within the limits of elasticity. 54. Uniformly varying stress is illustrated by fig. 26. It occurs Uni- in each case for stresses within the elastic limit) in a bent beam, formly n a tie subjected to non-axial pull, and ^_ jr . varying n a long strut or column where buckling ^-" "T- ~2 stress. makes the stress become non-axial. uniformly varying stress the intensity at any point P is roportional to the listance of P from line MN, called/ Fi - 26 the neutral axis, which lies in the plane of the stressed surface and it right angles to the direction AB, which is assumed to be that in which the intensity of stress varies most rapidly. There is no varia- io MN. If MN passes through the centre of gravity of the sur- Fig. 27. X, l>' k face, as in fig. 27, it may easily be shown that the total pull stress on one side of the neutral axis is equal to the total push stress on the other side, whatever be the form of the surface AB. 1e re- .ultant of the whole stress on A B is in that case a couple, whose moment may be found as follows. Let cZS be an indefinitely small part of the surface at a distance x from the neutral axis through C, and let p be the intensity of stress on rfS. The moment of the stress on dS is xpdS. But ^=^ r ^ 1 =^ 2 c/rc 2 (see fig. 27). The whole moment of the stress on AB is/xpdS = (p 1 /a: i ya; 2 rfS=p 1 I/ 1 or p. 2 l/x.,, where I is the moment of inertia of the surface AB about the neutral axis through C. 55. A stress such as that shown in fig. 26 or fig. 28 may be regarded as a uniformly distributed t stress of intensity j> (which is the intensity at the centre of gravity of the surface C) and a stress of the kind shown in fig. 27. The resultant is ^o^, where S is the whole area of the surface, and it acts at a distance CD from C such that the moment > S . CD = +p Q )l/x 2. Hence D/l), and PI = f -CD/I). 56. Simple bendingoccurswhen a Fig. 28. beam is in equilibrium under equal and opposite couples in the plane of the beam. Thus if a beam (fig. 29), supported at its ends, be loaded at two points so that "VVT 7 ; "^Y" 1 tllG T^OT" tion of the beam lying between W 1 and W 2 is subjected to a simple bending stress. On any section AB the VJJ, only stress consists of pull and push, and has Fig. 29. for its resultant a couple whose moment M = W,? 1 = "VV 2 7 2 . This is called the bending moment at the section. If the stress be within the elastic limits it will be distributed as in fig. 30, with the neutral axis at the centre of gravity of the section. The greatest intensities of push and of pull, at the top and bottom edge respect- ively, are (by 54) ^ 1 = Mj/,/I and p.^^ly.Jl, and the intensity at any point at a distance y above or below C is p = M?//I. 57. Let the bending moment now be increased ; non-elastic strain will begin as soon as either p l or ^ exceeds the corresponding limit of elasticity, and "the distribution of stress will be changed in consequence of the fact that the outer layers of the beam are taking set while the inner layers are still following Hooke's law. As a simple in- stance we may consider the case of a material strictly elastic up to a certain stress, and then so plastic that a relatively very large amount of strain is produced without further change of stress, a case not very far from being realized by soft wrought-iron and mild steel The diagram of stress will now take the form sketched in f> 31. If the elastic limit is (say) less for compression than for tension, the diagram will be as in fig. 32, with the neutral axis shifted towards the tension side. AVhcn the beam is relieved from external load it will be left in a state of internal stress, repre- sented, for the case of fig. 31, by the dotted lines in that figure 58 In consequence of the action which has been illustrated (in a somewhat crude fashion) by figs. 31 and 32, the moment required Simple bending. W Bending beyond elastic limits.
 * ion of stress along lines parallel