The pins which join the links of suspension bridges, and the rivets which join the wrought iron plates of girder bridges, are subject to shearing stress, and the area to be shorn through must be made sufficient to bear the total shearing stress on that part of the structure. Wood is .strong to resist tension, and would be much employed for ties but for the difficulty of taking hold of the ends of the tie in such a way that these ends shall not fail by being shorn.
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Fig. 4.
Fig 4 shows the end of a balk of wood with a strap bolted to it. This strap would be torn off by the shearing of the wood along the dotted lines ab and cd, with a stress which would be much less than that required to overcome the tenacity of the wood ; for if the dimensions of the balk of pine are 6 in. by 2 in. with an inch hole, its tensile strength will be 10 x 5 or 50 tons, while if the bolt be 1 inch diameter, and be placed 4 inches from one end, it could be torn out by shearing 1 G inches of the wood ; now each inch will only resist a shearing stress of say 600 fi>, so that the bolt would be torn out with only 4 - 3 tons. Thus at least eleven such bolts, each 4 inches from its neighbour, and occupying 3 feet 8 inches in length of this balk, with an iron strap of corresponding length, would be required to render the full strength of the balk as a tie available. A similar difficulty is met with when timber is joined, as in fig. 5, where shearing would take place along the dotted lines AB or CD.
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Fig. 5.
Table III.—Ultimate Strength to resist Shearing ?f,
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Name of Material. Tons per sq. inch. Cast-iron 14 5 Wrought Iron 22 Steel Rivets 30 to 36 Red Tine -22 to "35 Larch -43 to 75 Oak (British) 1
The strengths given here for wood are those obtained by testing the resistance to shearing along the grain.
The strength of wrought iron to resist shearing may be taken as practically equal to its strength to resist tension an assumption which facilitates the design of rivetted and other joints.
The strength of steel to resist shearing is less than its strength to resist tearing in the ratio of three to four approximately. Hi vets employed to joint steel plates require, therefore, to be larger or more numerous than those em ployed for iron plates of equal dimensions.
§7. Elasticity.—When a piece of any material is under tension or compression, it is lengthened or shortened by the stress, and the amount of extension or compression for the same length and stress varies with different materials. Approximately correct results are obtained by assuming that the extension or compression of a given piece of the material of uniform cross section, under a uniformly distri buted stress constant throughout its length, is proportional to the length of the piece and to the intensity of the stress.
Let p be the intensity of the stress in tons per square inch, let I be the length of a specimen of a given material in any unit, and let e be the extension or compression observed in the same unit. Then the expression is, on the above assumption, a constant quantity, and this ratio is experimentally found for many materials to be sensibly constant for stresses which do not approach the ultimate strength. This constant ratio E has received the name of modulus of elasticity, and is generally expressed in tons per square inch, this being the unit in which p is measured. Thomson and Tait (Elements of Nat. Phil.) call E the "measure of longitudinal rigidity," a better name than the modulus of elasticity. The astual extension or compression e of any piece of a structure is given by the expression—
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where p is in tons per square inch ; e will be given in terms of the unit in which I is expressed.
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Name of Material Tons per sq. inch, j Name of Material. Tons per sq. inch. Wrought Iron. .10,000 to 13,000 Iron Wire ....... 11,500 6700 Wire Ropes Steel Bars Cast-iron... .13,000 to 19,000 5250 to 10,000 Slate 5800 to 7100 Red Pine 650 to 850 Larch 400 to 600 Oak 535 to 780 Teak... 1070
The modulus of elasticity is very generally assumed as equal for extension and compression, which is nearly true for wrought iron and steel under any stress to which these materials can be safely subjected. The principle of con tinuity shows that there can be no difference in the value of E for positive and negative stresses so long as these are small. The law according to which E varies as p changes is not accurately known for any material. When the stress is small the value of E appears to be more nearly constant in wrought iron than in cast-iron, but the change in the value of E corresponding to a change from a small stress to the ultimate stress is greater in wrought iron than in cast-iron. The values in the table cannot be depended upon to give very exact results, as the elasticity of different speci mens of the same material varies considerably ; the theory is wholly inapplicable when the breaking strain is ap proached ; the engineer, however, seldom requires to calcu late the extensions or compressions when the stress is even so great as one-third of the ultimate strength. For the same reason the engineer need seldom take account of the permanent set caused by stresses exceeding those within which the material may be considered perfectly elastic. B. D. Stouey, in his work on the theory of strains on girders, gives an excellent summary of what is known experimentally concerning set and the modulus of elasticity.
the centre of surface of the cross section of the piece of a structure, and is normal to the cross section, the stress is called axial, and it is usually assumed that this stress will be borne uniformly by all the elements into which the surface of the cross section may be conceived as divided. This is not necessarily true, but it is approximately true for the forms usually employed by engineers. When the stress is not axial it cannot be considered, even approximately, as uniformly distributed ; the greatest intensity of stress will
occur towards that edge of the cross section which is nearest
