. The action of materials under stress; . l axes. Hence the above distortion resultsfrom two shears at right angles, and necessarily equal; andsuch shears cause no change of volume. Fig. 80 shows the distorted cube. A square traced onthe side of the original cube will become a rhombus, the angles CHANGE OF FORM. 199 of which are greater and less than a right angle by the equalamount 0. Now one-half the angle \~ — d has for its tangenti(i — ^) -i- 4(i + ^05 hence I I -\-k = tan 1(1- I — tan ^ 6 I + tan 1 e or A = tan -J 6. But as d is small, I :=z ^ d, ox d ^ 2X. Therefore a stretch and an equa
. The action of materials under stress; . l axes. Hence the above distortion resultsfrom two shears at right angles, and necessarily equal; andsuch shears cause no change of volume. Fig. 80 shows the distorted cube. A square traced onthe side of the original cube will become a rhombus, the angles CHANGE OF FORM. 199 of which are greater and less than a right angle by the equalamount 0. Now one-half the angle \~ — d has for its tangenti(i — ^) -i- 4(i + ^05 hence I I -\-k = tan 1(1- I — tan ^ 6 I + tan 1 e or A = tan -J 6. But as d is small, I :=z ^ d, ox d ^ 2X. Therefore a stretch and an equal shortening, along a pairof rectangular axes, are equivalent to a simple distortion rela-tively to a pair of axes making anglesof 45° with the original axes; andthe amount of the distortion isdouble that of either of the directchanges of length which composeit. This fact also appears from theconsideration that a distortion of asquare is equivalent to an elongation of one diagonal and ashortening of the other in equal For steel, as before, I =Pi = — ^2 = 20,000 lbs. 1,450 725 = 4 45, if 203. Modulus of Shearing Elasticity.—Similarly, equalshearing stresses q on two pairs of faces of a cube, in direc-tions parallel to the third face, will distort that third face intoa rhombus, each angle being altered an amount 6, there being Oi > distortion of shape only, and not change of ; volume. Fig. 81. f Under the law which has been proved ; true within the elastic limit, and the definition/^ of the modulus of elasticity, § 10, a modulusof transverse (or shearing) elasticity, C, alsocalled co-efficient of rigidity, as E may be called co-efficient ofstiffness, may be written, C = q -^ d. As these two unit shears are equivalent to a unit pull andthrust of the same magnitude per square inch, at right angles
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Keywords: ., bookcentury1800, bookdecade1890, booksubjectstrengt, bookyear1897
