. The Bell System technical journal . X = , X;, = , e^, = Xj = \y = By = Equation (13a), sin \p = , whence xP = ° tan 1/ = From Equation (9), From Equation (4a), sin 2(^1 = 01 = °tan 01 = tan 01 = 01 = ° FromEquation (9), it is obvious that the maximum value of tan ^ occursat 01 = 45°, and is in this case equal to 528 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 This example is illustrated in Fig. 3. The question of direction of the x and y axes is simply settled by draw-ing a line through either of the acut
. The Bell System technical journal . X = , X;, = , e^, = Xj = \y = By = Equation (13a), sin \p = , whence xP = ° tan 1/ = From Equation (9), From Equation (4a), sin 2(^1 = 01 = °tan 01 = tan 01 = 01 = ° FromEquation (9), it is obvious that the maximum value of tan ^ occursat 01 = 45°, and is in this case equal to 528 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 This example is illustrated in Fig. 3. The question of direction of the x and y axes is simply settled by draw-ing a line through either of the acute angles of the parallelogram, crossingthe parallelogram at an angle <^i with the longer side. This line will beparallel to the x direction, which is, according to the convention, thatof greatest strain. So far no mention has been made of strain in the third dimension;that is, a change in thickness of the sheath. In plastic deformation, thevolume change is generally negligible. This requires that whence KKK = 11 X. = AxAu XmI. Fig. 3—A square and the parallelogram resulting from stretching to lengthratios \x = Al in the x-direction and Xj, = in the y-direction. Table I Principal Strains, per cent Degrees of twist in 3 feet Parallel to Surface Perpendicular to Surface Max. Min. 180 16 06 -19 270 26 09 -27 360 33 14 -34 450 36 20 -39 540 42 19 -41 630 43 22 -43 720 46 24 -45 PRINCIPAL STRAINS IN BUCKLED SURFACES 529 In the example given, X. = , e, = Polyothyloiie sheaths of cabh^ specimens 3 feet long buckled severelyover their entire length when the cables were twisted 720° and showedthe strains given in Table I at steps up to the final twist . The ratio of maximum to minimum strain parallel to the surface isabout 2:1. Tests with a 1:1 ratio , a more severe condition, have shownthat the principal strains at rupture will be of the order of 300 per it is evident that the strains incidental to the most severetypes of handling will not,
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