. The Bell System technical journal . ons (4) and (5), r (« +1) tan (.i,; - *;) = ^-^ ,^^l (6) - fe) Let ^ = 90° - (0i - 0o) then 1 - ftan (90° - (01 - 02)) = tan ^ = —-. ^^^^ (7) H^ + s. which is the shear between Li and Li . si) 2 {Sx + l/^Si) = tan 01 + cot 0i = ^^— (8) sin Z01 and substituting this in equation (7), . 2K\y tan ^p «^ 201 = -^^^r^ (9) 526 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 whence cos 201 = a/i - 4^^v tanV (Xj — XyY Remembering that and 2, 1 + COS 201 , . cos 01 = (11a) . 2 , 1 - cos 201 , . sm 01 = (lib) and substitutmg Equation (10) in Equation (11), Equation (11) i
. The Bell System technical journal . ons (4) and (5), r (« +1) tan (.i,; - *;) = ^-^ ,^^l (6) - fe) Let ^ = 90° - (0i - 0o) then 1 - ftan (90° - (01 - 02)) = tan ^ = —-. ^^^^ (7) H^ + s. which is the shear between Li and Li . si) 2 {Sx + l/^Si) = tan 01 + cot 0i = ^^— (8) sin Z01 and substituting this in equation (7), . 2K\y tan ^p «^ 201 = -^^^r^ (9) 526 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 whence cos 201 = a/i - 4^^v tanV (Xj — XyY Remembering that and 2, 1 + COS 201 , . cos 01 = (11a) . 2 , 1 - cos 201 , . sm 01 = (lib) and substitutmg Equation (10) in Equation (11), Equation (11) inEquation (3), and then sohdng the quadratic equation thus formed forXx and X„ , we have ,2 ,2 {L? + l7) ± V (L? + L,r - 4L? L? cos^ ^ ,^.^. Ax, Ay — 2 ^^-^ Referring to Fig. 2, and using the law of cosines, and remembering thatLz is the ratio of the strained to the unstrained length of the diagonal, ^ a • t 2-^3 ~ (^1 + -^2 ) r 1 o \ — cos d = sni \p = j—j (13a) 2/viL/2 whence cos lA = ,r>T> ——— ^^^^. Fig. 2—A parallelogram formed by straining a square. L/, La and L3 are theratios of the lengths of the indicated lines to their original lengths. PRINCIPAL STRAINS IN BUCKLED SURFACES 527 This expression, substituted in Equation (12) and reduced, gives Ai, Aj, — \i.^) It may be noted here that a property of the parallelogram, namely, inthe notation used here, //f + L? = Li + Li (15) makes it immaterial which diagonal is used. This may be readily seenby substituting L3 = Li -\- L2 — Li in Equation (14). The effect is merely that of substituting L4 for L3 .In Equation (13a), howe\er, the result is a change in the sign of \l/. As an example of the application of these equations, the measurementsof one specimen were: li = L2 = L3 = From Equation (14), X = , X;, = , e^, = Xj = \y = By = Equation (13a), sin \p = , whence xP = ° tan 1/ = From Equation (9), From Equation (4a), sin 2(
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