. The strength of materials; a text-book for engineers and architects. espec-tively, but such rule is not reliable. It is more nearly truefor I beams used as stanchions than for built-up sections. Alternative Approximate Method.—Where the bendingstress is large compared with the direct stress it seems reason-able to allow that instead of the previous treatment we shallsubtract the bending stress from the value of /, used in thestrut formula. The compressive bending stress for an effective eccentricity . using the Rankine formula we shall have , P e d, i =^ = ~ ~^^A 1 + ac2 P /, , c, ^ ed,. AV
. The strength of materials; a text-book for engineers and architects. espec-tively, but such rule is not reliable. It is more nearly truefor I beams used as stanchions than for built-up sections. Alternative Approximate Method.—Where the bendingstress is large compared with the direct stress it seems reason-able to allow that instead of the previous treatment we shallsubtract the bending stress from the value of /, used in thestrut formula. The compressive bending stress for an effective eccentricity . using the Rankine formula we shall have , P e d, i =^ = ~ ~^^A 1 + ac2 P /, , c, ^ ed,. AV ¥^ Cast-iron Struts Eccentrically Loaded.—In dealingwith cast-iron struts with eccentric loads it must be remem-bered that they will probably fail by tension. The safe load P from the tension standpoint X dt , 306 THE STRENGTH OF :\L\TERIALS when /, is the safe tensile stress, and this should be comparedwith the safe load from the compression standpoint, and thelower value adopted. * Modified Euler Theory for Eccentric Loading.—In this case we have, Fig. 137,. or putting Fig . 137. —Eccent ric Loading of Columns P (e + x) = M = - Elfdy^ •*• d-xdy~ m = _ P ( EI^ /P \ E I X + e) d^-xdy~ — m {x + e) ^j The general solution of this is (a- 4- e) = A cos my -\- 3 cos m y* Cf. p. 282, equation (1). (1) = COLUMNS, STANCHIONS AND STRUTS 307 Since x = 0 for y ^ ± ^, B = 0 as before .•. (x + e) = A cos my (2) . • ^ when .X = 0, e = A cos ^^ . m L .. A = e sec ^ m L . •. X = e sec —^— . cos my — e = e (sec —^ . cos my — 1 j (3) At point o where y = 0 eccentricity = x„ + e = e^ / m L ,\ , m L = e ( sec —^ I j + e = e sec -„ - =^^2^7© ••?•^^ where f^ ^ ~\ . •. stress at o = . I 1 + -\-o~ A\ F = (putting ^ = c) A. (l +^f sec ^ ^g (6) Now let 0 = r^ ^ — and call it the Eulerian angle. 2 Ve ^ Then, stress at o =/,.(! + ,2 ^^^ ^) C^) Values of sec 6 are given in the table on p. 308, taken from Professor Basquins paper * previously referred to. c.
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