. The action of materials under stress; . s as the length /; the maximum slopevaries as /^; and the maximum deflection as /^ The slopeand deflection also vary inversely as I, or inversely as thebreadth and the cube of the depth of the beam. The maxi-mum safe deflection, however, consistent with the workingunit stress/, varies as /^ and inversely as j/j, or the depth ofthe beam. These relationships are true for other cases, aswill be seen in what follows. The ease with which problems regarding deflection aresolved depends greatly upon the point taken for the origin, asit influences the value of
. The action of materials under stress; . s as the length /; the maximum slopevaries as /^; and the maximum deflection as /^ The slopeand deflection also vary inversely as I, or inversely as thebreadth and the cube of the depth of the beam. The maxi-mum safe deflection, however, consistent with the workingunit stress/, varies as /^ and inversely as j/j, or the depth ofthe beam. These relationships are true for other cases, aswill be seen in what follows. The ease with which problems regarding deflection aresolved depends greatly upon the point taken for the origin, asit influences the value of the constants of integration. M max. — —{wl) . i/ = —^wP. {wl)P fPV max. — — ^ \ . Vi = ~~^. -WL X i 105. Beam fixed at one end; uniform load of zv per unitover the whole length /; origin at the wall. Let B = ^. Slope X = ^- = —B/(/^ — 2/x 4- x^) =ax ^ _ B {^Px — Ix + \x^ 4- C). When x ^^ o, ^~ = o; . •. C = v^ = _B(|/V — llx + tV^* -f C).When a: = o, X = o; . •. C == o. ?^Cancel factors before I IT lOO STRUCTURAL MECHANICS. For X ?= l, tan. max. slope — — B(/^ — /^ 4- y*) = — \..^. \and V max. := — B(i/^ — \l -f iV/) = — ^^^^^^ 8EI Again, for maximum safe deflection, consistent with unitstress/in the extreme fibre at the dangerous section, M max. = — {wl) },l = ^. 2/1 , {wl)P fP . •. wl =^ -; and v^ - ? — y,r 8 EI 4Eji io6. Combination of Uniform Load and Single Load at one end of a beam -fixed at the other end. Add the corres-ponding values of the two cases preceding. M max. - — [W/ + \{wl)l\,tan. max. slope = — -:— [-J W/^ -|- \ (^wl)P\\ Note, in the expression forzmax., the relative deflectionsdue to a load at the end and to the same load distributedalong the beam; and compare with the respective maximumbending moments. ^li the preceding beam weighs 50 lbs., the addi- 50 . 60* . 3 tional deflection will be ^—- = ii^-» too 8 . 1,400,000 .512 small a quantity to be of importance. In the maj
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Keywords: ., bookcentury1800, bookdecade1890, booksubjectstrengt, bookyear1897
