. The strength of materials; a text-book for engineers and architects. es respectively, assum-ing the inside to be in tension and the outside to be incompression, we haveMaximum compressive stress _ / _ M f d, 1 Maximum tensile stress = ft = ir^^l r~-/^~—yr ~ 1 < • • (1^) A R \^Li (R — dt) I Position of Neutral Axis.—The value of y to make /, = 0gives the distance y,, of the neutral axis from d e. ? • L (R + 2/.) Vo = — L R — L t/„ LR R 1 + L , R2 This enables us to find the position of the neutral axis. Alternative Formula.—The stress formula on Winklersassumption that yi = y can be put in
. The strength of materials; a text-book for engineers and architects. es respectively, assum-ing the inside to be in tension and the outside to be incompression, we haveMaximum compressive stress _ / _ M f d, 1 Maximum tensile stress = ft = ir^^l r~-/^~—yr ~ 1 < • • (1^) A R \^Li (R — dt) I Position of Neutral Axis.—The value of y to make /, = 0gives the distance y,, of the neutral axis from d e. ? • L (R + 2/.) Vo = — L R — L t/„ LR R 1 + L , R2 This enables us to find the position of the neutral axis. Alternative Formula.—The stress formula on Winklersassumption that yi = y can be put in a number of alternativeforms. Suppose, for instance, that the neutral axis is at distance y^below the centroid line c c-^ (Fig. 250). Then total strain atp Q = Pi Qi — p Q will be proportional to {y + y„) the distancefrom the •. We write P^ Q^ — p q = m (?/ + y„). Moreover, p q = (?/ + R) x /e o d CURVED BEAMS 537 h _ JPiQi - P„9 _ (y + y.) mE P Q (?/ + R) * /e O D /. ^ • V T^{ where ti is a constant(2/ + ^) (2/ + I^) (13). Fig. 252. Since 2 /„ . a = 0 Eii y + y ^ + R = 0 1. e. y. y «? ^+R a .V + 1/ + R (14) 538 THE STRENGTH OF MATERIALS Again, taking moments about the neutral axis we have Zj y y -^ Bo J. / = M (y + y,.) ? i»+«{2i^}- Resals Construction.—^According to this constructionwe proceed as before and find the link rigidity line n n passing through the centroid g of the curveA R Ri B r/ r is then found by graphical or other methods,and the moment of inertia I^ of the link rigidity curve is thenfound about the line n n. In practice it is sufficient to apply the construction toone half only. Then N N is the neutral axis line and _MR {y + y„)^!~ I, (y + R) ^^ The rule for the line n n passing through the centroid ofthe link rigidity curve is 2/o %a^y Now aj a X R Q a R .R + y yo ^ <^y _Z/R + y ja^ +y This is the relation required by equation (14) Inn = S 1 (2/ + yo? /inN i MR (1/ + yo).•. In equation (16) /, =
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