PROPOSITIONS RELATING to FLEXURE.
A. The steles: V a cylinder is to that of its en monneribing rectangular prune as three times the bulk e the cylinder is tofaur tines that of the prism.
We may consider the different strata of deeds stance as acting on levers equal in length to the die twee of each from the EMS ;, for although these is DO fixed Menai at the axis, yet the whole fame e the same as if such a fulcrum existed, since the op posite maims of the opposite parts would relieve the fulcrum from all pressure. Then the tension deeds stratum being also as 'the same distance x, sad the breadth of the stratum being called ey, the fluxion of the force oar dikes side of the axis will he es'yde while that of the forest of the many the radius being r, ie ersidx. Now x being the area of hid the por tion included between the stratum and the azis, of which the fluxion is gde, the &Kama' of z be ydx or since = 1 rr rr rr or :ad: y +lag rr r k fr 4angds , consequently the fluent of is rr which, when '5= 0 becomes rfx, or one ourth of the product of the square of the radius by the area of the section, while the fluent of that is,' # re,. the force. of the prism, becomes I e or I r= x one third of the product of the same square into the area of the section of the prism.
Hence the radius of curvature of a cylindrical Maa column, instead of (Art. Baum; Prop. El), will 12A Maa ? be - 11/4 the weight of the modulus M' decreasing in the same proportion as the bulk, when 'the prism is reduced to a cylinder. The force is supposed in this proposition to be either, transverse or applied. at a considerable distance from the aids.; but the error will not be material in any other case.
The curvature eing proportional to the distance from the line of direct= of the force,. or to the ordi nate, when that line, is considered ,as the abscise, the elastic' curve must, in this case, initially coincide with a portion of the harmonic curve, well known for its utility in the resolution of a variety of problems of this kind.. Now if the half length of the comeient
curve be called A, corresponding to a quadrant of the generating circle, and the greatest ordinate y, c being the quadrant of a circle of which the radius is unity, the radius. of, ournature r corresponding to y will be ckcy k , that is, a third proportional toy and - the ra dius of the generating circle ; consequently 124 librium without flexure, but sinee.it will be totter, ing, it cannot exist in nature.
By applying this.determination to the. strength 434 wood and iron,, compared. with; the modulus of else, deity, it. appears, that a round column or a square pillar of either of these substances cannot be bent by) any longitudinal force applied to. the axis, which it can withstand without being crushed, unless its length be greater.than 12 or IS times its thickness, respectively: nor.a column or pillar of.stone, mime, it be 40 or 45 times as long as it is thick. Hence we may. infer, as a practical rule, that every piece of timber or iron, intended to withstand any consider. able compressing force, should be at least as many inches in thickness as it is feet in length, in order tth avoid the lolls of force:which necessarily arises.fconx curvature.