The elasticity and resistance of the materials of engineering . ral axes of the different sections, and it iscalled the neutral surface of the bent beam. The neutral axisof any section, therefore, is the line of intersection of the planeof section and neutral surface. Hereafter the axis of X will be so taken as to traverse thecentres of gravity of the different normal sections before flex-ure. The origin of co-ordinates will then be taken at thecentre of gravity of the fixed end of the beam, as shown inFig. I. The value of the expression {a^ -\- b^x), in terms of the ex-ternal bending moment,
The elasticity and resistance of the materials of engineering . ral axes of the different sections, and it iscalled the neutral surface of the bent beam. The neutral axisof any section, therefore, is the line of intersection of the planeof section and neutral surface. Hereafter the axis of X will be so taken as to traverse thecentres of gravity of the different normal sections before flex-ure. The origin of co-ordinates will then be taken at thecentre of gravity of the fixed end of the beam, as shown inFig. I. The value of the expression {a^ -\- b^x), in terms of the ex-ternal bending moment, is yet to be determined. Considerany normal section of the beam located at the distance x fromO, Fig. I, and let OA = I. Also let Fig. 2 rep-resent the section considered, in which BC isthe neutral axis and d and ^^ the distances ofthe most remote fibres from BC. Let momentsof all the forces acting upon the portion {l—x)of the beam be taken about the neutral axisBC. If, again, d is the variable width of beam, the internal resisting moment will be :. d iVj bz dz —E(a^ -f ^i-^) z^. b dz. — d. But the integral expression in this equation is the momentof ijiertia of the normal section about the neutral axis^ whichwill hereafter be represented by /. The moment of theexternal force, or forces, /% will be F{l—x) and it will beequal, but opposite in sign, to the internal resisting : Art. 17.] GENERAL FORMULy^. 115 F{l-x) = Af = - E {a, + b,x)I . . (23) M .-. - {a, J^ b,x) = ^j- . • . . (24) Substituting this quantity in Eq. (16) : d^w M dx^ EI (25) It will hereafter be seen that Eq. (25) is one of the mostimportant equations in the whole subject of the ^Resistance ofMaterials y An equation exactly similar to (25) may, of course, bewritten from Eq. (16); but in such an expression J/will repre-sent the external bending moment about an axis parallel tothe axis of Z, No attempt has hitherto been made to determine the com-plete values of u^ v^ and w^ for the
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