The elasticity and resistance of the materials of engineering . ¢ ⢠(3) EI ^ = ^. (4) Hence : ^\j:-7^ = - (5) 124 THEORY OF FLEXURE. [Art, 18. The second member shows that a bending moment: M, - M r= M, â applied to a curved beam whose radius of curvature at anysection is p, will produce a change of curvature expressed by : Pi P In other words : f/ie common theory of ficxttre is applicable tocurved beams of slight curvature. In such a case â, Eq. (2), expresses the variation (increase or decrease) of curvature caused by the M. It is tobe distinctly borne in mind, however, that Eq. (2)
The elasticity and resistance of the materials of engineering . ¢ ⢠(3) EI ^ = ^. (4) Hence : ^\j:-7^ = - (5) 124 THEORY OF FLEXURE. [Art, 18. The second member shows that a bending moment: M, - M r= M, â applied to a curved beam whose radius of curvature at anysection is p, will produce a change of curvature expressed by : Pi P In other words : f/ie common theory of ficxttre is applicable tocurved beams of slight curvature. In such a case â, Eq. (2), expresses the variation (increase or decrease) of curvature caused by the M. It is tobe distinctly borne in mind, however, that Eq. (2) itself ismade approximately true only by considering the curvaturevery small. The limits within which the common theory is applicable to curved beams, and the degreeof approximation of the appli-cation, will be shown by thefollowing investigations, inwhich the longitudinal com-pression and extension, due tothe external forces, w^ill beneglected. In the figure let a portionof any curved beam, whoselateral dimensions are smallcompared with its length, be. represented. Let AB represent an indefinitely short length,ds, of the neutral surface. C is the centre of curvature of dsbefore flexure, and C the same point after flexure. Since the Art. 18.] THE COMMON THEORY OF FLEXURE. 125 lateral dimensions are small compared with the length, if thestrains are not great, any normal cross section may, withoutessential error, be taken as plane after flexure, and such planespassing through A and B will then contain the radii of curva-ture at the points A and B. Let : AC = p and AC = palso : Aa = Ab = Be â Bd = unity. « Aa and Bd are the positions taken by Ad and Be after flexure. The angle, before flexure, between two radii A C and BC, in- dsdefinitely near to each other, is â ; after flexure, as the figure shows, the same angle becomes ây. Hence the change in curvature (or change of angle between consecutive radii)caused by flexure is : ds i-j- \p P Now let the amount of sho
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