§12.1 Miscellaneous Topics 511 Neutral axis Centroidal Beom X 0x15 section (a)Initially stroight beam-Linear (b)Initially curved beam-Non-linear stress stress dis封ribution distribution Fig.12.2.Stress distributions across beams in bending.(a)Initially straight beam linear stress distribution; (b)initially curved deep beam-non-linear stress distribution. i.e. 2·dA=0 (12.6) (R,+y) Unlike the case of bending of straight beams,therefore,it will be seen by inspection that the above integral no longer represents the first moment of area of the section about the centroid.Thus,the centroid and the neutral axis can no longer coincide. The bending moment on the section will be given by: M=a.dA.y=E(R-R) ·dA (12.7) R (R1+y) but (R+y) =dA-/ (R1+y) and from eqn.(12.5)the second integral term reduces to zero for equilibrium of transverse forces. ∫0朵)A=dA=A=M where h is the distance of the neutral axis from the centroid axis,see Fig.12.3.Substituting in egn.(12.7)we have: M E(R1-R2).hA R2 (12.8) From egn.(12.4) (R1+》= E (R1-R2) y R2 M=(R+yhA (12.9) M i.e. (12.10) y hA(R1+y) My My or 0= (12.11) hA(R1+y)hARo512.1 Miscellaneous Topics 51 1 +--f+. Centroidol Beom X axis section ( a) Initially straight beam -Linear ( b Initially curved beam - Non-linear stress stress distribution distribution Fig. 12.2. Stress distributions across beams in bending. (a) Initially straight beam linear stress distribution; (b) initially curved deep beam-non-linear stress distribution. i.e. *dA=O J& (12.6) Unlike the case of bending of straight beams, therefore, it will be seen by inspection that the above integral no longer represents the first moment of area of the section about the centroid. Thus, the centroid and the neutral axis can no longer coincide. The bending moment on the section will be given by: but (12.7) and from eqn. (12.5) the second integral term reduces to zero for equilibrium of transverse forces. .. where h is the distance of the neutral axis from the centroid axis, see Fig. 12.3. Substituting in ean. (12.7) we have: From eqn. (12.4) i.e. or (12.8) (1 2.9) (12.10) (12.1 1)