512 Mechanics of Materials 2 §12.1 Axis of curvature R Neutrai oxis Beam cross-section Fig.12.3.Relative positions of neutral axis and centroidal axis. On the opposite side of the neutral axis,where y will be negative,the stress becomes: My My =-hA(R1-y)hAR; (12.12) These equations show that the stress distribution follows a hyperbolic form.Equation(12.12) can be seen to be similar in form to the"simple bending"equationT. o M y=7 with the term hA(RI+y)replacing the second moment of area 1. Thus in order to be able to calculate stresses in deep-section beams with high initial curvature,it is necessary to evaluate h and Ri,i.e.to locate the position of the neutral axis relative to the centroid or centroidal axis.This was shown above to be given by the condition: y ·dA=0. J(R1+y) Now fibres distance y from the neutral axis will be some distance ye from the centroidal axis as shown in Figs.12.3 and 12.4 such that,in relation to the axis of curvature, R+y=Rc+yc with y=yc+h .'from egn.(12.5) (+h) (Rc yc) ·dA=0 Re-writing yc+h (Rc+yc)-Rc+h=(Rc +yc)-(Rc-h). Timoshenko and Roark both give details of correction factors which may be applied for standard cross- sectional shapes to be used in association with the simple straight beam equation.(S.Timoshenko.Theory of Plates and Shells,McGraw Hill,New York;R.J.Roark and W.C.Young,Formulas for Stress and Strain,McGraw Hill,New York).512 Mechanics of Materials 2 912.1 IRc -. 1-11 ! iY‘ Beam cross-section Fig. 12.3. Relative positions of neutral axis and centroidal axis. On the opposite side of the neutral axis, where y will be negative, the stress becomes: (12.12) These equations show that the stress distribution follows a hyperbolic form. Equation (12.12) can be seen to be similar in form to the “simple bending” equationt. aM __ - - YI with the term hA(R1 + y) replacing the second moment of area I. Thus in order to be able to calculate stresses in deep-section beams with high initial curvature, it is necessary to evaluate h and RI, Le. to locate the position of the neutral axis relative to the centroid or centroidal axis. This was shown above to be given by the condition: Now fibres distance y from the neutral axis will be some distance y, from the centroidal axis as shown in Figs. 12.3 and 12.4 such that, in relation to the axis of curvature, R1 + y = R, + yc with Y=Yc+h :. from eqn. (12.5) TTimoshenko and Roark both give details of correction factors which may be applied for standard cross￾sectional shapes to be used in association with the simple straight beam equation. (S. Timoshenko, Theory of Plares and Shells, McGraw Hill, New York; R. J. Roark and W.C. Young, Formulas for Stress and Strain, McGraw Hill, New York)