§12.1 Miscellaneous Topics 513 一.Axi5 of curvoture R R =Rc+yc dy Fig.12.4. (ye+h) ·dA (Rc+yc) =∫+对-- (Rc yc) (Rc+yc) =A-(R-h) 1 ·dA=0 (Rc+yc) A h=Rc- A =Rc- (12.13) (Rc+yc) A A and R1=Rc-h= dA (12.14) (Rc+yc) Examples 12.1 and 12.2 show how the theory may be applied and Table 12.1 gives some dA usefulqio forfor standard shapes of beam cros-section Note Before applying the above theory for bending of initially curved members it is perhaps appropriate to consider the benefits to be gained over that of an approximate solution using the simple bending theory. Provided that the curvature is not large then the simple theory is reasonably accurate;for example,for a radius to beam depth ratio R/d of as low as 5 the error introduced in the maximum stress value is only of the order of 7%.The error then rises steeply,however,as curvature increases to a figure of approx.30%at Rc/d=1.5. (c)Initially curved beams subjected to bending and additional direct load In many practical engineering applications such as chain links,crane hooks,G-clamps etc.,the component cross-sections will be subjected to both bending and additional direct load,whereas the equations derived in the previous sections have all been derived on the assumption of pure bending only.It is therefore necessary in such cases to obtain a solution by the application of the principle of superposition i.e.by resolving the loading system into$12.1 Miscellaneous Topics 513 .. Axis of curvature 11-1- -.__ --~ - I Fig. 12.4. P1 = A - (R, - h) .dA =O A =Re-- (12.13) A h= R, - (12.14) A A and R1= Re - h = J (Rc dA +Ye) =E Examples 12.1 and 12.2 show how the theory may be applied and Table 12.1 gives some useful equations for J - for standard shapes of beam cross-section. Note Before applying the above theory for bending of initially curved members it is perhaps appropriate to consider the benefits to be gained over that of an approximate solution using the simple bending theory. Provided that the curvature is not large then the simple theory is reasonably accurate; for example, for a radius to beam depth ratio R,/d of as low as 5 the error introduced in the maximum stress value is only of the order of 7%. The error then rises steeply, however, as curvature increases to a figure of approx. 30% at R,/d = 1.5. dA r (c) Initially curved beams subjected to bending and additional direct load In many practical engineering applications such as chain links, crane hooks, G-clamps etc., the component cross-sections will be subjected to both bending and additional direct load, whereas the equations derived in the previous sections have all been derived on the assumption of pure bending only. It is therefore necessary in such cases to obtain a solution by the application of the principle of superposition i.e. by resolving the loading system into