Oblique bending of a bimetallic bar with a cross section bounded by circular arcs

Abstract

The p re sen t paper concerns the bending of a twol ayer p r i s m a t i c ba r whose c ross sect ion is bounded by c i r c u l a r a r c s . The solution to the problem is cons t ruc ted with the aid of funct ions s i m i l a r to the bending s t r e s s function proposed by Timoshenko [1] with r e f e r ence to the bending of homogeneous rods . w Let us cons ider the e las t i c bending of the twol aye r p r i s m a t i c bar whose c ross sect ion is shown in the f igure . The side sur faces 71 and 7z as well as the boundary 3 0 between the two layers are all cy l ind r i ca l su r faces , The or igin of the r ec t angu la r coordinate sy s t em is placed in the fixed end of the ba r in such a way that the y axis coincides with the axis of symmet ry , while the x axis coincides with the c r o s s s e c t i o n chord. The force is applied to the f ree end of the bar and passes through the bend center , so that there is no to r s ion . Let the l aye r s be isot ropie , the e las t ic moduIi E~ and E 2 of the m a t e r i a l s dis t inct , and the Po isson coeff ic ients #j and P2 equal . Then, as in the bending of homogeneous rods, only three components of the s t r e s s ) t e n s o r a r e d i f ferent f rom zero, namely 0"~1), T(zlx ) , T~ly ) for layer I and the analogous components for l ayer II. Henceforth the analogous exp r e s s ions for both reg ions will contain the index i which has the value i =1 in region I and i = 2 in region II. The indices of p wil l be omit ted . The n o r m a l s t r e s s atzL)" is given by the fo rmula