MT-1620 al.2002 Assume the following geometrical behavior a) Each cross-section(@ each z) rotates as a rigid body( no distortion"of cross-section shape in x, y) b Rate of twist, k= constant c)Cross-sections are free to warp in the z-direction but the warping is the same for all cross-sections This is the" St. Venant Hypothesis warping"= extensional deformation in the direction of the axis about which the torque is applied Given these assumptions, we see if we can satisfy the equations of elasticity and B.C.·s. SEMI-INVERSE METHOD Consider the deflections Assumptions imply that at any cross-section location Z do z=kz dz careful a constant Rivello uses o! rate of twist (define as0@z=0) Paul A. Lagace @2001 Unit 10-p. 6MIT - 16.20 Fall, 2002 Assume the following geometrical behavior: a) Each cross-section (@ each z) rotates as a rigid body (No “distortion” of cross-section shape in x, y) b) Rate of twist, k = constant c) Cross-sections are free to warp in the z-direction but the This is the “St. Venant Hypothesis” warping is the same for all cross-sections “warping” = extensional deformation in the direction of the axis about which the torque is applied Given these assumptions, we see if we can satisfy the equations of elasticity and B.C.’s. ⇒ SEMI-INVERSE METHOD Consider the deflections: Assumptions imply that at any cross-section location z:  dα α =  dz  z = k z (careful! a constant Rivello uses φ!) rate of twist (define as 0 @ z = 0) Paul A. Lagace © 2001 Unit 10 - p. 6