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where we note the presence of the stiffness in torsion (G)defined by Equation 16.3.Thus, 16.2 LOCATION OF THE TORSION CENTER Consider the cantilever beam that is clamped at its left end as shown schematically in Figure 16.2,and more particularly the segment limited by the cross sections denoted by Do and D.In the section D,0 is the elastic center and C is the torsion center the position of which we wish to determine. With this objective,we will apply on the cross section D the two following successive loadings: Loading No.1:One applies on the torsion center C of the cross section D a force F situated in the plane of the section. Loading No.2:One applies on the same cross section D a torsional moment denoted as M(see Figure 16.2). When one applies these two loads successively,the final state is independent of the order of the application.As a consequence for the external forces acting on the isolated segment (Do D,the work corresponding to loading No.1 on the displacements created by loading No.2 is equal to the work corresponding to loading No.2 on the displacements created by loading No.1.This can be written in the following form: W(loading Ix displacement 2)W(loading x displacement 1 Now we evaluate these works: a)W (loading 1 x displacement 2) On D creates the bending moments M and thus a normal stress distribution given in the principal axes by Equation 15.17 as: My M Do Do case n°1 case°2 Figure 16.2 Cantilever Beam with Two Successive Loadings 2003 by CRC Press LLCwhere we note the presence of the stiffness in torsion ·GJÒ defined by Equation 16.3. Thus, 16.2 LOCATION OF THE TORSION CENTER Consider the cantilever beam that is clamped at its left end as shown schematically in Figure 16.2, and more particularly the segment limited by the cross sections denoted by D0 and D1. In the section D1,0 is the elastic center and C is the torsion center the position of which we wish to determine. With this objective, we will apply on the cross section D1 the two following successive loadings:  Loading No. 1: One applies on the torsion center C of the cross section D1 a force situated in the plane of the section.  Loading No. 2: One applies on the same cross section D1 a torsional moment denoted as Mx (see Figure 16.2). When one applies these two loads successively, the final state is independent of the order of the application. As a consequence for the external forces acting on the isolated segment (D0 D1), the work corresponding to loading No. 1 on the displacements created by loading No. 2 is equal to the work corresponding to loading No. 2 on the displacements created by loading No. 1. This can be written in the following form: W (loading 1¥ displacement 2) = W (loading 2¥ displacement 1) Now we evaluate these works: a) W (loading 1 ¥ displacement 2)  On D0: creates the bending moments Mz and My, thus a normal stress distribution given in the principal axes by Equation 15.17 as: Figure 16.2 Cantilever Beam with Two Successive Loadings dW dx -------- 1 2 -- · Ò GJ dqx dx -------- Ë ¯ Ê ˆ 2 or 1 2 -- Mx 2 · Ò GJ = = ------------ F F sxx ( )1 Ei Mz EIz · Ò ------------ y Ei My EIy · Ò = – ¥ + ------------ ¥ z TX846_Frame_C16 Page 312 Monday, November 18, 2002 12:32 PM © 2003 by CRC Press LLC
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