Then,taking into account the displacement field in Equation 16.1,the work done on Do is On D:The torsion center C does not move in the plane of the cross section during torsion.The work done by the force F in the displacement field of torsion is nil. b)W (loading 2 X displacement 1) Forceas applied to the torson center does not lead to the rotation of the cross sections around the longitudinal axis x.From this the torsional moment M does not work on the bending displacement field due to F. The equality of the two works is then written as: {高×y+备×@-%+as=0 then: 告w-:t+gus +高J+Ak=0 This relation has to be verified when the force applied at C varies in magnitude and direction in the plane of the section.One can deduce from there that the relation is valid no matter what the values of M,and M are.Both the above integrals are then nil.One extracts from this property the coordinates of the torsion center: 1 立=而Bw电ds In summary,the uniform torsion of a cylindrical composite beam made of perfectly bonded isotropic phases can be characterized by a homogenized 2003 by CRC Press LLCThen, taking into account the displacement field in Equation 16.1, the work done on D0 is
On D1: The torsion center C does not move in the plane of the cross section during torsion. The work done by the force in the displacement field of torsion is nil. b) W (loading 2 ¥ displacement 1) Force as applied to the torsion center C does not lead to the rotation of the cross sections around the longitudinal axis x. From this the torsional moment Mx does not work on the bending displacement field due to . The equality of the two works is then written as: then: This relation has to be verified when the force applied at C varies in magnitude and direction in the plane of the section. One can deduce from there that the relation is valid no matter what the values of Mz and My are. Both the above integrals are then nil. One extracts from this property the coordinates of the torsion center: In summary, the uniform torsion of a cylindrical composite beam made of perfectly bonded isotropic phases can be characterized by a homogenized sxx ( )1 ux ( )2 ¥ dS DÚ Ei Mz EIz · Ò – ------------ ¥ y Ei My EIy · Ò + ------------ ¥ z Ó ˛ Ì ˝ Ï ¸ dqx dx --------j dS DÚ = dqx dx -------- Ei Mz EIz · Ò – ------------ ¥ y Ei My EIy · Ò + ------------ ¥ z Ó ˛ Ì ˝ Ï ¸ F – yzc + zyc ( )dS DÚ = F F F dqx dx -------- Ei Mz EIz · Ò – ------------ ¥ y Ei My EIy · Ò + ------------ ¥ z Ó ˛ Ì ˝ Ï ¸ F – yzc + zyc ( )dS DÚ = 0 Mz EIz · Ò ------------ Eiy F Eiy 2 – zc + Eiyzyc ( )dSº DÚ º My EIy · Ò ------------ Eiz F Eiz2 + yc – Eiyzzc ( )ds DÚ + = 0 yc 1 EIy · Ò ------------ Eiz F dS DÚ = – zc 1 EIz · Ò ------------ Ei y F dS DÚ = TX846_Frame_C16 Page 313 Monday, November 18, 2002 12:32 PM © 2003 by CRC Press LLC