15.1.1.2 Longitudinal Displacement By definition,longitudinal displacement is denoted by u(x)and written as: u(0)=7 (.y.zy which consists of a mean displacement ux)and an incremental displacement △uxas: ux(x,y,z)=u(x)+Aux(x,y,z) where one notes that: E,△uxdS=0 (15.2) JD 15.1.1.3 Rotations of the Sections By definition,this is the fictitious rotation-or equivalent-given by the following expression: 0(x) 品 Eux(x,y,z)×ydS Or,with the above: 15.1.1.4 Elastic Center Origin 0 of the coordinate y is chosen such that the following integral is zero: Eyds =0 We call elastic center the corresponding point 0 that is located as in the expression above.Then Auy takes the form: Aux(x,y,z)=-ye(x)+nx(x,y,z) with: ∫and∫En.s=09 The displacement ux(,y,2)can then take the form: ux(x,y,2)=u(x)-ye(x)+n(x,y,z). 3 In what follows,the second property is the consequence of Equation 15.2. 2003 by CRC Press LLC15.1.1.2 Longitudinal Displacement By definition, longitudinal displacement is denoted by u(x) and written as: which consists of a mean displacement u(x) and an incremental displacement Dux as: where one notes that: (15.2) 15.1.1.3 Rotations of the Sections By definition, this is the fictitious rotation—or equivalent—given by the following expression: Or, with the above: 15.1.1.4 Elastic Center Origin 0 of the coordinate y is chosen such that the following integral is zero: We call elastic center the corresponding point 0 that is located as in the expression above. Then Dux takes the form: with3 : The displacement ux (x,y,z) can then take the form: 3 In what follows, the second property is the consequence of Equation 15.2. u x( ) 1 · Ò ES ----------- Eiux( ) x, y, z dS DÚ = ux( ) x, y, z = u x() D + ux( ) x, y, z Ei Dux dS DÚ = 0 qz( ) x –1 EIz · Ò ------------ Eiux( ) x, y, z ¥ y dS DÚ = qz( ) x –1 EIz · Ò ------------ u x( ) Ei ydS DÚ Ei DÚ + Dux( ) x, y, z y dS Ó ˛ Ì ˝ Ï ¸ = Ei ydS DÚ = 0 Dux( ) x, y, z = –yqz( ) x + hx( ) x, y, z Eihx ydS DÚ and Eihx dS = 0 *( ) DÚ ux( ) x, y, z = u x( ) – yqz( ) x + hx( ) x, y, z . TX846_Frame_C15 Page 285 Monday, November 18, 2002 12:30 PM © 2003 by CRC Press LLC