Wo (a) (b) Figure 17.2 Flexural Displacements In this figure,(a)represents a section before and after bending and (b)shows the evolution of the section as a rigid displacement(parameters vo,wo,and e) to which are associated increments ny and n in the plane y,z.Note:Due to the existence of midplane symmetry,the antisymmetric manner with respect to z, these increments are small,but they cannot be neglected a priori.(At this stage, we do not have a definition for the equivalent rotation,noted as e in b). This justifies the interest in the definition of the displacement field relating these increments.A supplementary interest rests in the possibility,during the study,to look after the necessary approximations more closely to lead to a useful technical formulation. 17.2 DISPLACEMENT FIELD The elastic displacement at each point of the laminate has the components u(x,y,z),v(,y,z)and w(x,y,z).With simplified description of Paragraph 12.2.1, one sees in Figure 17.2 (b),average translations denoted as vo and wo.and a rotation of the section denoted as e,to which one superimposes the supple- mentary displacements n,and n.We will define these averages in integral forms as follows: Translation along x direction:By definition,this is uo such that: 1r2 uo =p5 u(x,y,z)dz b/2 Rotation about the y axis:By definition,this is (x,y)such that': .b2 E12 u(x,y,z)x zdz -b/2E1 Approximations that do not appear neatly in the specialized literature. 3 Such a definition of the"average rotation"will be fundamental in the following to ensure the energetic coherence of the formulation for the transverse shears (see Section 17.6.6). 2003 by CRC Press LLCIn this figure, (a) represents a section before and after bending and (b) shows the evolution of the section as a rigid displacement (parameters v0, w0, and qx) to which are associated increments hy and hz in the plane y,z. Note: Due to the existence of midplane symmetry, the antisymmetric manner with respect to z, these increments are small, but they cannot be neglected a priori. (At this stage, we do not have a definition for the equivalent rotation, noted as qx in b). This justifies the interest in the definition of the displacement field relating these increments. A supplementary interest rests in the possibility, during the study, to look after the necessary approximations more closely to lead to a useful technical formulation. 2 17.2 DISPLACEMENT FIELD The elastic displacement at each point of the laminate has the components u(x, y, z), v (x, y, z) and w (x, y, z). With simplified description of Paragraph 12.2.1, one sees in Figure 17.2 (b), average translations denoted as v0 and w0, and a rotation of the section denoted as qx, to which one superimposes the supple￾mentary displacements hy and hz. We will define these averages in integral forms as follows:
Translation along x direction: By definition, this is u0 such that:
Rotation about the y axis: By definition, this is qy(x,y) such that 3 : Figure 17.2 Flexural Displacements 2 Approximations that do not appear neatly in the specialized literature. 3 Such a definition of the “average rotation” qy will be fundamental in the following to ensure the energetic coherence of the formulation for the transverse shears (see Section 17.6.6). u0 1 h -- u x( ) , y, z dz –h/2 h/2 Ú = qy E11 EI11 --------- E12 EI12 + --------- Ë ¯ Ê ˆ u x( ) , y, z ¥ zdz –h/2 h/2 Ú = TX846_Frame_C17 Page 320 Monday, November 18, 2002 12:33 PM © 2003 by CRC Press LLC