Neil McCartney

Properties for Design of Composite Structures

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x 1 squared minus one-half ModifyingAbove epsilon With caret Subscript upper T Baseline x 2 squared period"/>(2.229)

      where the displacement component has been selected to be zero at the origin.

      The through-thickness displacement of the top surface of the beam, at x3=0, can be defined in terms of two lengths, R1 and R2, which are the radii of curvature of this surface in the x1–x3 plane and the x2–x3 plane, respectively. The exact relationships are given by the well-known formulae

      StartFraction 1 Over upper R 1 EndFraction equals minus StartStartFraction StartFraction partial-differential squared u 3 Over partial-differential x 1 squared EndFraction OverOver left-bracket 1 plus left-parenthesis StartFraction partial-differential u 3 Over partial-differential x 1 EndFraction right-parenthesis squared right-bracket Superscript 3 slash 2 Baseline EndEndFraction comma StartFraction 1 Over upper R 2 EndFraction equals minus StartStartFraction StartFraction partial-differential squared u 3 Over partial-differential x 2 squared EndFraction OverOver left-bracket 1 plus left-parenthesis StartFraction partial-differential u 3 Over partial-differential x 2 EndFraction right-parenthesis squared right-bracket Superscript 3 slash 2 Baseline EndEndFraction period(2.230)

      For small deflections

      StartFraction 1 Over upper R 1 EndFraction equals minus StartFraction partial-differential squared u 3 Over partial-differential x 1 squared EndFraction comma StartFraction 1 Over upper R 2 EndFraction equals minus StartFraction partial-differential squared u 3 Over partial-differential x 2 squared EndFraction period(2.231)

      Thus, it follows from (2.229) that

      upper R 1 equals StartFraction 1 Over ModifyingAbove epsilon With caret Subscript normal upper A Baseline EndFraction comma upper R 2 equals StartFraction 1 Over ModifyingAbove epsilon With caret Subscript normal upper T Baseline EndFraction comma(2.232)

      providing a useful physical interpretation of the strain parameters ε^A and ε^T.

      The final requirement is to determine the loading state that is consistent with the various strain parameter values. It is assumed that stresses within the beam can arise from an applied in-plane loading that is equivalent to an applied axial force FA and a transverse force FT acting in the mid-plane between the upper and lower surfaces of the beam, and an axial applied bending moment per unit area of cross section MA and a transverse applied bending moment per unit area of cross section MT. From mechanical equilibrium

      where σA and σT are the effective axial and transverse applied stresses. On substituting (2.220) and (2.221) into (2.233), the following effective axial and transverse stresses are obtained

      The relations (2.234) are now expressed in the form

      On substituting (2.220) and (2.221) into (2.237) the following relations, enabling the determination of the effective axial and transverse bending moments per unit area of cross section, are obtained

      upper M Subscript upper A Baseline plus one-half h zero width space sigma Subscript upper A Baseline equals h left-bracket upper E overTilde Subscript upper A Baseline left-parenthesis one-half epsilon overbar Subscript upper A Baseline plus one-third h ModifyingAbove epsilon With caret Subscript upper A Baseline right-parenthesis plus nu Subscript upper A Baseline upper E overTilde Subscript upper T Baseline left-parenthesis one-half epsilon overbar Subscript upper T Baseline plus one-third h ModifyingAbove epsilon With caret Subscript upper T Baseline right-parenthesis plus one-half nu overTilde Subscript a Baseline sigma Subscript t Baseline minus one-half upper E overTilde Subscript upper A Baseline alpha overTilde Subscript upper A Baseline upper Delta upper T right-bracket comma(2.238)

      upper M Subscript normal upper T Baseline plus one-half h sigma Subscript normal upper T Baseline equals h left-bracket nu Subscript normal upper A Baseline upper E overTilde Subscript normal upper T Baseline left-parenthesis one-half epsilon overbar Subscript normal upper A Baseline plus one-third h ModifyingAbove epsilon With caret Subscript normal upper A Baseline right-parenthesis plus upper E overTilde Subscript normal upper T Baseline left-parenthesis one-half epsilon overbar </p> </div><hr> <div class= Скачать книгу