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6.6: Off–Axis Loading of a Lamina

  • Page ID
    8204
  • Under off-axis loading of a single lamina the applied stress state can be resolved to give the stresses along the laminar principal axes. The stress state of a body can be described by a stress tensor, shown schematically below, which can be related to the strain tensor by the equation:

    \[\sigma_{i j}=C_{i j k l} e_{k l}\]

    where Cijkl is a fourth rank stiffness tensor containing 81 components.

    Equation 1 can be rearranged to give:

    \[e_{i j}=S_{i j k l} \sigma_{k l}\]

    Equation for stress tensor

    where Sijkl is the compliance tensor. If the body is in equilibrium both the stress tensor and the stiffness tensor must be symmetric about the diagonal. Then writing Equation 11 as a matrix equation (See Nye 1985) and taking into account the symmetry of the composite itself, the number of independent terms in Cpq reduces to a reasonably small number. A few examples are shown here:

    Matrix maths

    Resolving Stresses within a Lamina

    For a single lamina it is reasonable to assume that all the stresses acting are in the laminar plane, so that σ3 = τ23 = τ31 = 0. Assuming orthotropic symmetry (likely for a lamina) equation 2 becomes

    \[\left[\begin{array}{c}
    \varepsilon_{1} \\
    \varepsilon_{2} \\
    \gamma_{12}
    \end{array}\right]=[S]\left[\begin{array}{c}
    \sigma_{1} \\
    \sigma_{2} \\
    \tau_{12}
    \end{array}\right]=\left[\begin{array}{ccc}
    S_{11} & S_{12} & 0 \\
    S_{21} & S_{22} & 0 \\
    0 & 0 & S_{66}
    \end{array}\right]\left[\begin{array}{c}
    \sigma_{1} \\
    \sigma_{2} \\
    \tau_{12}
    \end{array}\right]\]

    when stresses are applied along the principal axes of the lamina.

    Clearly, when σ2 = τ12 = 0.

    \[S_{11}=\frac{1}{E_1}\]

    Similar considerations give

    \[S_{22}=\frac{1}{E_{2}}, \quad S_{66}=\frac{1}{G_{12}}, \quad S_{12}=\frac{-\nu_{12}}{E_{1}}=\frac{-\nu_{21}}{E_{2}}\]

    where

    \[\nu_{21}=\left[f \nu_{f}+(1-f) \nu_{m}\right] \frac{E_{2}}{E_{1}}\]

    and

    \[v_{12}=\left[f v_{\mathrm{f}}+(1-f) v_{\mathrm{m}}\right]\]

    Click here for derivation of Poisson's ratio.

    We can now find elastic constants for a lamina whose fibres are at an angle θ to the loading direction by the following resolving procedure:

    \[\left[\begin{array}{l}
    \varepsilon_{x} \\
    \varepsilon_{y} \\
    \gamma_{x y}
    \end{array}\right]=[\bar S]\left[\begin{array}{l}
    \sigma_{x} \\
    \sigma_{y} \\
    \tau_{x y}
    \end{array}\right]\]

    The result is that under an arbitrary planar loading system, the transformed compliance tensor replaces the compliance tensor in equation 3. Similar to before,

    \[E_{x}=\frac{1}{\bar{S}_{22}}, \quad E_{y}=\frac{1}{\bar{S}_{22}}, \quad G_{x y}=\frac{1}{\bar{S}_{66}}, \quad \nu_{x y}=-E_{x} \bar{S}_{12}, \quad \nu_{y x}=-E_{y} \bar{S}_{12}\]

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