6.8: Tensile–Shear Interactions and Balanced Laminates

Last updated
Save as PDF

Page ID: 35951

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

From equation 15, the 'interaction' terms S₁₆ and S₂₆ are both non-zero and this indicates that, under off-axis loading, normal stresses produce shear strains (as well as normal strains) and shear stresses produce normal strains (as well as shear strains). This tensile-shear interaction is also present in laminates, but does not occur if the loading system is applied along the principal axes of a single isolated lamina, in which case S₁₆ = S₂₆ = 0 as in equation 13.

\[\eta_{x y x}=E_{x} \bar{S}_{16} \text { and } \eta_{x y y}=E_{y} \bar{S}_{26}\]

The extent of this tensile-shear interaction is quantified by the parameters η_xyx and η_xyy (Click to open pop-up)

Balanced laminates

Tensile-shear interactions are undesirable as they lead to distortions and local microstructural damage and failure. A laminate whose interaction ratios are zero is said to be 'balanced' . Use the model below to investigate the variation of η_xyx with loading angle.