Composite Laminates

Abstract

Lamination theory for the stress in the individual layers of a composite laminate is reviewed, and appropriate failure criteria are introduced. Optimization of lay-up, based on lamination theory, involves the orientation of the different layers and the number of plies per layer to best tailor the material to the specific application. Different methods of optimization are introduced, including use of a simplified netting analysis for an initial estimate of the required lay-up. Starting from this, an iterative procedure using full lamination theory is described, making step-by-step adjustments to the number of plies to reach a minimum. The discrete ply thickness is a significant complication in formal optimization methods. With integer variables for the number of plies, the ‘branch-and-bound’ method progressively eliminates combinations of variables that cannot lead to a solution in the search for an optimum. Alternatively, a genetic algorithm—a semi-random process—retains a family of designs and repeatedly combines the best of them to arrive at an optimum. A spreadsheet program is presented for the analysis and optimization of a composite laminate under in-plane load, with discrete ply thickness and internal stresses due to change in temperature, based on either the Tsai–Hill or the Tsai–Wu failure criterion.

Exercises

Use the material data in Table 8.1 in the exercises below.

• 8.1 Calculate the $A_{{11}}$ and $A_{33}$ terms of the A-matrix for a symmetric laminate consisting of 4 plies at +60°, 4 plies at −60° and 8 plies at 0°, with a ply thickness of 0.125 mm. Use the spreadsheet ‘Composite Laminate’ to verify the values obtained. With the full A-matrix in the spreadsheet, calculate the elastic constants of the laminate.

Follow the procedure in Example 8.1.

• 8.2 If the laminate in Exercise 8.1 carries a tensile load of 1000 N/mm in the 0° direction, calculate the stresses in the 0° and ±60° layers. Use the spreadsheet to verify the values obtained.

Follow the procedure in Example 8.2.

• 8.3 Calculate the values of the Tsai–Hill criterion for each layer of the laminate in Exercise 8.2, under the given tensile load. If the load is increased, at what load does first ply failure occur according to the Tsai–Hill criterion?

Identify the stress component making the largest contribution to the Tsai–Hill criterion in the most critical layer.

• 8.4 Compare the failure envelopes of the Tsai–Hill and Tsai–Wu criteria for different combinations of stress $\upsigma_{1}$ and $\upsigma_{2}$ in a single layer of a composite laminate.

Use the appropriate tensile or compressive strength data in Table 8.1 . For a series of values of $\upsigma_{1}$ , calculate the corresponding pair of values of $\upsigma_{2}$ to just satisfy each criterion. (This may more easily be done by Goal Seek in Excel.) To compare the two criteria, make a plot of the different combinations of $\upsigma_{1}$ and $\upsigma_{2}$ at failure on the same polar plot with axes $\upsigma_{1}$ and $\upsigma_{2}$ . Notice the discontinuity in the Tsai–Hill curve as $\upsigma_{1}$ or $\upsigma_{2}$ becomes negative.

• 8.5 Derive the first row of the ${\mathbf{T}}^{T}$(transpose) matrix in the second of Eq. (8.5) to transform stresses in layer axes into stresses in laminate axes.

Sketch a right-angled triangle with sides normal to stresses $\upsigma_{1}$ and $\upsigma_{2}$ (at angle $\uptheta$ to the laminate axes) and a third side normal to stress $\upsigma_{x}$ . Calculate the resulting forces on all three sides of the triangle under stresses $\upsigma_{1}$, $\upsigma_{2}$ and shear stress $\uptau_{12}$ . Assume some arbitrary size and thickness of the triangle. Calculate the horizontal components of the resulting forces on all three sides of the triangle. By equilibrium of the three horizontal force components, deduce the three required terms of the ${\mathbf{T}}^{T}$ matrix.

• 8.6 Use the method in Example 8.4 for the initial design of a 0°, ±45°, 90° laminate under the following alternative loading cases:

  $$ \begin{array}{*{20}l} {N_{x} = 2000\,{\text{N/mm}}\, ,} \hfill & {N_{xy} = 1000\,{\text{N/mm,}}} \hfill \\ {N_{y} = 500\,{\text{N/mm}}\, ,} \hfill & {N_{y} = 1500\,{\text{N/mm,}}} \hfill \\ {N_{xy} = 1000\,{\text{N/mm}}\, ,} \hfill & {N_{xy} = 500\,{\text{N/mm}}.} \hfill \\ \end{array} $$

The laminate should be balanced and symmetric, with a ply thickness of 0.125 mm.

Select the larger number of plies in each fibre direction from the two loading cases. From the solution choose an actual lay-up, observing the practical restrictions in Table 8.2.

• 8.7 Use the iterative redesign procedure in Sect. 8.2.2 to improve the design in Exercise 8.6 above.

Use the program ‘Composite Laminate’ to calculate Tsai–Hill values at each step.

• 8.8 Use the program ‘Composite Laminate’ with the Tsai–Hill criterion to optimize the 0°, ±45°, 90° balanced, symmetric laminate in Exercise 8.6, under the same loading and with the same ply thickness.

Use the GRG Nonlinear method in Solver. Add constraints for the fixed ply angles. Repeat the optimization from a few different starting points.

• 8.9 Repeat Exercise 8.8 using the Evolutionary method in Solver.

Repeat the optimization a few times with ‘Population Size’ (in Options/Evolutionary) set to 10 and ‘Seed’ set to zero to generate different starting points.

• 8.10 Calculate the minimum thickness of the laminate in Exercises 8.8 and 8.9 above if there is no restriction on ply angles and the discrete ply thickness is ignored.

Use the GRG Nonlinear method in Solver. Delete the integer constraints on numbers of plies in Solver and constraints on fixed ply angles (if previously added). Select ‘Multistart’ (in Options/GRG Nonlinear) to assist in locating a true optimum. Set ‘Population Size’ to 10 and ‘Random Seed’ to zero.