Laminate Design Charts

Polar plots and carpet plots: two ways to see a whole design space at once instead of one laminate at a time.

Why a chart instead of a number

Analysing one laminate gives you one answer. It tells you whether that layup works, but not whether a slightly different one would work better, or how much margin you have against getting the fibre angles a few degrees off in manufacture.

A design chart answers the surrounding question by sweeping one input across its range and plotting the result. Both charts on this page are built from the same ABD matrix machinery as a single analysis; they just run it many times. They differ in what they sweep: the polar plot sweeps the direction you measure in, the carpet plot sweeps how much of each fibre orientation the laminate contains.

Polar plots: stiffness in every direction

Laminate engineering constants are always quoted in a coordinate system. E_x means "stiffness along x", and x is wherever you decided to put it. Rotate the loading direction and you get different numbers, because a laminate is anisotropic.¹ A polar plot makes that dependence visible by plotting the constant as a radius against direction as an angle.

Constructing it takes no new theory. Measuring in direction θ is the same as leaving the measurement along x and rotating the laminate by -θ instead, so every ply angle has θ subtracted, the ABD matrix is rebuilt, and E_x, E_y, G_xy and ν_xy are read off exactly as they are for a single analysis. Doing it this way means a value on the rose and the same value in the results table can never disagree.

Only 0 to 180 degrees needs computing. A direction and the direction 180 degrees away are the same line in the plane, so the second half of the rose is a mirror of the first.

The same plot for a quasi-isotropic [0/±45/90]_s layup is a circle: E_x is the same whichever direction you load it in.

Reading the rose

• A circle means quasi-isotropic. A layup such as [0/±45/90]_s is equally stiff in every in-plane direction, so its E_x rose is a circle. That is usually the point of choosing it. A circle is also the boring case: there is nothing left to optimise by rotating the part.
• Lobes mean directionality. A unidirectional laminate has two long lobes along the fibres and a narrow waist across them. The narrower the waist, the more the part depends on being loaded in the direction you assumed.
• The steepness matters as much as the peak. If E_x falls off sharply within a few degrees of the peak, a small ply misalignment in manufacture costs real stiffness. Off-axis stiffness drops surprisingly fast: even a modest angle away from the fibres loses a large fraction of E₁.¹
• Different constants peak in different places. A [±45] angle-ply is the clearest example: E_x is at its lowest along x, where the fibres are 45 degrees away from the load, while G_xy is at its highest there. Its two roses are effectively rotated 45 degrees from each other, so plot the constant your load case actually cares about.

Carpet plots: stiffness for a family of layups

Most real laminates are built from three orientations: 0 degrees for axial load, ±45 for shear, 90 for transverse. The design question is then how much of each. A carpet plot answers that in one picture by plotting a property against the percentage of ±45 plies, drawing one curve for each percentage of 0 degree plies. The 90 degree content is whatever is left over. The result is a grid of curves that looks a little like a carpet, which is where the name comes from; the format goes back to the composite design handbooks.²

What makes the chart possible is a specific property of the A matrix. It is a sum over plies weighted only by thickness:

No z-position appears, so A does not care what order the plies are stacked in, nor how thick the laminate is overall. It depends only on the proportion of each orientation present. That is exactly why a chart of percentages is meaningful: two laminates with the same percentages have the same in-plane stiffness even if their stacking sequences are completely different. The membrane constants follow from inverting A:³

The ±45 content is always split evenly between +45 and -45. That even split is what makes the laminate balanced, driving A₁₆ and A₂₆ to zero so that tension produces no shear and G_xy means what you expect it to mean.

Reading the carpet

Work backwards from the requirement. Find the horizontal line for the E_x your design needs, see which curves reach it, and read off the percentage combinations that do. Usually several will, and you then choose between them on the other criteria: shear stiffness, transverse stiffness, weight, or how many plies the percentages round to.

The trade is visible directly in the slope of the curves. Adding ±45 plies buys shear stiffness and costs axial stiffness, so the E_x curves slope down to the right while the G_xy curves slope up. Where a curve is flat, you are getting the extra shear stiffness almost for free; where it is steep, you are paying for it.

What a carpet plot cannot tell you

• Anything about bending. The independence from stacking order that makes the chart work applies to A only. The B and D matrices weight each ply by its distance from the midplane, so they depend on sequence entirely. Two laminates that share a point on the carpet can have very different flexural stiffness, and one may be symmetric while the other warps.
• Whole plies. Percentages are continuous, laminates are not. A carpet point at 33% zeros has to become an actual integer number of plies, and the rounding shifts the real percentages slightly off the chart.
• Strength. The chart is a stiffness chart. Two layups with equal E_x can fail at very different loads, which is what the failure criteria and progressive failure tools are for.

Neither chart replaces analysing the laminate you actually intend to build. They narrow the field, and then you analyse the two or three candidates that survive.

A third sweep: fibre volume fraction

There is one more variable worth sweeping, and it sits a level below the laminate. Before you choose a stacking sequence, the ply itself has properties that depend on how much fibre is packed into it. The volume fraction sweep covers that chart, since it belongs with the micromechanics models it plots.

In the ABD Composites dashboard

The Engineering Constants tool has a Polar tab that sweeps any saved laminate at one degree resolution and lets you switch between E_x, E_y, G_xy and ν_xy. The numbers on the rose come from the same code path as the table above it.

The Carpet Plots tool takes a single UD lamina material and draws the 0 / ±45 / 90 family for the property you select. Because the chart needs only percentages, you do not have to build a laminate first, which makes it the right tool for the earliest stage of a layup decision.

References

1. "Off-Axis Loading of a Lamina," Mechanics of Fibre-Reinforced Composites, DoITPoMS / TLP Library, University of Cambridge. Open access
2. Composite Materials Handbook, CMH-17 (formerly MIL-HDBK-17), vol. 3: Polymer Matrix Composites Materials Usage, Design and Analysis. Carpet plots as a laminate design format.
3. "Stiffness of Laminates," Mechanics of Fibre-Reinforced Composites, DoITPoMS / TLP Library, University of Cambridge. Open access
4. Jones, R.M. Mechanics of Composite Materials, 2nd ed., Taylor & Francis, 1999. Chapter 4.
5. Tsai, S.W. and Hahn, H.T. Introduction to Composite Materials, Technomic, 1980.

Frequently Asked Questions