Laminate Response to Applied Loads

How a set of applied forces and moments becomes six numbers that describe the whole laminate, how to read them, and what the deformed shape picture is actually showing you.

The ABD matrix tells you how stiff a laminate is. This page is about the next question: given a load, how much does it actually deform? That answer is the hinge of the whole analysis, because every ply stress and every failure check downstream is derived from it.

What "load" means for a laminate

A laminate in CLT has no length and no width. It is a stacking sequence, not a panel. So loads cannot be expressed in newtons: there is no edge length to spread them over. Instead they are expressed per unit width, as six quantities called the force and moment resultants:

• N_x, N_y - in-plane normal forces per unit width, in N/mm. Positive is tension.
• N_xy - in-plane shear force per unit width, in N/mm.
• M_x, M_y - bending moments per unit width, in N·mm/mm. Positive bends the laminate so the top surface goes into compression.
• M_xy - twisting moment per unit width, in N·mm/mm.

The practical consequence is that you convert to resultants before you analyse. A 5 kN tensile load carried by a 250 mm wide strip is N_x = 20 N/mm. A pressure load on a panel is turned into resultants by the plate or beam formulas covered in structural analysis, which is exactly what those tools exist to do.

Solving for the response

CLT's constitutive law runs from deformation to load: stack the three 3x3 stiffness blocks into one 6x6 matrix and multiply.¹

In design you almost always know the load and want the deformation, so the equation is used the other way round. Inverting the 6x6 matrix gives the response directly:

Two things are worth noticing about that inversion. First, it must be done on the full 6x6 matrix, not on A and D separately. Inverting them separately quietly assumes B = 0, which is only true for a symmetric laminate. Second, the response is linear in the load: double the load and you double every strain and curvature. That single property is what makes the rest of this page, and much of the rest of the toolset, work.

Reading the six numbers

The result is six values. The first three describe stretching of the midplane, the last three describe how the laminate curves.

Strains are dimensionless, which makes them awkward to read as decimals: a realistic value is 0.0021. Engineers therefore usually quote microstrain, µε, where 1 µε = 10^-6. That same 0.0021 reads as 2100 µε, which is much easier to compare against a strain allowable or a strain gauge reading.

Watch for responses you did not ask for. Pull on a laminate along x only and you may still get shear strain, curvature, or both. Shear strain from pure tension means the laminate is not balanced (A₁₆ and A₂₆ are non-zero). Curvature from pure tension means it is not symmetric (B is non-zero). Neither is an error; both are consequences of the stacking sequence that the six numbers make visible.

Where the load comes from: three contributions

Mechanical load is not the only thing that deforms a laminate. A temperature change and a change in absorbed moisture each produce their own equivalent force and moment resultants, which add to the mechanical ones before the solve:

Because the solve is linear, you can either add the three load vectors and solve once, or solve three times and add the responses. Both give the same total, exactly. That is what makes a contribution breakdown honest rather than an approximation: the mechanical, thermal and moisture columns are each a real full solve, and they sum to the total column with nothing left over.

It is also what makes the breakdown useful. If a laminate is deforming more than expected, the breakdown tells you which of the three is responsible, and a fix aimed at the wrong one will not help. Cure-induced curvature, for instance, is entirely a thermal contribution, so no amount of reducing the mechanical load will remove it. See hygrothermal effects for where N^T and N^M come from.

Each group is one response component. The bars are the three contributions; the marker is their net total. Bars pointing in opposite directions are contributions that cancel.

Contributions that cancel are worth looking for specifically. A thermal contribution that opposes the mechanical one is carrying part of your load for you at the temperature you analysed, and will stop doing so at a different temperature. A load case that looks comfortable only because two contributions happen to offset is a load case to re-run across the full temperature range.

The deformed shape

The three curvatures are hard to picture as numbers. They are much easier to picture as a surface. Under the Kirchhoff assumption the curvatures are the second derivatives of the out-of-plane deflection w,¹ which integrates to a simple quadratic surface:

Which of the familiar shapes that surface takes is decided by one number, the Gaussian curvature K:²

• K < 0, saddle: anticlastic. It curves up one way and down the other. Pure twist, with only κ_xy non-zero, is a saddle.
• K ≈ 0, cylindrical: curves one way only, and can be rolled from a flat sheet without stretching it.
• K > 0, dome or bowl: synclastic. It curves the same way in both directions, like a shallow cap.
• All curvatures near zero, flat: no bending. Note that a flat midplane can still be stretched, so flat is not the same as undeformed.

The shape is the information; the size is not. CLT has no in-plane dimensions, so there is no true deflection magnitude to draw. What carries meaning is the shape and the relative sizes of the three curvatures, which is why the view exaggerates deflection by an amount you control rather than by a hidden constant.

One case where the picture is knowingly wrong

A thin unsymmetric laminate cooled from its cure temperature does not settle into the saddle that linear CLT predicts. Hyer showed in 1981 that such laminates snap into one of two cylindrical shapes instead, because the room-temperature deflection is large compared with the laminate thickness and the saddle turns out to be an unstable equilibrium.³ Capturing that needs geometrically nonlinear analysis, which CLT is not. If you are looking at an unsymmetric layup with a large temperature change, treat the predicted saddle as an indication that the laminate warps, not as its final shape.

In the ABD Composites dashboard

The Laminate Load Analysis tool takes the six resultants plus an optional temperature and moisture change, and reports the response two ways from the same solve. The Table view gives exact numbers, one column per contribution plus a Total column. The Graph view gives the same values as grouped bars, which makes relative size and cancellation obvious at a glance. A contribution with no data behind it, a laminate whose plies carry no CTE values for example, is shown as unavailable rather than as a misleading zero.

The Deformed Shape tab draws the surface above from the same curvatures, names the shape, and lets you switch between the total response and any single contribution. Viewing the thermal contribution alone is the quickest way to see cure warpage on its own, with the mechanical load taken out of the picture.

References

1. Jones, R.M. Mechanics of Composite Materials, 2nd ed., Taylor & Francis, 1999. Chapter 4 (laminate constitutive equations, Kirchhoff kinematics).
2. Roylance, D. "Laminated Composite Plates," in Mechanics of Materials, MIT / Engineering LibreTexts. Open access
3. Hyer, M.W. "Some Observations on the Cured Shape of Thin Unsymmetric Laminates," Journal of Composite Materials, vol. 15, no. 2, pp. 175-194, 1981. doi:10.1177/002199838101500207
4. Barbero, E.J. Introduction to Composite Materials Design, 3rd ed., CRC Press, 2018. doi:10.1201/9781315296494

Frequently Asked Questions