Why Do The Boeing 787 Dreamliner Wings Curve When Flying?
The wings of the Boeing 787 are so flexible because its carbon fiber material can be stretched more, and the high aspect ratio of 11 will magnify this effect. In flight, all you will feel is less shaking due to gusts, because the wing will dampen load changes more effectively. On the ground, the wing might have less tip clearance, because less in-built dihedral is needed – the rest is supplied by the wing’s elasticity in flight.
The influence on performance is slightly negative, but this is a very weak effect. It can be compared to the rolling resistance of a stiff bicycle versus one with a spring-loaded frame.
The amount of bending for a given bending moment depends on three factors:
- Wing span: A given curvature of the wing due to bending at the wing root will cause a tip displacement which is proportional to the distance of that tip from the root.
- Spar heigt: This curvature grows with the inverse of the square of the spar height. A lower relative thickness of the wing will produce more bending.
- Spar material: The of the material describes how much it stretches for a given stress. More important, however, is the elastic elongation at yield stress. Carbon fiber has a higher Young’s modulus than aluminium, but is elastic until rupture, so it can be stretched more and produces more bending at yield stress.
The numbers: The Young’s modulus of aluminium is fairly constant for a wide range of alloys and normally 70,000 MPa or N/mm². The modulus of graphite fibers depends on their manufacturing process and varies between 200,000 and 700,000 MPa or N/mm². However, this value cannot be compared directly to that of aluminium. The final modulus of the composite depends on fiber orientation and resin content.
It is safe to assume that Boeing (or more precisely, Mitsubishi Heavy Industries) uses a modern, high-strength fiber like (IM stands for intermediate modulus), which has a modulus of 276,000 MPa. It is also safe to assume that most of the fibers are oriented in span direction, so they can contribute fully to taking the bending loads. If we assume a conservative fiber content of 60%, the resulting modulus of the spar material should be 164,000 MPa. However, the spar is not a discrete component, but part of the wing box which also has to take torsion loads. While Aluminium is anisotropic material (it has the same properties in all directions), CFRP is highly anisotropic, and adding torsional strength will require additional fibers in other directions. Consequence: The effective modulus of the wing box in bending direction might be as low as 110,000 MPa.
In the end, what counts is how much material is there to carry the bending loads. Here the yield stress of the material comes into play: The more stress a material can tolerate before it shows plastic deformation, the less of it is needed to carry a given bending moment. To arrive directly at the maximum deformation, it is enough to look at the maximum elastic strain. With IM7, this is 1.9%, and with high-strength 7068aluminium, it is less than 1% before the material suffers permanent elongation. This means that, even though CFRP is stiffer than aluminium, it can be loaded more and will stretch more before it reaches its limits.