**UFRJ**

# Flexural stiffness in insect wings

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Fig.·4. Flexural stiffnessversusspan/chord length in 16 insect species. Individuals of each species are plotted in the same color. Axes are on a logarithmic scale. (A) Spanwise flexural stiffness EI versuswing span; for log–log transformed data, y=2.97x+0.08, r2=0.95 (using species averages, y=2.93x+0.03, r2=0.96). Measured flexural stiffness of a glass coverslip (at varying lengths) is shown in black. (B) Chordwise flexural stiffness EI versuschord length; for log–log transformed data y=2.08x–1.73, r2=0.91 (using species averages, y=2.01x–1.8, r2=0.96).

2985Wing venation and scaling of flexural stiffness spanning the size and flexural stiffness range measured in real wings, applied point forces at the free end, and recorded the displacement of the model plate. We then calculated EI using Equation·1 and compared this to the known EIof the plate (based on its cross-sectional dimensions and material stiffness). We found that Equation·1 slightly overestimates flexural stiffness (by up to 12%) when the plate is longer than it is wide (i.e. for wings measured in the spanwise direction), and underestimates flexural stiffness (by up to 80%) when plates are wider than they are long. However, there is no overall trend in wing shape with increasing span or chord length, and these changes in EIare relatively small compared to the large range of flexural stiffness measured both within and between species (Fig.·4).

We also used the finite element model of an insect wing to explore the effects of inherent wing camber on flexural stiffness estimates. The maximum camber measured in Manduca wings with a laser ranging technique (see Combes and Daniel, 2003a) was 5% in the chordwise direction and 4% in the spanwise direction. Applying these levels of camber to the model wing had almost no effect on displacement when the wing was cambered parallel to the wing attachment (i.e. when the model wing was cambered in the chordwise direction and chordwise flexural stiffness was measured). When the wing was cambered in the direction perpendicular to the attachment (i.e. when the wing was cambered in the spanwise direction and chordwise flexural stiffness was measured), displacement varied up to a maximum of 40% from the value measured in a flat plate, indicating a relatively minor effect on flexural stiffness.

Finally, the assumption in Equation·1 that flexural stiffness is homogeneous across a wing may lead to a systematic error in the reported values, which could potentially underlie the observed scaling relationships. If flexural stiffness varies along the wing span or chord, the reported values of overall flexural stiffness represent some weighted integration of EIalong the length measured. If this integral varies systematically as beam length increases, it could account for part of the size scaling inthe data. We explored this hypothesis numerically by integrating various simple functions (that represent how stiffness might vary in the wing) over increasing beam length. We found that the integral of EIover the wing may vary slightly with length depending on the function used, but this variation is far smaller than the range of values measured in real wings, and is therefore unlikely to cause the observed scaling relationships (Combes, 2002).

Scaling of flexural stiffness

The strong correlations between wing size and flexural stiffness suggest that size scaling is the dominant factor determining overall flexural stiffness in insect wings. Because EIis a composite measure that incorporates the second moment of area as well as the material stiffness of a wing, it is not surprising that spanwise and chordwise flexural stiffness increase with wing size; wings with larger spans generally also have larger average chord lengths (and thus I, the second moment of area, is higher). Iis proportional to a beam’s width times its thickness cubed (Equation·2). For measurements of spanwise flexural stiffness, the width of the beam is the average chord length, and in the species tested average chord length isdirectly proportional to wing span (y=0.2546x–0.0004, r2=0.8574). The average thickness of the wings tested, however, is unknown. Average thickness may be proportional to span (if wings grow isometrically), or could be independent of span (since all cuticle consists of a single cell layer with extracellular deposits). If we assume that E, the material stiffness of wing cuticle, does not vary with wing size, we would predict that flexural stiffness should scale with length (if thickness is independent of span) or with length to the fourth power (if thickness is directly proportional to span). The results of this study do not agree with either of these predictions; spanwise EIscales with the cube of chord length, whereas chordwise EIscales with the square of span. Thus, increased second moment of area alone cannot account for the observed scaling of spanwise and chordwise flexural stiffness.

Scaling of flexural stiffness has been examined previously in dried butterfly wings (Steppan, 2000) and in the primary flight feathers of birds (Worcester, 1996), both of which show a positive correlation between flexural stiffness and size. The authors compared the observed scaling relationships with the theory of geometric similarity (see Alexander et al., 1979), where structures maintain a similar shape regardless of size, as well as with the theory of elastic similarity (see McMahon, 1973), in which loaded structures maintain a similar angular deflection regardless of size. Neither study (Steppan, 2000; Worcester, 1996) found scaling patterns that could be explained by geometric similarity, and the results of our work appear to agree with these conclusions. The exponents of the relationship between insect wing flexural stiffness and body mass (0.91 spanwise and 0.61 chordwise; Combes, 2002) are far from the expected value of 1.67 proposed by Worcester (1996) for geometric similarity, and the exponents of the relationship between flexural stiffness and wing area (1.50 spanwise and 0.9 chordwise; Combes, 2002) are also far from the expected value of 2.0 proposed by Steppan (2000).

If the scaling of wing flexural stiffness provides functional, rather than geometric similarity across a range of body sizes, wing angular deflection should remain constant, as in elastic similarity (McMahon, 1973). If we take tip displacement divided by wing span (or trailing edge displacement divided by chord length) as a rough measure of strain or curvature in the wing, we can rearrange Equation·1 as:

d / L= FL2 / EI·.(3)

Spanwise flexural stiffness scales with L3and chordwise flexural stiffness with L2. This suggests that in the chordwise direction, d/Lis directly proportional to force, regardless of the chord length of the wing. In the spanwise direction, d/Lis proportional to F/L, so spanwise curvature would be smaller in large wings for a given force.

To assess whether d/Lremains constant in flying insects over a range of sizes, we need to know how the forces on flapping insect wings scale with size. If we assume that the primary forces on an insect’s wings are aerodynamic, then force is proportional to body mass. However, several studies have suggested that the inertial forces generated by flapping wings may be considerably larger than the aerodynamic forces (Combes and Daniel, 2003b; Daniel and Combes, 2002; Ellington, 1984; Ennos, 1989; Lehmann and Dickinson, 1997; Zanker and Gotz, 1990), and therefore inertial forces may bemore important in determining wing deformations. A generalized scaling argument for inertial force in insect wings is difficult to derive because wingbeat frequency does not scale strongly with size in the insects studied here. However, small insects often have significantly higher wingbeat frequencies, so the ratio of inertial to aerodynamic forces acting on their wings may be as high or higher than in large insects with heavier (but slower) wings (Combes and Daniel, 2003b; Daniel and Combes, 2002).

Effects of wing venation on flexural stiffness

Although both spanwise and chordwise flexural stiffness scale with wing length, the magnitude of flexural stiffness in these directions differs greatly; spanwise EIis approximately 1–2 orders of magnitude higher than chordwise EIin all species tested (Fig.·4). Because spanwise flexural stiffness increases as L3and chordwise flexural stiffness only as L2, this anisotropy is generally bigger in larger-winged insects.

The finite element analysis of an insect wing shows that this structural anisotropy is due to a common venation feature of insect wings: leading edge veins. The model without any strengthening veins demonstrates that the basic planform shape of the wing would lead to similar spanwise and chordwise flexural stiffness if no veins were present (Fig.·3B). Adding leading edge veins to the model increases spanwise flexural stiffness dramatically, generating spanwise–chordwise anisotropy (Fig.·3B).

Clustered or thickened veins in the leading edge of the wing are found in nearly all insects, even insects that have lost all other wing veins (such as some hymenopterans and small dipterans). Thus, spanwise–chordwise anisotropy may be a universal trait among insects. This anisotropy would serve to strengthen the wing from bending in the spanwise direction while allowing chordwise bending to generate camber. It could also facilitate spanwise torsion, which is seen in many species during supination (Ennos, 1988; Wootton, 1981).

Although leading edge veins appear to play a crucial role in determining the relative magnitudes of spanwise and chordwise flexural stiffness, the details of venation pattern measured in this study do not appear to affect the overall flexural stiffness of the wing. We did find, however, that the residuals of spanwise flexural stiffness are correlated with the residuals of chordwise flexural stiffness. This indicates that some insects have wings that are generally stiffer (in both directions) than expected for their size, while others have wings that are more flexible than expected. The residuals from the original data show that dragonflies, hawkmoths, flies (except for craneflies) and bumblebees all have wings that are stiffer than expected for their size. Damselflies, craneflies and lacewings have more flexible wings than expected, while butterflies and wasps are intermediate.

The functional significance of phylogenetic changes in wing venation (such as loss of cross veins and increased vein thickness) remains unclear. Perhaps more derived groups of insects have simply evolved a venation pattern that allows them to maintain the essential scaling of wing stiffness in a more economical way (e.g. using less vein material), or perhaps the venation patterns are related to something entirely different, such as the distribution of sensory receptors on the wing (Kammer, 1985).

Alternatively, venation pattern may in fact affect wing stiffness, but in ways that could not be detected in this study. For example, venation pattern may not affect overall stiffness, but could influence how stiffness varies throughout the wing (see Combes and Daniel, 2003a). In addition, the stiffness measurements in this study exclude the outer 30% of the wing, which is likely to be the most flexible region. Differences in wing stiffness between insects with veins that extend to and delineate the trailing edge (such as odonates; see Fig.·1) and insects with primarily unsupported membrane in the trailing edge (such as hymenopterans) would most likely be found in this region. How the spatial distribution of stiffness contributes to the instantaneous shape of a dynamically moving wing is a subject of further study, and will be crucial to understanding the implications of mechanical design of wings to insect flight performance.

List of symbols

EI flexural stiffness Ematerial stiffness (Young’s modulus) Isecond moment of area Leffective beam length Fapplied force dwing displacement at the point of force application w width t thickness

R. Sugg and J. Edwards graciously assisted in identifying the insects used in this study. D. O’Carroll, T. Morse and J. Kingsolver provided specimens, and D. Combes assisted in collecting and transporting the spiderwasps. T. Garland provided the program for calculating independent contrasts, as well as helpful advice on its use. D. Grunbaum and R. Huey contributed useful comments on both the project and drafts of the paper. This work was supported by NSF grant F094801 to T.D., the John D. and Catherine T. MacArthur Foundation, an NSF graduate fellowship to S.C. and an ARCS fellowship to S.C.

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