**UFRJ**

# Flexural stiffness in insect wings

(Parte **2** de 4)

From the digitized vein images, we counted the number of vein intersections in each wing as a measure of the complexity of vein branching. We imported images of the wing silhouette and veins into Matlab and measured the planform area of the whole wing and of the wing veins. From these measurements, we calculated the vein density (proportion of planform wing area occupied by veins) as the planform vein area divided by total wing area. We also determined average vein thickness by finding the total length of veins in the wing (see Combes, 2002) and dividing planform vein area by total vein length. We divided this value by wing span to scale for overall wing size.

Finally, we measured two venation characteristics related to the leading edge of the wing: the proportion of veins in the leading edge and the density of veins in the leading edge. We defined the leading edge visually as a cohesive unit of veins running spanwise along the anterior edge of each wing (Combes, 2002). We imported digitized images of these leading edge veins and a silhouette of the leading edge area (leading edge veins plus the area they surround) into Matlab. We then calculated the proportion of veins in the leading edge as the leading edge vein area divided by total planform vein area, and calculated leading edge vein density as the leading edge vein area divided by the area of the leading edge silhouette.

Phylogenetic analysis of correlated characters

Because the species tested share a phylogenetic history, some characters (e.g. wing venation and flexural stiffness) may be correlated in closely related groups simply because these species share a common ancestor that possessed these characters, and not because the characters are related in any functional way. As a result, the species used in this study cannot be treated as independent data points for statistical analysis. However, any differences in traits between two adjacent (closely related) groups can be assumed to have occurred independently, after the two groups diverged.

Therefore, we can calculate the independent contrasts (or differences) between values of a trait in adjacent groups, and plot the standardized contrasts of one trait against those of another to see if a relationship exists between the traits when the effect of phylogeny has been removed (Felsenstein, 1985; Garland et al., 1999).

We calculated independent contrasts of flexural stiffness and wing venation data using the Phenotypic Diversity Analysis Programs (PDAP), version 6.0, developed by Garland, Midford, Jones, Dickerman and Diaz-Uriarte [originally from Garland et al. (1993), with modifications in Garland et al. (1999) and Garland and Ives (2000)]. This program allows one to construct a phylogenetic tree and enter tip values for traits; it then calculates independent contrasts and performs various diagnostics on the output.

Some aspects of the phylogenetic relationships of the pterygotes (winged insects) are under debate, and the divergence times of groups and families are far from certain. However, we constructed a phylogeny of the species used in this study by combining the available information, and in some cases averaging or choosing midpoints between estimates of branching times in the phylogeny (Fig.·1). To determine how sensitive the analysis is to phylogenetic branch length, we also calculated independent contrasts using arbitrary branch lengths (Pagel, 1992), in which all internode branches are equal to one, but the tips of the tree are contemporaneous. We tested for correlations between standardized contrasts in JMP 4 on a Macintosh computer.

Finite element modeling

To explore how wing veins contribute to the flexural stiffness of a wing, we used MSC Marc/Mentat to create a simplified finite element model of an insect wing based on the Manducawing shown in Fig.·1. Our goal was not to reproduce the behavior of a real Manducawing, but rather to create a general model of a wing to explore how simply adding or strengthening veins in certain regions of the wing affects the overall flexural stiffness of the structure. Therefore, we did not attempt to recreate the precise three-dimensional shape of a Manduca wing (including changes in membrane thickness, vein cross-sectional shape and vein/membrane attachments), but instead modeled the wing as a flat plate of uniform thickness, composed of thin shell elements arranged to mimic the planform shape and vein configuration of a Manducawing (Fig.·3A).

This generalized model allowed us to increase the stiffness of various groups of veins beyond that of the surrounding membrane to determine how these veins contribute to overall wing flexural stiffness. The flexural stiffness of individual veins in real insect wings is determined by both their geometric properties (generally those of hollow tubes) and the material properties of their walls. A hollow tube has a second moment of area Ithat can be several orders of magnitude higher than that of a flat plate of the same width (and equivalent wall thickness). Because our model is composed solely of flat elements (with a lower second moment of area), we adjusted

S. A. Combes and T. L. Daniel

2983Wing venation and scaling of flexural stiffness

the flexural stiffness of the veins by increasing the material stiffness (E, Young’s modulus) of these elements beyond that of the surrounding membrane elements.

We chose an element density of 1200·kg·m–3(as measured in insect wings; Wainwright et al., 1982) and an element thickness of 45·mm. We used a Poisson’s ratio of 0.49, as measured in some biological materials (Wainwright et al., 1982); because the Poisson’s ratio of insect wings is unknown, we tested the effects of using a Poisson’s ratio of 0.3 and found that the difference in model behavior was negligible. To determine the minimum number of elements needed, we performed a sensitivity analysis with models composed of 200, 350, 865 and 2300 total elements, and found that 865 elements are sufficient to ensure asymptotic performance of the model.

We subjected the model wing to virtual static bending tests that mimic the tests performed on actual wings, fixing the base with zero displacement or rotation and applying a point force to the wing tip (Fig.·3A, blue dot). The model calculates the tip displacement due to this point force (and given the material properties of the membrane and veins). We then used the applied force, displacement, and wing span to calculate overall spanwise EIfor the model wing as above (Equation·1). Similarly, we fixed the model wing at the leading edge and applied a point force at the trailing edge (Fig.·3A, orange dot) to calculate chordwise flexural stiffness.

In all simulations, we used a material stiffness of 1·109·Nm–2 for the membrane elements (as measured in locust wings; Smith et al., 2000). In the first set of simulations, we used a material stiffness of 1·109·Nm–2 for the vein elements as well (thus the wing was essentially veinless in these simulations). We then increased the material stiffness of the vein elements by orders of magnitude up to 1·1015·Nm–2, while membrane stiffness remained at 1·109·Nm–2. This increased vein material stiffness represents not only potential differences in the material properties of wing veins and membrane, but also differences in the second moment of area caused by the three-dimensional shape of veins. To test the effect of leading edge veins alone, we increased the material stiffness of the leading edge veins (in pink, Fig.·3A) by orders of magnitude to 1·1015·Nm–2, while fixing the material stiffness of the remaining vein and membrane

To validate our finite element modeling approach, we created a model of the glass coverslip used as an experimental control. We attached the coverslip at its base, with 4.85·cm of the coverslip extending past the point of attachment, and used a Young’s modulus for glass of 7·1010·Nm–2 (Gordon, 1978). We applied a series of point forces to the tip of the coverslip (equivalent to those applied in the experiment) and compared the tip displacements predicted by the model to those measured in the experiment.

Results

Flexural stiffness measurements

The measurements of overall wing flexural stiffness in this study did not reveal a significant dorsal–ventral difference in the spanwise or chordwise direction in any species tested; we therefore averaged dorsal and ventral meaurements of EIin each direction. These flexural stiffness measurements were significantly correlated with all size variables tested (see Combes, 2002). However, spanwise stiffness was more strongly correlated with wing span (r2=0.95; Fig.·4A) than

Vein material stiffness (N m–2)

Spanwise EI, lead only

Chordwise EI, all veins

Chordwise EI, lead only

Spanwise EI, all veins

Fig.·3. Finite element model of an insect wing and results of virtual bending experiments. (A) Finite element model (FEM) based on the forewing of Manduca sexta, with leading edge vein elements in pink, other vein elements in green and membrane elements in yellow. (B) FEM wing flexural stiffness EI (calculated from applied force and wing displacement) versusmaterial stiffness Eof vein elements. In all simulations, the material stiffness of the membrane elements was 1·109·Nm–2. For results shown in green, the material stiffness of all vein elements was increased; for results shown in pink, the material stiffness of only leading edge vein elements was increased (while other vein elements remained at 1·109·Nm–2). Filled symbols, spanwise EI; open symbols, chordwise EI.

with any other size variable, or with the principal components of wing or body size. Similarly, chordwise stiffness was most strongly correlated with chord length (r2=0.91; Fig.·4B). The measurements of flexural stiffness also revealed a large anisotropy in all species tested, with spanwise flexural stiffness approximately 1–2 orders of magnitude larger than chordwise flexural stiffness (Fig.·4).

Phylogenetic analysis of correlated characters

To verify that the correlations between flexural stiffness and size remain significant when the effect of phylogeny is removed, we calculated standardized independent contrasts of log-transformed spanwise EI, wing span, chordwise EI andchord length. The relationships between standardized independent contrasts of these wing size and flexural stiffness traits were significant, and the slopes were nearly the same as in the original data (span: y=3.04x, r2=0.96; chord: y=2.02x, r2=0.96). The residuals from these relationships can be used as an estimate of flexural stiffness with the effect of size (and phylogeny) removed (for examples of similar uses in correcting for body size, see Garland and Janis, 1993; Rezende et al.,2002).

These size-corrected estimates of flexural stiffness were not significantly correlated with standardized independent contrasts of any of the five wing venation characters measured. However, the residual of spanwise flexural stiffness was positively correlated with the residual of chordwise flexural stiffness (r2=0.37, P=0.012), and several contrasts of vein characters were correlated with each other (Combes, 2002). When Pagel’s arbitrary phylogenetic branch lengths (Pagel, 1992) were used, the residuals of flexural stiffness remained uncorrelated with wing venation characters, and the residuals of spanwise and chordwise flexural stiffness remained positively correlated (Combes, 2002).

Finite element modeling

The model of the glass coverslip provided estimates of tip displacement that were within 5% of those measured in the experiments, validating the finite element method. Virtual static bending tests of the model wing showed that chordwise flexural stiffness was higher than spanwise flexural stiffness when no veins were present (when membrane and vein material stiffness were the same; Fig.·3B). However, when veins were added and their material stiffness was increased, spanwise flexural stiffness increased beyond chordwise flexural stiffness. When only leading edge veins were added, spanwise flexural stiffness increased as above, while chordwise flexural stiffness did not change significantly from the veinless model (Fig.·3B).

Discussion

Our measurements of overall flexural stiffness in insect wings demonstrate that flexural stiffness varies widely in the species tested, but is strongly correlated with wing size. In fact, the span or chord length of a wing accounts for over 95% of the variability in spanwise and chordwise flexural stiffness, when the effects of phylogeny have been removed. Because flexural stiffness is a length-independent measure (for example, EIdoes not change significantly in glass coverslips over a range of lengths; Fig.·4A), these results suggest that there is a strong scaling relationship between wing size and flexural stiffness.

This strong relationship is apparent despite several assumptions inherent in the use of a homogeneous beam equation (Equation·1). Because wings resemble plates more closely than beams, we used simple finite element models to assess how robust the beam equation is to variations in the shape of the beam. We created models of rectangular plates

S. A. Combes and T. L. Daniel

Wing span (m) Chord length (m)

S pan w is e EI (

Odonata

Hymenoptera

Diptera Lepidoptera

Isoptera Neuroptera

Aeshna Pachydiplax Lestes Ischnura Zootermopsis

Hemerobius

Pepsis

Sceliphron Bombus Tipula

Villa

Eristalis Calliphora

Manduca Ochlodes Pieris Glass slide

Chord w is e EI (

(Parte **2** de 4)