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First published online July 23, 2003

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Flexural stiffness in insect wings I. Scaling and the influence of wing venation

S. A. Combes^* and T. L. Daniel

Department of Biology, University of Washington, Seattle, WA 98195, USA

View larger version (38K): [in a new window]   Fig. 1. Drawings of forewings from insects used in this study, arranged on the phylogenetic tree used to calculate independent contrasts. Veins are drawn at actual thickness; wings are not shown to scale. Genus and species names (when known) are shown under each wing, and orders are listed at their branching points. Branching and divergence dates of orders were derived from Kristensen (1991), Kukalova-Peck (1991), Whiting et al. (1997) and Wootton (1990b). Branching patterns and divergence dates within orders were derived from Benton (1993), Maddison (1995a,b), Trueman and Rowe (2001) and Wiegmann and Yeates (1996).

View larger version (15K): [in a new window]   Fig. 2. Method used to measure insect wing displacement in response to an applied force. (A) For spanwise measurements, point forces were applied at approximately 70% of wing span, near the leading edge (top). For chordwise measurements, point forces were applied at 70% of chord length, midway between the wing base and tip (bottom). (B) Wings were fixed to the left side of the apparatus, and the right side was lowered until a pin attached to a flexible beam contacted the wing. A micrometer was then used to lower the right arm by a known distance, applying a point force that moved the wing down and the flexible beam up. The motion of the beam was recorded via an optical sensor, and used to calculate the applied force and displacement of the wing.

View larger version (34K): [in a new window]   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) versus material stiffness E of vein elements. In all simulations, the material stiffness of the membrane elements was 1x109 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 1x109 Nm-2). Filled symbols, spanwise EI; open symbols, chordwise EI.

View larger version (14K): [in a new window]   Fig. 4. Flexural stiffness versus span/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 versus wing 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 versus chord 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).