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. 2018 Apr 24;8:6444. doi: 10.1038/s41598-018-24123-6

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© The Author(s) 2018

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

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Theoretical and finite element models for membrane bending. (A) Membrane bending model modified from beam bending theory. A section of the membrane spanning the grating groove of length L is represented as a linear elastic beam with both ends (points A and B) freely supported by the adjacent grating ridges. The beam bends (blue line) under a uniformly distributed load q and the tangents to the membrane at point A and B make angles θ_A and θ_B respectively with horizontal line AB. The vertical displacement of any point x along the beam is denoted by v and is maximum when x = L/2 and θ_A = θ_B. (B) Three dimensional finite element model simulation that was used to validate the results from the theoretical model for beam bending.