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AOE2024: Thin Walled Structures
Module 6: Flexure due to Bidirectional Bending in
Thin-Walled Cross Sections
Food for Thought
• Consider the frame structure inside a wing as a beam and consider the load due to lift
and drag.
Bishay, P.L.; Burg, E.; Akinwunmi, A.; Phan, R.;
Sepulveda, K. Development of a New Span-Morphing
Wing Core Design. Designs 2019, 3, 12.
Jan R. Wright and Jonathan E. Cooper,
Introduction to Aircraft Aeroelasticity and Loads, Second
Edition, 2015 John Wiley & Sons, Ltd.
Jan R. Wright and Jonathan E. Cooper,
Introduction to Aircraft Aeroelasticity and Loads,
Second Edition, 2015 John Wiley & Sons, Ltd.
A. Kabir, M. S. Chowdhury, M. J. Islam and M. Islam,
”Numerical Assessment of the Backward Facing Step for
NACA 0015 Airfoil using Computational Fluid
Dynamics,” 2019 1st International Conference on
Advances in Science, Engineering and Robotics
Technology (ICASERT), Dhaka, Bangladesh, 2019, pp.
1-6, doi: 10.1109/ICASERT.2019.8934501.
• Due to bending about the z-axis (from lift), and bending
about the y-axis (from drag), the wing (modeled as a
beam) is subjected to bidirectional bending.
• Furthermore, due to the potential of the beam cross
section being asymmetric, this problem becomes even
more complex.
AOE 2024: Thin Walled Structures Module 6: Flexure due to Bidirectional Bending Gilbert 1 / 45
AOE 2024
Prof. Gilbert
Introduction to
Flexure
Deriving the
General Flexure
Formula
Principal
Moment of
Inertia
Orientation
Plane of Loading
Orientation
Plane of Bending
Orientation
Example
Problems
Overview
20 Introduction to Flexure
Deriving the General Flexure Formula
Principal Moment of Inertia Orientation
Plane of Loading Orientation
Plane of Bending Orientation
Example Problems
AOE 2024: Thin Walled Structures Module 6: Flexure due to Bidirectional Bending Gilbert 2 / 45
AOE 2024
Prof. Gilbert
Introduction to
Flexure
Deriving the
General Flexure
Formula
Principal
Moment of
Inertia
Orientation
Plane of Loading
Orientation
Plane of Bending
Orientation
Example
Problems
Deriving the General Flexure Formula
Bidirectional Beam Bending
• Consider a Beam with arbitrarily shaped cross section with the x-axis passing through
the bending axis at point O of the cross section (i.e. the neutral axis for bending)
AOE 2024: Thin Walled Structures Module 6: Flexure due to Bidirectional Bending Gilbert 3 / 45
AOE 2024
Prof. Gilbert
Introduction to
Flexure
Deriving the
General Flexure
Formula
Principal
Moment of
Inertia
Orientation
Plane of Loading
Orientation
Plane of Bending
Orientation
Example
Problems
Deriving the General Flexure Formula
• To derive the flexure formula, we start by assuming an x-displacement field consistent
with axial extension and pure bending about the z- and y- axis:
ux(x, y, z) = ux0(x)︸ ︷︷ ︸
Axial Extension
−y duy0
dx
+ c1(y, z)︸ ︷︷ ︸
Bending about z-axis
−z duz0
dx
+ c2(y, z)︸ ︷︷ ︸
Bending about y-axis
,
where ux0, uy0, and uz0 are the x-displacement, y-deflection, and z-deflection of the
neutral axis, respectively, and are functions of x only.