Bending of Thin-Walled Beams. Introduction

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1 Introduction Beams are essential in aircraft construction Thin walled beams are often used to lower the weight of aircraft There is a need to determine that loads applied do not lead to beams having ecessive Stresses Deflections DouglasFDLanding

2 Wing Spar Design Since the wing essentially operates in the full cantilever mode, it is subjected to a high degree of bending. The main structures supporting the load are the spars.

3 Wing Bo Beam Design In many commercial airplanes today, the wing is also used to contain fuel. The boed beam design makes it convenient for this. The added weight from the fuel is an important in ensuring that the wing is able to sustain the bending stresses developed.

4 Beam bending analysis using singularity functions Singularity function form Integration of singularity function Singularity functions Concentrated moment Concentrated load Uniformly distributed load Linearly increasing load > < = a a a a n n ) ( = + n a a n n 1 ) ( = a C V 0 ) ( a C V = 1 ) ( a C V = 2 ) ( a C V =

5 Relationship between shear force and bending moment The shear force distribution can be determined by differentiating the bending moment distribution The bending moment distribution can be obtained by integrating the shear force diagram M = V V = M

6 Fleural Stress Fleural stress eqn. σ = My I M : bending moment y : distance from the beam ais I : second moment of area +M y 2 A Compression Tension Beam ais Fleural stress is zero along the beam ais

7 Bending Failure of Push-Pull Tube The push-pull tube is part of a mechanical linkage in a Cessna plane to etend the nose gear that failed by bending failure. The push-pull tube was believed to have been constructed with an unintended slight bend in it. From cyclical usage, the tube bent to approimately 30 before finally fracturing into two pieces

8 Bending Failure of Push-Pull Tube Landing gears that do not deploy properly can cause nasty incidents. Poor maintenance or design can lead to strong bending stresses developing during actuation. Coupled with the forces during touch down, it is possible for member failure to occur.

9 Honeycomb Sandwiched Structures The leading edge of the wing is typically made of honeycomb sandwich structures due to the high fleural strength to weight ratio. The leading edge is subject to strong surface forces during flight that can cause deformation if not strengthened.

10 Shear Stress Shear stress eqn. τ = VQ It V : shear force Q : first moment of area y A I : second moment of area y 2 A t : thickness of the beam cross-section Q is zero at top and bottom of beam section Shear stress is zero at top and bottom of beam section

11 Cross Section of the Spar A solid spar must normally cope with compression stresses along the upper edge, tension stresses along the lower edge and shear stresses in between. The web can be thinned to reduce weight. The shear web has to be of sufficient thickness to resist the shear stresses. The bo beam design also applies the same strategy where there is now two vertical webs instead on one.

12 Boeing 787 Wing Test The first Boeing 787 Dreamliner demonstration wing bo shown represents two thirds of the airplane's wing span and is full-scale in size. It was tested to demonstrate the structural integrity of the design, gather data required for certification, and validate the repair methods for the materials being used Boeing787WingTest

13 Beam Slope & Deflection Bending moment of beam is related to the second derivative of beam deflection 2 d y EI = 2 d M I : second moment of area E : modulus of elasticity y 2 A Integrating once gives the slope of the beam Integrating again gives the deflection of the beam EI dy d = Md + C 0 EIy = d Md + C + C HelicopterRefuelingAccident

14 Bend Testing the Wright Flyer In 1993, NASA undertook a project to eamine the design of the airplane that the Wright brothers used in They reconstructed the airplane and tested the deflection of the wing using sand weights.

15 Unsymmetrical Bending Stress at a point in the cross-section of a beam subjected to bending depends on Position of point Applied loading Geometric properties of the cross section This applies regardless of whether the cross-section is opened or closed It is important to establish the notation and sign conventions in advance

16 Unsymmetrical Bending of Tail Structure The tail structure of a Viking includes struts subject to inboard/outboard loads which caused the vertical horizontal stabilizer support tubes to be loaded in combined unsymmetrical bending (dominant) and torsion. The resultant bending stresses are maimum at the top of the tubes resulting in the cracks.

17 Resolution of Bending Moments Internal force system Resolution of bending moments M M y = M sin θ = M cosθ

18 Direct Stress from Bending (1) Suppose a beam supports bending moments M & M y and bends about the neutral ais NA which coincides with centroid C NA will pass through C Eg06-01 If inclination of NA to C is α The bending stress is Epressed using the resolved bending moments M, My, second moments of area I, I y, and product moment of area I y Eg PlaneWingTest

19 Direct Stress from Bending (2) If the beam cross section has either C or Cy as ais of symmetry, I y = 0, or If M y = 0 If M = 0 When M y = 0, -ais becomes the NA. When M = 0, y-ais becomes the NA. The position of NA depends on the form of loading applied & the geometrical properties of the cross-section.

20 Position of the neutral ais The neutral ais always passes through the centroid of the cross-section but the inclination α depends on the loading and geometrical properties of cross-section. At the neutral ais, σ z = 0. Since tan α = -y/ Eg06-03

21 Deflections due to bending (1) Suppose at some section of unsymmetrical beam the deflection normal to the neutral ais is ζ. Centroid is displaced from C I to C F. As ρ is the radius of curvature Components u & v of ζ are in the ve directions of and y Differentiating twice and substituting for ζ If u = d 2 u/dz 2, v = d 2 v/dz 2 Eg TubeBending AutomatedTubeBending

22 Deflections due to bending (2) For u = d 2 u/dz 2, v = d 2 v/dz 2 Even if M y = 0, M produces curvatures in z and yz planes. The same happens when M = 0. An unsymmetrical beam will deflect both vertically and horizontally even though loading is entirely in the vertical plane If the beam cross section has either C or C y as ais of symmetry, I y = 0, or SuperJumboPlanes_LandingGear

23 Corrugated Structures One method of improving bending stiffness is by having corrugated surfaces.

24 Corrugated Aircraft Structures Some aircraft constructed before 1920 had corrugated skins, with the grooves parallel to the line of flight. Unfortunately, this affected the airflow and increased drag. Furthermore, the stiffness was only improved in one direction.

25 Area properties of thin-walled sections (1) Thickness t is assumed to be small compared with their cross-sectional dimensions Stresses are assumed constant across the thickness Squares and higher powers of t are neglected in the computation of sectional properties I = ( b + t / 2) t b + t th t( h t 12 / 2) 3 Epanding and removing powers of t 2 and upwards I = 2bth + t(2 ) 12 2 h I yy can be obtained in a similar manner 3

26 Area properties of thin-walled sections (2) Thin-walled sections need not have components in the and y ais Inclined thin-walled sections can complicate calculation of section properties I a / 2 = 2 ty ds = a / a β 0 t( s sin β) ds = t sin 12 Similarly I yy = a t cos β I y a / 2 = 2 tyds = 2 0 a / 2 0 t( s sin β)( s cosβ) ds = a 3 t sin 2β 24 This is an approimation as powers of t 2 and upward are neglected.

27 Bending Failure of a Hollow Tube During bending, one side of the cross section is subject to tension and the other compression. In thin circular hollow tubes, the compression can cause localized buckling that leads to kinks forming at one spot. In some cases, multiple kinks form and leads to a wrinkling behaviour of the tube.