Bending of Thin-Walled Beams Introduction - Biofuturexbiofuturex.com/mae3407/AircraftStruct6.pdf · Bending of Thin-Walled Beams Introduction ... The resultant bending stresses are

Bending of Thin-Walled Beams

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 excessive– Stresses– Deflections

06-01-DouglasFDLanding


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.


Wing – Box Beam Design

In many commercial airplanes today, the wing is also used to contain fuel. The boxed 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.


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

⎩⎨⎧

>−<

=−axaxax

ax nn

)(0

1

1

+−

=−+

∫ nax

axn

n

1)( −−= axCxV0)( axCxV −=

1)( axCxV −=2)( axCxV −=


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

xMV∂∂

=∫ ∂= xVM


Flexural Stress

• Flexural stress eqn.

– M : bending moment– y : distance from the beam axis– I : second moment of area

• Flexural stress is zero along the beam axis

IMy

=σBeam axis

+M

Tension

Compression

∫ ∂Ay2


Bending Failure of Push-Pull Tube

The push-pull tube is part of a mechanical linkage in a Cessna

plane to extend 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 approximately 30° before finally

fracturing into two pieces


Bending Failure of Push-Pull TubeLanding 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.


Honeycomb Sandwiched Structures

The leading edge of the wing is typically made of honeycomb sandwich structures due to the high flexural strength to weight ratio.

The leading edge is subject to strong surface forces during flight that can cause deformation if not strengthened.


Shear Stress• Shear stress eqn.

– V : shear force– Q : first moment of area– I : second moment of area– 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

ItVQ

=τ

∫ ∂Ay∫ ∂Ay2


Cross Section of the Spar

The box beam design also applies the same strategy where there is now two vertical webs instead on one.

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.


Boeing 787 Wing Test

The first Boeing 787 Dreamliner demonstration wing box 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.

06-02-Boeing787WingTest


Beam Slope & Deflection• Bending moment of beam is related

to the second derivative of beam deflection

I : second moment of areaE : modulus of elasticity

• Integrating once gives the slope of the beam

• Integrating again gives the deflection of the beam

Mdx

ydEI =2

2

∫ ∂Ay2

∫ +=x

CMdxdxdyEI

0 1

210 0CxCMdxdxEIy

x x++= ∫ ∫

06-03-HelicopterRefuelingAccident


Bend Testing the Wright Flyer

In 1993, NASA undertook a project to examine the design of the airplane that the Wright brothers used in 1903.

They reconstructed the airplane and tested the deflection of the wing using sand weights.


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


Unsymmetrical Bending of Tail Structure

The tail structure of a Viking 17-30 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 maximum at the top of the tubes resulting in the cracks.


Resolution of Bending Moments

θ= cosMM y

θ= sinMM x

Internal force system

Resolution of bending moments


Direct Stress from Bending (1)Suppose a beam supports bending moments Mx & My and bends about the neutral axis NA which coincides with centroid C

If inclination of NA to Cx is α

The bending stress is

Expressed using the resolved bending moments Mx, My, second moments of area Ix, Iy, and product moment of area Ixy

Eg06-01

Eg06-02

NA will pass through C

06-04-PlaneWingTest


Direct Stress from Bending (2)

If the beam cross section has either Cx or Cy as axis of symmetry, Ixy = 0, or

If My = 0

When My = 0, x-axis becomes the NA. When Mx = 0, y-axis becomes the NA. The position of NA depends on the form of loading applied & the geometrical properties of the cross-section.

If Mx = 0


Position of the neutral axis

At the neutral axis, σz = 0. Since tan α = -y/x

The neutral axis always passes through the centroid of the cross-section but the inclination α depends on the loading and geometrical properties of cross-section.

Eg06-03


Deflections due to bending (1)

Components u & v of ζ are in the –ve directions of x and y

Suppose at some section of unsymmetrical beam the deflection normal to the neutral axis is ζ. Centroid is displaced from CI to CF. As ρ is the radius of curvature

Differentiating twice and substituting for ζ

If u” = d2u/dz2, v” = d2v/dz2 Eg06-04

06-05-TubeBending06-06-AutomatedTubeBending


Deflections due to bending (2)

Even if My = 0, Mx produces curvatures in xz and yz planes. The same happens when Mx = 0.

An unsymmetrical beam will deflect both vertically and horizontally even though loading is entirely in the vertical plane

For u” = d2u/dz2, v” = d2v/dz2

If the beam cross section has either Cx or Cy as axis of symmetry, Ixy = 0, or

06-07-SuperJumboPlanes_LandingGear


Corrugated StructuresOne method of improving bending stiffness is by having corrugated surfaces.


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.


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

12)2/(2

212)2/(2

32

3 thtthtbttbI xx−

+⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ ++

+=

12)2(2

32 htbthI xx +=

Expanding and removing powers of t2 and upwards

Iyy can be obtained in a similar manner


Area properties of thin-walled sections (2)• Thin-walled sections need not have components in the x

and y axis • Inclined thin-walled sections can complicate calculation

of section properties

12sin)sin(22

232/

0

22/

0

2 β=β== ∫∫

tadsstdstyIaa

xx

12cos23 β

=taI yySimilarly

This is an approximation as powers of t2 and upward are neglected.

242sin)cos)(sin(22

32/

0

2/

0

β=ββ== ∫∫

tadsssttxydsIaa

xy


Bending Failure of a Hollow TubeDuring 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.

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Bending of Thin-Walled Beams Introduction - Biofuturexbiofuturex.com/mae3407/AircraftStruct6.pdf · Bending of Thin-Walled Beams Introduction ... The resultant bending stresses are