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Shear of Thin Walled Beams

Introduction

• Apart from bending, shear is another potential structural failure mode of beams in aircraft

• For thin-walled beams subjected to shear, beam theory is based on assumptions applicable only to thin-walled sections

• Solid and thick sections will not be considered • Relationships between the following are retained

– Bending moments– Shear forces– Load intensities

07-01-ShearCutter


Sandwich Honeycomb PanelsAircraft construction uses sandwich honeycomb panels extensively.A honeycomb sandwich panel 2 times thicker than aluminum plate is seven times stiffer.One that is 4 times thicker is 37 times more stiff.The weight remains the same.


Sandwich Honeycomb Panel Failure

Despite their stiffness, sandwich honeycomb panels can still fail.When subjected to excessive bending, the plates will fracture.When subjected to excessive shear, the matrix will tend to tear.


Carbon-fiber-reinforced polymer failure

The matrix tends to tear off when excessive shear stress is created. This will cause hackles to form, which are rough features on the fractured surface.

On closer examination, it is possible to observe the fracture

fibers sticking out, which is often called fiber pullout.


Stress, Strain, Displacement Relationships (1)

Shear flow q is shear force per unit length

The element of the beam wall is maintained in equilibrium by system of flexural and shear stresses.

General stress on element of closed or open section beam

Flexural stress and shear flow system on the element



It is possible to derive the direct strain εs in terms of vt, vn, s, and 1/r

For equilibrium in s direction

For equilibrium in the z direction

Axial, tangential & normal components of a point in beam wall

Axial strain in the z direction can be related to w through

Eg07-01



Shear strain γ is found in terms of w& vt by considering the shear distortion of element δs x δz

Since γ = φ1 + φ2Eg07-02

07-02-ShearSimulation07-03-ShearWall07-04-EarthquakeTest



Differentiating

Displacement relationships & position of center of twist (open or closed)

The displacements u, v, θ is equivalent to pure rotation about R (center of twist)

Differentiating

Comparing coefficients, we have Eg07-03


Shear of an Open Section (1)

From bending theory

Taking the derivative and using the relationship dMy/dz = Sx

The open section supports shear loads Sx, SyThere is no twisting or bending of cross sectionShear loads must both pass through the shear center


Shear of an Open Section (2)

If the origin for s is taken at the edge of the cross section q = 0, when s = 0

If a section has either Cx or Cy as axis of symmetry, Ixy = 0

Integrating wrt s from some origin

Eg07-04


Shear Center of Open Section

• Shear center is the point in the cross-section where shear loads produce no twisting

• This is also the center of twist of sections subjected to torsion

• Stresses produced by the separate actions of shear & torsion can be added by superposition

• When a cross-section has an axis of symmetry, the shear center must lie on the axis


Open section flexure

Open section tubes can behave as highly controlled springs.This patent describes a precision positioning device designed with open section tubes.

- “Split tube flexure” United States Patent 6585445


DaVinci’s Flying MachineLeornado DaVinci’s interests extend beyond art.

About 1485 he drew detailed plans for a human-powered ornithopter (a wing-flapping device intended to fly).

DaVinci was the first to point out that flying without feathered wings was a possibility.


Ornithopter (1)The first ornithopters capable of flight were constructed in France in the 1870s.

They started from unmanned and finally to manned flights.

The wings must passively twist in response to the flapping.


Ornithopter (2)

The wing design uses the well-known fact that a circular tube loses a lot of its torsional rigidity if it is slit lengthwise.

This is possible using slit tube slipped over a supporting shaft, the mating edges of the slit slide smoothly along each other with a shearing action, giving torsional deflections. 07-05-OrnithopterSeagull

07-06-OrnithopterDragonfly


Shear of a Closed Section (1)

Similar to open section analysis

If we choose an origin for s where shear flow is qs,0

Same general conditions of open section apply. Some differences:(a) Shear loads may be applied through points in the cross-section other than shear center(b) It is generally not possible to choose origin for s where q is known

Note: the first two terms on right represent shear flow in open section loaded through shear center

Eg07-05



Convenient way to obtain qb by supposing closed beam is cut at some point

Rewriting for qb = open section shear



If moment center is chosen to coincide with lines of action Sx & Sy

If A is the area enclosed by the mid-line of beam section wall

Eg07-06


Foldable Bicycle

Foldable bicycles permit easy transport and storage when not used.The frame mainly comprises tubular close beams.The height of the seat can be set by compressing one tubular beam over another to fix it.


Aerial Refueling

Air refueling is the process of transferring fuel from one aircraft (the tanker) to another (the receiver) during flight. This allows the receiving aircraft to remain airborne longer.The refueling boom is a long telescoping rigid hollow shaft and can fail by bending and shear.The valve at the boom tip has shear rivets that ensures clean break-off during emergencies.


911 Critical Damage

When the airplanes struck the WTC, critical damage that led to collapse was brought about by the destruction of the interior column.The interior column can be thought of as a closed section that failed by shearing.

07-08-WTCFEM07-09-WTCCollapse


Twist & Warping (1)

As q = τt & τ = Gγ,

Shear loads not applied through the shear center of a closed section will cause section to twist & warp. From the shear strain relation

From

Substituting dvt / dz

Integrating wrt s from the chosen origin and G is also a function of s

AOs is the area swept by generator from O07-10-BirdStrike07-11/12-AlaskaAirlines


Twist & Warping (2)

Continuing integration completely around the cross-section

Rearranging

From

The last two terms represent the effect of relating warping displacement to an arbitrary origin

Eg07-07


Twist & Warping (3)

If the origin coincides with the center of twist REg07-08


Toothpick Glider

It is possible to create “toothpick” size gliders that can fly well.The wings are usually slightly warped because of the force applied to the paper when the wing is cut out. The sharper the scissors, the less warp you will find. If the wing surface is twisted or warped, your aircraft will spin or turn to the left or right. Reducing this twist or warp reduces drag, and will allow the aircraft to use the lift more equally for the left and right wings.


Engine fire warpage

A Boeing 767 parked inside a hanger at LAX experiences a spectacular engine failure and fire while conducting an engine test on June 2, 2006.The extreme heat causes the flap just to the left of the engine mount to be warped.


Shear Center of Closed Section Beam

Using the condition that a shear load acting through the shear center of a section produces zero twist (dθ / dz = 0)

If Gt is constant Eg07-09

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Shear of Thin Walled Beams Introduction - Biofuturex · Shear of Thin Walled Beams Introduction ... human-powered ornithopter (a wing- ... Microsoft PowerPoint - AircraftStruct7.ppt