Shear of Thin Walled Beams. Introduction

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1 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 ShearCutter

2 Sandwich Honeycomb Panels Aircraft 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.

3 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.

4 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.

5 Stress, Strain, Displacement Relationships (1) General stress on element of closed or open section beam Flexural stress and shear flow system on the element The element of the beam wall is maintained in equilibrium by system of flexural and shear stresses. Shear flow q is shear force per unit length

6 Stress, Strain, Displacement Relationships (2) For equilibrium in the z direction For equilibrium in s direction Eg07-01 Axial strain in the z direction can be related to w through Axial, tangential & normal components of a point in beam wall It is possible to derive the direct strain ε s in terms of v t, v n, s, and 1/r

7 Stress, Strain, Displacement Relationships (3) Shear strain γ is found in terms of w & v t by considering the shear distortion of element δs x δz Since γ = φ 1 + φ 2 Eg ShearSimulation ShearWall EarthquakeTest

8 Stress, Strain, Displacement Relationships (4) Displacement relationships & position of center of twist (open or closed) Differentiating Differentiating The displacements u, v, θ is equivalent to pure rotation about R (center of twist) Comparing coefficients, we have Eg07-03

9 Shear of an Open Section (1) The open section supports shear loads S x, S y There is no twisting or bending of cross section Shear loads must both pass through the shear center From bending theory Taking the derivative and using the relationship dm y /dz = S x

10 Shear of an Open Section (2) Integrating wrt s from some origin If the origin for s is taken at the edge of the cross section q = 0, when s = 0 If a section has either C x or C y as axis of symmetry, I xy = 0 Eg07-04

11 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

12 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

13 DaVinci s Flying Machine Leornado DaVinci s interests extend beyond art. About 1485 he drew detailed plans for a human-powered ornithopter (a wingflapping device intended to fly). DaVinci was the first to point out that flying without feathered wings was a possibility.

14 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.

15 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 OrnithopterSeagull OrnithopterDragonfly

16 Shear of a Closed Section (1) 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 Similar to open section analysis If we choose an origin for s where shear flow is q s,0 Eg07-05 Note: the first two terms on right represent shear flow in open section loaded through shear center

17 Shear of a Closed Section (2) Rewriting for q b = open section shear Convenient way to obtain q b by supposing closed beam is cut at some point

18 Shear of a Closed Section (3) If A is the area enclosed by the mid-line of beam section wall If moment center is chosen to coincide with lines of action S x & S y Eg07-06

19 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.

20 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.

21 911 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 WTCFEM WTCCollapse

22 Twist & Warping (1) Shear loads not applied through the shear center of a closed section will cause section to twist & warp. From the shear strain relation As q = τt & τ = Gγ, From Substituting dv t / dz Integrating wrt s from the chosen origin and G is also a function of s A Os is the area swept by generator from O BirdStrike 07-11/12-AlaskaAirlines

23 Twist & Warping (2) Continuing integration completely around the cross-section Eg07-07 Rearranging From The last two terms represent the effect of relating warping displacement to an arbitrary origin

24 Twist & Warping (3) If the origin coincides with the center of twist R Eg07-08

25 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.

26 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, The extreme heat causes the flap just to the left of the engine mount to be warped.

27 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