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Unformatted text preview: Structural Mechanics and Dynamics AE213 Professor Ferri M H Aliabadi Course Layout
Main Topics covered are:
 Shear flow analysis for thin walled sections  Structural failure assessment
 Buckling analysis Shear flow analysis for thin walled sections This is the extension of beam theory (15‘ year) to thin walled
sections. The topics covered include:
Loading . Shear flow in open sections M 1 t _.
' Shear flow in closed sections i? A  Shear Centre On completion oithis par_t_you should have gained an understanding of thin walled aircraft structures
under general loading conditions. It W!” allow you to assess the effect of aerodynamic loading on the
performance of an aircraft design. Text book: Aircraft Structures by THGMegson Structural Failure Assessment °Two dimensional stress ﬁelds
vPrinoiple stresses °Mohr‘s circle Two dimensional strain ﬁelds
~StressStrain relationship ~Failure assessment Buckling Analysis On Completion of this part you will
be able to detem'iine the Critical
Buckling load and deﬂection of
Struts with different end conditions
and eccentricity Bending
Stiffness
E: Short Comp rasacm Ductile
Member Material Aircraft Structures An aircraft structure performs two essential functions: 1. It maintains the aerodynamic shape and contain the
content 2. it transmits and resists loads applied to its surface Aircraft structures are usually made of stiffened shells. An
unstiffened shell monocoque shell of large crosssection
would require fairly thick walls to avoid buckling of the skin.
Hence a Semimonocoque construction plays is usually
used in which stiffeners play an important role, allowing a
thinner skin to be used. _ Traiting
Ex‘le rIor La "d'n‘J ed ge
piale gear / Inboard ﬂap Wing doubler 51111“? Han vane
Spoiler In board aileron Outboard ﬂap
Flap vane
Outboard aileron Canter Longitudinal members are Open Section Fuselage Review of Classical Beam Theory Classical beam theory assumes that a beam is a member that is long in one direction
and short in the other two directions, the member has a high aspect ratio. Before Defamation i After Deformation Compress Horizontal lines 5 " become curved ‘ 1 2 Vertical lines
remain straight, yet
rotate a. The shape oflhe Review of Classical Beam Theory
crosssection of V
the beam is not changed by anyr
loading. b. Plane sections
remain plane. To show how the distortion will strain the
material, let us isolate a segment ofthe
beam at a distance 2 along the beam’s
length and has an underformed thickness Undeformed stats Deform stats Strain variation : — 8 I11 ax
ﬂ Radius of :I I! a. curvature Longitudinal axis . ‘ﬁAZF— AZ”, c. tt follows that the strain in the tongitudinal direction of the beam can only very linearly over
the beam cross—section. 11 y y For a
— Emax symmetrical
Neutr Neutral axis Strain variation Stress variation d. From assumption c it follows that the direct stress can only vary linearly over
the cross—section al axis c72=a0+alx+a2y (1) Where an. al and 492 are constants for a speciﬁc feeding. The stresses are referred
to the centre of area of the direct stress carrying material on the section‘ the centroid
of the section The bar over the x and y denote that these are centroidal axes. 12 Beam and Wall Tube Theory Sign Conventions The positive forces for beam and tube theory is shown below. The origin of the
geometrical coordinate axes is located at the centroid of the crosssection. For a beam the centroid is the centre of area of ail of the direct stress carrying
material on the section. This is important for some
idealisations used in thin walled
tube analysis. if some material is
assumed to only carry shear
stress than the area of that
material is not inctuded in the
calculation of the centroid or
second moments of area ofthe
section. (fut Face B To determine the internal stresses the beam section must be ‘cut' in order to expose
the internal forces. However, the out has two faces and the set of forces on one of
these faces has to be chosen for deriving the stresses. For Cut Face A there is a
toadis‘zress equivatence stress restztteots = spotted toads I
Applied I
Land
For Cot Face B the toads and the stresses are in equéiibrium
Stress resuitsnts + spoiied loads =
zero. SK .
the stresses on the two out feces ' are equai En magnitude but have x“
opposite signs.  ‘
1 Cut Face a
By convention the stresses are I _ found for Cut Face A This means that the stresses
are then eq1,iivaient to the
sootied toads rather than being
in eooéiibréom with them Relationship between Shear forces and
Moments We can derive the relationship between the shear forces and moments by considering
equilibrium over an element of iength dZ Beam Equilibrium Equations Consider a length of beam, dz. lying in the yz plane. For equilibrium in the ydirection:
)‘ {2} Taking moments about the left hand
end of the element dz gives Equilibrium in the YZPlane Similarly. considering equilibrium in the DroPpmg second order “arms and X2 plane gives the pair of equations cancelling gives (3) (4) 1? Loadings and Stress Resultants The general direct stress distribution over the crosssection of the beam is completely deﬁned for a beam by equation (1). The total and load N is found by integrating the direct
stress over the material on the crosssection that carries direct stress Similarly the bending moments are Bending Moments Thesereduoeto
M =al i—aglxx 3: xy (8) I‘d}I =a]I +3131“ Where the second moments of area about the centroid are
I“ =ly2dA ; 1”, = [£sz : I“ =jsrydA (10)
A A A 3’)” I“ and I“. are always positive num bers' but {X}, can be positive, negative or zero. The
case where is zero is a special case. The oentroidai axes in this case are called principal
axes and are denoted by the symbols X and . 19 1 Direct Stress Distribution on the CrossSection Rewriting the equations for the moments in matrix form 1 xx xy 8’2
+aI xy lyy at ny Similarly for the direct stress 01 = a1)? + 612? can be written in matrix form as A a
52:5 i] a2 (12)
1 lnvei‘ting the square matrix of equation (‘I ‘1} and substituting this into equation (12) gives (13} 10 Expanding (13) gives 1
[MK—13M}. Y." I“. I x}, XX For a symmetric section I”. = 0, hence M _ My
cr,= y+ _ —)_c
I 1”, xx
If only Mjs applied Mx_
or_,= I xx It is convenient to rewrite the direct stress — moment relationship in terms of
‘=a='_l*:'.’::t§ve moments With the effective bending moments _ 1 I.
anxumyj/{lﬂ
1W 1”:
_ In, l“
My: My— MK 1—
Ixx '3)“ After all of the algebraic manipulation equation (15) show how the direct
stress can be found for beams with arbitrary shaped crosssections ...
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This note was uploaded on 03/24/2010 for the course AE A.213 taught by Professor S.pinhoandm.aliabadi during the Winter '09 term at Imperial College.
 Winter '09
 S.PinhoandM.Aliabadi

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