Course Layout Slides

Course Layout Slides - Structural Mechanics and Dynamics...

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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 ~Stress-Strain 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 cross-section would require fairly thick walls to avoid buckling of the skin. Hence a Semi-monocoque 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 cross-section 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 cross-section. 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 y-direction: )‘ {2} Taking moments about the left hand end of the element dz gives Equilibrium in the YZ-Plane 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 cross-section 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 cross-section 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 Cross-Section 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 cross-sections ...
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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.

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Course Layout Slides - Structural Mechanics and Dynamics...

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