Lecture M1 Slender (one dimensional) StructuresReading: Crandall, Dahl and Lardner 3.1, 7.2This semester we are going to utilize the principles we learnt last semester (i.e the 3 greatprinciples and their embodiment in the 15 continuum equations of elasticity) in order tobe able to analyze simple structural members. These members are: Rods, Beams, Shaftsand Columns. The key feature of all these structures is that one dimension is longer thanthe others (i.e. they are one dimensional).Understanding how these structural members carry loads and undergo deformations willalso take us a step nearer being able to design and analyze structures typically found inaerospace applications. Slender wings behave much like beams, rockets for launchvehicles carry axial compressive loads like columns, gas turbine engines and helicopterrotors have shafts to transmit the torque between the components andspace structuresconsist of trusses containing rods. You should also be aware that real aerospacestructures are more complicated than these simple idealizations, but at the same time, agood understanding of these idealizations is an important starting point for furtherprogress.There is a basic logical set of steps that we will follow for each in turn.1) We will make general modeling assumptions for the particular class of structuralmemberIn general these will be on:a) Geometryb) Loading/Stress Statec) Deformation/Strain State2) We will make problem-specific modeling assumptions on the boundary conditions thatapply (idealized supports, such as pins, clamps, rollers that we encountered with trussstructures last semester)a.) On stressesb.) On displacements3) We will apply an appropriate solution method:a) Exact/analytical (Unified, 16.20)b Approximate (often numerical) (16.21). Such as energy methods (finiteelements, finite difference - use computers)Let us see how this works:Applied at specifiedlocations in structureRods (bars)The first 1-D structure that we will analyzes is that of a rod (or bar), such as weencountered when we analyzed trusses. We are interested in analyzing for the stressesand deflections in a rod.First start with a working definition - from which we will derive our modelingassumptions:"A rod (or bar) is a structural member which is long and slender and is capable ofcarrying load along its axis via elongation"Modeling assumptionsa.) GeometryL = length (x1 dimension)b = width (x3 dimension)h = thickness (x2 dimension)Cross-section A (=bh)assumption: L much greater than b, h (i.e it is a slender structural member)(think about the implications of this - what does it imply about the magnitudes of stressesand strains?)b.) Loading - loaded in x1 direction onlyResults in a number of assumptions on the boundary conditionsSimilarly on the x2 face - no force is applied† s21=s12= 0s32=s23= 0s22= 0on x1 face - take section perpendicular to x1and c.) deformation† s31= 0s32= 0s33= 0s12= 0s13= 0s11dA = PAÚs11dx2dx3ÚÚ= Pfis11=Pbh=PARod cross-section deforms uniformly (is this assumption justified? - yes, there are noshear stresses, no changes in angle)So much for modeling assumptions, Now let's apply governing equations and solve.1. Equilibrium† ∂smn∂xn+ fn= 0only s11 is non-zero† ∂s11∂x1+ f1= 0† ∂s11∂x1= 0 fis11= constant =PAConstitutive Lawsstress - strain equations:† e11= S1111s11e22= S2211s11e33= S3311s11f1 = body force =0 for this caseSo long as not fullyanisotropic - this is allthat is requiredCompliance Form† S1111=1ExS2211= -nxyExS3311= -nxzExFor isotropic material gives: † e11=1Es11e22=-nEs11e33=-nEs11Now apply strain – displacement relations:† emn=12∂um∂xn+∂un∂xmÊ Ë Á ˆ ¯ ˜ e11=∂u1∂x1,e22=∂u2∂x2,e33=∂u3∂x3Hence (for isotropic material):† s11E=∂u1∂x1fiPAE=∂u1∂x1integration gives:† u1=Px1AE1+ g(x2,x3)Apply B. C. † u1= 0 @ † x1= 0 fi g(x2,x3) = 0i.e. uniaxial extension only, fixed at root† fi u1=Px1AE.similarly † u2=-nPAEx2† u3=-nPAEx3check:† e12=12∂u1∂x2+∂u2∂x1Ê Ë Á ˆ ¯ ˜ = 0÷Assessment of assumptions(Closer inspection reveals that our solutions are not exact.)1) Cross section changes shape slightly. A is not a constant.If we solved the equations of elasticity simultaneously, we would account for this.Solving them sequentially is ok so long as deformations are small. (dA is secondorder.)2) At attachment point boundary conditions are different from those elsewhere onthe rod.† u1= 0, u2= 0, u3= 0(Remember recitation examplelast term – materials axiallyloaded in a rigid container. Alsoproblem set question about thinadhesive joint.)We deal with this by invoking St. Venant's principle:"Remote from the boundary conditions internal stresses and deformations will beinsensitive to the exact form of the boundary condition."And the boundary condition can be replaced by a statically equivalent condition(equipollent) without loss of accuracy.How far is remote?This is the importance of the "long slender" wording of the rod definition.This should have been all fairly obvious. Next time we will start an equivalent processfor beams - which will require a little more thought.M2 Statics of BeamsReading: Crandall, Dahl and Lardner, 3.2-3.5, 3.6, 3.8A beam is a structural member which is long and slender and is capable of carryingbending loads. I.e loads applied transverse to its long axis.Obvious examples of aerospace interest are wings and other aerodynamic surfaces. Liftand weight act in a transverse direction to a long slender axis of the wing (think of gliderwings as our prototype beam). Note, even a glider wing is not a pure beam – it will haveto carry torsional loads (aerodynamic moments).1.) Modeling assumptionsa.) Geometry, slender member, L>>b, hAt this stage, will assume arbitrary, symmetric cross-sections, i.e.:b.) Loading• Similar to rod (traction free surfaces) but applied loads can be in the zdirection c.) Deformation• We will talk about this later2.) Boundary conditionsAs for rod, trussesPinned, simply supported CantileverDraw FBD, apply equilibrium to determine reactions.3.) Governing equations• Equations of elasticity4.) Solution Method• Exact (exactly solve governing differential equations)• Approximate (use numerical solution)But first need to look at how beams transmit load.Internal ForcesApply methods of sections to beam (also change coordinate system – to x, y, z –consistent with CDL). Method exactly as for trusses. Cut structure at
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