MIT 16 01  Slender Structures (27 pages)
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Slender Structures
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 27
 School:
 Massachusetts Institute of Technology
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 16 01  Unified Engineering I, II, III, & IV
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Lecture M1 Slender one dimensional Structures Reading Crandall Dahl and Lardner 3 1 7 2 This semester we are going to utilize the principles we learnt last semester i e the 3 great principles and their embodiment in the 15 continuum equations of elasticity in order to be able to analyze simple structural members These members are Rods Beams Shafts and Columns The key feature of all these structures is that one dimension is longer than the others i e they are one dimensional Understanding how these structural members carry loads and undergo deformations will also take us a step nearer being able to design and analyze structures typically found in aerospace applications Slender wings behave much like beams rockets for launch vehicles carry axial compressive loads like columns gas turbine engines and helicopter rotors have shafts to transmit the torque between the components andspace structures consist of trusses containing rods You should also be aware that real aerospace structures are more complicated than these simple idealizations but at the same time a good understanding of these idealizations is an important starting point for further progress 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 structural member In general these will be on a Geometry b Loading Stress State c Deformation Strain State 2 We will make problem specific modeling assumptions on the boundary conditions that apply idealized supports such as pins clamps rollers that we encountered with truss structures last semester a b On stresses On displacements Applied at specified locations in structure 3 We will apply an appropriate solution method a b Exact analytical Unified 16 20 Approximate often numerical 16 21 Such as energy methods finite elements finite difference use computers Let us see how this works Rods bars The first 1 D structure that we will analyzes is that of a rod or bar such as we encountered when we analyzed trusses We are interested in analyzing for the stresses and deflections in a rod First start with a working definition from which we will derive our modeling assumptions A rod or bar is a structural member which is long and slender and is capable of carrying load along its axis via elongation Modeling assumptions a Geometry L 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 stresses and strains b Loading loaded in x1 direction only Results in a number of assumptions on the boundary conditions s 31 0 s 32 0 s 33 0 Similarly on the x2 face no force is applied s 21 s 12 0 s 32 s 23 0 s 22 0 on x1 face take section perpendicular to x1 s12 0 s13 0 and s 11dA P A s 11dx2 dx3 P fi s 11 c deformation P P bh A Rod cross section deforms uniformly is this assumption justified yes there are no shear stresses no changes in angle So much for modeling assumptions Now let s apply governing equations and solve 1 Equilibrium s mn fn 0 x n only s11 is non zero s 11 f1 0 x1 f1 body force 0 for this case s 11 P 0 fi s 11 constant x A 1 Constitutive Laws stress strain equations e11 S1111s 11 e22 S2211s 11 e 33 S 3311s 11 Compliance Form So long as not fully anisotropic this is all that is required 1 Ex n xy S2211 Ex n S 3311 xz Ex S1111 1 s 11 E n s 11 For isotropic material gives e22 E n e 33 s 11 E e11 Now apply strain displacement relations 1 u u e mn m n 2 xn x m u u u e11 1 e22 2 e 33 3 x1 x2 x 3 Hence for isotropic material s 11 u1 E x1 fi P u1 AE x1 integration gives u1 Px1 g x2 x 3 AE1 Apply B C u1 0 x1 0 fi g x2 x 3 0 i e uniaxial extension only fixed at root fi u1 Px1 AE similarly u2 u3 check e12 nP x2 AE nP x3 AE 1 u1 u2 0 2 x2 x1 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 second order 2 At attachment point boundary conditions are different from those elsewhere on the rod u1 0 u2 0 u3 0 Remember recitation example last term materials axially loaded in a rigid container Also problem set question about thin adhesive joint We deal with this by invoking St Venant s principle Remote from the boundary conditions internal stresses and deformations will be insensitive 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 process for beams which will require a little more thought M2 Statics of Beams Reading Crandall Dahl and Lardner 3 2 3 5 3 6 3 8 A beam is a structural member which is long and slender and is capable of carrying bending loads I e loads applied transverse to its long axis Obvious examples of aerospace interest are wings and other aerodynamic surfaces Lift and weight act in a transverse direction to a long slender axis of the wing think of glider wings as our prototype beam Note even a glider wing is not a pure beam it will have to carry torsional loads aerodynamic moments 1 Modeling assumptions a Geometry slender member L b h At 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 z direction c Deformation We will talk about this later Distributed load 2 Boundary conditions As for rod trusses Pinned simply supported Cantilever Draw FBD apply equilibrium to determine reactions 3 Governing equations Equations of elasticity 4 Solution Method Exact exactly solve governing differential equations Approximate use numerical solution But first need to look at how beams transmit load Internal Forces Apply methods of sections to beam also change coordinate system to x y z consistent with CDL Method exactly as for trusses Cut structure at location where we wish to find internal forces apply equilibrium obtain forces In the case of a beam the structure is continuous rather than consisting of discrete bars so we will find that the internal forces and moments are in general a continuous function of position Internal
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