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gm1_4_notes - Lecture M1 Slender(one dimensional Structures...

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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.) On stresses b.) On displacements 3) We will apply an appropriate solution method: a) Exact/analytical (Unified, 16.20) b Approximate (often numerical) (16.21). Such as energy methods (finite elements, finite difference - use computers) Let us see how this works: Applied at specified locations in structure
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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 (x 1 dimension) b = width (x 3 dimension) h = thickness (x 2 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 x 1 direction only Results in a number of assumptions on the boundary conditions
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Similarly on the x 2 face - no force is applied s 21 = s 12 = 0 s 32 = s 23 = 0 s 22 = 0 on x 1 face - take section perpendicular to x 1 and c.) deformation s 31 = 0 s 32 = 0 s 33 = 0 s 12 = 0 s 13 = 0 s 11 dA = P A s 11 dx 2 dx 3 = P s 11 = P bh = P A
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Rod cross-section deforms uniformly (is this assumption justified? - yes, there are no
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