Lecture M1: Slender (One Dimensional) Structures

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 Ú s 11dA = P A ÚÚ 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 Ê∂u ∂u ˆ e mn = Á m + n ˜ 2 Ë ∂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 Ê ∂u1 ∂u2 ˆ + Á ˜=0 2 Ë∂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.