University:
Massachusetts Institute of Technology
Course:
16.01 | Unified Engineering I, II, III, & IV
Academic year:

2021
Views:

413

Pages:

7
Author:

Alma Stephenson

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 forces Opposite directions on two faces - equilibrium (Note Crandall Dahl and Lardner use V = -S) where M = Bending movement S = Shear force F = Axial force Beam Bending bar, rod "beam bar" Also drawn as Example of calculating shear force and bending moment distribution along a beam. Example 1. Cantilever beam. Free Body Diagram (note moment reaction at root) Equilibrium: H A = 0, VA = P, M A = -PL Take cut at point X ,distance x from left hand end (root). 0< x < L. Replace the effect of the (discarded) right hand side of beam by an equivalent set of forces and moments † (M, S, F) which vary as a function of position, x. Apply equilibrium + Â Fx = 0 Æ F(x) = 0 Â Fz = 0 ↑ + P - S(x) = 0 S(x) = P ÂMX =0 † + Pl + M (x) - Px = 0 M (x) = -P(l - x) Draw "sketches" - bending moment, shear force, loading diagrams Axial Force diagram (zero everywhere in this case) Shear Force Diagram Bending moment diagram: NOTE: At boundaries values go to reactions (moment at root, applied load at tip). These representations are useful because they provide us with a visual indication of where the internal forces on the beam are highest, which will play a role in determining where failure might occur and how we should design the internal structure of the beam (put more material where the forces are higher).

Lecture M2: Statics of Beams

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