MT-1620 al.2002 Unit 14 Behavior of General (including Unsymmetric Cross-section ) Beams Readings Rivello 7.1-7.5,77,7.8 T&g Paul A Lagace, Ph. D Professor of aeronautics Astronautics and Engineering Systems Paul A Lagace @2001
Paul A. Lagace © 2001 MIT - 16.20 Fall, 2002 Paul A. Lagace, Ph.D. Professor of Aeronautics & Astronautics and Engineering Systems Unit 14 Behavior of General (including Unsymmetric Cross-section) Beams Readings: Rivello 7.1 - 7.5, 7.7, 7.8 T & G 126
MT-1620 al.2002 Earlier looked at Simple Beam Theory in which one considers a beam in the x-z plane with the beam along the x-direction and the load in the z-direction Figure 14.1 Representation of Simple Beam Now look at a more general case Loading can be in any direction Can resolve the loading to consider transverse loadings p,(x) ind p,(x); and axial loading px X Include a temperature distribution T(x, y, z) Paul A Lagace @2001 Unit 14-2
Unit 14 - 2 Paul A. Lagace © 2001 MIT - 16.20 Fall, 2002 Earlier looked at Simple Beam Theory in which one considers a beam in the x-z plane with the beam along the x-direction and the load in the z-direction: Figure 14.1 Representation of Simple Beam • Loading can be in any direction • Can resolve the loading to consider transverse loadings py(x) and pz(x); and axial loading px(x) • Include a temperature distribution T(x, y, z) Now look at a more general case:
MT-1620 al.2002 Figure 14.2 Representation of General Beam force/leng Maintain several of the same definitions for a beam and basic assumptions Geometry: length of beam(x-dimension) greater than y and Z dimensions Stress State Oxx is the only "important stress; Ox and ox found from equilibrium equations, but are secondary in importance Deformation: plane sections remain plane and perpendicular to the midplane after deformation( Bernouilli-Euler Hypothesis) Paul A Lagace @2001 Unit 14-3
Unit 14 - 3 Paul A. Lagace © 2001 MIT - 16.20 Fall, 2002 Figure 14.2 Representation of General Beam Maintain several of the same definitions for a beam and basic assumptions. • Geometry: length of beam (x-dimension) greater than y and z dimensions • Stress State: σxx is the only “important” stress; σxy and σxz found from equilibrium equations, but are secondary in importance • Deformation: plane sections remain plane and perpendicular to the midplane after deformation (Bernouilli-Euler Hypothesis)
MT-1620 al.2002 Definition of stress resultants Consider a cross-section along x Figure 14.3 Representation of cross-section of general beam Place axis@ center of gravity of section ● Where O dA M,=Jon- da M y These are resultants l Paul A Lagace @2001 Unit 14-4
Unit 14 - 4 Paul A. Lagace © 2001 MIT - 16.20 Fall, 2002 Definition of stress resultants Consider a cross-section along x: Figure 14.3 Representation of cross-section of general beam Place axis @ center of gravity of section where: These are resultants! S dA z xz = − ∫∫ σ F dA = ∫∫ σ xx S dA y xy = − ∫∫ σ M z dA y xx = − ∫∫ σ M y dA z xx = − ∫∫ σ
MT-1620 al.2002 The values of these resultants are found from statics in terms of the loading px, py, pz, and applying the boundary conditions of the problem Deformation Look at the deformation. In the case of Simple Beam Theory, had dx where u is the displacement along the X-axis This comes from the picture Figure 14.4 Representation of deformation in Simple Beam Theory 2 dx s for small angles Now must add two other contributions Paul A Lagace @2001 Unit 14-5
Unit 14 - 5 Paul A. Lagace © 2001 MIT - 16.20 Fall, 2002 The values of these resultants are found from statics in terms of the loading px, py, pz, and applying the boundary conditions of the problem Deformation Look at the deformation. In the case of Simple Beam Theory, had: u z d w d x = − where u is the displacement along the x-axis. Now must add two other contributions….. Figure 14.4 Representation of deformation in Simple Beam Theory This comes from the picture: for small angles