unit13

# unit13 - MIT 16.20 Fall 2002 Unit 13 Review of Simple Beam...

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MIT - 16.20 Fall, 2002 Unit 13 Review of Simple Beam Theory Readings : Review Unified Engineering notes on Beam Theory BMP 3.8, 3.9, 3.10 T & G 120-125 Paul A. Lagace, Ph.D. Professor of Aeronautics & Astronautics and Engineering Systems Paul A. Lagace © 2001

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MIT - 16.20 Fall, 2002 IV. General Beam Theory Paul A. Lagace © 2001 Unit 13 - 2
MIT - 16.20 Fall, 2002 We have thus far looked at: in-plane loads torsional loads In addition, structures can carry loads by bending . The 2-D case is a plate , the simple 1-D case is a beam . Let’s first review what you learned in Unified as Simple Beam Theory (review of) Simple Beam Theory A beam is a bar capable of carrying loads in bending. The loads are applied transverse to its longest dimension. Assumptions : 1. Geometry Paul A. Lagace © 2001 Unit 13 - 3

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MIT - 16.20 Fall, 2002 Figure 13.1 General Geometry of a Beam a) long & thin l >> b, h b) loading is in z-direction c) loading passes through “shear center” no torsion/twist (we’ll define this term later and relax this constraint.) d) cross-section can vary along x 2. Stress state a) σ yy , σ yz , σ xy = 0 no stress in y-direction b) σ xx >> σ zz σ xz >> σ zz only significant stresses are σ xx and σ xz Paul A. Lagace © 2001 Unit 13 - 4
MIT - 16.20 Fall, 2002 Note : there is a load in the z-direction to cause these stresses, but generated σ xx is much larger (similar to pressurized cylinder example) Why is this valid? Look at moment arms: Figure 13.2 Representation of force applied in beam σ xx moment arm is order of (h) σ zz moment arm is order of ( l ) and l >> h ⇒ σ xx >> σ zz for equilibrium Paul A. Lagace © 2001 Unit 13 - 5

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MIT - 16.20 Fall, 2002 3. Deformation Figure 13.3 Representation of deformation of cross-section of a beam deformed state (capital letters) undeformed state (small letters) o is at midplane define: w = deflection of midplane (function of x only) Paul A. Lagace © 2001 Unit 13 - 6
MIT - 16.20

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