MT-1620 al.2002 Unit 13 Review of Simple Beam Theory Readings Review Unified Engineering notes on Beam Theory BMP 38.3.9,3.10 t&G 120-125 Paul A Lagace, Ph. D Professor of aeronautics Astronautics and Engineering Systems Paul A Lagace @2001
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
MT-1620 al.2002 IV. General Beam Theory Paul A Lagace @2001 Unit 13-2
MIT - 16.20 Fall, 2002 IV. General Beam Theory Paul A. Lagace © 2001 Unit 13 - 2
MT-1620 al.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
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
MT-1620 al.2002 Figure 13.1 General Geometry of a Beam C2oss-5ACT/oA a)long&thin→>b,h b)loading is in z-direction c)loading passes through "shear center= no torsion/twist (we'l define this term later and relax this constraint d)cross-section can vary along X 2. Stress state a)Oy, Oyz, Oxy=0 =no stress in y-direction ZZ = only significant stresses are ox and o Paul A Lagace @2001 Unit 13-4
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) σyy, σyz, σxy = 0 ⇒ no stress in y-direction b) σxx >> σzz σxz >> σzz ⇒ only significant stresses are σxx and σxz • Paul A. Lagace © 2001 Unit 13 - 4
MT-1620 al.2002 Note: there is a load in the z-direction to cause these stresses, but generated o is much larger(similar to pressurized cylinder exampl Why is this valid? ook at moment arms Figure 13.2 Representation of force applied in beam fo 是2 Oxx moment arm is order of (h o,, moment arm is order of() ZZ and e>>h >0, for equilibrium Paul A Lagace @2001 Unit 13-5
MIT - 16.20 Fall, 2002 Note: there is a load in the z-direction to cause these stresses, but generated σxx 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 σxx moment arm is order of (h) σzz moment arm is order of (l) and l >> h ⇒ σxx >> σzz for equilibrium Paul A. Lagace © 2001 Unit 13 - 5