LECTURE+35+COE-3001-A

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Unformatted text preview: •  THIS LEADS US ADDITONAL AND VERY USEFUL DIFFERENTIAL EQUATIONS FOR THE SHEAR FORCE (V) AND LOAD INTENSITY (q) 4/9/13 M. Mello/Georgia Tech Aerospace 11 GENERALIZE TO DIFFERENTIAL EQUATIONS FOR DEFLECTION OF NONPRISMATIC BEAMS y y x z If z is neutral axis Z y 2 dA then Iz = d2 v EIz 2 = M (8- 9a) dx ☞ Restate (8- 7) ☞ ☞ this accounts for cases when moment of iner?a varies as a func?on of x (along the length of the beam), i.e., Iz = Iz(x) ²་  NOW, IF WE DIFFERENTIATE (8- 9a) WITH REPSECT TO (x) THIS LEADS TO OTHER USEFUL DIFFERENTIAL EQUATIONS FOR THE DEFLECTION OF NONPRISMATIC BEAMS ✓ ✓ ◆ ◆ ✓ ◆ 2 2 2 2 d dv dM d dv d2 M dV d EI d v = dM = V EIz 2 = EIx 2 = =q = x z dx dx dx 2 dx dx dx dx2 dx dx2 dx ✓ ◆ d d2 v EIz 2 = V (8- 9b) dx dx 2 ✓ 2 d dv EIx 2 dx2 dx ◆ = q (8- 9c) NEED TO INVOKE PRODUCT RULE 4/9/13 M. Mello/Georgia Tech Aerospace 12 SPECIALIZE TO DIFFERENTIAL EQUATIONS for DEFLECTION OF PRISMATIC BEAMS •  NOW WE CAN SPECIALIZE TO THE CASE WHERE THE FLEXURAL RIGIDITY (EI) REMAINS CONSTANT ALONG THE LENGTH OF THE BEAM (I.e. prisma?c beams) •  FACTOR OUT THE FLEXURAL RIGIDITY (EI) TO OBTAIN: ✓3◆ 2 dv dv EI = V (8- 10b) EI 2 = M (8- 10a) dx3 dx Bending- moment equaBon EIv ” = M (8- 12a) Shear- force equaBon EIv 000 =V (8- 12b) ☞ Integrate 2 ?mes to get v(x) ☞ Integrate 3 ?mes to get v(x) ☞ obtain 2 integra?on constants ☞ obtain 3 integra?on constants 4/9/13 M. Mello/Georgia Tech Aerospace ✓ d4 v EI dx4 ◆ = q (8- 10C) Load equaBon EIv 0000 = q (8- 12c) ☞ Integrate 4 ?mes to get v(x) ☞ obtain 4 integra?on constants 13 QUICK RECAP OF ASSUMPTIONS WHICH LEAD US TO THESE DIFFERENTIAL EQUATIONS ☞  MATERIAL OBEY’S HOOKE’S LAW (LIINEAR ELASTIC MATERIAL) ☞  SLOPES OF THE DEFLECTION CURVE ARE SMALL ☞  SHEAR DEFORMATIONS ARE NEGLIGIBLE ☞  THEORY IS THEN ECHNICALLY RESTRICTED TO PURE BENDING BUT AS WE SAW BEFORE, WE MAY IGNORE THE EFFECT OF SHEAR DEFORMATIONS FOR SMALL BEAM DEFLECTIONS 4/9/13 M. Mello/Georgia Tech Aerospace 14 A NOTE ON CURVATURE IN CASES OF BEAM DEFLECTION INVOLVING LARGE SLOPES •  ANY CALCULUS BOOK CAN BE CONSULTED TO SHOW THAT CURVATURE IS ACTUALLY (EXACTLY) DESCRIBED BY THE FOLLOWING RELATION: Strictly speaking this must be used in cases Involving large rota?ons 00 1 ⌫ = = (8- 13) Gere and Goodno rederive it… ⇢ [1 + (⌫ 0 )...
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## This note was uploaded on 09/19/2013 for the course CEE 3001 taught by Professor Zhu during the Spring '09 term at Georgia Tech.

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