LECTURE+35+COE-3001-A

8 7 is referred

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 )...
View Full Document

Ask a homework question - tutors are online