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LECTURE+36+COE-3001-A

# d4 v ei dx4 q 8 10c load equabon

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Unformatted text preview: EGLIGIBLE ☞  THEORY IS THEN TECHNICALLY 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 5 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 )2 ]3/2 (Euler used it for large deﬂec=ons) •  As we have seen, in the case of the small beam rota=ons, the curvature of the beam is given by: 1 d2 v = = 2 (8- 5) ⇢ dx •  The assump=on of small beam rota=ons is therefore equivalent to assuming that (ν’)2 <<< 1 in (8- 13). 4/9/13 M. Mello/Georgia Tech Aerospace 6 BEGIN LECTURE 36 4/9/13 M. Mello/Georgia Tech Aerospace 7 EXAMPLE 8- 3 Problem: Simple beam supports a concentrated load P Determine: 1.  equa=on of the deﬂec=on curve 2.  angle of rota=on ΘA and ΘB at the supports 3.  maximum deﬂec=on δmax 4.  Deﬂec=on δC at the midpoint C of the beam SOLUTION: We must ﬁrst obtain bending- moment rela=ons 0<x<a M= P bx L 0<x<a a<x<b M= P bx L P (x 4/9/13 M. Mello/Georgia Tech Aerospace a) a<x<L 8 EXAMPLE 8- 3 (cont.) •  APPLY BENDING- MOMENT RELATION: 00 EIv = 00 EIv = P bx L (0 x a) P bx L P (x a) ( a x L) •  INTEGRATE TO OBTAIN BEAM SLOPE EQUATIONS: P bx2 EIv = + C1 2L 0 P bx2 EIv = 2L 0 4/9/13 (0 x a) a)2 P (x 2 + C2 ( a x L) M. Mello/Georgia Tech Aerospace 9 EXAMPLE 8- 3 (cont.) •  INTEGRATE SLOPE SOLUTIONS TO OBTAIN BEAM DEFLECTION EQUATIONS: P bx3 EIv = + C1 x + C3 6L P bx3 EIv = 6L a) 3 P (x 6 (0 x a) + C2 x + C4 ( a x L) •  DETERMINE CONSTANTS OF INTEGRATION: Apply con=nuity condi=ons at x=a ☞ condi=on 1: slope from to lem (x=a- ) must equal slope to the right (x=a+) 2 2 P (x a)2 EIv 0 = P bx + C (0 x a) EIv 0 = P bx + C 2 ( a x L) 1 2L 2L 2 P ba2 P ba2 C1 = C2 + C1 = + C2 2L 2L ☞ condi=on 2: deﬂec=on to the lem (x=a- ) must equal deﬂec=on to the right (x=a+) P ba3 P ba3 + C1 a + C3 = + C2 a + C4 C3 = C4 6L 6L 4/9/13 M. Mello/Georgia Tech Aerospace 10 EXAMPLE 8- 3 (cont.) •  DETERMINE CONSTANTS OF INTEGRATION (cont.): Apply boundary condi=ons at x=a and x=L...
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