Is given by the 2nd derivave when the

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Unformatted text preview: N CONVENTIONS FOR BENDING MOMENT (M), SHEAR FORCE (V), AND DISTRIBUTED LOADS (q) ☞ IMPORTANT SIGN CONVENTIONS: •  x,y axes are posi?ve to the right and upward •  Deflec?on v is posi?ve upward •  Slope dv/dx and angle of rota?on Θ are posi?ve when counterclockwise w.r.t pos. x axis •  Curvature κ is pos. when beam is bent upward •  Bending moment M> 0 when it produces Compression in upper part of the beam 4/9/13 M. Mello/Georgia Tech Aerospace 8 BEGIN LECTURE 35 Euler- Bernoulli beam theory Historical note: The basic rela?onship sta?ng that the curvature of a beam is propor?onal the bending moment (8- 6) was first obtained by Jacob Bernoulli, although he obtained an Incorrect constant of propor?onality… in other words he didn’t get (EI)! The rela?onship was used later by Euler, who solved the ODE of the deflec?on curve for both large and small deflec?ons… Euler was the man. 4/9/13 M. Mello/Georgia Tech Aerospace 9 DIFFERENTIAL EQUATIONS OF THE BEAM DEFLECTION CURVE Here are the big results from last lecture: 1 d2 v curvature is the essenBally equal to the 2nd derivaBve of the = = (8- 5) beam profile for any material provided rotaBons are small ⇢ dx2 •  Next we recall from module 5 (stresses in beams) that curvature is given by: = 1 = M (8- 6) ⇢ EI •  Hence, d2 v M (8- 7) Note the flexural rigidity (EI) in the denominator and how it this dx 2 = EI controls curvature for a given bending moment •  (8- 7) IS REFERRED TO AS THE BENDING- MOMENT EQUATION •  If you know the bending moment M(x) then you can integrate twice and obtain the profile of the deformed beam! 4/9/13 M. Mello/Georgia Tech Aerospace 10 DIFFERENTIAL EQUATIONS OF THE BEAM DEFLECTION CURVE (cont.) d2 v EI 2 = M (8- 7) BEGIN (AGAIN) BENDING- MOMENT EQUATION dx dV •  RECALL, = q (8- 8a) dx FROM MODULE 4 SHEAR FORCES AND BENDING MOMENTS dM •  and = (8- 8b) V dx •  Note: Capital “V” is the shear force (of course)… and lowercase “v” in (8- 7) is the beam deflec?on… •  (8- 8A) AND (8- 8B) MOTIVATE US TO TAKE ADDITIONAL DERIVATIVES OF EQUATION (8- 7)...
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