L20 - Chapter 9 Deflection of Beams What is the elastic...

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Chapter 9 Deflection of Beams What is the elastic curve? What is the slope and deflection at point B .
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From Paramount Roll & Forming Inc.
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Deformation of a Beam Under Transverse Loading • Bending under general transverse loadings. EI x M ) ( 1 = ρ • Cantilever beam subjected to concentrated load at the free end, EI Px = 1 • Curvature varies linearly with x • At the free end A , = = A A ρ ρ , 0 1 • At the support B , PL EI B B = , 0 1 1/ρ =curvature at location x
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Deformation of a Beam Under Transverse Loading • M=0 -> Curvature is zero i.e., at each end and at E . EI x M ) ( 1 = ρ • M>0 concave upwards • M<0 concave downwards • Maximum curvature occurs where the moment magnitude is a maximum. • determine maximum deflection and slope.
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Equation of the Elastic Curve • From elementary calculus, simplified for beam parameters, 2 2 2 3 2 2 2 1 1 dx y d dx dy dx y d + = ρ • Substituting and integrating, () 2 1 0 0 1 0 2 2 1 C x C dx x M dx y EI C dx x M dx
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L20 - Chapter 9 Deflection of Beams What is the elastic...

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