EGM 3520 Chapter 9 Deflection of Beams Contents...

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EGM 3520 Chapter 9 Deflection of Beams
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EGM 3520 Department of Mechanical and Aerospace Engineering Contents Introduction 9.1 Deformation of a Beam under Transverse Loading 9.1A Equation of the Elastic Curve 9.1B Determination of the Elastic Curve from the Load Distribution 9.2 Statically Indeterminate Beams 9.3 Using Singularity Functions to Determine the Slope and Deflection of a Beam 9.4 Method of Superposition 2 9.5 Moment Area Theorems 9.5A General Principles 9.5B Application to Beams with Symmetric Loading 9.5C Bending-Moment Diagrams by Parts 9.6 Application of Moment-Area Theorems to Beams with Unsymmetric Loadings 9.6A General Principles 9.6B Maximum Deflection 9.6C Use of Area-Moment Theorems with Statically Indeterminate Beams
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EGM 3520 Department of Mechanical and Aerospace Engineering From Chapter 4 3 deformation of beam due to bending I E M 1
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EGM 3520 Department of Mechanical and Aerospace Engineering 9.1 Deformation of a Beam Under Transverse Loading 4 Relationship between bending moment and curvature for pure bending remains valid for general transverse loadings. Cantilever beam subjected to concentrated load at the free end, Curvature varies linearly with x At the free end A , At the support B , P x M(x) = -Px EI x M ) ( 1 EI Px 1 A A ρ ρ , 0 1 PL EI B B , 0 1
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EGM 3520 Department of Mechanical and Aerospace Engineering 9.1 Deformation of a Beam Under Transverse Loading 5 Overhanging beam Reactions at A and C Bending moment diagram Curvature is zero at points where the bending moment is zero, i.e., at each end and at E . Beam is concave upwards where the bending moment is positive and concave downwards where it is negative. Maximum curvature occurs where the moment magnitude is a maximum. An equation for the beam shape or elastic curve is required to determine maximum deflection and slope. EI x M ) ( 1
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EGM 3520 Department of Mechanical and Aerospace Engineering 9.1A Equation of the Elastic Curve 6 Substituting and integrating, From elementary calculus, simplified for beam parameters, slope, , is small EI x M ) ( 1 2 2 1 d y EI EI M x dx 2 2 M x d y dx EI 2 2 2 3 2 2 2 1 1 dx y d dx dy dx y d dy dx
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EGM 3520 Department of Mechanical and Aerospace Engineering 9.1A Equation of the Elastic Curve 7 From elementary calculus, simplified for beam parameters, tan θ ≈ θ, for small θ Substituting and integrating, EI x M ) ( 1 2 2 2 3 2 2 2 1 1 dx y d dx dy dx y d tan dx dy dx dy 2 2 1 d y EI EI M x dx 1 0 x dy EI EI M x dx C dx
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EGM 3520 Department of Mechanical and Aerospace Engineering 8 11.5 11.5 Error in tan(x)=x Approximation x tan(x) - x
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EGM 3520 Department of Mechanical and Aerospace Engineering 9 5.73
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EGM 3520 Department of Mechanical and Aerospace Engineering 9.1A
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  • Summer '08
  • DICKRELL
  • Mechanical and Aerospace Engineering, 0

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