Bending Deflectionmi@see.cn
Bending Deflection mi@seu.edu.cn
Contents·TheElasticCurve,Deflection&Slope（挠曲线、挠度和转角）·DifferentialEquationoftheElasticCurve（挠曲线微分方程）·Deflection&SlopebyIntegration（积分法求挠度和转角）·BoundaryConditions（边界条件）·SymmetryConditions（对称性条件）·ContinuityConditions（连续性条件）·DirectIntegrationfromDistributedLoads（直接由分布荷载积分求挠度和转角）·DirectIntegrationfromTransverseLoads（直接由剪力积分求挠度和转角）·DeformationsinaTransverseCrossSection（梁横截面内的变形）·CurvatureShortening（梁由于弯曲造成的轴向位移）2
• The Elastic Curve, Deflection & Slope （挠曲线、挠度和转角） • Differential Equation of the Elastic Curve（挠曲线微分方程） • Deflection & Slope by Integration（积分法求挠度和转角） • Boundary Conditions（边界条件） • Symmetry Conditions（对称性条件） • Continuity Conditions（连续性条件） • Direct Integration from Distributed Loads（直接由分布荷载积分求 挠度和转角） • Direct Integration from Transverse Loads（直接由剪力积分求挠度 和转角） • Deformations in a Transverse Cross Section（梁横截面内的变形） • Curvature Shortening（梁由于弯曲造成的轴向位移） Contents 2
Contents·Deflection&SlopebySuperposition（叠加法求挠度和转角）·SuperpositionofLoads（荷载叠加法）·SuperpositionofRigidized Structures（刚化叠加法）·CombinedSuperposition（荷载和变形组合叠加法）·Deflection&SlopebySingularFunctions（奇异函数法求挠度和转角）·Deflection&SlopebyMoment-AreaTheorems（图乘法求挠度和转角）·StiffnessCondition（刚度条件）·WaystoIncreaseFlexuralRigidity（梁的刚度优化设计）·BendingStrainEnergy（弯曲应变能）3
• Deflection & Slope by Superposition（叠加法求挠度和转角） • Superposition of Loads（荷载叠加法） • Superposition of Rigidized Structures（刚化叠加法） • Combined Superposition（荷载和变形组合叠加法） • Deflection & Slope by Singular Functions（奇异函数法求挠度和转 角） • Deflection & Slope by Moment-Area Theorems（图乘法求挠度和转 角） • Stiffness Condition（刚度条件） • Ways to Increase Flexural Rigidity（梁的刚度优化设计） • Bending Strain Energy（弯曲应变能） Contents 3
The Elastic Curve, Deflection and Slope. The elastic curve: beam axis under bending, required todetermine beam deflection and slope: Bending deflections (w = f(x)): vertical deflection of the neutralsurface, defined as downward positive / upward negative: Slope (0 = 0(x) ~ tan(0) = dw/dx): rotation of cross-sectionsdefined as clockwise positive / counter clockwise negative0wxDeflectioncurvewI4
• The elastic curve: beam axis under bending, required to determine beam deflection and slope. x w w Deflection curve The Elastic Curve, Deflection and Slope • Bending deflections (w = f(x)): vertical deflection of the neutral surface, defined as downward positive / upward negative. • Slope (θ = θ(x) ≈ tan(θ) = dw/dx): rotation of cross-sections, defined as clockwise positive / counter clockwise negative 4
Differential Eguation of the Elastic Curve.Curvature of the neutral surface11wM(x)K2)3/2p(x)(l + wEIp(x)xxM<0M>0MMM w">O Mw"<0WWEIw"=-MEl: flexural rigidity: The negative sign is due to the particular choice of the w-axis5
• Curvature of the neutral surface EIz M x x ( ) ( ) 1 2 3/2 1 ( ) (1 ) w w x w EIw M w x M M M 0 w 0 w x M 0 M w 0 M EI: flexural rigidity Differential Equation of the Elastic Curve 5 • The negative sign is due to the particular choice of the w-axis