东南大学：《建筑力学 Architectural Mechanics》课程教学课件（英文讲稿）A14 Bending Deflection

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)xxM0MMM 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

Deflection and Slope by IntegrationEIw" = -M(x)EIw' =-/ M(x)dx +CElw = -{ [ M(x)dxdx +Cx + D: Conventionally assuming constant flexural rigidity (E) Integration constants C and D can be determined from boundaryconditions, symmetry conditions, and continuity conditions6

EIw   M(x) EIw    M x x C  ( )d EIw   M(x)dxdx Cx  D • Integration constants C and D can be determined from boundary conditions, symmetry conditions, and continuity conditions. Deflection and Slope by Integration • Conventionally assuming constant flexural rigidity (EI) 6

Boundary Conditions - Simple Beams30kN/m30kN/m30kN·mBCA2m2m11. Deflections are restrained at the hinged/rolled supports=W^=0; W=01

0; 0    w w A B Boundary Conditions – Simple Beams • Deflections are restrained at the hinged/rolled supports 7

Boundary Conditions- Cantilever BeamsmAT. Both the deflection and rotation are restrained at theclamped end=W=0; =08

0; 0    wA A  8 Boundary Conditions- Cantilever Beams • Both the deflection and rotation are restrained at the clamped end

Symmetry Conditions? Both the geometry andloads are symmetric aboutBthe mid-section (x = L/2)L=→=02L2l2qM123EBDCAB219

Symmetry Conditions • Both the geometry and loads are symmetric about the mid-section (x = L/2) 0   C 9

Continuity ConditionswPAB1aX2LL0≤ ≤a,α≤x≤L,O≤≤L-αw(x =a)=w(x2 =a); Φ(x =a)=0(x2 =a)w(x =a)=w(x =L-a); (xi =α)=-0(x =L-α10

Continuity Conditions                 1 2 3 1 2 1 2 1 3 1 3 0 , , 0 ; ; x a a x L x L a w x a w x a x a x a w x a w x L a x a x L a                           10 P